Stories About Us

By Lore Segal

在 The New Yorker 杂志上有全文，点击这里阅读。

In the Mail

Once [writers have] finished a new manuscript and put it in the mail, they exist in a state of suspended emotional and psychic animation . . . and it’s cruelty to animals to keep them waiting.

—Robert Gottlieb, The Paris Review

“Let’s get the complaining out of the way,” proposed Hope. “I’ve got me a pacemaker.”

Farah said, “I’m losing my vision.”

Bessie said, “I lost my husband.”

And Bridget said, “I sent my story to a friend from my old writing class.”

“And how is that a complaint?” Bessie asked her.

Bridget said, “Because it feels—maybe it’s something like the actor’s stage fright.”

“What I felt,” Lucinella said to Bridget, “when I sent you my water-bug story. Embarrassed.”

“Water bugs! Yuck!” everybody said, and Bessie said, “But why is it embarrassing?”

Bridget said, “There’s a kind of shyness or shame to be demanding another person’s time and attention.”

“Of exposing oneself,” Lucinella said. “I begged Bridget for her honest opinion, and Bridget wrote me that there is not one of us who doesn’t conceal from herself the hope, the expectation, that the honest opinion will be that we have written a masterpiece.”

“And the fear,” Bridget added, “that we will be found out to be a jackass.”

“What’s your story about?” Farah asked Bridget.

“About a group of women friends getting old together.”

“About us, you mean,” Hope said.

“About us, but you understand that I write stories. About us sitting round the table and talking, but not necessarily what we say and not identifiably any one of us.”

“But why not identifiably us?” asked Farah.

“Maybe so as not to make public what we feel free to tell each other in private; to not offend with what I understand or misunderstand. And because I’ve already killed off two of us on the page. But really it’s about how we make up the people in our stories.”

“Only we’re right here. You don’t have to make us up.”

“Yes, I do. I don’t have to invent, but I have to imagine us. People, pacemakers, and glaucoma are not the stuff that can be pasted into a Microsoft document or onto a sheet of paper. Remember ‘Star Trek’? You’re beamed to a different dimension by being decimated and then reassembled on arrival. I turn us into the words that would allow my friend Anna to imagine us.”

“So what did your Anna think of us?”

“She didn’t.”

“How do you mean?”

“I haven’t heard from her.”

“When did you send her your story?”

“Tomorrow will be four weeks to the day.”

“So what will you do?”

“Lie in bed at night and stew. Dream vengeful dreams. Or imagine Anna putting off the obligation to read my story. I picture her putting off the reading in order to put off the barely thinkable prospect of having to say something to me if she doesn’t like it or doesn’t get the point, and so she puts it off and puts it off until she forgets about it and, eventually, forgets the cause of the pinpoint of guilt in the lower left back side of her brain.”

“Poor Anna,” said everybody.

Grandmother Mole

Hope said, “My agenda? Falling asleep at the movies.”

Farah said, “Now that it’s hard for me to read, I lie on my bed and listen to YouTube philosophy which I don’t understand.”

“What don’t you understand?” Bridget asked her. They were talking on Zoom.

“What is ‘the excluded middle’?” Farah said. “And how does the inability of the finite to imagine the infinite prove the existence of God?”

Bridget said, “I’ve been reading ‘The Peaceable Kingdom’ with great-granddaughter Libby. ‘The lion shall lie down with the lamb and little Libby will lead them.’ ”

Bessie said, “Lucky you to still have a little Libby to read to. My Eve’s Johnny used to like it when I read him the story of Mole who shouts and shouts and doesn’t stop shouting until his Grandmother Mole figures out that what he means is ‘Notice me.’

“Yesterday,” continued Bessie, “I was coming out of the building, when who should be walking up the street but Johnny. ‘You are coming to see me!’ I said. Why, he said—wasn’t I feeling well? I was O.K., I told him. He told me that he was on his way up the block to study with his friend in his friend’s new place, and he said goodbye and I said goodbye. Is it unreasonable of me to think he might have some feelings for his grandmother now that I have lost Colin?”

“Colin wasn’t his real grandfather,” Hope said.

“Maybe not, but he was just lovely with the boy. Used to take him sailing, with me, terrified, watching from the deck of the house.”

“When they turn teens,” Farah said, “we know it isn’t grandmothers they have on their souls. If I live long enough, Hami will become a grownup and we may be friends again.”

Ilka said, “Not necessarily. Maggie is looking through my papers and found—wait!” Ilka disappeared from the screen and returned holding up a fragile postcard. “It’s dated 1889, from the grandmother I never knew to my father. A loose translation: ‘My dearest, so very beloved Hansl, How is it possible that you have not in a whole week found the smallest little minute to send one line to your mama . . .’ ”

“The ur-complaint. Right out of Nichols and May,” said Lucinella. “Listen!” She manipulated her iPad.

A woman’s voice said, “This is your mother. Do you remember me?”

A male voice said, “I was just going to call you. . . . Do you know that I had my finger on the—”

The mother said, “I sat by that phone all day Friday, and all day Friday night, all day Saturday, and all day Sunday. Finally, your father said to me, Phyllis, eat something, you’ll faint. I said, Harold, no, I don’t want to have my mouth full when my son calls me. . . .”

The son’s voice: “I feel awful.” The mother: “Oh, honey, if I could believe that, I’d be the happiest mother in the world.”

•

At the following ladies’ lunch, Bridget read the friends the Nichols-and-May Mole story she had written.

She read: ‘‘When Grandmother Mole met Mole outside her tunnel, she said, ‘I’ve been waiting for you.’ ‘Why, Grandmother Mole, aren’t you feeling well?’ Mole asked her, but Grandmother Mole was feeling well enough. She said, ‘It’s just I want to see you.’ ‘Grandmother Mole, do you need me to go hunting for you?’ ‘No, thank you,’ Grandmother Mole said.’ ”

Bridget interrupted her story to say, “I was going to check Wikipedia. What do moles hunt? Do moles hunt? So, anyway. ‘Grandmother Mole,’ Mole asked her, ‘do you need me to come and dig you a new tunnel?’ But there was nothing wrong with Grandmother Mole’s old tunnel. ‘What I want,’ Grandmother Mole said to her Mole, ‘is for you to want to visit me, to want to talk and to be with me.’ ”

“She means ‘Notice me,’ ” commented Farah.

Bridget returned to her story. “ ‘So goodbye, Grandmother Mole,’ Mole said. And he walked up the block to his friend’s tunnel to study.”

“To hang out,” amended Farah.

Bessie said, “Oh, leave him be. Let him hang out or study or do whatever kids do.”

“What has happened to you?” Lucinella asked Bessie. “What has changed?”

Bessie said, “My grandson e-mailed me. I thought all the kids ever did was ‘text’ each other.”

She pulled out a piece of paper. “Johnny wrote me: ‘I’m sorry I haven’t been available recently!!! How are you? Let’s have lunch or dinner soon. I miss you.’ ”

Why with an Exclamation

At one Old Rockingham lunch before Colin’s illness—so it must have been some three or more years ago, Bridget thought—Colin had invited everyone out on his boat. Bessie had stayed in to deal with the complications of lunch. Hope and Farah opted to help her, but Bridget and Ruth went sailing with Colin. Ruth soon wanted to get back to firm land, but for Bridget it was bliss, the revelation of what she felt she had been born to do, to be.

Today, they were seated at Bridget’s table. Handing around plates and passing the salad, she was reluctant to say what was going to take too long and be too cumbersome. The friends were all readers, but she alone kept returning to Proust. Still, she set sail. “The payoff,” she said, “is that you come across improbable behaviors and recognize your friends—recognize yourself.”

“Like what, for instance?” her friends obliged her by asking.

Bridget said, “Marcel’s father has been more than hinting—has been nudging his excellent friend, whose name I’m not even going to try to recall, for membership in the Institute, to which the friend has the means to get him elected. Well, why does the friend not act for him?

“So Proust tells another story,” continued Bridget, feeling that she was getting farther and farther from the home shore. “Marcel’s father invites a family friend to dinner. Knowing dear old Swann’s desire to meet a certain young woman, he does not invite the young woman.”

“And how does that explain anything?” asked Hope.

“It illustrates the ancient truth that ‘To those who have, shall be given; from those who have not, shall be taken even that they have,’ ” Bridget explained. “But Proust is saying that the trick of generally well-constituted natures is to not give a friend what the friend wants.”

“But why do we?” Bessie said. “That doesn’t explain why we do that.”

Bridget said, “This is not the ‘why’ that expects a ‘because.’ It’s the ‘Why, how curious,’ with an exclamation mark!”

Left Shoulders

Ilka said, “You remember how we said ‘No more planes, no more trains’?”

“And no more movies,” added Farah. “No more going to the theatre.”

“Or going anywhere. Going out,” Bessie said. “Eve asks me don’t I want to take a turn down on Riverside Drive, and I say no. I want to sit where I am sitting.”

“There have been times, lately,” Ilka said, “when I’ve thought, No more talking. There’s something I want to say, but my mouth doesn’t open to say it, or not in the moment when there is a gap in the conversation. I remember my mother sitting beside me at our dinner parties—it was she who had usually cooked. Was I aware, in the heat of what I was saying to whoever sat on my right, of her sitting behind my left shoulder? Did I understand that she no longer had what it takes to make her way into the conversation?”

“The energy to self-start,” Hope said, “to insert oneself.”

Ilka said, “I would go on and on, talking and talking.”

“So many left shoulders,” Bessie said.

“I used to have an Indian friend, Padma,” said Ilka. “Padma might be sitting on the sofa talking with me, and if my husband walked in she would minimally adjust herself to also face him, and if my mother came in, Padma, with the most natural, littlest movement, turned to include her also. It was the prettiest thing to observe. We in the West seem not to have that instinct or that skill. My friend John comes to see me and we are talking. The bell rings and it’s my friend Joe. In another minute, John and Joe are talking and I am behind their two left shoulders.”

Bessie said, “Old people seem always, at any table, to be sitting on the wrong side of left shoulders.”

Hope said, “Lotte used to make us laugh at her old-people stories where old meant being forty or thirty, she sitting in the passenger seat and the driver not flirting with her.”

Bridget said, “Nobody ever flirted with me. I don’t think I knew how. But I used to congratulate myself that, having been a plain girl, I had become a rather well-looking old person. Then I read Proust’s description of the party where he almost mistakes his first love for her mother. She has grown old, all his friends are old. He, Marcel, is old. I was amused that it hurt my feelings when he says that the woman who smiles at him is old and ugly.”

“But we have the blessed Zoom,” Bessie said, “where we are small and at a distance. Zoom hides more than our wrinkles. Much to be said for Zoom.”

“But I was surprised,” Bridget said, “that it hurt my feelings when he said old and ugly.”

Who is Outside?

Ladies’ movie night. The friends, who have become less and less willing to leave their homes, agree to watch a movie on the TV and then meet on Zoom to talk. But here Ilka proves, as she herself says, a dead loss. “I turned it off before the tall guy, who doesn’t know, is about to walk into the room.”

“But that’s the payoff for all that good suspense!” Bessie says. “That’s the edge-of-the-chair moment we have been waiting for.”

“That’s why I turned it off,” Ilka says.

“You don’t like suspense?”

“Hate it. Can’t stomach suspense. I mean that my physical stomach does a number on me.”

“And last week you didn’t finish watching ‘The Quiet Place,’ and that was a nice suspense, a happy ending.”

Ilka says, “There are no happy endings.”

“It’s odd,” Bessie says, “because you always seem the reasonable one of us.”

“I’ve been having nasty nights,” says Ilka.

Here is something they all understand: “You don’t sleep and you have horrible dreams?”

“A revolving dream,” Ilka says, “in which I must find a rhyme, like an algebra problem going around and around and around. It’s this poem I’m trying to translate. Theodor Kramer was a poet in Vienna in the thirties, a Jew. My Uncle Paul’s favorite writer. I’ve got the first verse:

Who is outside ringing at the door?
And we not even out of bed?
I’ll go, love, and take a look.
Only the boy who came and left the bread.

“But the second verse:

Who is outside ringing at the door?
You stay, dear.
It was a man talking with the
neighbors asking who we are?

“When the word doesn’t meet its own rhyme, we are puzzled and then we doubt,” says Ilka.

Lucinella suggests, “How about ‘You stay, my dear’ and ‘asking who we were’—a false rhyme but . . . ”

Ilka says, “No, or maybe yes? Let it stand for the moment.” She adds, “It’s not the night so much as the morning hour, the hideous reëntry into the day—the day, mind you, which I rather like. I do. I like my old lady’s life quite well enough. What I mean is the ugly hour before I’m awake—the necessity to wake up. Remember Lotte, the everlasting questioner, asking, ‘So what is that all about?’ ‘Why do you? Why did you?’ she would ask. And I understood that I was not so interested, that I don’t much believe in my explanations.

“But may I read you the rest of the poem? And maybe you can find me a title:

Who is outside ringing at the door?
Go, love, and run the bath.
The mail has come but not the letter we were waiting for.

Who is outside ringing at the door?
Go, my darling, turn the beds about.
It was the super.
First of next month we have to be out.

Who is outside ringing at the door?
How the fuchsia blooms so near.
Dearest, pack me my toothbrush,
and don’t cry,
They are here.

Vienna, 1938 ♦

Theodor Kramer’s poem is taken from “Gesammelte Gedichte” (“Collected Poems”), in three volumes, edited by Erwin Chvojka. © Paul Zsolnay Verlag, Vienna, 1997-2005.

Owing to a production error, a previous version of this story omitted a line from Theodor Kramer’s poem.

Published in the print edition of the October 7, 2024, issue, with the headline “Stories About Us.”

Lore Segal is the author of several books, including “Half the Kingdom,” “Her First American,” and “Ladies’ Lunch and Other Stories.”

