General Black-Scholes Models Accounting for Increased Market

Home , Asset pricing, Options spread

General Black-Scholes mo dels accounting for increased

market volatility from hedging strategies

K. Ronnie Sircar George Papanicolaou

June 1996, revised April 1997

Abstract

Increases in market volatility of asset prices have b een observed and analyzed in recentyears and their

cause has generally b een attributed to the p opularity of p ortfolio insurance strategies for derivative securities.

The basis of derivative pricing is the Black-Scholes mo del and its use is so extensive that it is likely to in uence

the market itself. In particular it has b een suggested that this is a factor in the rise in volatilities. In this

work we present a class of pricing mo dels that account for the feedback e ect from the Black-Scholes dynamic

hedging strategies on the price of the asset, and from there backonto the price of the derivative. These

mo dels do predict increased implied volatilities with minimal assumptions b eyond those of the Black-Scholes

theory. They are characterized by a nonlinear partial di erential equation that reduces to the Black-Scholes

equation when the feedback is removed.

We b egin with a mo del economy consisting of two distinct groups of traders: Reference traders who are

the ma jorityinvesting in the asset exp ecting gain, and program traders who trade the asset following a

Black-Scholes typ e dynamic hedging strategy, which is not known a priori, in order to insure against the risk

of a derivative security. The interaction of these groups leads to a sto chastic pro cess for the price of the asset

which dep ends on the hedging strategy of the program traders. Then following a Black-Scholes argument, we

derive nonlinear partial di erential equations for the derivative price and the hedging strategy. Consistency

with the traditional Black-Scholes mo del characterizes the class of feedback mo dels that we analyze in detail.

We study the nonlinear partial di erential equation for the price of the derivativeby p erturbation metho ds

when the program traders are a small fraction of the economy,bynumerical metho ds, which are easy to use

and can b e implemented eciently, and by analytical metho ds. The results clearly supp ort the observed

increasing volatility phenomenon and provide a quantitative explanation for it.

Keywords: Black-Scholes mo del, Dynamic hedging, Feedback e ects, Option pricing, Volatility.

Submitted to Applied Mathematical Finance

Scienti c Computing and Computational Mathematics Program, Gates Building 2B, Stanford University, Stan-

ford CA 94305-9025; [email protected]; work supp orted by NSF grant DMS96-22854.

Department of Mathematics, Stanford University, Stanford CA 94305-2125; [email protected] .edu; work

supp orted by NSF grant DMS96-22854. 1

Contents

1 Intro duction 2

1.1 The Black-Scholes Mo del : ::: ::: ::: ::: :: ::: ::: ::: ::: ::: :: : 3

1.2 Feedback E ects : :: ::: ::: ::: ::: ::: :: ::: ::: ::: ::: ::: :: : 3

2 Derivation of the Mo del 5

2.1 The Generalized Black-Scholes Pricing Mo del :: :: ::: ::: ::: ::: ::: :: : 5

2.2 Framework of Mo del Incorp orating Feedback :: :: ::: ::: ::: ::: ::: :: : 6

2.3 Asset Price under Feedback ::: ::: ::: ::: :: ::: ::: ::: ::: ::: :: : 7

2.4 Mo di ed Black-Scholes under Feedback :: ::: :: ::: ::: ::: ::: ::: :: : 8

2.5 Consistency and Reduction to Black-Scholes ::: :: ::: ::: ::: ::: ::: :: : 10

2.5.1 Classical Black-Scholes under Feedback : :: ::: ::: ::: ::: ::: :: : 11

2.6 Multiple Derivative Securities : ::: ::: ::: :: ::: ::: ::: ::: ::: :: : 13

2.7 Europ ean Options Pricing and Smo othing Requirements : ::: ::: ::: ::: :: : 13

2.7.1 The Smo othing Parameter :: ::: ::: :: ::: ::: ::: ::: ::: :: : 14

2.7.2 Smo othing by Distribution of Strike Prices : ::: ::: ::: ::: ::: :: : 15

2.7.3 The Full Mo del :: ::: ::: ::: ::: :: ::: ::: ::: ::: ::: :: : 16

3 Asymptotic Results for small 16

3.1 Regular Perturbation Series Solution ::: ::: :: ::: ::: ::: ::: ::: :: : 17

3.2 Least Squares Approximation by the Black-Scholes Formula : ::: ::: ::: :: : 19

3.2.1 Adjusted Volatility as a function of time :: ::: ::: ::: ::: ::: :: : 20

3.2.2 Adjusted Volatility as a function of Asset Price :: ::: ::: ::: ::: :: : 21

3.3 Extension to Multiple Options : ::: ::: ::: :: ::: ::: ::: ::: ::: :: : 22

4 General Asymptotic Results 24

5 Numerical Solutions and Data Simulation 26

5.1 Data Simulation : :: ::: ::: ::: ::: ::: :: ::: ::: ::: ::: ::: :: : 27

5.2 Numerical Solution for many options : ::: ::: :: ::: ::: ::: ::: ::: :: : 29

5.2.1 Volatility Implications :: ::: ::: ::: :: ::: ::: ::: ::: ::: :: : 31

5.2.2 Varying strike times ::: ::: ::: ::: :: ::: ::: ::: ::: ::: :: : 32

6 Summary and conclusions 33

1 Intro duction

One of mo dern nancial theory's biggest successes in terms of b oth approach and applicability has

