实践导向的随机波动率模型应用指南

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《洛伦佐·贝戈米的随机波动率建模》一书旨在将复杂的金融数学理论与实际金融衍生品建模中的实用问题相结合。作者明确指出，这本书并非关于随机波动率的全面理论教程，也不是纯粹的数学金融教科书，因为此类书籍在书店和大学图书馆中早已丰富多样。作者的目标是向读者传授如何运用特定类型的随机波动率模型来解决实际问题，尤其是在金融衍生工具定价和风险管理中的应用。在金融工程领域，模型构建被视为一门实践导向的学科。作为工程师，我们的任务是将金融市场的问题转化为数学模型，并在必要时利用高级数学工具求解。然而，这些问题往往源于实际交易中的紧迫问题，如期权定价、风险评估和策略制定。书中强调了与金融数学系列——查普曼与霍尔/CRC金融数学系列——的联系，这个系列致力于捕捉金融领域的最新发展，包括对学术界和从业者都有吸引力的教科书、参考文献和手册。系列编辑们，如来自剑桥大学、马里兰大学和帝国理工学院的学者，共同确保了内容的深度和实用性。书中特别提到了一些已出版的作品，例如Jerome Detemple的《美式衍生品：估值与计算》，Pierre Henry-Labordère的《分析、几何与金融中的建模方法：高级期权定价方法》，以及M.A.H.Dempster和Ke Tang合作的《商品金融计算方法》等。这些书籍展示了该系列广泛的主题覆盖，不仅包括理论讲解，还包含丰富的实例和编程代码，以便读者能在实践中理解和应用随机波动率模型。《洛伦佐·贝戈米的随机波动率建模》是一本实用型的金融数学指南，它深入浅出地探讨了如何将随机波动率模型应用于解决实际金融问题，对于希望将理论知识转化为实践技能的金融从业人员和学生来说，具有很高的参考价值。

Preface

Pour soulever un poids si lourd

Sisyphe, il faudrait ton courage!

Bien qu’on ait du cœur à l’ouvrage,

L’Art est long et le Temps est court.

Baudelaire, Les Fleurs du mal

Tu quid? ousque sub alio moveris? Impera et dic quod memoriae tradatur,

aliquid et de tuo profer.

Seneca, Letters to Lucilius, XXXIII

C’est icy un livre de bonne foy, lecteur.

Montaigne, Essais

– This, Reader, is an honest book. It warns you at the outset that it is not a

treatise on stochastic volatility.

Nor is it a mathematical nance textbook – there are treatises and textbooks

galore on the shelves of bookshops and university libraries.

Rather, my intention has been to explain how stochastic volatility – and which

kind of stochastic volatility – can be used to address practical issues arising in the

modeling of derivatives.

Modeling in nance is an engineering eld: while our task as engineers is to

frame problems in mathematical terms and solve them using sophisticated machinery

whenever necessary, the problems themselves originate in the form of embarrassingly

practical trading questions.

I have been fortunate to have spent my career as a quant in an institution – Société

Générale – and in a eld – equity derivatives – that the derivatives community has

come to associate. I have tried to convey the experience thus gained. My main

objective has been to clearly state the motivation for each question I address and to

represent the thought process one follows when designing or using a model.

A model’s (ir)relevance is measured by its (in)ability to account for all non-

linearities of a derivative payo with respect to hedging instruments, and to ade-

quately reect the former in the prices it produces.

Thus have I often lamented the propensity of “pricing quants” or “expectation

calculators” to envision their task as that of producing (real) numbers auspiciously

called prices, and to favor analytical tractability or computational speed at the

expense of model relevance. What is the point in ultrafast mispricing?

Kind souls will allege it is still preferable to ultraslow mispricing.

xvi Preface

This book is intended to be read in sequence. It assumes familiarity with the

basic concepts and models of quantitative nance. The motivation for stochastic

volatility is the subject of the rst chapter; this is followed by a chapter on local

volatility, both a special breed of stochastic volatility and a market model, which I

survey as such.

I urge you, Reader, to read it fully, as I introduce notions and discuss issues that

are referred to repeatedly throughout the book. After a warm-up with forward-start

options we embark on stochastic volatility. Jump and Lévy models are briey dealt

with at the end of Chapter 10.