Greensleeves

By Sigrid Nunez

在 The New Yorker 杂志上有全文，点击这里阅读。作者有华人血统。

***

“What I want to know,” the woman said to the therapist, “is why the voices always say mean, terrible things. Why don’t they ever say things like ‘You’re a good person. You’re a great, smart, wonderful guy, your life matters, and you deserve to be happy’? I mean, instead of saying, ‘You’re no good, your life is worthless, everyone hates you, you should hurt yourself, you deserve to be hurt, you deserve to die.’

“Even worse,” the woman went on, “why do the voices always say things like ‘Go shove some innocent stranger in front of an oncoming train’? Instead of, like, ‘How about helping that little old lady with her bags?’ ”

He wanted to laugh, but the woman was being earnest. She was young—early thirties, he guessed—with an unremarkable face except for her eyes, so dark you could barely distinguish iris from pupil. She stared at him from under thick bangs, the only part of her black hair that had been streaked blond. Kiss Me Deadly red lipstick, and a long-sleeved forest-green dress of some suède-like fabric that looked vintage. His gaze kept being drawn to her gleaming manicure, each copper-colored nail like a Japanese beetle.

He could have told her that what she was saying wasn’t true. The voices didn’t always bully or suggest evil acts. Sometimes their words were impersonal, and might even be kind. Sometimes they didn’t speak at all but only breathed heavily—which could, he supposed, be as sinister as threats or curses. Some hummed, or chanted, or sang. “I hear lullabies,” one patient had told him.

If he’d wanted to get into a conversation with Lady Greensleeves, he might have said all this. He might have added the obvious: non-negative voices were not necessarily a positive thing. The problem with the lullabies was that they drove a woman of fifty to rock back and forth and suck her thumb. And, of course, there was a certain kind of person, one of the worst kinds of person, who seemed to live with a voice continually telling them how great they were, and who felt victimized because their perfection was not universally acknowledged.

He’s bored, the woman thought. He isn’t listening. He’s just being polite. No, he wasn’t even bothering to pretend. Not polite!

In fact, he was interested in one aspect of what she said: the way she imagined that it was all arranged. Somebody had arranged for the voices to be hateful, when they could just as easily have been loving. Her frustration with the way things are. Interesting, but hardly new. It was something he dealt with every day. He made a living off it. God could have created whatever kind of world He wished to create—so why had He chosen this fucked-up one?

If the therapist had wanted to get into a conversation with the woman, and if he hadn’t been afraid of sounding pretentious, or as if he were mansplaining, he might have brought up the Gnostic belief that the Supreme Being had not made this world and did not rule it; the one responsible was another, lesser spirit, not a benevolent or all-knowing one. She probably would not have heard of Gnosticism, though. This was not patriarchal condescension (raised by his mother, who had been eulogized as one of the finest historians of her generation, and whose own mother had been renowned for her work in biochemistry, he would not have made that mistake) but a nod to her youth. His sister, who taught English to undergrads, had told him about how a discussion of the idiom “to wash one’s hands of something” had revealed that her students had no idea who Pontius Pilate was.

The woman was sitting too close to him. He was reminded of the old joke, from his residency days: “Will you kiss me, Doctor?” “Kiss you! I shouldn’t even be on this couch with you!”

People curious about illness, seeking advice about symptoms, maybe rolling up a sleeve to expose a mole—what doctor didn’t get some of that outside the office? Like the surgeon he knew who’d once been asked by a woman at a party if he’d slip into a bathroom with her and check out a breast lump. Many of the questions put to him began with “Is it normal/abnormal . . .” Is it normal for a person to want to stay single forever? Abnormal for someone to love their partner and yet be turned off by them in bed?

But he didn’t have to get into a conversation with Greensleeves. He didn’t have to say what he knew about psychosis and auditory hallucinations. He didn’t have to stay on the couch where she was sitting too close to him. He could excuse himself and—

“Is it my imagination, or are there more crazy people out there these days, especially in the subway?”

A man was passing behind them, in the narrow space between the couch and a wall of books. He was carrying an empty wineglass, which he’d been on his way to refill when he overheard the woman. Earlier, this man and his wife and the therapist had all arrived at the party at the same moment and had been introduced to one another by the host when she greeted them. Maybe because he didn’t expect to see any of these people again, the therapist had immediately forgotten the names of everyone he’d met that evening. Really, he didn’t know what he was doing there. Though he and the host had been neighbors for decades, with apartments at opposite ends of the same floor, they hardly knew each other. She lived alone, an elderly widow, a minor socialite who entertained often but had never before invited him to one of her soirées. Tonight’s occasion was the publication of a book by a journalist she knew (as did he, though only by name). She was dressed theatrically in a flowing black-and-orange wrap and the black snood turban she preferred to a wig. (The hair lost during her treatment for cancer some years ago had grown back only in sparse wisps.)

The therapist had a feeling he knew why he’d been invited, which had happened at the last minute. The day before, he and the host had run into each other by the elevator. In the bit of small talk they’d shared going down—about what, he couldn’t remember—he’d noted something in her attitude toward him, something subdued, searching. An air of sympathy, it seemed. He thought at first it was for the loss of his mother, but that had happened half a year ago, and how, in any case, would this woman have known about that? Then, as they were about to go their separate ways, she said, “You know, if you’re free tomorrow night, I’m having a little party.”

Nothing about his wife.

He guessed that it had been the building gossip, the landlord’s mother, who watched the world from her ground-floor window or an armchair in the lobby: she must have passed on to the widow in 14-F that the wife in 14-A—the tall one with the nice clothes and her nose in the air—had moved out.

Taken by surprise, he hadn’t known how to decline. An act of neighborly kindness, after all, of sympathy. But immediately he regretted it. He’d never been one for parties, especially not when they were full of strangers. He’d promised himself that he would not stay long; it was enough just to put in an appearance. These were not people he needed to get to know. And he, too, was moving out of the building soon, though he hadn’t mentioned this to the host.

She sailed about the living room like a giant monarch butterfly, briefly landing on this or that knot of guests. From time to time, she exchanged a look with the young woman sitting beside the therapist on the couch. Once the host’s personal assistant, the young woman had then attended film school and was now working on her first docufiction. The two women were close—May-December besties, they called themselves. They had the kind of bond that enabled them to communicate telepathically.

Very attractive, yes, but too old for you.

It’s not like that. We’re not flirting. We’re just talking.

Well, maybe you should change the subject. He doesn’t look like he’s having a very good time.

He’s not really listening. His head is off somewhere in the clouds.

And those are probably dark clouds. As I told you, his wife just walked out on him. But he’s not your responsibility. If he’s being a bore, why don’t you get up and mingle?

I will soon. You look great, by the way.

Thanks, my dear. So do you. Love the nail color!

***

“You’re not supposed to say ‘crazy’ anymore,” a woman sitting in an armchair on the other side of the coffee table admonished. She was holding a canvas tote bag out of which peeked the pert snout of a miniature Yorkie. Every few minutes, the dog had a fit of shuddering, as if some ghastly thought had occurred to it.

“Why’s that?” the man behind the couch asked.

“Because it’s offensive. It’s what they call ableist.” At this, everyone listening did something with their eyes—rolled them or scrunched them or looked down at the floor. “You’re supposed to say ‘mentally ill.’ ”

“But what if you mean to be offensive?” the man said. “As in ‘What are you, fucking crazy?’ ”

Ignoring this, the woman went on, “It’s not just in the subways. I don’t know about you, but I find that people are behaving like they’re mentally ill everywhere I go. I was in the supermarket the other day, and a security guard pointed out that I’d dropped a glove. I picked it up and thanked him, and, what do you know, he lit into me! ‘I don’t need your thanks, lady,’ he said. ‘I am doing my job. I see everything. That is my job. I don’t do it for your thanks.’ I was stunned. I would have been afraid if we’d been alone—he was that hostile. And all because I said ‘Thank you’!”

“I had something like that happen to me, too,” another woman said. “An Uber driver yelled at me because I said, ‘It’s right there,’ as we were driving up to the restaurant. He said, ‘I know where it is! I have the G.P.S.! I don’t need you to tell me!’ I never got out of a car so fast.”

A convergence of people at the center of the room now hemmed the therapist in, making it awkward for him to try to escape. There was a copy of the journalist’s new book on the coffee table. The journalist himself was standing nearby, talking with a woman who bore a strong resemblance to the therapist’s wife. Same chin-length brown hair, similar elfin profile, and what looked like an identical navy-blue tailored suit.

This was just an observation, not cause for pain. He did not miss his wife. He did not want her back. Although they had never been enemies, they had not been friends for a long time. Their love story had followed a familiar trajectory: hot, warm, cool, cold. By the end, they had often felt invisible to each other. Yet they might have gone on like that for years, as spouses do, had she not, in the course of her travels as an art curator, met someone else. Once she’d moved into her artist lover’s loft, the therapist was eager to downsize. He was about to close on a place that was both smaller and less elegant than where he lived now but also conveniently situated in the building where he had his office.

He picked up the book and started flipping through it. He already knew what it was about. The author was part of that league of current pundits who were trying to master the trick of candidly laying out the existential perils facing humanity without totally devastating the reader. The book was “a brilliant balance of fear and hope,” according to the publisher. And its subject matter was increasingly relevant to psychotherapy. The last conference the therapist had attended was on the effects of climate anxiety on mental health. For years now, he’d been seeing patients with symptoms of pre-traumatic stress disorder, emotional problems associated with uncertainty—if not outright dread—about the future. And recently this had also hit close to home.

***

If his head was lost in dark clouds, it was not because of his wife but because of his daughter. Their only child, a girl with an ebullient, affectionate personality whose behavior had never given her parents a day’s grief, she had been the marriage’s perpetual blessing. Astonished that their flawed selves could have got this one thing so right, they attributed it to her own essential goodness. Because she had grown to be both self-accepting and self-reliant, genuinely contented with life, they hadn’t been dismayed that, unlike the two of them, she had not been an outstanding student, or that, once out of school, she had shown no compelling desire to pursue any particular career. But, though she lacked ambition and focus, she was an avid reader, she was intellectually curious, and she seemed incapable of laziness or boredom. She was also undeniably happier than most of her striving, perfectionist peers, and she did not envy them. She did not appear to envy anyone. She was good at making and keeping friends. She was diligent at the various part-time jobs she took on to patch together a living. Neither of them minded when she couldn’t make the rent and was forced to move back home for a time. Husband and wife always got along better when their daughter was around. They gave thanks when they compared themselves with other parents, many of whom struggled in their relationships with their children or had even become alienated from them. And so much for the old canard about how children of shrinks grow up to be emotional wrecks.

It was at a friend’s wedding that their daughter met the man with whom she herself hoped to one day be joined in marriage, and to have at least one child. Motherhood was something she had always wanted and assumed would be part—surely the most wonderful part—of her life. She and her boyfriend moved in together a few weeks before the emergence of COVID. Months of lockdown deepened their attachment, but now the therapist’s daughter had begun questioning the morality of bringing children into such a broken world. What mother today could be assured that her child would have a good life? Its birth would occur in the midst of numerous ongoing global crises, and the therapist’s daughter was haunted by her vision of the damage that a child might have to endure after she was gone.

Her boyfriend respected her feelings, though he did not share her pessimism. He reminded her that some of the smartest, best-informed people—scientists and environmental and political activists among them—had not forgone having kids. There were steps that could be taken to prevent the worst effects of climate change, to save democracy, to force the arc of the moral universe toward justice.

But the only way for any of that to happen, and for disaster to be averted, she said, was for people around the world to come together, to work together. And what did she see? Everywhere, escalating division and strife and a hardening of the belief that the way forward lay not in communal problem-solving but in selfishness, demagoguery, and violence. Enraged mobs of people who hated one another so much that they would sooner die than work together.

If she came to regret not having a child, she could live with that. What she never wanted to face was regret for having had one.

Her father had been very careful not to say anything that might influence her decision—as he was very careful not to influence the decisions of several of his patients who were wrestling with the same issue. Her mother, on the other hand, was having none of it. She accused her daughter of being “defeatist.” She did not want to be cheated of grandchildren. “I’ll never forgive you,” she said, unforgivably.

The therapist did not see the situation as hopeless. First of all, his daughter could change her mind again (there was still some time for that), and, if she did not, perhaps she’d consider adoption. What concerned him more was the dramatic alteration in her temperament. She who had never before suffered from depression now went spiralling into the abyss. Dutifully, she carried on, uncomplaining, but in a wan, robotic version of herself, indifferent to pleasure, looking forward to nothing. She dismissed the idea of going into therapy or taking any type of medication. “I’ll be all right,” she kept saying. But a year passed, and she was the same.

When the therapist learned that she had given a cousin some of her clothes, including a brown leather jacket that he knew she loved, he was alarmed. Giving away possessions was something people contemplating suicide were known to do.

His daughter laughed in exasperation. “For God’s sake, Daddy. You’ve never heard of decluttering? I was just tired of those old things.”

Then, suddenly, she decided to take a trip. She wanted to go somewhere beautiful. And she wanted to go alone. She and her boyfriend needed some time apart. That the relationship had been undergoing strain was hardly surprising. Though her boyfriend insisted that he wished to spend his life with her, with or without children, she was torn and could not help feeling guilty.

Again, the therapist saw this as a warning sign. People who took their lives often went away from home to do so. Hotel suicides were not uncommon. And again his daughter scoffed. “Seriously? You really think I’d go halfway around the world, and put you through all the trouble of shipping my corpse?”

***

It was one of those moments which sometimes happen at a gathering: in the midst of a lively conversation comes a lull, when for no apparent reason everyone falls silent.

The woman who’d been talking with the journalist spoke first. “Someone once told me that in Turkey, I believe it was, when a room goes quiet all at once, people say, ‘Somewhere a girl child is born.’ Meaning, when a son is born it’s an occasion for cheers and celebration, but when it’s a daughter no one knows what to say. Thus the awkward silence.”

Everyone groaned, the Yorkie shuddered, and in a loud, displeased-sounding voice the young woman on the couch said, “My family is Turkish, and I’ve never heard that.”