b een Black-Scholes pricing, which allows investors to calculate the `fair' price of a derivative security

whose value dep ends on the value of another security, known as the underlying, based on a small

set of assumptions on the price b ehavior of that underlying. Indeed, b efore the metho d existed,

pricing of derivatives was a rather mysterious task due to their often complex dep endencies on the

underlying, and they were traded mainly over-the-counter rather than in large markets, usually

with high transaction costs. Publication of the Black-Scholes mo del [4] in 1973 roughly coincided

with the op ening of the Chicago Board of Trade and since then, trading in derivatives has b ecome

proli c.

Furthermore, use of the mo del is so extensive, that it is likely that the market is in uenced by 2

it to some extent. It is this feedback e ect of Black-Scholes pricing on the underlying's price and

thence backonto the price of the derivatives that we study in this work. Thus we shall relax one of

the ma jor assumptions of Black and Scholes: that the market in the underlying asset is p erfectly

elastic so that large trades do not a ect prices in equilibrium.

1.1 The Black-Scholes Mo del

The primary strength of the Black-Scholes mo del, in its simplest form, is that it requires the estima-

tion of only one parameter, namely the market volatility of the underlying asset price in general

as a function of the price and time, without direct reference to sp eci c investor characteristics

such as exp ected yield, utility function or measures of risk aversion. Later work by Kreps [6] and

Bick[2,3] has placed b oth the classical and the generalized Black-Scholes formulations within the

framework of a consistent economic mo del of market equilibrium with interacting agents having

very sp eci c investmentcharacteristics.

In addition, the Black-Scholes analysis yields an explicit trading strategy in the underlying as-

set and riskless b onds whose terminal payo is equal to the payo of the derivative securityat

maturity.Thus selling the derivative and buying and selling according to this strategy `covers' an

investor against all risk of eventual loss, for a loss incurred at the nal date as a result of one half

of this p ortfolio will b e exactly comp ensated by a gain in the other half. This replicating strategy,

as it is known, therefore provides an insurance p olicy against the risk of holding or selling the

derivative: it is called a dynamic hedging strategy since it involves continual trading, where to

hedge means to reduce risk. That investors can hedge and knowhow to do so is a second ma jor

strength of Black-Scholes pricing, and it is the proliferation of these hedging strategies causing a

feedbackonto the underlying pricing mo del that we shall study here.

The missing ingredients in this brief sketch of the Black-Scholes argument are, rstly, the exis-

tence of such replicating strategies, which is a problem of market completeness that is resolved in

this setting by allowing continuous trading or, equivalently, in nite trading opp ortunities; and

secondly, that the price of running the dynamic hedging strategy should equal the price of the

derivative if there is not to b e an opp ortunity for some investor to make `money for nothing', an

arbitrage. Enforcement of no-arbitrage p ointwise in time leads to the Black-Scholes pricing equa-

tion; more globally, it can b e considered as a mo del of the observation that market trading causes

prices to change so as to eliminate risk-free pro t-making opp ortunities.

1.2 Feedback E ects

There has b een muchwork in recentyears to explain the precise interaction b etween dynamic hedg-

ing strategies and market volatility. As Miller [17] notes, \the widespread view, expressed almost

daily in the nancial press ... is that sto ck market volatility has b een rising in recentyears and

that the intro duction of low-cost sp eculativevehicles such as sto ck index futures and options has

b een mainly resp onsible." Mo dellin g of this phenomenon typically b egins with an economyoftwo

typ es of investors, the rst whose b ehavior upholds the Black-Scholes hyp otheses, and the second

who trade to insure other p ortfolios. Peters [18] has `smart money traders' who invest according to

value and `noise traders' who follow fashions and fads; the latter overreact to news that may a ect

future dividends, to the pro t of the former. Follmer and Schweizer [9]have `information traders'

who b elieve in a fundamental value of the asset and that the asset price will take that value, and

`noise traders' whose demands come from hedging; they derive equilibrium di usion mo dels for the

asset price based on interaction b etween these two. 3

Brennan and Schwartz [5] construct a single-p erio d mo del in which a fraction of the wealth is

held by an exp ected utility maximizing investor, and the rest byaninvestor following a simple

p ortfolio insurance strategy that is a priori known to all. They assume the rst investor has a

CRRA utility function and obtain b etween 1 and 7 increases in Black-Scholes implied market

volatility for values of the fraction of the market p ortfolio sub ject to p ortfolio insurance varying

between 1 and 20.