This work was written mostly at night, during hours normally devoted to rest

and sleep. Thus, dear Reader, I beg your forgiveness. You will do me a great service

by reporting typos, inaccuracies, downright errors, to lb.svbook@gmail.com.

For everything I have learned so far I have to be thankful to more people than

my memory can remember. I wish to thank the practitioners and academics that I

have met regularly at conferences and seminars, and also fellow workers at Société

Générale: quants, traders, structurers.

I am especially indebted to my coworkers in the Quantitative Research team,

past and present. They will nd here a reection of the very many discussions we

have had over the years.

The generous help of colleagues – quants and traders – in proofreading the

manuscript is also gratefully acknowledged, chief among them Florent Bersani

and Pierre Henry-Labordère whose eagle eyes have helped clear the text of many

imprecisions.

Finally, I would like to express my gratitude to Rena, Elisa and Chiara for their

patience.

Thank you Rena for occasionally lling the loneliness of nightly writing sessions

with music. This has resulted for the author in some unlikely pairings, e.g. local

volatility and the piano part of the Kreutzer sonata, the Heston model and the Chopin

Barcarolle, Rachmanino’s Third and options on variance, etc.

Paris, Summer 2015

Cover art: in Albrecht Dürer’s famous Melencolia I print (detail), a disgruntled quant sits amid the tools of his trade –

some unfamiliar to the author. The scale and hourglass are symbols, respectively, for no-arbitrage and the convex order.

Rows and columns of the magic square all sum up to 34, as do numbers in the four quadrants, as well as in the

center quadrant. The middle cells of the bottom row spell out the year the engraving was made: 1514.

Chapter 1

Introduction

Why would a trader use a stochastic volatility model? What for? Which issues does

one address by using a stochastic volatility model? Why aren’t practitioners content

with just delta-hedging their derivative books? These are the questions we address

in this introduction.

We begin our analysis by reviewing the Black-Scholes model and how it is

used on trading desks. It may come as a surprise to many that, despite the widely

publicized inconsistency between the actual dynamics of nancial securities, as

observed in reality, and the idealized lognormal dynamics that the Black-Scholes

model postulates, it is still used daily in banks to risk-manage derivative books.

One may think that a model derives its legitimacy and usefulness from the

accuracy with which it captures the historical dynamics of the underlying security

– hence the scorn demonstrated by econometricians and econophysicists for the

Black-Scholes model and its simplistic assumptions, upon rst encounter. With

regard to models, many are used to a normative thought process. Given the behavior

of securities’ prices – as specied for example by an xx-GARCH or xxx-GARCH

model – this is what the price of a derivative should be. Models not conforming to

such type of specication – or to some canonical set of stylized facts – are deemed

“wrong”.

This would be suitable if (a) the realized dynamics of securities benevolently

complied with the model’s specication and (b) if practitioners only engaged in delta-

hedging. The dynamics of real securities, however, is not regular enough, nor can it

be characterized with sucient accuracy that the normative stance is appropriate.

Moreover, such an approach skirts the issue of dynamical trading in options – at

market prices – and of marking to market.

Rather than calibrating their favorite model to historical data for the spot process,

and, armed with it and trusting its seaworthiness, endeavor to ride out the rough seas

of nancial markets, derivatives practitioners will be content with barely oating

safely and making as few assumptions as possible about future market conditions.

Still, this requires some modeling infrastructure – hence this book: while they do

not use the models’ predictive power and may have little condence in the reliability

of the models’ underlying assumptions, practitioners do need and make use of the

models’ pricing equations. This is an important distinction: while the Black-Scholes

model is not used on derivatives desks, everybody uses the Black-Scholes pricing

equation.

2 Stochastic volatility modeling

Indeed, a pricing equation is essentially an analytical accounting device: rather

than predicting anything about the future dynamics of the underlying securities,

a model’s pricing equation supplies a decomposition of the prot and loss (P&L)

experienced on a derivative position as time elapses and securities’ prices move

about. It allows its user to anticipate the sign and size of the dierent pieces in

his/her P&L. We will illustrate this below with the example of the Black-Scholes

equation, which could be motivated by elementary break-even accounting criteria.

More sophisticated models enable their users to characterize more precisely

their P&L and the conditions under which it vanishes, for example by separating

contributions from dierent eects that may be lumped together in simpler models.