Just then the therapist’s phone, which was resting on the coffee table, pinged. He glanced at the message: “I’m still depressed, but now I’m depressed with a view.”

A laugh escaped him, and, feeling the need to explain, he said simply, “My girl child.”

Everyone else laughed then, too, and the host raised her glass in the air.

“To daughters!” she said.

“To daughters!”

***

Was he surprised that she followed him? Probably not. For the hour or so more that he’d stayed at the party, she had either sat beside or shadowed him, and when, at last, he found a suitable moment to leave, there she was, apparently ready to leave, too. Instead of stopping at the elevator and pressing the button, she kept right on walking with him to his door.

He thought better of offering her any alcohol, even though that meant that he couldn’t have any, either. They’d both had enough. He made them each a cup of ginger tea instead, which they drank sitting in his kitchen.

She had a brother who’d heard voices for years, she said, and for years he’d hidden it, even from her, his big sister, with whom he had always been close. Until recently, he’d been in law school, where, according to the voices—which had gone from speaking only occasionally to almost never shutting up, from speaking softly to shouting—he was being programmed to fight in an imminent battle for planetary control. His professors were all in on it, a Latin-speaking cabal with secret but decidedly malign intent. To escape them, he had quit school and holed up at his parents’, where, despite its being the house he’d grown up in, he now felt like too much of a stranger to stay. He’d settled into a tree trunk in the town park, though he didn’t entirely trust the pigeons there: some might be carrying the enemy’s messages.

After a fistfight with another vagrant, he’d sought shelter in a private garden, where he frightened the owners by refusing to leave. An empty soda can he’d been holding had flown through the air, and, though it missed its mark, he was arrested for trespassing and attempted assault. He had landed not in jail but in a hospital, where he was diagnosed with schizophrenia.

“I’ve been trying to learn as much as I can,” she said. “Everyone says it’s incurable, but to me that feels like giving up. It’s like the doctors have just written him off. Pump him full of drugs, and they’re done. They definitely calmed him down, but they haven’t brought him back to anything like his true self.”

And what if the diagnosis was wrong? According to her research, misdiagnoses of mental illness happened all the time. She’d also learned that other diseases—of the body, not of the mind—could trigger psychosis.

This was the subject of the film she had started working on, a docufiction about her brother and his illness. More docu than fiction, she said. An investigation.

“He could have something metabolic,” she said. “It could be a reaction to some kind of toxin he was exposed to.” Or what if he was afflicted by some mental illness other than schizophrenia, one that was curable, or at least less severe?

He couldn’t tell her what she wanted to hear. He could only follow the rule for discussing catastrophic diagnoses with loved ones: Be honest, but not brutally so. Do not encourage false hope.

“Anything can happen,” she said staunchly. “Like those people trapped for years in a coma who suddenly one day wake up.”

There was an intensity, almost a radiance, about her that the therapist had not observed earlier, and which he took as a sign of love. Her fierce love for her poor brother, against whose demons she was girded to fight.

He hadn’t eaten at the party, and when his stomach growled, immediately, as if in response, hers growled back. For the first time, he saw her smile, a generous smile that showed her fine teeth and beautified her face.

In the fridge was some leftover chicken and broccoli he’d ordered from a Chinese restaurant the previous night, which he now heated in the microwave. Small though the portions were, she did not finish hers.

He had admired her eyes before, but not until now her long eyelashes, which his heart kept catching on.

He could not offer an opinion about a patient he’d never seen, he explained, and he apologized for not being able to be of more help. But not only was he not an expert in the treatment of schizophrenia; he wasn’t taking on any new patients. (This was true, though he didn’t add that he’d been thinking of retiring from practice altogether and limiting himself to teaching.)

He had assumed that she lived somewhere in town, but in fact her home was in another state; she had taken a train to come to the party. Were it not so late, she would have stayed over in 14-F, as she had often done before. But she thought that her friend had probably gone to bed by now.

There were two spare bedrooms in the therapist’s apartment. She chose his daughter’s.

After they’d said good night, he lay awake in bed, waiting for her to come to him.

***

He watched her tiny figure far below, getting into the taxi the doorman had hailed for her. It had rained in the night, and now a weak light cast a pearlescent sheen over all, softening the city’s sharp edges. A world deceptively at peace.

He wandered through the apartment with his coffee mug. To think that in another month he’d be gone, never again to see these rooms that once brimmed with family life. The happy first years with his wife and a baby girl whose birth could not have been celebrated more. Yet it had come to this: his wife estranged, his precious only child in anguish.

In his daughter’s room, the bed was still made, but there was an indentation in the duvet, where he pictured his lover sitting, head in hands, debating.

***

Down the hall, his neighbor had woken up early, too, as was her habit. The night before, after she’d closed the door on the last guest, she had sat down in the living room and started to cry. It was what she always did after a party. Whether she judged the party to have been a success or a failure, it always ended in tears. She had a good cry, and then she had a cigarette—one of the three a day that she permitted herself—and when she had finished smoking she took a pill and went to bed. The next day, as always the morning after a party, she woke up feeling blue, a mood that could be expected to persist till nightfall.

Her P.P.D., she called it. Post-party depression.

Whenever she had trouble getting out of bed, she remembered what an analyst from many years ago had told her: Put your feet on the floor.

Her feet were still dangling when the phone rang: her young friend, calling from the train.

They chatted about the party (definitely a success, this one) before turning to what had happened later, in 14-A, upon which she became aware of a disagreeable sensation, like a bout of reflux.

“So you really think he can help?”

“I don’t know how much he can do, but it would be great to have someone with his expertise as an ally. He could be so useful—not just for my brother but for the film. Maybe I could even get him to do an onscreen interview.”

“Speaking of the film, be sure to let me know if you need more money.” (She had already agreed to finance the whole project.)

“I will, thanks.”

“You want to see him again, but does he want to see you?”

A spurt of completely inappropriate jealousy was what it was. And apparently not to be tamped down.

“He didn’t say anything about a next time. To be honest, he seemed hesitant. Which is understandable, I guess, given that he and his wife just separated. But I do think he likes me. Not to be graphic, but he was super passionate. And he obviously wanted to please me.”

An image of the two of them, unnerving in its obscene splendor, flickered on the bedroom wall.

“So, what are you going to do? Ask him on a date?”

“I’d rather not be the one to get in touch first. I mean, I’ve already come on pretty strong. I don’t want him to think I’m desperate, you know, or looking to catch a rich husband, or whatever.”

“Ah, yes. That hasn’t changed, has it? The woman always has to play it cool.”

“I don’t suppose you’re thinking of throwing another party anytime soon?”

“No. But what I could do is invite you both to dinner.”

This was what made them besties.

***

He decided to walk the two miles to his office that day. He needed the exercise, and he was a firm believer in the usefulness of walking for reflection, for generating good ideas or clearing the head of bad ones. He’d still get there in time to call his daughter before he had to see his first patient. He’d made her promise that they’d text or talk every day while she was abroad.

It was not an idea but a tune that came to him as he walked: an old English folk song that, in college, he’d learned to play on the guitar.

When he got to his office, he turned on his computer and typed “Greensleeves” in the Google search bar. He’d always thought of it as a romantic song, a poignant expression of yearning love—how had it ever become a popular carol about the birth of Christ?

Now he learned that it was only a legend that it had been written by Henry VIII for Anne Boleyn, during the time he was courting her. In fact, the true composer remained unknown.

There were numerous versions online. Guitar, piano, flute, lute, violin, sax, orchestra, rock band, a cappella. Even a solo whistler.

James Taylor. James Galway. The King’s Singers. Olivia Newton-John. Lynyrd Skynyrd. Vaughan Williams. John Coltrane. Everybody loved that song.

He chose a recording by Marianne Faithfull that had been made in 1964.

At one point in the four centuries of her existence, Lady Greensleeves was thought to have been a prostitute, or at least a woman of loose morals. A reference to the color green was said to have been code for sex when the song was written. A green gown, for example, was suggestive of hanky-panky in the grass.

Quite a stretch, if you asked him.

It was named by many the world’s most beautiful love song.

He played it again.

And again. ♦

Day 121

By Solvej Balle
Translated by Barbara J. Haveland

The first two volumes of Solvej Balle's septology, On the Calculation of Volume, from which this extract is taken, will be published next month by New Directions, Barbara J. Haveland lives in Copenhagen.

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***

There is someone in the house. Heard as he moves around room upstairs. When he gets out of bed or when he goes down the stairs and into the kitchen. There's the gush of water through the pipes when he fills a kettle. The sound of metal on metal when he sets the kettle on the stove and the very faint click when he turns on the gas. Then there's a pause until the water comes to a boil. There's the rustle of tea leaves and paper as first one, then another spoonful of tea leaves is taken from a paper bag and poured into the teapot, then the sound of water being poured over the tea leaves, but such sounds can be heard only in the kitchen. I can hear the fridge being opened, because the door bumps against a corner of the countertop. Then there's another pause, while the tea steeps, and in a moment I'll hear the chink of a cup and saucer being taken from the cupboard. I don't hear the sound of the tea being poured into the cup, but I can hear footsteps moving from the kitchen to the living room as he carries the cup of tea through the house. His name is Thomas Selter. The house is a two-story stone cottage on the outskirts of the town of Clairon-sous-Bois in northern France, No one enters the back room overlooking the garden and the woodpile.

It is the eighteenth of November. I have got used to that thought. I have got used to the sounds, to the gray morning light and to the rain that will soon start to fall in the garden. I have got used to footsteps on the floor and doors being opened and closed. I can hear Thomas going from the living room to the kitchen and putting the cup down on the countertop and before long I hear him in the hall. I hear him take his coat from its peg and I hear him drop his umbrella on the floor and pick it up.

Once Thomas has gone out into the November rain, there is silence in the house. Broken only by my own sounds and the soft patter of rain outside. There is the scratch of pencil on paper and the scrape of the chair on the floor when I push it back and get up from the table. There are the sounds of my footsteps as I cross the room and the very slight creak of the door handle when I open the door into the hall.

While Thomas is out, I usually wander around the house. I go to the toilet, get water from the kitchen, but I soon go back to the room. I close the door and sit down on the bed or the chair in the corner, so I won't be seen from the garden path should anyone look in.

When Thomas returns carrying two thin plastic bags, the sounds start up again. The key in the door and shoes being wiped on the mat. The crinkle of the plastic bags when he sets down his shopping. The sound of the rolled umbrella, which he lays on the chair in the hall, and a moment later that of his coat being hung on the rack by the door. I hear the repeated crinkle of plastic as he places his shopping bags on the countertop and puts things away. He puts cheese in the fridge, pops two cans of tomatoes into a cupboard, and leaves a bar of chocolate on the countertop. When the bags are empty, he balls them up and stows them in the cupboard under the sink. Then he closes the door and leaves them to carry on crinkling in there.

During the day I hear him in the office upstairs. I hear a swivel chair rolling across the floor and the printer churning out labels and letters. I hear footsteps on the stairs and the gentle thud on the floorboards as Thomas sets down packages and letters in the hall. I hear him in the kitchen and the living room. I hear the sound of a hand or a sleeve brushing the wall as he goes back upstairs, I hear him in the bathroom and I hear a sound from the toilet that can only be made by a man peeing standing upright.

Presently I hear him on the stairs and in the hall again, then he goes into the living room and sits down in an armchair by the window facing the road. And there he waits, reading and watching the November rain.

He is waiting for me. My name is Tara Selter. I am sitting in the back room overlooking the garden and the woodpile. It is the eighteenth of November. Every night, when I lie down to sleep in the bed in the guest room, it is the eighteenth of November and every morning, when I wake up, it is the eighteenth of November. I no longer expect to wake up to the nineteenth of November and I no longer remember the seventeenth of November as if it were yesterday.

I open the window and throw out some bread for the birds that will soon be gathering in the garden. They show up when there is a break in the rain. First the blackbirds, who peck at the last apples on the apple tree or the bread I've thrown out, and then a lone robin. Moments later a long-tailed tit flies down, closely followed by some great tits who are promptly seen off by the blackbirds. Shortly afterward the rain comes on again. The blackbirds carry on feeding for a little while longer, but when the rain gets heavier they fly off to take shelter in the hedge.

Thomas has lit a fire in the living room. He has brought in wood from the garden shed, and I soon start to feel the house growing warmer. I heard the sounds from the hall and the living room, but now that Thomas is sitting with his reading, all I can hear is my pencil on the paper, a whisper soon eclipsed by the sound of the rain.

I have counted the days and, if my calculations are correct, today is November 18, No. 121. I keep track of the days. I keep track of the sounds in the house. When it is quiet I do nothing. I lie down and rest on the bed or I read a book, but I make no sound. Or hardly any. I breathe. I get up and tiptoe about the room. The sounds carry me around. I sit on the bed or gently pull the chair out from the table by the window.

In the middle of the afternoon, Thomas puts on some music in the living room. First I hear him in the hall and the kitchen. I hear him putting a kettle on the stove, hear his footsteps on the floor as he goes back through to the living room and puts on the music. Then I know it will soon clear up. The clouds will pass and there will be a glimmer of sunshine.

I usually get ready to go out as soon as the music starts. I get up and put on my coat and my boots. I stand by the door for a few moments until the music is so loud that I can leave the house without being heard, the strains issuing from the living room masking the sound of doors being opened, of footsteps on the floor and doors being closed.

I leave the house by the garden door. I slip my bag over my shoulder, softly open the guest-room door, step into the hall and close it behind me. On the floor are three medium-size envelopes and four brown cardboard packages with our name on them: T. & T. Selter. That's us. We deal in antiquarian books, specializing in illustrated works from the eighteenth century. We buy the books at auction, from private collectors or other booksellers and then sell them on, sending them off in brown packages bearing our name. I slip silently past the packages on the floor, open the door, and step outside. I don't need an umbrella. It is still raining slightly, but it won't be long before it stops completely. I don't take the garden path leading up to the gate, instead I turn left and go around the side of the house, past the shed and down to a corner of the garden that can't be seen from the house. I pass a plot of leeks and two rows of Swiss chard and come to a gap in the fence, which brings me out onto the road. I glance back briefly. I see a thin trail of smoke coiling up from the chimney, hear the very faint sound of music, but I hurry on. A few steps further and I can hear neither music nor rain because the rain has stopped, the music has been left behind me, and the only sounds are my footsteps on the sidewalk, the rumble of a few cars and the distant voices of children at a school some blocks away.