In similar vein, Frey and Stremme [11] present a discrete time and then a continuous time econ-

omy of reference traders Black-Scholes upholders and program traders p ortfolio insurers. They

derive an explicit expression for the p erturbation of the It^o di usion equation for the price of the

underlying asset by feedback from the program traders' hedging strategies. In particular, they

consider the case of a `Delta Hedging' strategy for a Europ ean option whose price cx; t is given

by a classical Black-Scholes formula, that is one with constantvolatility . It is describ ed bya

function x; t which is the numb er of units of the underlying asset to b e held when the asset price

is x at time t, and it is a result of Black-Scholes theory that this numb er is the `delta' of the option

price: = @ c=@ x.Thus they examine the e ect of b eing completely sp eci ed but for the constant

, and ask the questions i howvalid is it for the program traders to continue to use the classical

Black-Scholes formula when they also know ab out feedback e ects on the price of the underlying

sto ck? and ii given the single degree of freedom allowed for the choice of , what is the b est value

to cho ose for ? The latter question leads them to a family of xed-p oint problems and thence

to lower and upp er b ounds for suitable . They rep ort up to 5 increases in market volatility

when program traders have a 10 market share. This framework is also studied bySchonbucher

and Wilmott [22,23] who analyse p erturbations of asset prices by exogenously given Black-Scholes

hedging strategies and, in particular, induced price jumps as expiration is approached.

A full description of the extent and typ e of program trading that o ccurs in practice is given by

Du ee et al. [7]. They write that \the New York Sto ck Exchange NYSE has de ned program

trading as the purchase or sale of at least fteen sto cks with a value of the trade exceeding $1

million. Program trading has averaged ab out 10 p ercent of the NYSE volume or 10 to 20 million

shares p er day in the last year-and-a-half that these data have b een collected. This activity has

fallen to ab out 3 p ercentofvolume since the NYSE encouraged rms to restrict some program-

related trades after Octob er 1989." In addition, they rep ort that 10 30 of all program trading

o ccurs on foreign exchanges, particularly in London.

In Section 2, we follow the framework of the Frey-Stremme mo del, but consider the hedging strat-

egy as unknown and derive equations for it using the mo di ed underlying asset di usion pro cess.

The key ingredients are a sto chastic income pro cess, a demand function for the reference traders,

and a parameter equal to the ratio of the numb er of options b eing hedged to the total number of

units of the asset in supply. In addition to the equations arising from an arbitrary demand function

and a general It^o income pro cess, we givetwo families of pricing mo dels that are consistent with

the generalized and classical Black-Scholes pricing equations, and which reduce to these when the

program traders are removed. Enforcing this consistency places a restriction on the form of the

reference traders' demand function, whichwe derive.

Avon Neumann-Morgenstern utility function with Constant Relative Risk Aversion is of the form ux=x =

for some >0.

In the sense of the instantaneous variance rate of the prices of the underlying asset resulting from their mo del. 4

We concentrate on the feedback mo del for a Europ ean call option consistent with classical Black-

Scholes in Section 3, and give asymptotic results for when the volume of assets traded by the

program traders is small compared to the total numb er of units of the asset. Here, the program

traders cause a small p erturbation to the classical Black-Scholes economy, and we measure the

e ects in terms of implied Black-Scholes volatility: the adjusted volatility parameter that should

b e used in the Black-Scholes formula to b est approximate, in a sense that is made precise, the

feedback-adjusted price of the option. We nd that the market volatility do es indeed increase as

anticipated, and by a greater degree than found in the Brennan-Schwartz or Frey-Stremme studies.

We also give an extension to the more realistic case when feedback comes from hedging a number

of di erent options on the underlying asset, pro ducing similar results.

In Section 4, we present general analytical results for an arbitrary Europ ean derivative security

with a convex payo function, again when program trading is small compared to reference trading.

We nd the mo del predicts higher implied market volatilities. Then in Section 5, we outline an ef-

cientnumerical scheme for solution of the mo del equations. We also presenttypical discrepancies

between historical volatilities and feedbackvolatilities as predicted by our mo del. These illustrate

that mispricings of up to 30 can o ccur if traders attempt to account for feedback-induced increased

market volatilityby constantly re-calibrating the Black-Scholes volatility parameter, instead of us-

ing the full theory.

If we assume that increased volatility is due primarily to feedback e ects from program trading

then we can use our mo del to estimate the fraction of the market that is b eing traded for p ortfolio

insurance purp oses. Jacklin et al. [15] argue that one of the causes of the crash of Octob er 19,

1987 was information ab out the extent of p ortfolio insurance-motivated trading suddenly b ecoming

known to the rest of the market. This prompted the realization that assets had b een overvalued

b ecause the information content of trades induced by hedging concerns had b een misinterpreted.

Consequently, general price levels fell sharply. Similar conclusions are reached by Du ee et al. [7],

Gennotte and Leland [12], and Grossman [13].

We summarize and give conclusions and plans for future work in Section 6.

2 Derivation of the Mo del

We state for reference the generalized Black-Scholes pricing partial di erential equation whose

derivation is detailed in [8] for example, and then extend it to incorp orate feedback e ects from

p ortfolio insurance.