Again, the issue, from a practitioner’s perspective, is not to be able to predict anything,

but rather to be able to dierentiate risks generated by these dierent contributions

to his/her P&L and to ensure that the model oers the capability of pricing these

dierent types of risk consistently across the book at levels that can be individually

controlled.

It is then a trading decision to either hedge away some of these risks, by taking

osetting positions in more liquid – say vanilla – options or by taking osetting

positions in other exotic derivatives, or to keep these risks on the book.

1.1 Characterizing a usable model – the Black-Scholes

equation

Imagine we are sitting on a trading desk and are tasked with pricing and risk-

managing a short position in an option – say a European option of maturity

whose

payo at t = T is f(S

), where S is the underlying.

The bank quants have coded up a pricing function:

P (t, S)

is the option’s price

in the library model. Assume we don’t know anything about what was implemented.

How can we assess whether using the black-box pricing function

P (t, S)

for risk-

managing a derivative position is safe, that is whether the library model is usable?

We assume here that the underlying is the only hedging instrument we use. The

case of multiple hedging instruments is examined next.

•

The rst sanity check we perform is set

t = T

and check that

equals the

payo:

P (t = T, S) = f (S), ∀S. (1.1)

Provided (1.1) holds, we proceed to consider the P&L of a delta-hedged position.

For the purpose of splitting the total P&L incurred over the option’s lifetime into

pieces that can be ascribed to each time interval in between two successive delta

rehedges, we can assume that we sell the option at time

, buy it back at

t + δt

then

Introduction 3

start over again.

δt

is typically 1 day. Let

∆

be the number of shares we buy at

delta-hedge.

Our P&L consists of two pieces: the P&L of the option itself, of which we are

short, which comprises interest earned on the premium received at

, and the P&L

generated by the delta-hedge, which incorporates interest we pay on money we

have borrowed to buy

∆

shares, as well as money we make by lending shares out

during δt:

P &L = −[P (t + δt, S + δS) − P (t, S)] + rP (t, S) δt + ∆(δS −rSδt + qSδt)

where

δS

is the amount by which

moves during

δt

is the interest rate and

the

repo rate, inclusive of dividend yield.

How should we choose

∆

? We pick

∆ =

so as to cancel the rst-order term

in δS in the P&L above.

We now expand the P&L in powers of

δS

and

δt

. We would like to stop at the

lowest non-trivial orders for

δt

and

δS

: order one in

δt

, and order two in

δS

, as the

order one contribution is canceled by the delta-hedge. What about cross-terms such

as δSδt?

In practice, this term, as well as higher order terms in

δS

, are smaller than

δS

and

δt

terms. Indeed, to a good approximation, the variance of returns scales

linearly with their time scale, thus



δS



is of order

δt

and

δS

is of order

√

δt

The

contributions at order one in

δt

and order two in

δS

are then both of order

δt

while

the cross-term

δSδt

and terms of higher order in

δS

are of higher order in

δt

, thus

become negligible as δt → 0.

We then get the following expression for our carry P&L – the standard denomi-

nation for the P&L of a hedged option position:

P &L = −



− rP + (r − q) S



δt −



δS



(1.2)

•

The rst piece – called the theta portion – is deterministic. It is given by the

time derivative of the option’s price (sometimes theta is used to denote

only), corrected for the nancing cost/gain during

δt

of the delta hedge and

the premium.

The property that the variance of returns scales linearly with their time scale is equivalent to the

property that returns have no serial correlation. Securities’ returns do in fact exhibit some amount of

serial correlation at varying time scales, of the order of several days down to shorter time scales and this is

manifested in the existence of “statistical arbitrage” desks. Serial correlation in itself is of no consequence

for the pricing of derivatives, however the measure of realized volatility will depend on the time scale of

returns used for its estimation. As will be clear shortly, for a derivatives book, the relevant time scale is

that of the delta-hedging frequency.

How small should

δt

be so that this is indeed the case? The order of magnitude of

δS

Sσ

√

δt

where

is the volatility of

. It turns out that for equities, volatility levels are such that for

δt = 1

day,

higher order terms can usually be ignored. There is nothing special about daily delta rebalancing; for

much higher volatility levels, intra-day delta re-hedging would be mandatory.

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