Not long afterward, when Thomas sees that the rain has stopped, he turns off the music. He puts on his coat and picks up the pile of letters and packages from the hall floor. He leaves the house at 3:24. Carrying letters and packages. T. & T. Selter. That's us. But time has come between us. We walk along the narrow roads into town or back to the house. We are outside, we walk around in a break in the rain, but we do not take the same roads. He does not expect to meet me along the way, nor will he. I know another route, and by the time he returns to the house I am back in the room looking out on the garden.

If there is anything I need, I buy it in a small supermarket a few blocks away. I allow plenty of time and usually take a roundabout way home. I come through the gate and up the garden path to the back door and let myself in. The house is quiet. Thomas is out and it is no longer raining. He is on his way into town and by the time he has dropped off his packages the sun will have broken through. He will take a walk through the woods and down to the river and won't return until late in the afternoon, after the rain has started again, because there is no one waiting for him in the house and there is nothing he has to do.

Usually, when I return, I leave my shopping bags in the guest room. I hang my coat over the back of the chair, I take off my boots and go through to the kitchen. There's a cup next to the sink and the kettle on the stove is still slightly warm. I can follow Thomas's tracks through the house. I go upstairs and into the office. There are piles of books on the desk and a scattering of papers. There are books on the shelves and in boxes on the floor. One of the boxes is open because Thomas has been searching through it for something and hasn't closed it again. In the bedroom next door to the office, it looks as if someone has just got up, but only one side of the bed has been slept on.

I have an hour and a half in the house before Thomas gets back. I have time to have a bath or wash some clothes in the sink, I have time to take a book from the shelf and sit down with it in one of the armchairs by the window.

If I spend the time in the living room, I usually listen to music or read until it starts to get dark, but today I am staying in here, in the room overlooking the garden and the woodpile. I heard Thomas take his coat off the peg and I heard him leave the house. I opened the door into the hall, the packages are gone from the floor, and now I am sitting at the table by the window. It is the eighteenth of November. I am becoming used to that thought.

***

On the morning of the seventeenth of November, I said goodbye to Thomas at the door of the house. The time was a quarter to eight, the taxi was waiting outside, and I caught the 8:17 train from Clairon-sous-Bois. I was going to Bordeaux for the annual auction of illustrated works from the eighteenth century. The sky was gray, there was moisture in the air, but it didn't come to rain.

From Clairon station I traveled to Lille-Flandres, switched to Lille-Europe, and went from there to Paris, where I changed trains for Bordeaux. I got to the station in Bordeaux just before two o'clock and after a moment's confusion due to road work outside the station, with lots of barriers, signs, and closed walkways, I found my way to the exhibition center where the auction was to be held, arriving there a few minutes later. I registered and was given a program and a badge, which read 7ÈME SALON LUMIÈRES, followed by my name and, below that, our company name, T. & T. Selter.

I was in good time for the main auction of illustrated books, which was due to start at three o'clock. A couple of auctions had already been held, and I could see from the program that again this year there would be talks and panel discussions, but none that I was planning to attend.

Once inside, I faltered for a moment, slightly disoriented by a scene that spoke of a conference in progress, all closed doors and abandoned coffee cups, until I spotted the signs and arrows pointing to the auction room and the adjoining exhibition hall, where scores of antiquarian booksellers had, as always, set up their stands displaying books and scientific illustrations. I had a fairly good idea of which books I wanted to bid for in the auction, and once I had taken a look at the most important of these, I did a round of the exhibition hall. I said hello to some booksellers I knew from before, and then, at a few minutes to three, I took my seat in the auction hall, which soon began to fill with people streaming out of the conference rooms.

I managed to buy twelve works at the auction. Five for which we had already had requests and seven others for which I thought we could get decent prices. We deal mainly in moderately priced books and sell to a mixed bunch of collectors, most of them in Europe, although we do have a few customers in other parts of the world. As a rule, I am the one who travels to auctions and visits antiquarian bookshops while Thomas takes care of cataloguing and shipping. To begin with we did everything together, but we have gradually split the responsibilities between us. I'm not sure why it fell to me to do the traveling. Maybe because I don't mind traveling so much and maybe because I very quickly developed a certain instinct for the books, a feel for the paper, an eye for the quality of the printing, for a well-crafted binding. I don't know what it is, but it's almost physical, like an inchworm testing whether a leaf is worth creeping across, or a bird listening to insects moving in the bark of a tree. It might be a detail: the sound when you flick through the pages, the feel of the lettering, the depth of the imprint, the saturation of the colors in an illustration, the precision of the details in a plate, the hues of the edges, I don't know what it is that seals it for me, because even though I generally know which works I'm interested in, usually I'm not absolutely sure whether I want to buy a book until I have it in my hand.

After the auction, I went back to the exhibition hall, paid for some books I had had put aside, and found another six works that I had been looking for, as well as a few I hadn't known about. I usually arrange for the heaviest and most valuable books to be sent straight to Clairon-sous-Bois, but I often take a few of my purchases away with me in my bag: on this occasion, a pocket encyclopedia of birdcalls arranged by note, new to me; a second edition of Harcard's book on the anatomy of animals; and a fine copy of Boisot's renowned book on spiders, Atlas des Araignées, which one of our regular customers wished to give as a present to a friend and which we had therefore promised to look out for.

Late in the afternoon on the seventeenth of November, I boarded the train to Paris and arrived late that evening at the Hôtel du Lison, where we usually stay when we're in town. The Hôtel du Lison sits on the corner of the Rue Almageste, where several of the antiquarian booksellers with whom we deal have their shops and where our good friend Philip Maurel has his antique coin shop. My plan for the next two days, apart from buying books for our business and seeing Philip, was to visit the research library Bibliothèque 18, in Clichy, where had an appointment the following day, the nineteenth of November, with a librarian by the name of Nami Charet, who had written an article on some hitherto overlooked variations on eighteenth-century printing techniques. Her discovery of certain changes to engravers' tools and working methods had made it possible to date illustrations from the end of that period with great accuracy, thereby shedding light on discrepancies between their genesis and year of publication.

Once at the hotel, I called Thomas. It wasn't a long conversation. I told him about the day's finds and asked him if he had any titles to add to my list for the next day. He had come up with a couple of works that he thought would be worth looking for, and he had also just arranged with two of our fellow dealers in the Rue Renart, a side street off the Rue Almageste, to have a couple of books set aside. He asked me to examine them and to buy them if they were in decent condition. I added these titles to my list and promised to go take a look at them. I think we chatted a little more about the auction and a bit about the November weather before we said our good nights and hung up.

We try to avoid long phone calls when we're apart. Not only because otherwise we would just end up having long, detailed dialogues on the condition of the books, publication years, the illustrations, and the pricing, but also because such conversations seem to increase the distance between us. The minute we deviate from simple, practical matters, the conversation lapses imperceptibly into a kind of audio link, a muted love mumble. Our communication, initially meaningful and coherent, turns into a series of fitful exchanges containing neither sentences nor information: little words and sounds meant—I suppose—to keep the line between us open, but that, instead, make all too clear how far apart we are. So we have learned to spit the work between us, stick to practical matters, and speak to each other only when necessary.

I've forgotten many of the details of our conversation, but Thomas-who remembers the seventeenth of November as if it were yesterday—has told me that I had spoken in glowing terms of my day's haul and that I had been wondering whether T. & T. Selter should expand its operations to include scientific plates and illustrations. We talked mostly about the practical problems associated with my suggestion, particularly regarding shipping, which would probably be Thomas's job. I felt the idea was worth considering, but Thomas was not so sure.

I don't recall the rest of the conversation, but I do remember that shortly afterward I had a bath and then I settled myself on the bed in my hotel room and ran an eye over my book list. I also remember that I was quite tired from all the traveling, that I set the alarm on my phone, got undressed, and climbed into bed.

***

I still don't know whether it would be a good idea to for T. G. T. Selter to expand into plates and illustrations, but I do know that such considerations no longer have any meaning. I know too that Thomas left his packages at the post office a while ago, that he has been down by the river and past the old water mill, and that he has walked through the woods and will soon be back.

I keep an eye on the rain clouds. The clouds tell me when time is up. The light fades, and I see the sky change to dark gray. If I'm sitting in the living room with a book, it grows too dark for me to read and I prepare to retreat to the guest room. I sit for a moment listening to the rain and, when it intensifies, I know that Thomas will soon be back. I get up from the armchair by the window. I go to the kitchen, rinse my cup in the sink, put it in the cupboard, and come into the guest room. I usually remember to turn the heat up before I leave the living room. The embers in the grate have long gone cold, and when Thomas gets back he'll be drenched.

But I am not sitting in the living room today. I am sitting at the table in the guest room and the rain clouds are gathering once more. I gaze out at the garden and the apple tree. A couple of apples have fallen onto the grass, a light breeze has blown the autumn leaves almost dry while I've been sitting here, but the tree will soon be wet with rain again. I can still see the birds when they dart about in the faint light. They can tell that rain is on the way, but they haven't yet gone to roost in the hedge.

Darkness falls as I wait for Thomas to go by outside. The writing on the page in front of me becomes hard to read. I've shut the door to the hall and moved away from the window. I usually sit on the bed while I wait for Thomas to get back, and I know that first one and then another dark shadow will come along the road at the bottom of the garden. The first shadow is our neighbor. The second is Thomas making his way through the rain. This is the only time I see him: a wet silhouette down by the fence. The rest of the day he is just sounds in the rooms of the house.

Not until Thomas has once again become sounds in the rooms of the house do I turn on a light. I heard his footsteps on the garden path, his key in the lock, and the door being opened and closed. I heard him wipe his feet on the mat and the faint click of the switch when he turned on the hall light. I can see light filtering in under the door, and I have turned on the lamp on the table. My light fills this room and filters out under the door, but it cannot be seen from the hall, mingling as it does to invisibility with the light out there.

I have moved back to the table by the window, and before long I hear Thomas's feet on the stairs and in the passage again. I hear him in the kitchen and the hall. I hear him open the door facing the road and go out to fetch a leek from the garden and some onions from the shed. I can hear him pulling on the pair of rubber boots by the door. I can hear him walking down the side of the house, and then nothing until he returns with his vegetables. I hear him chopping vegetables for soup. Hear the rattle of the pot on the stove and, once the soup is ready, the scrape of chair legs on the kitchen floor. A little later, I hear the gush of water through the pipes as Thomas washes his plate in the kitchen sink, then I hear him putting the plate back in the cupboard before going to the living room. He spends his evening reading Jocelyn Miron's Lucid Investigations, and it's almost midnight before he switches off the hall light and goes upstairs, but that is a while off yet, the evening is just beginning. Thomas is getting changed in the bedroom above, and I am remembering a long succession of November days that have begun to run together in my mind. There are 121 days to remember. If I can.

***

To begin with, the eighteenth of November was in no way unusual. I woke up in my hotel room at 7:30 and went down for breakfast half an hour later. I spent the day visiting various antiquarian bookshops in the area around Rue Almageste, calling in along the way at Philip Maurel's shop at No. 31. Philip's new assistant, whom I had not met before, thought he would be back toward the end of the afternoon, so I said I would pop in again around five o'clock. At the different bookshops where we normally buy things when we are in town, I found a number of the works I was looking for. I stopped in at one dealer's in the Rue Renart to look at the copy of Histoire des eaux potables that Thomas had found for sale and that one of our customers had made numerous inquiries about. It was a very nice copy, so I bought it on the spot and put it in my bag, to be taken back with me to Clairon-sous-Bois the very next day for Thomas to forward to our impatient client. At the same dealer's, I found a couple of other works, which I bought and asked to be sent to Clairon, and in another I picked up a copy of Thornton's Heavenly Bodies in good condition and containing two plates included only in this one edition, published in two small printings in 1767.

Just before five o'clock, I descended the steps to Philip Maurel's shop again. It had been a while since I had seen him, six months, maybe more, and we sat for a while chatting at the big desk in the front of the shop, with Philip getting up every now and again to serve a customer or to answer the phone. I told him about the house in Clairon-sous-Bois, which he hasn't yet visited, although we've been there for a couple of years now. I spoke about love, about the apple tree in the garden, about leeks and Swiss chard. I told him about the autumn floods, about the river that occasionally overflows its banks, and about our business, from which we were at last able to make a living, about the growing demand for illustrated works from the eighteenth century, about the auction and my latest finds. Philip told me about life on the Rue Almageste and his new girlfriend, Marie, whom I had met in the shop earlier that day, about the political unrest that autumn, about the high demand for rare and not-so-rare relics of the past— a trend he had also observed in his own business.

Philip specializes in coins from imperial Rome, a business that, when he first set up shop as a very young man, most of his friends regarded as a bit of a joke, but which in recent years has proved to be a lucrative concern. He told me in wonder how, on several occasions, he had been invited to dinner by one of his regular customers only to find himself at the center of an interested crowd consisting not only of elderly gentlemen, but also of younger men and women, all eager to hear about imperial mint regulations, the details of coin-pressing techniques, or about somewhat suspect Belarusian coin discoveries. He spoke of hosts who would suddenly whisk visitors off through their vast Parisian apartments to show off a coin collection, not to the surprise or embarrassment of their guests at their host's weird hobby, but to their delight, studying their host's latest acquisitions intently through powerful magnifying glasses. Philip confessed that he had been rather surprised to see his own quiet passion receiving so much attention. These small metal tokens from a bygone age had apparently become the subject of a new collecting mania—not exactly a wave, perhaps, but certainly a growing interest, of which he was keenly aware, in his business.