2.1 The Generalized Black-Scholes Pricing Mo del

Supp ose there is a mo del economy in which traders create a continuous-time market for a particular

n o

asset whose equilibrium price pro cess is denoted by X ;t 0 : There are two other securities in

the economy: a riskless b ond with price pro cess = e , where r is the constant `sp ot' interest

t 0

n o

rate, and a derivative security with price pro cess P ;t 0 whose payo at some terminal date

~ ~ ~

T>0 is contingent on the price X of the underlying asset on that date: P = hX , for some

T T T

function h. The asset is assumed to pay no dividends in 0 t T .

We use the word generalized in the sense that the underlying asset price is a general It^o pro cess rather than the

sp eci c case of Geometric Brownian Motion as in the classical Black-Scholes derivation. 5

An It^o pro cess for the price of the underlying is taken as given:

~ ~ ~ ~

dX = X ;tdt + X ;tdW ; 2:1

t t t t

o n

W ;t 0 is a standard Brownian Motion on a probability space ; F ; P , and and where

satisfy Lipschitz and growth conditions sucient for the existence of a continuous solution to 2.1.

In practice, these functions are often calibrated from past data to make such a mo del tractable.

~ ~ ~ ~

Then we supp ose the price of the derivative is given by P = C X ;t for some function C x; t

t t

that is assumed suciently smo oth for the following to b e valid. Well-known arguments involving

construction of a self- nancing replicating strategy in the underlying asset and b ond lead to the

generalized Black-Scholes partial di erential equation for the function C x; t

~ ~ ~

@ C 1 @ @ C C

+ r + x; t x C =0; 2:2

@t 2 @x @x

~ ~

for each x> 0;t > 0. The terminal condition is C x; T = hx. Furthermore, C x; t is indep en-

dent of the drift function x; t in 2.1.

Later, we shall refer to the classical Black-Scholes equation which comes from taking the un-

derlying's price to b e a Geometric Brownian Motion,

~ ~ ~ ~

X dt + X dW ; 2:3 dX =

t t t t

for constants and . Substituting x; t x in 2.2, gives

~ ~ ~

@ C 1 @ C @ C

2 2

+ x + r x C =0; 2:4

@t 2 @x @x

whichisknown as the classical Black-Scholes equation. It do es not feature the drift parameter

in 2.3.

The dynamic hedging strategy to neutralize the risk inherent in writing the derivative is, ex-

~ ~

plicitly, to hold the amount C X ;t of the underlying asset at time t, continually trading to

x t

~ ~ ~ ~ ~

maintain this numb er of the asset, and invest the amount C X ;t X C X ;t in b onds at time

t t x t

t. The cost of doing this is C x; t, which is the `fair' price of the derivative. Equivalently, holding

~ ~

the derivative and C X ;t of the underlying is a risk-free investment.

x t

2.2 Framework of Mo del Incorp orating Feedback

We next describ e the framework of the continuous-time version of the mo del economy prop osed by

Frey and Stremme [11].

In this economy there are two distinct groups of traders: reference traders and program traders

whose characteristics are outlined b elow. These investors create a continuous-time market for a

particular asset whose equilibrium price pro cess in this setting is denoted by fX ;t 0g : The price

pro cess of the b ond is denoted = e as b efore, and the price of the derivative securityis

t 0

The party that sells the derivative contract is called the writer, and the buyer is called the holder. 6

fP ;t 0g.

Wecharacterize the larger of the two groups of traders, the reference traders, as investors who

buy and sell the asset in suchaway that, were they the only agents in the economy, the equilib-

rium asset price would exactly follow a solution tra jectory of 2.1. Furthermore, this price would

b e indep endent of the distribution of wealth amongst the reference traders, so we can consider all

the reference traders' market activities by de ning a single aggregate reference trader who repre-

sents the actions of all the reference traders together in the market. Alternatively, they can b e

describ ed as traders who blindly follow this asset pricing mo del, and act accordingly.To derive the

mo del for the asset price incorp orating feedback e ects, rather than taking X as given, we supp ose

the aggregate reference trader has two primitives:

1. an aggregate sto chastic income equal to the total income of all the reference traders mo delled

by an It^o pro cess fY ;t 0g satisfying

dY = Y ;tdt + Y ;tdW ; 2:5

t t t t

where fW g is a Brownian motion, and and are exogenously given functions satisfying the

usual technical conditions for existence and uniqueness of a continuous solution to 2.5. We

note at this stage that the functions y; t and y; t will not app ear in the options pricing

equations that we shall derive, so that the incomes pro cess, which is not directly observable,

need not b e known for our mo del.

2. a demand function D X ;Y ;t dep ending on the income and the equilibrium price pro cess.

t t

Examples for D will b e argued from a consistency criterion with the Black-Scholes mo del, as

in the sp ecial case treated in the bulk of this pap er.

The second group, consisting of program traders, is characterized by the dynamic hedging strategies

they follow for purp oses of p ortfolio insurance. Their sole reason for trading in the asset is to hedge

against the risk of some other p ortfolio eg. against the risk incurred in writing a Europ ean option

on that asset. Their aggregate demand function is given by X ;t which is the amountofthe

asset those traders want to hold at time t given the price X ;we do not allow to dep end on

knowledge of the representative reference trader's income pro cess Y to which they have no access.