We reflected a little on this demand for trophies from the past, and I showed Philip my copy of Histoire des eaux potables. We talked about how keen the buyer had been to get hold of it and wondered why someone would want to buy a book like this. Who buys an over-two-hundred-year-old book on the history of drinking water? A collector of some sort, but what exactly his collection consisted of and how the book was to figure in it was hard to guess. I knew nothing about him except his name, his address, and the sound of his voice on the phone when he had called for the third time, a few weeks earlier, to inquire about the book. I assumed him to be a middle-aged man, and I knew that he had purchased two or three other books from us before, although I couldn't remember which ones.

I still remember how we discussed with a certain irony and detachment this burgeoning interest in the treasures of the past. Although we too could be said to suffer from this same nostalgia or hunger for history or whatever you want to call it, we were both slightly surprised that it was becoming more widespread. I think we almost felt the urge to apologize for our strange interests, which we now evidently shared with more and more people. Well, at least we had managed to make a living out of them. They were our businesses, not a hobby or a fleeting fad. We had our companies, T. & T. Selter and Maurel Numismatique, to manage, and we felt, I suppose, that we had a somewhat more pragmatic relationship with our inner nostalgist than our customers.

Fortunately, Marie turned up in the middle of this conversation. Philip introduced us, we exchanged a few words about the current conversation and also about our having met each other earlier in the day. After a little while, Marie pulled up a chair and sat down at the counter while Philip went out to pick up some food and a couple of bottles of wine from a nearby shop.

***

It was one of the first cold November evenings. There had been a rain shower in the morning, the rest of the day had been cloudy with the occasional sunny spell, and it was starting to feel quite chilly in the shop. In the little kitchen at the back, where Philip had also slept for the first years after opening the shop, he had an old gas heater, which Marie and I decided to try to light. Marie wiped a layer of dust off the top and together we managed to maneuver the large blue gas cylinder sitting next to it into the back of the heater, attach it, and push the heavy appliance through to the front of the shop and over to the counter. We found a box of matches in a kitchen drawer and lit the gas. By the time Philip got back, the shop was warming up nicely. We sat around the counter and spent the next few hours eating, drinking, and talking.

What I remember most clearly from that evening is how much l enjoyed sitting there between Philip and Marie. They had a closeness I could not help but notice. Not the sort of unspoken awareness that shuts other people out, the self-absorption of a couple, in the first throes of love, who need constantly to make contact by look or touch, or the fragile intimacy that makes an outsider feel like a disruptive element and gives you the urge to simply leave the lovers alone with their delicate alliance. They had an air of peace about them, Marie and Philip, which reminded me of the time, five years ago, when I first met Thomas. The sudden feeling of sharing something inexplicable, a sense of wonder at the existence of the other—the one person who makes everything simple—a feeling of being calmed down and thrown into turmoil at the same time. Philip and Marie had clearly decided to spend the rest of their lives together, it was as simple as that, so what could they do but see what the future would bring. Seen in that light, my visit was a perfectly normal occurrence, I was an ingredient in their life together, a colleague of sorts and a friend, married to one of Philip's friends from his early teens. I was not in the way, I was neither a help nor a hindrance, neither an opponent nor a supporter, neither a buyer nor a seller; I was simply a fact in their lives.

I no longer remember the details of the evening's conversation, but I remember the atmosphere in the room. I remember at one point sitting there, studying a corner of the big old oak desk we were sitting around, which functioned as the shop's counter. I remember the marks in the wood and a little display of coins in transparent boxes, which we had to push to one side to make space for the plates and glasses. I also remember the warmth of the room and my accident with the gas heater. It happened some time after we had finished eating. I had pushed my chair back a bit, but the heat was almost overpowering, so I got up to move the heater a little farther away. I remember Philip had picked up our plates and gone out to the little kitchen at the back of the shop for a corkscrew to open another bottle of wine, and I remember saying something about how hot it was and that I would just move the heater slightly. Marie got up, but I had already placed a hand on its upper edge and given it a firm push away from the desk.

While we'd been sitting there, the heater had, of course, become red hot, and the metal edge on which I had laid my hand was scorching, so just as the heavy appliance started to shift, I felt a sudden, searing pain in my hand. I let out a cry, an expletive probably. Marie came over and managed to move the heater while I stood there, paralyzed for a moment by the pain. Having deposited our plates in the kitchen, Philip promptly reappeared with a bowl of cold water into which I plunged my hand, and for the rest of the evening I sat like that, with my hand immersed in a bowl of water, although this did nothing to ease the pain. That was the only unusual thing to happen that night.

I got back to the hotel just before eleven o'clock and shortly after that I called Thomas. He had been engrossed in Jocelyn Miron's book and sounded as though he hadn't been expecting me to call. I don't know whether I had disturbed his reading or whether he was glad of the interruption, but I remember how he filled me in on the central themes of the book and explained the different forms of Enlightenment thinking outlined by Miron. I remember too that we discussed the book's rather odd subtitle, "Rises and Falls of Enlightenment Projects," and then I told him about my visit to Philip Maurel, about his girlfriend, Marie, and their obvious love for each other, about the growing demand for Roman coins and my accident with the heater in Philip's shop. Thomas told me that it had rained for most of the day, apart from a break of a few hours in the afternoon. He had taken letters and packages to the post office and later, when the sun had broken through, he had decided to take a walk through the woods and along the river. He had gone as far as the old water mill, thinking that the rain had passed, but on the way home he had been caught in a heavy shower and got absolutely drenched. I remember how we talked about the risk of the river overflowing. He had said that, judging by the height of the water, if the weather forecast for the next few days held good then the danger of flooding was over for now. I don't recall all the details of our conversation, but I'm sure I reported on the day's finds, on prices and delivery arrangements, and I know we discussed my plans for the following day, the nineteenth, when I was due to meet Nami Charet at Bibliotheque 18.

Once or twice during our conversation, I complained about the still-painful burn on my hand and we had a little laugh at my thoughtlessness; it wasn't the first time I had come to grief for not heeding, as Thomas put it, the basic principles of cause and effect. He suggested I get some ice cubes to cool the wound, and then we probably exchanged a few inconsequential remarks. I remember one of us observing that we were slipping back into our old familiar audio-link mode, murmurings that accentuated the distance between us, and before we said good night a couple of times and hung up I think I promised Thomas I would go straight down to the hotel lobby for some ice cubes. That was the last time I spoke to Thomas before time fell apart.

I lay awake for a while, half an hour, maybe longer. I had got hold of a bag of ice cubes, laid the ice on the burn, and wrapped a towel around my hand. The pain and the cold kept me awake, and I tossed and turned a bit, feeling strangely restless in bed, but gradually the chilling effect or the tiredness or whatever it was caused the pain to recede and when I eventually fell asleep it was with a towel full of melting ice cubes around my hand and a small stack of books on the desk in the room: some notes on birdcalls, an atlas of spiders, a book on heavenly bodies, a work on the history of drinking water, and an encyclopedia of animal anatomy. On a chair next to the bed lay my phone, with the alarm set for 7:30 the next morning. Over the back of the chair hung a dress, a sweater, and a pair of tights, and on the floor were a pair of ankle boots and a large shoulder bag containing an extra dress, tights, underwear, a purse, a bunch of keys, an almost-empty water bottle, and a rolled-up umbrella.

I still have the bag beside me on the floor and the books are on the table next to the spare bed. I am no longer in the hotel, but in our house in Clairon-sous-Bois, in the room overlooking the garden. It is evening and it is still the eighteenth of November. Thomas is in the living room, reading, and soon he will switch off the light and check that the doors are locked. He will go upstairs and when he wakes up tomorrow morning his eighteenth of November will have been forgotten.

This is the 121st time I have lived through the eighteenth of November, and the burn is still visible as a slender scar on my hand. It started out as an angry, puffy weal. This soon began to weep, then a long, brownish scab formed over it. Little by little the scab loosened and fell off, leaving a shiny pink mark. It can no longer be felt and, in a moment, when I turn the light off, it will no longer be seen.

如何用 GIMP 拉直和摆正图片

我在学习《Option Volatility and Pricing: Advanced Trading Strategies and Techniques》，并扫描了书中的图片。扫描出来的图片歪歪扭扭的，需要把他们摆正。

在互联网上找到了一个教程——Straighten a Crooked Photo with GIMP，方法是先设置参考线（guide），然后用旋转（Rotate）工具进行调整。好在扫描的图片里有直线，设置参考线十分方便，旋转起来就比较容易。

随后，我又在 Reddit 上找到一个帖子——How to resquare an image?，介绍了如何使用测量（Measure）工具的拉直（Straighten）功能来快速校正。在回复中，有人提到可以使用透视（Perspective）工具来校正更为扭曲的图片。不过幸运的是，我扫描的图片没有太大的变形，因此无需使用这个工具。

大多数的图片可以直接用测量工具拉直就够了，但是有些图片还需要用旋转工具稍作精细调整。

第九章和第十章的图片都进行了校正，效果不错，我很满意。

Chapter 10 Bull and Bear Spreads

Chapter 10 Bull and Bear Spreads
第十章 牛市价差和熊市价差

While delta neutral strategies are perhaps the most popular among active option traders, many traders prefer to trade with a bullish or bearish perspective in the underlying instrument. The trader who wishes to take a directional position in the underlying instrument has the choice of doing so in either the instrument itself, buying or selling a futures contract or stock, or in the option market. If he chooses the option market, the trader has the opportunity to integrate option pricing theory into a bull or bear strategy in order to take advantage of theoretically mispriced options.

虽然 delta 中性策略在活跃的期权交易者中很受欢迎，但许多交易者还是更倾向于对标的资产采取看涨或看跌的立场。希望对标的资产进行方向性操作的交易者可以选择直接交易该资产——买入或卖出期货合约或股票，或在期权市场中操作。如果选择期权市场，交易者便有机会将期权定价理论融入看涨或看跌策略中，以利用理论上被错误定价的期权。

NAKED POSITIONS
裸头寸

Since the purchase of calls or the sale of puts will create a positive delta position, and the sale of calls or purchase of puts will create a negative delta position, we can always take a directional position in a market by taking an appropriate naked position in either calls or puts. If all options are overpriced (high implied volatility), we might sell puts to create a bullish position, or sell calls to create a bearish position. If all options are underpriced (low implied volatility), we might buy calls to create a bullish position, or buy puts to create a bearish position.

由于买入看涨期权或卖出看跌期权会形成正 delta 头寸，而卖出看涨期权或买入看跌期权会形成负 delta 头寸，因此我们可以通过适当的裸头寸来在市场中采取方向性操作。如果所有期权被高估（隐含波动率较高），我们可以卖出看跌期权构建看涨头寸，或卖出看涨期权构建看跌头寸。如果所有期权被低估（隐含波动率较低），则可以买入看涨期权构建看涨头寸，或买入看跌期权构建看跌头寸。

The problem with this approach is that, as with all non-hedged positions, there is very little margin for error. If we purchase options, we will lose money not only if the market moves in the wrong direction, but also if the market fails to move enough in our favor to offset the time premium in the option. If we sell options, we face the prospect of unlimited risk if the market moves violently against us. An experienced trader will always look for a way to improve the risk/reward characteristics of his position by looking for positions with the greatest possible margin for error. This philosophy applies no less to directional strategies than to volatility strategies. In directional strategies, as in volatility strategies, this can often be done by finding an appropriate spread.

这种方式的问题在于，与所有非对冲头寸一样，容错空间非常小。如果买入期权，不仅市场走势不利会造成亏损，即使市场未能足够有利于我们以抵消期权的时间价值也会亏损。而如果卖出期权，则市场剧烈波动可能带来无限风险。有经验的交易者总会寻找改善头寸风险回报的方式，确保头寸有尽可能大的容错空间。这一理念同样适用于方向性策略和波动率策略。在方向性策略中，也可以通过寻找合适的价差来实现。

BULL AND BEAR RATIO SPREADS
牛市和熊市比例价差

If a trader believes that implied volatility is too high, one sensible strategy is a ratio vertical spread. For example, with the underlying market at 100 suppose a June 100 call has a delta of 50 and a June 110 call has a delta of 25. A delta neutral spreader can:

buy 1 June 100 call (50)
sell 2 June 110 calls (25)

当交易者认为隐含波动率过高时，可以选择使用比例垂直价差策略。例如，在标的市场价格为 100 的情况下，假设 6 月 100 看涨期权的 delta 为 50，而 6 月 110 看涨期权的 delta 为 25。此时，采用 delta 中性策略的交易者可以：

买入 1 份 6 月 100 看涨期权（50）
卖出 2 份 6 月 110 看涨期权（25）

Since the spread is delta neutral, it has no particular preference for upward or downward movement in the underlying market.

由于该价差为 delta 中性，对标的市场的上涨或下跌没有特别倾向。

Now suppose the same trader believes that this ratio vertical spread is a sensible strategy, but at the same time he is also bullish on the market. There is no law that requires him to do this spread in a delta neutral ratio. If he wants this spread to reflect his bullish sentiment, he might change the ratio slightly:

buy 2 June 100 calls (50)
sell 3 June 110 calls (25)

现在假设同一交易者认为这个比例垂直价差是合理的策略，同时他对市场持看涨观点。没有规定必须按照 delta 中性的比例来进行价差操作。如果他希望反映自己的看涨情绪，可以稍微调整比例：

买入 2 份 6 月 100 看涨期权（50）
卖出 3 份 6 月 110 看涨期权（25）

The trader has essentially the same ratio vertical spread, but with a bullish bias. This is reflected in the total delta for the position of +25.

该交易者的比例垂直价差现在带有看涨倾向，整体 delta 为 +25。

There is, however, an important limitation using this type of ratio strategy to create a bullish or bearish position. In our example the trader is initially bullish, but the position is still a ratio vertical spread. As such, it has a negative gamma. If the trader has underestimated volatility, and the underlying market moves up too quickly, the spread can invert from a positive to a negative delta. If the market rises far enough, to 130 or 140, eventually all options will go deeply into-the-money and the deltas of both the June 100 and June 110 calls will approach 100. Eventually, the trader will be left with a delta position of -100. Even though the trader was correct in his bullish sentiment, the position was primarily a volatility spread, so that the volatility characteristics of the position eventually outweighed any considerations of market direction.