We shall assume for the moment that the program traders are hedging against the risk of having

written identical derivative securities, and indicate the extension to the general case where the

insurance is for varying numb ers of di erent derivatives in section 2.6. For convenience, we shall

write X ;t= X ;t, where can b e thought of as the demand p er security b eing hedged.

t t

In this work, we do not assume that is given, but go on to derive equations it must satisfy.

2.3 Asset Price under Feedback

Wenow consider how market equilibrium and Y determine the price pro cess of the asset, X . Let

t t

us assume that the supply of the asset S is constant and that D x; y ; t= S D x; y ; t, so that D

0 0

is the demand of the reference traders relative to the supply. Then we de ne the relative demand

of the representative reference trader and the program traders at time t to b e GX ;Y ;t, where

t t

Gx; y ; t= D x; y ; t+ x; t; 2:6

and := =S is the ratio of the volume of options b eing hedged to the total supply of the asset.

The normalization by the total supply has b een incorp orated into the de nition of D , and is 7

the prop ortion of the total supply of sto ck that is b eing traded by the program traders.

Then, setting demand supply = 1 at each p oint in time to enforce market equilibrium gives

GX ;Y ;t 1; 2:7

t t

which is the determining relationship b etween the tra jectory X and the tra jectory Y whichis

t t

known. We assume that Gx; y ; t is strictly monotonic in its rst two arguments, and has continu-

ous rst partial derivatives in x and y , so that we can invert 2.7 to obtain X = Y ;t, for some

t t

smo oth function y; t. This tells us that the pro cess X must b e driven by the same Brownian

Motion as Y .

Using It^o's lemma, we obtain

@ 1 @ @ @

+ + Y ;t Y ;tdW ; 2:8 dt + Y ;t dX = Y ;t

t t t t t t

@y @t 2 @y @y

and, di erentiating the constraint G y; t;y;t 1, we obtain

@ @ G=@ y

= ;

@y @ G=@ x

where @ G=@ x 6=0by strict monotonicity. Thus the asset price pro cess X satis es under the

e ects of feedback the sto chastic di erential equation

dX = X ;Y ;t dt + v X ;Y ;t Y ;tdW ; 2:9

t t t t t t t

where the adjusted volatilityis

D X ;Y ;t+ X ;t

y t t y t

v X ;Y ;t= ; 2:10

t t

D X ;Y ;t+ X ;t

x t t x t

which, we note, is indep endent of the scaling by the supply.

The adjusted drift X ;Y ;tis

t t

G G

G 1 G G G G

t xy y y yy

: 2:11 + + + 2 =

2 3

G G 2 G G G

x x x

x x

2.4 Mo di ed Black-Scholes under Feedback

Next we examine how the mo di ed volatility whichisnow a function of e ects the derivation

of the Black-Scholes equation for P from the p oint of view of the program traders. We shall follow

the Black-Scholes derivation of [8] with the crucial di erence that the price of the underlying is

driven not by 2.1, but by 2.9 which dep ends on a second It^o pro cess Y . The calculations are

similar to those for sto chastic volatility mo dels see [14] and [8, Chapter 8] in that such mo dels

typically start with a sto chastic di erential equation like 2.9 for the underlying asset price whose

volatility is driven by an exogenous sto chastic pro cess given by an expression like 2.5, where Y

would represent a non-traded source of risk.

Since the derivation will tell us howmuch of the asset the program traders should buy or sell 8

to cover the risk of the derivative, we shall b e able to obtain an expression for in terms of the

price of the derivative. Generalizing the usual argument, we supp ose the price of one unit of the

derivative securityisgiven by P = C X ;t for some suciently smo oth function C x; t. Then we

t t

construct a self- nancing replicating strategy a ;b in the underlying asset and the riskless b ond:

t t

a X + b = P ;

T T T T T

with

Z Z

t t

a X + b = a X + b + a dX + b d ; 0 t T , 2:12

t t t t 0 0 0 0 s s s s

0 0

and a =X ;t b ecause it is exactly the amount of the underlying asset that the trader must

t t

hold to insure against the risk of the derivative security, and it is therefore the time t demand for

the asset p er derivative security b eing hedged. Excluding the p ossibility of arbitrage opp ortunities

gives

a X + b = P ; 0 t T , 2:13

t t t t t

and by It^o's lemma and 2.12,

dP = a dX + b d

t t t t t

= [a X ;Y ;t+ b r ] dt + a v X ;Y ;t Y ;t dW :

t t t t t t t t t t

Comparing this expression with

@ C @C 1 @C

2 2

dt + dX + v dt; dP =

@t @x 2 @x

and equating co ecients of dW gives

a =X ;t= : 2:14

t t

From 2.13,

P a X

t t t

; 2:15 b =

and matching the dt terms in the two expressions for dP and using 2.14 and 2.15 we nd

@C @ C @C 1

2 2

+ + v = +r C x : 2:16

@t @x 2 @x

Wenow exploit the fact that the volatility term v comes from feedback from hedging strategies.