然而，用此类比例策略构建看涨或看跌头寸有一个重要限制。在我们的例子中，交易者最初持看涨立场，但该头寸仍为比例垂直价差，因此具有负 gamma。如果交易者低估了波动性，而标的市场上涨速度过快，该价差的 delta 可能会从正变为负。若市场上涨到 130 或 140，所有期权将变为深度实值状态，6 月 100 和 6 月 110 的看涨期权的 delta 将接近 100，最终交易者将持有-100 的 delta 头寸。尽管交易者的看涨预期是正确的，但由于该头寸主要是波动率价差，波动率的影响最终超过了市场方向的考虑。

The delta can also invert with a backspread. Unlike a negative gamma position, where the inversion is caused by swift price movement in the underlying contract, a backspread can invert when the market is less volatile than expected. For example, suppose that with the underlying market at 100 a trader believes that implied volatility is too low. He might decide on a delta neutral backspread:

buy 2 June 110 calls (25)
sell 1 June 100 call (50)

delta 也可能在反向价差中发生倒置。与负 gamma 头寸因标的合约价格快速变动而导致倒置不同，反向价差在市场波动性低于预期时可能发生倒置。例如，在标的市场价格为 100 的情况下，交易者认为隐含波动率过低，可能选择 delta 中性反向价差：

买入 2 份 6 月 110 看涨期权（25）
卖出 1 份 6 月 100 看涨期权（50）

If, however, he is bullish on the market, he can, as in the last example, change the ratio to reflect this sentiment:

buy 3 June 110 calls (25)
sell 1 June 100 call (50)

如果他对市场持看涨立场，可以如前例调整比例以反映该情绪：

买入 3 份 6 月 110 看涨期权（25）
卖出 1 份 6 月 100 看涨期权（50）

His delta position of +25 reflects this bullish bias.

此时的 delta 头寸为 +25，带有看涨倾向。

We know that as time passes, or as volatility declines, all deltas move away from 50. If time begins to pass with no movement in the underlying contract, the delta of the June 100 call will remain at 50, but the delta of the June 110 call will start to decline. If, after a period of time, the delta of the June 100 call declines to 10, the delta of the position will no longer be +25, but will instead be -20. Since this spread is a volatility spread, the primary consideration, as before, is the volatility of the market. Only secondarily are we concerned with the direction of movement. If the trader overestimates volatility, and the market moves more slowly than expected, the spread which was initially delta positive can instead become delta negative.

随着时间推移或波动率下降，所有 delta 值都会偏离 50。如果标的合约没有波动，6 月 100 看涨期权的 delta 会保持在 50，而 6 月 110 看涨期权的 delta 则会开始下降。假设一段时间后，6 月 100 看涨期权的 delta 降至 10，此时头寸 delta 将不再是 +25，而是-20。由于该价差是波动率价差，波动率仍然是主要考量因素，市场方向则其次。如果交易者高估了波动性，而市场移动比预期更缓慢，最初为正 delta 的价差可能转为负 delta。

BULL AND BEAR BUTTERFLIES AND TIME SPREADS
牛市和熊市蝶式价差及时间价差

Butterflies and one-to-one (non-ratioed) time spreads can also be chosen in such a way as to reflect a trader's bullsh or barsh blad like ratio spreads, however, their delta characteristics can also invert as market conditions change.

与比例价差类似，蝶式价差和一对一（非比例）时间价差也可以反映交易者的牛市或熊市观点。不过，这类策略的 delta 也会随市场条件变化而反转。

With the underlying market at 100, a delta neutral trader might buy the June 95/100/105 call butterfly (buy a 95 call, sell two 100 calls, buy a 105 call). The trader hopes that the market will sit still at 100 so that at expiration the butterfly will widen to its maximum value of five points. If, however, a trader wants to buy a butterfly, but is also bullish on the market, he can choose a butterfly where the inside exercise price is above the current price of the underlying contract. If the underlying market is currently at 100, he might choose to buy the June 105/110/115 call butterfly. Since this position wants the underlying market at 110 at expiration, and it is currently at 100, the position is a bull butterfly. This will be reflected in the position having a positive delta.

在标的市场价格为 100 时，一位 delta 中性交易者可能买入 6 月 95/100/105 看涨期权蝶式价差（买入 95 看涨期权，卖出两个 100 看涨期权，买入 105 看涨期权）。该交易者希望市场停留在 100 附近，以便在到期时蝶式价差扩大到最大值五点。然而，如果交易者看涨市场，也可以选择行权价高于当前标的价格的蝶式价差。如果当前标的市场为 100，交易者可能选择买入 6 月 105/110/115 看涨蝶式价差。由于该头寸希望标的市场在到期时达到 110，而目前价格为 100，因此该头寸属于牛市蝶式，表现为正 delta。

Unfortunately, if the underlying market moves too swiftly, say to 120, the butterfly can invert from a positive to a negative delta position. Since at expiration the butterfly always has its maximum value with the underlying contract right at the inside exercise price, the trader will now want the market to fall back from 120 to 110. Whenever the underlying market is below 110, the position will be bullish; whenever the underlying market is above 110, the position will be bearish.

如果标的市场快速上涨至 120，蝶式价差可能会从正 delta 反转为负 delta。由于在到期时蝶式价差的最大值总是出现在标的合约正好位于中间行权价时，此时交易者会希望市场从 120 回落至 110。当标的市场低于 110 时，该头寸呈现多头；当标的市场高于 110 时，该头寸呈现空头。

Conversely, if the trader is bearish, he can always choose to buy a butterfly where the inside exercise price is below the current price of the underlying market. But again, if the market moves down too quickly and goes through the inside exercise price, the position will invert from a negative to a positive delta.

相反，如果交易者看空，可以选择行权价低于当前标的价格的蝶式价差。不过，如果市场快速下跌并穿过中间行权价，则该头寸会从负 delta 反转为正 delta。

In a similar manner, a trader can choose time spreads that are either bullish or bearish. A long time spread always wants the near-term contract to expire exactly at-the-money. A long time spread will be initially bullish if the exercise price of the time spread is above the current price of the underlying market. (footnote 1: In the futures market the situation may be complicated by the fact that different futures months may be trading at different prices. Instead of choosing a traditional time spread where both options have the Same exercise price, the trader may have to choose a diagonal spread to ensure that the position is either bullish (delta positive) or bearish (delta negative).) With the underlying market at 100, the June/March 110 call time spread (buy the June 110 call/sell the March 110 call) will be bullish since the trader will want the underlying market to rise to 110 by March expiration. The June/March 90 call time spread (buy the June 90 call/sell the March 90 call) will be bearish, since the trader will want the underlying market to fall to 90 by March expiration. However, like a long butterfly, a long time spread has a negative gamma, so that if the market moves through the exercise price, the delta can invert. If the market moves from 100 to 120, the June/March 110 call time spread, which was initially bullish, will become bearish. If the market moves from 100 to 80, the June/March 90 call time spread, which was initially bearish, will become bullish.

同样，交易者可以选择偏向牛市或熊市的时间价差。长时间价差总是希望近期合约在到期时恰好位于平值。如果时间价差的行权价高于当前标的价格，长时间价差会呈现初始看涨立场。（脚注 1：在期货市场中，由于不同月份的期货价格可能不同，交易者可能需选择对角价差，以确保头寸偏多（正 delta）或偏空（负 delta）。）在标的市场价格为 100 时，6 月 /3 月 110 看涨时间价差（买入 6 月 110 看涨期权 / 卖出 3 月 110 看涨期权）为看涨头寸，因为交易者希望标的市场在 3 月到期时上涨至 110。6 月 /3 月 90 看涨时间价差（买入 6 月 90 看涨期权 / 卖出 3 月 90 看涨期权）则为看跌头寸，因为交易者希望标的市场在 3 月到期时下跌至 90。然而，像长蝶式价差一样，长时间价差具有负 gamma，因此如果市场穿越行权价，delta 可能会反转。如果市场从 100 上升至 120，最初看涨的 6 月 /3 月 110 看涨时间价差将转为空头。如果市场从 100 跌至 80，最初看跌的 6 月 /3 月 90 看涨时间价差将转为多头。

VERTICAL SPREADS
垂直价差

It is common for traders to take bull or bear market positions by choosing appropriate ratio spreads, butterflies, or time spreads, but in each of these positions volatility is still the primary concern. A trader can be right about market direction, but if he is wrong about volatility, the spread may not retain the directional characteristics that the trader intended.

交易者通常会通过选择适当的比例价差、蝶式价差或时间价差来采取牛市或熊市头寸，但这些策略中波动性仍然是主要关注点。即使交易者对市场方向判断正确，但如果对波动性判断错误，该价差可能无法维持预期的方向性特征。

If a trader focuses initially on the direction of the underlying market, he might look for a spread where the directional characteristics of the spread are the primary concern, and volatility is of secondary importance. He would like to be certain that if the spread is initially bullish (delta positive) it will remain bullish under all possible market conditions. If it is initially bearish (delta negative) it will remain bearish under all possible market conditions.

如果交易者更关注标的市场的方向性特征，可能会选择一类策略，以方向性为主要考量，波动性为次要考量。他希望确保价差初始为看涨（正 delta）时，在任何市场条件下仍保持看涨；若初始为看跌（负 delta），在任何市场条件下也仍为看跌。

The class of spreads which meet the above requirements are known as vertical spreads. A vertical spread always consists of one long (purchased) option and one short (sold) option, where both options are of the same type (either both calls or both puts) and expire at the same time. The options are distinguished only by their different exercise prices. Typical vertical spreads might be:

buy 1 June 100 call
sell 1 June 105 call

or

buy 1 March 105 put
sell 1 March 95 put

符合上述条件的策略被称为垂直价差。垂直价差始终由一个买入期权和一个卖出期权构成，两个期权类型相同（均为看涨或看跌期权），且具有相同到期日，仅行权价不同。典型的垂直价差包括：

买入 1 份 6 月 100 看涨期权
卖出 1 份 6 月 105 看涨期权

或

买入 1 份 3 月 105 看跌期权
卖出 1 份 3 月 95 看跌期权

Vertical spreads are not only initially bullish or bearish, but they remain bullish or bearish no matter how market conditions change. Two options which have different exercise prices but which are identical in every other respect cannot have identical deltas. In the first example, where the trader is long a June 100 call and short a June 105 call, the June 100 call will always have a delta greater than the June 105 call. If both options go deeply into-the-money or very far out-of-the-money, the deltas may be almost identical. Even then, the June 100 call will have a delta fractionally greater than the June 105 call. In the second example, no matter how market conditions change, the March 105 put will always have a delta greater than the March 95 put.

垂直价差不仅在初始时为看涨或看跌，并且无论市场条件如何变化，方向性特征都不变。具有不同行权价但其他条件相同的两个期权，其 delta 不可能相同。在第一个例子中，交易者持有 6 月 100 看涨期权并卖出 6 月 105 看涨期权时，6 月 100 看涨期权的 delta 始终大于 6 月 105 看涨期权。如果两个期权都深度价内或深度价外，delta 可能几乎相同，但 6 月 100 看涨期权的 delta 始终会比 6 月 105 看涨期权略大。在第二个例子中，无论市场如何变化，3 月 105 看跌期权的 delta 始终大于 3 月 95 看跌期权。

At expiration, a vertical spread will have a minimum value of zero, if both options are out-of-the-money, and a maximum value of the amount between exercise prices, if both options are in-the-money. If the underlying market is below 100 at expiration, the June 100/105 call spread will be worthless because both options will be worthless. If the market is above 105, the same spread will be worth five points because the June 100 call will be worth exactly five points more than the June 105 call. Similarly, the March 95/105 put spread will be worthless if the underlying market is above 105 at expiration, and it will be worth 10 points if the market is below 95. The expiration values of typical bull and bear vertical spreads are shown in Figures 10-1 and 10-2.

到期时，若两个期权均处于价外，垂直价差的最低价值为零；若两个期权均处于价内，则最大价值为行权价之差。若到期时标的市场低于 100，6 月 100/105 看涨价差将一文不值，因为两个期权均无价值；若市场高于 105，该价差将价值五点，因为 6 月 100 看涨期权的价值比 6 月 105 看涨期权多五点。同样，若到期时标的市场高于 105，3 月 95/105 看跌价差将无价值；若市场低于 95，该价差将价值 10 点。典型的牛市和熊市垂直价差的到期价值如图 10-1 和图 10-2 所示。

Since a vertical spread at expiration will always have a value between zero and the amount between exercise prices, a trader can expect the price of such a spread to be somewhere within this range. A 100/105 call vertical spread will trade for some amount between zero and five points; a 95/105 put vertical spread will trade for some amount between zero and 10 points. The exact value will depend on the likelihood of the underlying market finishing below the lower exercise price, above the higher exercise price, or somewhere in between. If the market is currently at 80 and gives little indication of rising, the price of the 100/105 call vertical will be close to zero, while the price of the 95/105 puts vertical will be close to 10. If the market is currently at 120 with little likelihood that it will fall, the price of the 100/105 call vertical will be close to five points, while the price of the 95/105 put vertical will be close to zero.