From 2.10, the adjusted volatility for X is a function of and its derivatives, so we can write

X ;t ; X ;t ;X ;Y ;t ; v X ;Y ;t= H

t t t t t t

for some function H . Then from 2.14 and 2.16, the function C x; tmust satisfy the nonlinear

partial di erential equation

2 2

@C @ C 1 @ C @C @C

2 2

+ H ;x;y;t C =0; 2:17 ; + r x

2 2

@t 2 @x @x @x @x 9

for x; y > 0, and 0 t

C x; T = hx;

x; T = h x;

C 0;t = 0;

0;t = 0:

The dep endence of H and on y can b e removed byinverting 2.6 to get a relation of the form

y = x; x; t;t.We note that as a global partial di erential equations problem, there is no

guarantee that the equation has a solution for a given demand function. Such situations need to

b e tackled on a case-by-case basis for a given demand function. However, we shall study these

equations in a vicinity of the Black-Scholes equation 2.2, and consider their global characteristics

at a later stage.

We can rewrite the equations in terms of the relative demand function G = D + . Since

H = G =G ,wehave

y x

1 D @C @ C @C

+ + r x C =0: 2:18

@t 2 D + C @x @x

x xx

This equation is also given bySchonbucher and Wilmott [24] when Y is a standard Brownian mo-

tion. However, in this general form, the mo del dep ends on mo delling of the unobservable pro cess

Y b ecause of the y; t factor. The nal step in the mo delling is to enforce consistency with the

Black-Scholes mo del as ! 0 and, as we shall see in the next subsection, this imp oses an imp ortant

restriction on the demand function.

Another approach is that of Frey [10] who examines the feedback e ect of the option replicat-

ing strategy of a large trader who prices and hedges according to the value of a \fundamental

state variable", analogous to Y in the present analysis. Thus Frey p erforms similar calculations

to those of this section using the p ortfolio evolving according to d= dC Y ;t Y ;tdX ,

t t t

and pro ceeds to eliminate X by the market-clearing condition. He obtains a quasilinear PDE for

y; t the hedging strategy dep ending on observed values of the pro cess Y , and proves existence

and uniqueness of a solution.

In this pap er, we view the feedback-p erturb ed pro cess X as the observable since feedbackis

o ccurring, and construct the mo del to reduce to Black-Scholes dep ending on the unp erturb ed X

when there is no program trading. This is an essential step in our approach.

2.5 Consistency and Reduction to Black-Scholes

Wenow complete the mo del so that it will reduce to the generalized or classical Black-Scholes

mo dels in the absence of program traders. In this case, the no-arbitrage pricing argument starting

with the function D x; y ; t and the income pro cess Y is carried out from the p oint of view of

an arbitrary investor in the derivative security rather than a program trader, with the following

mo di cations: = 0 in Section 2.3 so that 2.10 b ecomes

D x; y ; t

v x; y ; t= ;

D x; y ; t

This was p ointed out to us by a referee. 10

then everything in Section 2.4 up to 2.16 go es through as b efore, omitting the identi cation

a = after 2.12 and in 2.14. Equation 2.16 now b ecomes

2 2

y; tv x; y ; tC + r xC C =0: 2:19 C +

xx x t

We shall refer to 2.19 as the limiting partial di erential equation of 2.18 when there are no

program traders, and lo ok for conditions on the reference traders' demand function D such that

this reduces to the classical or generalized Black-Scholes equation.

2.5.1 Classical Black-Scholes under Feedback

We b egin with the sp ecial case that reduces to the classical Black-Scholes equation 2.4 with

volatility parameter to illustrate the metho d; the generalized version is a simple extension. We

also supp ose D do es not explicitly dep end on t, to simplify notation, and that reference traders

have the rational characteristics D < 0 their demand for the asset decreases when its price rises

and D > 0 their demand increases with their income.

Prop osition 1 If the incomes process Y isageometric Brownian motion with volatility parameter

, then the feedback model reduces to the classical Black-Scholes model in the absenceofprogram

traders if and only if the demand function is of the form

D x; y = U y =x; 2:20

for some smooth increasing function U , where = = .

Proof. By assumption, Y satis es

dY = Y dt + Y dW ; 2:21

t 1 t 1 t t

for constants and . Then 2.19 reduces to 2.4 if and only if

1 1

1 1 D x; y

2 2 2 2

= y x :

2 D x; y 2

Thus D must satisfy D =D = x=y , where wehave taken the negative square ro ot b ecause the

y x

left-hand side is negative under our hyp otheses for rational trading. The general solution of this

partial di erential equation is D x; y = U y =x for any di erentiable function U . Finally,by

direct di erentiation, D < 0 and D > 0 for x; y > 0 if and only if U > 0.

x y

Now, given a demand function of this form, we derive the pricing equation under feedback. The

di usion co ecientis

1 0

1 1 D x; y y =x U y =x

2 2 2 2 2

= y y :

1 1

2 0

2 D x; y + x; t 2 y =x U y =x

x x x

We can use the market-clearing equation U Y =X =1 X ;t from 2.7 to eliminate y . Let

t t

V b e the inverse function of U , whose existence is guaranteed by the strict monotonicityofU .