由于垂直价差到期时的价值始终介于零和行权价差之间，交易者可以预期其价格在这个范围内。例如，100/105 看涨垂直价差的交易价格将在零到五点之间；95/105 看跌垂直价差的交易价格将在零到十点之间。具体的价差值取决于标的市场最终收于下限行权价之下、上限行权价之上，或两者之间的概率。如果当前市场在 80，且上涨可能性较低，则 100/105 看涨垂直价差价格接近于零，而 95/105 看跌垂直价差则接近 10 点。如果当前市场在 120，且下跌可能性较低，则 100/105 看涨垂直价差价格接近五点，而 95/105 看跌垂直价差则接近于零。

If a trader wants to do a vertical spread, he has essentially four choices. If he is bullish he can choose a bull vertical call spread or a bull vertical put spread; if he is bearish he can choose a bear vertical call spread or a bear vertical put spread. For example:

bull call spread:

buy a June 100 call
sell a June 105 call

bull put spread:

buy a June 100 put
sell a June 105 put

bear call spread:

sell a June 100 call
buy a June 105 call

bear put spread:

sell a June 100 put
buy a June 105 put

交易者在做垂直价差时基本有四种选择。若看涨，可选择看涨垂直看涨价差或看涨垂直看跌价差；若看跌，可选择看跌垂直看涨价差或看跌垂直看跌价差。例如：

看涨看涨价差：

买入 6 月 100 看涨期权
卖出 6 月 105 看涨期权

看涨看跌价差：

买入 6 月 100 看跌期权
卖出 6 月 105 看跌期权

看跌看涨价差：

卖出 6 月 100 看涨期权
买入 6 月 105 看涨期权

看跌看跌价差：

卖出 6 月 100 看跌期权
买入 6 月 105 看跌期权

Note that a trader who is bullish can buy a 100 call and sell a 105 call, or buy a 100 put and sell a 105 put. And a trader who is bearish can buy a 105 call and sell a 100 call, or buy a 105 put and sell 100 put. This may seem counter intuitive, since one expects spreads which consist of puts to have characteristics which are the opposite of those which consist of calls. Regardless of whether a vertical spread consists of calls or puts, whenever a trader buys the lower exercise price and sells the higher exercise price, the position is bullish; whenever a trader buys the higher exercise price and sells the lower exercise price, the position is bearish. Call and put vertical spreads which expire at the same time and which consist of the same exercise prices have approximately the same delta, so they have approximately the same bullish or bearish characteristics. (footnote 2: We are assuming for the moment that all options are European, with no possibility of early exercise.)

需要注意的是，看涨的交易者可以买入 100 看涨期权并卖出 105 看涨期权，或者买入 100 看跌期权并卖出 105 看跌期权；而看跌的交易者可以买入 105 看涨期权并卖出 100 看涨期权，或者买入 105 看跌期权并卖出 100 看跌期权。看似反直觉，因为通常认为由看跌期权组成的价差特性应与看涨期权相反。无论垂直价差是由看涨期权还是看跌期权构成，只要买入低行权价、卖出高行权价，仓位就是看涨的；反之，买入高行权价、卖出低行权价，则仓位为看跌。到期时间相同且行权价格相同的看涨和看跌垂直价差的 delta 基本一致，因此它们的看涨或看跌特性也基本相同。（脚注 2：这里假设所有期权为欧式期权，不能提前行权。）

Given the many different exercise prices and expiration months available, how can a trader choose the vertical spread which best reflects his directional expectations and which gives him the best chance to profit from those expectations?

在多种行权价格和到期月份的期权选择中，交易者如何选择最符合其市场预期、并有望从中获利的垂直价差？

Since options have a limited life with a fixed expiration date, a trader who wants to use options to take advantage of an expected market move must first determine his time horizon. Is the movement likely to occur in the next month, in the next three months, in the next nine months? If it is currently May and the trader foresees upward movement, but believes the movement is unlikely to occur within the next two months, it does not make much sense to take a position in June options. If his expectations are long-term, he may have to take his position in September, or even December, options. Of course, as he moves further and further out in time, market liquidity may become more, rie wile co dete us are to aken in conside is is he very confident, and therefore willing to take a very large directional position? Or is he less certain and willing to take only a limited position? Two factors determine the total directional characteristics of a vertical spread:

1. The delta of the specific vertical spread
2. The size in which the spread is executed

由于期权有固定的到期日，想利用期权捕捉市场走势的交易者必须先确定时间预期。是预期价格在下个月、未来三个月，还是九个月内发生变化？假设当前为 5 月，交易者预期上涨，但认为在未来两个月内不太可能发生，那么选择 6 月期权意义不大。如果是长期预期，可能需要考虑 9 月甚至 12 月期权。当然，时间越长，市场流动性可能会有所下降。在确定仓位时，还需考虑两个因素：交易者是否非常自信，因此愿意承担较大的方向性仓位？还是不确定，只愿意承担有限的仓位？垂直价差的方向性特性取决于两个因素：

1. 特定垂直价差的 delta 值
2. 执行价差的规模

For example, a trader who wants to take a position which is 500 deltas long (equivalent to purchasing five underlying contracts) can either choose a vertical spread which is 50 deltas long and execute it 10 times, or choose a vertical spread which is 25 deltas long and execute it 20 times. Both positions will leave him long 500 deltas.

例如，若交易者希望建立 500 delta 的多头仓位（相当于买入五份标的合约），可以选择一个 50 delta 的垂直价差并执行 10 次，或选择一个 25 delta 的垂直价差并执行 20 次，最终都能使仓位达到 500 delta。

The delta value of a vertical spread is determined by various factors: time to expiration, volatility, and distance between exercise prices. Since a trader will be required to choose an expiration date which covers the period of expected directional movement, and since a trader will always make his best estimate of volatility over a given period, in practice the delta will be a function of the exercise prices which he chooses. The greater the distance between exercise prices, the greater the delta value associated with the spread. A 95/110 bull spread will be more bullish than a 100/110 bull spread, which will, in turn, be more bullish than a 100/105 bull spread. This is shown in Figure 10-3.

垂直价差的 delta 值受多个因素影响：到期时间、波动率和行权价之间的距离。交易者需要选择覆盖预期走势期间的到期日，并对该期间的波动率进行最佳估计，因此实际 delta 值主要取决于选择的行权价差。行权价差越大，价差的 delta 值越高。例如，95/110 的看涨价差比 100/110 的看涨价差更为看涨，而 100/110 看涨价差又比 100/105 更为看涨，如图 10-3 所示。

Once a trader decides on an expiration month in which to take his directional position, he must decide which specific spread is best. That is, he must decide which exercise prices to use. A common approach is to focus on the at-the-money options. If a trader does this, he will have the following choices:

Bull call spread:	buy an in-the-money call sell an at-the-money call	or	buy an at-the-money call sell an out-of-the-money call
Bear call spread:	buy an at-the-money call sell an in-the-money call	or	buy an out-of-the-money call sell an at-the-money call
Bull put spread:	buy an at-the-money put sell an in-the-money put	or	buy an out-of-the-money put sell an at-the-money put
Bear put spread:	buy an in-the-money put sell an at-the-money put	or	buy an at-the-money put sell an out-of-the-money put

当交易者确定了方向性仓位的到期月份后，需要选择具体的价差组合，即决定使用哪些行权价。通常的方法是关注平值期权。若采用这种方法，交易者将有以下选择：

牛市看涨价差：	买入一个价内看涨期权 卖出一个平价看涨期权	或	买入一个平价看涨期权 卖出一个价外看涨期权
熊市看涨价差：	买入一个平价看涨期权 卖出一个价内看涨期权	或	买入一个价外看涨期权 卖出一个平价看涨期权
牛市看跌价差：	买入一个平价看跌期权 卖出一个价内看跌期权	或	买入一个价外看跌期权 卖出一个平价看跌期权
熊市看跌价差：	买入一个价内看跌期权 卖出一个平价看跌期权	或	买入一个平价看跌期权 卖出一个价外看跌期权

With four different bull spreads and four different bear spreads available, how can the trader choose the spread which represents the best value? One way to do this is to use a theoretical pricing model to evaluate several different vertical spreads. This has been done in Figure 10-4 for six different options on an underlying futures contract: an in- themoney, at-the-money, and out-of-the-money call, and an in-the money, at-the-money, and out-of-the-money put. The assumptions include an underlying price of 100, 12 weeks to expiration, a volatility estimate of 20%, and an interest rate of 8%.

在四种不同的看涨价差和四种不同的看跌价差中，交易者如何选择最佳价差？一种方法是使用理论定价模型评估多种垂直价差组合。图 10-4 对六种不同的期权进行了此类评估：实值、平值和虚值的看涨期权，以及实值、平值和虚值的看跌期权。假设条件为标的价格 100、到期时间 12 周、波动率估值 20%、利率 8%。

Suppose we are interested in doing a bull call spread. Two possibilities are to buy a 95 call and sell a 100 call, or buy a 100 call and sell a 105 call. The spreads and their theoretical values are:

Spread	Theoretical Value	Delta
95/100 call spread	2.87	+20
100/105 call spread	1.88	+20

假设我们想做一个看涨认购价差，有两种选择：买入 95 看涨期权并卖出 100 看涨期权，或买入 100 看涨期权并卖出 105 看涨期权。两种价差及其理论价值如下：

价差组合	理论价值	Delta 值
95/100 看涨价差	2.87	+20
100/105 看涨价差	1.88	+20

With both spreads having the same delta of +20, it may appear that the 100/105 spread is a better value since it is likely to be less expensive. But should this be the sole determinant in choosing a strategy? As in all spreads, the option trader's goal is to create a position with positive theoretical edge, by either purchasing high value at a low price, or selling low value at a high price. In order to achieve this goal we need to know not only the theoretical values of the spreads, but also the prices of the spreads in the marketplace.

由于两种价差的 Delta 均为 +20，100/105 价差可能显得更划算，因为其成本较低。但是否应仅此作为选择策略的依据？在所有价差中，期权交易者的目标是通过低价买入高价值，或高价卖出低价值，以获得正向理论收益。为实现此目标，不仅需要了解价差的理论价值，还需要了解市场上的实际价格。

From an option trader's point of view, the relative prices of options in the marketplace is usually represented by the implied volatility. We know the value of the spreads based on our volatility input of 20%. What might be the prices of the spreads if implied volatility in the option market is something other than 20%? We can answer the question by using a theoretical pricing model with the same underlying price, time to expiration, and interest rate, but with volatilities lower and higher than 20%. This has been done in Figure 10-5 using volatilities of 16% and 24%, along with our estimated volatility of 20%.

从期权交易者的角度看，市场中期权的相对价格通常通过隐含波动率表示。我们知道在 20% 的波动率输入下的价差理论价值。如果市场隐含波动率不是 20%，价差价格可能会是多少？可以通过使用相同的标的价格、到期时间和利率，但设置低于和高于 20% 的波动率来计算。图 10-5 展示了在 16%、20% 和 24% 波动率下的理论价格。

Going back to our bull call spread, suppose implied volatility is lower than our estimate of 20%, say 16%. We can see that the price of the 95/100 spread will be approximately 3.01, while the price of the 100/105 spread will be approximately 1.78. We have two choices: we can pay 3.01 for a spread that is worth 2.87 (the 95/100 spread), or we can pay 1.78 for a spread that is worth 1.88 (the 100/105 spread). Clearly, we will prefer the 100/105 spread since it will result in a positive theoretical edge of .10. If we were to buy the 95/100 spread we would end up with a negative theoretical edge of .14.

回到我们的看涨价差，假设隐含波动率低于预估的 20%，为 16%。此时，95/100 价差约为 3.01，而 100/105 价差约为 1.78。我们有两个选择：要么花费 3.01 买入理论价值为 2.87 的 95/100 价差，要么花费 1.78 买入理论价值为 1.88 的 100/105 价差。显然，我们会选择 100/105 价差，因为这将带来 0.10 的正向理论收益。而若选择 95/100 价差，则会得到-0.14 的负向理论收益。

Now suppose implied volatility is higher than our estimate of 20%, say 24%. We can see that the price of the 95/100 spread will be approximately 2.77, while the price of the 100/105 spread will be approximately 1.94. Again, we have two choices: we can pay 2.77 for a spread that is worth 2.87 (the 95/100 spread), or we can pay 1.94 for a spread that is worth 1.88 (the 100/105 spread). Now we will prefer the 95/100 spread since it will result in a positive theoretical edge of .10. The 100/105 spread would result in a negative theoretical edge of .06.

假设隐含波动率高于预估的 20%，为 24%。此时，95/100 价差约为 2.77，而 100/105 价差约为 1.94。我们依然有两个选择：要么花费 2.77 买入理论价值为 2.87 的 95/100 价差，要么花费 1.94 买入理论价值为 1.88 的 100/105 价差。此时，我们会选择 95/100 价差，因为它将带来 0.10 的正向理论收益，而 100/105 价差则会产生-0.06 的负向理论收益。

What's happening here? Even though both spreads have the same delta values, under one set of circumstances we seem to prefer the 95/100 spread, while under different circumstances we seem to prefer the 100/105 spread. The reason becomes clear if we recall one of the characteristics of option evaluation introduced in Chapter 6:

这是什么原因？尽管两种价差的 Delta 值相同，但在不同情况下，我们的偏好却不同。这一现象的原因在于第六章介绍的一个期权定价特性：

If we consider three options, an in-the-money, at-the-money, and out-of-the-money option which are identical except for their exercise prices, the at-the-money option is always the most sensitive in total points to a change in volatility.

如果我们考虑三个相同到期日的期权，仅行权价不同，其中一个价内、一个平值、一个价外，那么平值期权对波动率变化最为敏感。

This means that when all options appear overpriced because we believe implied volatility is too high, in total points the at-the-money option will be the most overpriced. When all options appear underpriced because we believe implied volatility is too low, in total points the at-the-money option will be the most underpriced. This characteristic leads to a very simple rule for choosing bull and bear vertical spreads:

这意味着当我们认为隐含波动率偏高时，平值期权的总点数最为高估；而当我们认为隐含波动率偏低时，平值期权的总点数最为低估。这一特性引出了选择看涨和看跌垂直价差的简单规则：

If implied volatility is too low, vertical spreads should focus on purchasing the at-the-money option. If implied volatility is too high, vertical spreads should focus on selling the at-the-money options.

如果隐含波动率偏低，垂直价差应重点购买平值期权；如果隐含波动率偏高，垂直价差应侧重卖出平值期权。

Now we can see why the 100/105 call spread is a better value if implied volatility is 16%, while the 95/100 spread is a better value if implied volatility is 24%. If implied volatility is too low (10%), we want to buy the at-the-money (100) call. Having done this, we have only one choice in order to create a bull spread— sell the out-of-the-money (105) call. If implied volatility is too high (24%), we want to sell the at-the-money (100) call. Having done this, we again have only one choice in order to create a bull spread-buy the in-the-money (95) call.

因此，当隐含波动率为 16% 时，100/105 看涨价差性价比更高；而隐含波动率为 24% 时，95/100 价差更具优势。若隐含波动率过低（10%），我们应买入平值（100）看涨期权，然后卖出价外（105）看涨期权以形成看涨价差。若隐含波动率过高（24%），我们则应卖出平值（100）看涨期权，并买入价内（95）看涨期权。

The same principle is equally true for put vertical spreads. We always want to focus on the at-the-money option, buying it when implied volatility is too low, and selling it when implied volatility is too high. For example, suppose we want to do a bear put spread with implied volatility too low. In this case we want to buy the at-the-money (100) put. Having done this, we are forced to sell the out-of-the-money (95) put to create our bear spread. We can see from Figure 10-5 that we will pay approximately 1.90 for this spread, but the spread is worth 2.04. We will end up with a delta position of -20, and a positive theoretical edge of .14.

这一原则同样适用于看跌垂直价差。我们应优先关注平值期权，在隐含波动率低时买入，隐含波动率高时卖出。例如，当隐含波动率偏低时，我们可进行看跌价差，先买入平值（100）看跌期权，再卖出价外（95）看跌期权。根据图 10-5，我们将为此价差支付约 1.90，但其理论价值为 2.04，最终获得-20 的 Delta 头寸和 0.14 的正理论收益。

Of course, a trader is not required to execute any vertical spread by first buying or selling the at-the-money option. Such spreads always involve two options, and a trader can choose to either execute the complete spread in one transaction, or leg into the spread by trading one option at a time. In the latter case, he may decide to first trade the in-the-money or out-of-the-money option, and trade the at-the-money option at a later time. This is a practical trading decision he will have to make based on market conditions and the amount of risk he is willing to accept. Regardless of how the spread Is executed, the trader should focus on the at-the-money option, either buying it when implied volatility is too low, or selling it when implied volatility is too high.

当然，交易者在执行垂直价差时，不一定需要先买入或卖出平值期权。这种价差总是涉及两个期权，交易者可以选择一次性完成价差，或分步建仓，先交易价内或价外期权，再交易平值期权。这取决于市场状况和风险承受能力。但无论价差如何执行，交易者应聚焦平值期权，在隐含波动率低时买入，隐含波动率高时卖出。

In practice it is unlikely that one option will be exactly at-the-money, so that it may be difficult to decide which option to buy and which to sell. In such a case, it is usually best to focus on the option which is closest to at-the-money. If the underlying market is at 103, with 95, 100, 105, and 110 calls available, it is logical to focus on the 105 call since it is closest to at-the-money. If implied volatility is too low, a trader will want to buy the 105 call; if implied volatility is too high, a trader will want to sell the 105 call. He can then trade a different option in order to create a bull or bear vertical spread.

实际操作中，可能没有一个期权完全平值，此时应选择最接近平值的期权。例如，若标的市场在 103 点，有 95、100、105 和 110 的期权可选，则 105 期权最接近平值。若隐含波动率偏低，交易者应买入 105 期权；若隐含波动率偏高，则应卖出 105 期权，再交易另一个期权以形成看涨或看跌垂直价差。

Nor does a trader have to include the option which is closest-to-the-money as part of his spread. A trader who has a strong directional opinion can choose a vertical spread where both options are very far out-of-the-money or very deeply in-the-money. The delta values of such spreads will be very low, but a trader can create a highly leveraged position by executing each spread many times. For example, with the underlying market at 100, a trader who is strongly bullish might buy the 115/120 call spread (assuming such exercise prices are available). The cost of this spread will be very low since there is a high probability that the spread will expire worthless. Hence, the trader will be able to execute the spread many times at a relatively low cost. If he is right and the market does rise above 120, the spread will widen to its maximum value of five points, resulting in very large profits. But regardless of the exercise prices chosen, if implied volatility is low, the trader will attempt to buy the closer-to-the-money option, and if implied volatility is high, the trader will attempt to sell the closer-to-the-money option.

交易者并不一定非要在价差中包含最接近平值的期权。若交易者有较强的方向性观点，可以选择两个都处于深度价外或价内的期权来构建垂直价差。这类价差的 Delta 值会非常低，但交易者可以通过多次执行此类价差来获得高杠杆。例如，当标的市场价格为 100 时，若交易者非常看涨，可以购买 115/120 看涨期权价差（假设该执行价格可用）。由于价差很可能最终失效，该价差成本会很低，因此交易者可以以较低成本多次执行该策略。如果市场真的上涨至 120 以上，该价差会扩展至最大值 5 点，带来可观的利润。但无论选择何种执行价格，若隐含波动率较低，交易者会倾向于买入更接近平值的期权；若隐含波动率较高，则会倾向于卖出更接近平值的期权。

The choice of the at-the-money option is slightly different when we move to stock options. If we define the at-the-money option as the one whose delta is closest to 50, then we may find that the at-the-money option is not always the one whose exercise price is closest to the current price of the underlying contract. This is because the option with a delta closest to 50 will be the one whose exercise price is closest to the forward price of the underlying contract. In stock options, the forward price is the current price of the stock, plus carrying costs on the stock, less expected dividends. Assuming an underlying stock price of 99, six months to expiration, a volatility of 28%, interest rates of 10%, and no dividend, we can see from Figure 10-6 that the 105 call has a delta of 50 and therefore acts like the at-the-money option, even though its exercise price is six points higher than the current underlying price. Therefore, any vertical spread should focus on the 105 call or 105 put.

在股票期权中，平值期权的选择有所不同。如果将 Delta 最接近 50 的期权定义为平值期权，可能会发现该期权的执行价格并不总是最接近标的合约当前价格的那一个。这是因为 Delta 最接近 50 的期权执行价格通常更接近标的合约的远期价格。对于股票期权，远期价格为当前股价加上持有成本减去预期股息。例如，假设标的股票价格为 99，距到期时间为六个月，波动率为 28%，利率为 10%，且无股息。从图 10-6 可以看出，105 看涨期权的 Delta 为 50，因此可视为平值期权，尽管其执行价格比当前标的价格高出 6 点。因此，任何垂直价差都应关注 105 看涨或 105 看跌期权。

The approximate prices of vertical spreads with these stock options at implied volatilities of 23% and 33% are shown in Figure 10-7.

图 10-7 显示了这些股票期权在 23% 和 33% 隐含波动率下的垂直价差的近似价格。

As always, we can never be certain of the delta values of a spread because we can never be certain that we have the correct inputs into the model, in particular, volatility. If volatility turns out to be lower than our estimate, the deltas will move away from 50. If volatility turns out to be higher than our estimate, the deltas will move towards 50 This will change the delta values of the spreads. However, if we work around the at-the-money option, spreads with approximately the same amount between exercise prices will have approximately the same delta values. Our primary problem will be deciding whether implied volatility is too high or too low.

我们永远无法确定价差的确切 Delta 值，因为输入模型的各项数据（尤其是波动率）并不完全准确。如果波动率低于预期，Delta 值会偏离 50；如果波动率高于预期，Delta 值会趋向 50，从而改变价差的 Delta 值。然而，当我们围绕平值期权操作时，执行价格差不多的价差大致会拥有相同的 Delta 值。我们主要的问题是判断隐含波动率是过高还是过低。

Notice that in every case, whether with futures options or with stock options, whether using a low volatility or a high volatility, if one leg of our spread always involves an at-the-money option, the vertical spread which includes the in-the-money option always has a higher price than the spread which includes the out-of-the-money option. A new trader might take the view that since both spreads have approximately the same delta values, it must make sense to always purchase the cheaper spread or sell the more expensive. This ignores the purpose of option evaluation: to consider not only the initial cost of a strategy, but to compare that to the strategy's expected return. In our futures option example, a trader will always have to pay more for the 95/100 call spread than the 100/105 call spread. But if implied volatility is too high, the 95/100 spread will also return more to the trader. To see why, suppose a trader is choosing between the 95/100 bull call spread and the 100/105 bull call spread. Consider three possible cases. In case one, the trader is right and the market moves from 100 to 110. If that happens both spreads will be winners since they will both widen to their maximum value of five points. In case two, the trader is wrong and the market drops to 90. In that case both spreads will be losers since they will both collapse to zero. Now consider case three, where the trader is wrong because he expected the underlying market to rise and it didn't. But he wasn't terribly wrong, because it also didn't fall. It simply remained at 100. If that happens the 100/105 spread will collapse to zero, while the 95/100 spread will still widen to its maximum value of five points. The 95/100 spread is always more valuable than the 100/105 spread because it has time on its side. The 100/105 spread needs the market to rise to show a profit. The 95/100 spread doesn't really need the market to rise; it just needs for the market not to fall. The reader can confirm this by calculating the gamma and theta positions of the spreads in Figure 10-5. The 100/105 spread has a positive gamma and a negative theta, whereas the 95/100 spread has negative gamma and a positive theta.

无论是期货期权还是股票期权，无论波动率高低，如果价差的一条腿总是包含平值期权，那么包含价内期权的垂直价差价格始终高于包含价外期权的价差。新手可能认为，既然这两种价差的 Delta 值大致相同，选择购买更便宜的价差或卖出更贵的价差一定更合理。这种想法忽略了期权评估的真正目的：不仅要考虑策略的初始成本，还需对比策略的预期收益。在我们的期货期权示例中，交易者购买 95/100 看涨期权价差的花费总会高于购买 100/105 看涨期权价差的花费，但如果隐含波动率过高，95/100 价差的回报也会更高。假设交易者在 95/100 和 100/105 看涨价差之间选择，考虑以下三种情况：情况一，市场上升至 110，两种价差都将扩大至最大值 5 点，获利；情况二，市场下跌至 90，两种价差都会归零，亏损；情况三，市场停留在 100，100/105 价差会归零，而 95/100 价差则仍可扩大至 5 点。95/100 价差的优势在于它受时间影响较小，100/105 价差需要市场上涨才能获利，而 95/100 价差只需市场不下跌。图 10-5 显示了这种情况：100/105 价差的 Gamma 为正，Theta 为负，而 95/100 价差的 Gamma 为负，Theta 为正。

If a trader is considering a bull vertical spread, how likely is it that the market will rise? That depends on his ability to predict market direction. At the same time, he must ask himself how likely it is that the market will move. That depends on his ability to predict volatility. If a trader believes there is a good chance the market will move (high volatility), and at the same time believes the movement will be to the upside (bullish on direction), it makes sense to buy the 100/105 spread. On the other hand, if the trader believes the market is unlikely to move significantly (low volatility), but at the same time believes that whatever movement does occur is likely to come on the upside (also bullish), it makes sense to buy the 95/100 spread. In both cases the trader is trying to maximize the return on his investment when he is right and minimize the loss when he is wrong.

如果交易者考虑做多垂直价差，他需要判断市场是否会上涨，这取决于他对市场走势的预测能力。同时，他还需评估市场的波动性，这取决于他对波动率的判断能力。如果他认为市场可能大幅波动（高波动）并且预期方向向上（看涨），选择 100/105 价差较为合理；相反，如果他认为市场波动不大（低波动），但预期波动向上，则选择 95/100 价差更为合适。这样可以在预测正确时最大化收益，预测错误时将损失最小化。

Why might a trader with a directional opinion prefer a vertical spread to an outright long or short position in the underlying instrument? For one thing, a vertical spread is much less risky than an outright position. A trader who wants to take a position which is 500 deltas long can either buy five underlying contracts or buy 25 vertical call spreads with a delta of 20 each. The 25 vertical spreads may sound riskier than five underlying contracts, until we remember that a vertical spread has limited risk while the position in the underlying has open-ended risk. Of course, greater risk also means greater reward. A trader with a long or short position in the underlying market can reap huge rewards if the market makes a large move in his favor. By contrast, the vertical spreader's profits are limited, but he will also be much less bloodied if the market makes an unexpected move in the wrong direction.

为什么有方向观点的交易者会选择垂直价差而非标的合约的直接多头或空头头寸？首先，垂直价差比直接持仓风险更小。一个想要持有 500 Delta 多头头寸的交易者，可以选择买入五份标的合约，也可以买入 25 份 Delta 值为 20 的看涨垂直价差。尽管 25 份垂直价差看似风险更大，但垂直价差的风险是有限的，而标的合约的头寸则风险敞口无限。当然，风险大也意味着潜在收益更高。标的合约多头或空头在市场大幅变动时可能获得巨大回报，而垂直价差的收益是有限的，但当市场意外地向不利方向变动时，其损失也会更少。

Experienced traders accept the fact that they are human, and that their directional forecasts will occasionally be wrong. When that happens the vertical spreader has a distinct advantage over the trader with an outright position in the underlying market. By intelligently estimating volatility, an option trader can decide whether he wants time working for or against him. When he chooses to have time on his side (positive theta), he can sometimes show a profit when the outright trader either breaks even or snows a loss. When he chooses to have time working against him (positive gamma), the option trader's losses when he is wrong about market direction will often be less than the losses from an outright underlying position.

有经验的交易者会接受自己的预测有时可能出错的事实。在出错时，垂直价差策略与直接持仓相比更具优势。通过合理地估算波动率，期权交易者可以决定是否希望时间为己所用。当他选择让时间效应发挥正面作用（正 Theta）时，有时即便直接持仓者没有盈利或出现亏损，期权交易者仍可能盈利；当选择让时间效应对自己不利（正 Gamma）时，期权交易者在判断市场方向错误时的损失通常会低于直接持仓者的损失。