Then substituting y =x = V 1 and also using = , the di usion co ecient b ecomes

V 1 U V 1 1

2 2

x ; 2:22

2 V 1 U V 1 x

x 11

and we obtain a family of nonlinear feedback pricing equations that are consistent with and, in the

absence of program trading reduce to, the classical Black-Scholes equation:

V 1 C U V 1 C 1

x x

2 2

x C + r xC C =0: 2:23 C +

xx x t

2 V 1 C U V 1 C xC

x x xx

We note that these are independent of the parameters in the incomes process Y and dep end only

on the function U and , the observable market volatility of the underlying asset.

Setting = 0 in 2.23 immediately recovers the classical Black-Scholes partial di erential equa-

tion 2.4. Since is roughly the fraction of the asset market held by the program traders, it is

likely that it is a small numb er in practice. Thus as long as = C and C remain b ounded in

x xx

magnitude by a reasonable constant, we can study 2.23 as a small p erturbation of 2.4.

Schonbucher [22] also realises the need for consistency with classical Black-Scholes. He stud-

ied a linear in x and y demand function, which is not admissible in our formulation, and forced

consistency by allowing the drift and di usion co ecients of the pro cess Y in 2.5 to dep end on X .

t t

In the bulk of this pap er, we shall study the particular mo del arising from taking U as linear:

U z = z; > 0, to infer qualitative prop erties of feedback from program trading. The equation

b ecomes

3 2

@ C @C 1 @C

2 2

5 4

x + + r x C =0; 2:24

@C @ C

@t 2 @x @x

1 x

which is indep endentof .

If we supp ose that Y is the generalized It^o pro cess 2:5 then 2.19 reduces to the generalized

Black-Scholes equation 2.2 with volatility function x; t in the absence of program traders if

~ ~

and only if D is of the form D x; y ; t= U Qy; t Lx; t;t for some smo oth function U that is

strictly increasing in its rst argument, where

Z Z

x y

dv dv

;Lx; t= : Qy; t=

v; t v; t

This can b e shown easily as b efore, this time by nding the general solution of

D x; t

= :

D y; t

~ ~ ~

If V ;t denotes the inverse function of U ;t, and U its derivative with resp ect to the rst

argument, then, by straightforward calculations, we nd that the feedback pricing equation that

reduces to the generalized Black-Scholes mo del when there are no program traders is

2 3

~ ~

V 1 C ;t ;t U

x 1

4 5

C + x; t C + r xC C =0: 2:25

t xx x

~ ~

U V 1 C ;t ;t x; t C

1 x xx

We note that this equation is indep endentofy; t and y; t in 2.5 and, as in the generalized

Black-Scholes mo del, it is also indep endentof x; t in 2.1. 12

2.6 Multiple Derivative Securities

Wenow consider the case where the program traders create the aggregate demand function x; t

as a result of hedging strategies for n di erent derivative securities with expiration dates T , and

payo functions h x, i =1; 2; ;n: Let x; t b e the demand for the asset resulting from all

i i

the program traders' hedging strategies for insuring units of the i such option, so that

x; t= x; t:

i i

i=1

We assume the demand function of the reference traders is of the form U y =x for consistency

with classical Black-Scholes, and that U is linear to obtain an extension of 2.24. Then, if C x; t

is the price of the i option, following a Black-Scholes argument for each option, we nd

; 2:26 =

and

3 2

1 S

@ C @C 1 0 @C

i i i

2 2

5 4

x + + r x C = 0;t T ,

i i

1 1 2

@C @ C

@t 2 @x @x

1 S S x

0 0

C x; T = h x; 2.27

i i i

C = 0; t>T ,

i i

for each i =1; 2; ;n; where

C x; t: 2:28 C x; t=

j j

j =1

This is a system of n coupled nonlinear partial di erential equations for the functions C x; t; ;C x; t.

1 n

The pro cedure can easily b e applied to the case of a general demand function and Y a general It^o

pro cess to obtain a similar system of n equations for the functions C x; t; ;C x; t, analogous

1 n

to 2.18. The constants could b e generalized to dep end on time, or b e random pro cesses.

2.7 Europ ean Options Pricing and Smoothing Requirements

We fo cus on the problem of feedback caused by p ortfolio insurance against the risk of writing one

particular typ e of Europ ean call option which gives the holder the right, but not the obligation,

to buy the underlying asset at the strike price K at the expiration date T . The terminal payo

function is hx=x K , and this was the original derivative security priced by Black and

Scholes in [4], for which they obtained a closed-form solution known as the Black-Scholes formula:

r T t

C x; t:= xN d Ke N d ; 2:29

BS 1 2

where

T t log x=K + r +

; 2.30 d =

T t

T t; 2.31 d = d

2 1

We could also normalize the constants by := and de ne := =S as in the scalar case, but we shall

i j 0

j =1

keep the notation to a minimum. 13

and

s =2

N z = e ds:

As a terminal condition for the mo del in 2.24 that we shall study, this hx brings up the problem

of the denominator of the di usion co ecient,

3 2

5 4

@C @ C

1 x

7 00

b ecoming zero. This is b ecause h x= x K , so that at t = T , no matter how small is, the

denominator is negative in some neighb orho o d of K . Since wewould exp ect the di usion equation

to smo oth the terminal data as the equation is run backwards in time from T , the denominator

will go through zero as C and C b ecome smaller, causing the equation to b ecome meaningless.

x xx

Toavoid this situation, which arises solely b ecause of the kink in the option's payo function,

we imp ose a second consistency condition with the Black-Scholes mo del as the strike time is ap-

proached. That is, we ignore the feedback e ects predicted by this theory as t ! T b ecause of the

oversensitivity of the Black-Scholes dynamic hedging strategies to price uctuations around x = K ,

BS BS

which is re ected by the fact that C x; t Hx K and C x; t x K , as t ! T .In

x xx

practice, frenetic program trading close to expiration is temp ered by transaction costs, which can

b e regarded as a natural smo other.

Technically, this means that we set the feedback price C equal to the Black-Scholes price C

in some small interval T " t T , and then run the full equation backwards from T " instead

of T . It turns out that we can calculate " in terms of and to obtain sucient smo othing of the

data for well-p osedness of the nonlinear partial di erential equation. The smo othing parameter "

sp eci ed in this way, then completes our feedback pricing mo del.

Schonbucher and Wilmott [24,23] analyse jumps in the asset price that can b e induced by pro-

gram traders following the hedging strategy from the Black-Scholes formula as t ! T . They also

p ose a nonlinear PDE from a linear in x and y demand function with moving b oundaries aris-

ing from the jump conditions in the case of the hedging strategy pro duced endogenously by the

feedback theory.

2.7.1 The Smo othing Parameter

We de ne the smo othing parameter " to b e the minimum value of"> ^ 0 such that

min F x; T "^=0;

x>0

where

@C @ C

BS BS

x; t: F x; t:=1 x; t x

@x @x

Substituting the Black-Scholes formula 2.29 and calculating the minimum for each xed" ^, yields

the following nonlinear algebraic equation satis ed by ":

p 2

1 e

N = "+ :

Here, Hz denotes the Heaviside function Hz =0 if z<0 and Hz = 1 if z>0, and z is its derivative,

the Delta function. 14

This can b e solved numerically for given and , and a plot of " against is shown in Figure

2.1.

We note that " is indep endentofK and T as well as r , so that the smo othing do es not dep end

−3 x 10 8

0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

Fraction of Program Trading in Asset Market

Figure 2.1: Dimensionless Smo othing Parameter " as a function of

on the sp eci cs of the options contract. Indeed, the same " can b e used in the feedback mo del for

put option pricing as can b e seen from put-call parity for the Black-Scholes mo del, and also for

the system of equations in 2.27 when there is a uniform distribution of di erent call options with

the same strike time, but di erent strike prices ie. .

Furthermore, for small , " const. , which indicates that when program trading is small,

smo othing is a minor mo di cation to our mo del away from the strike time. Our numerical ex-

p eriments indicate that feedback prices away from T are quite insensitive to the choice of " even

more so the further t is from T .

2.7.2 Smo othing by Distribution of Strike Prices

In practice, the sp eci cs of options contracts b eing hedged by other program traders might not b e

known to each bank that is trading to insure its p ortfolio. It could, therefore, b e estimated bya

smo oth distribution function over a range of strike prices.

Let us supp ose there are M strike times T

1 2 M

monthly cycles, so it is natural to mo del these dates discretely. Then, if K , i =1; ;M,

are smo oth density functions describing the distribution of strike prices for options maturing at

T , the price C x; t; K of the option expiring at T with strike price K will satisfy 2.27 with

i i i

+ r T t

HT tW x; t, where C x; T ; K = x K , C 0;t; K = Ke and C x; t:=

i i i i i

i=1

W x; t:= C x; t; K K dK :

i i i

0 15

The denominator of the di usion co ecientisnow controlled b ecause C x; t satis es

3 2

1 S

1 @C 0 @ C @C

2 2

5 4

=0; + + r x C x

1 1

@C @ C

@t 2 @x @x

1 S S x

0 0

in eachinterval T

i1 i

K x K dK ; C x; T = W x; T +

i i p i

p=i+1

for i = M; M 1; ; 1 and T := 0. The b oundary condition at any t

0 M

r T t

C 0;t= HT te K K dK :

i i

i=1

So, in the `top' interval T ;T , the equation for C x; tiswell-p osed in the sense of the

M 1 M

denominator remaining p ositive provided that

1 1 1

1 S C x; T S xC x; T = 1 S K dK x x > 0;

M M M M

x xx

0 0 0

for all x> 0. This is a mild requirement on the distribution function . With this established,

the equation for C x; t; K iswell-p osed in T ;T , and we can pro ceed similarly to establish

M M 1 M

well-p osedness for C x; t, and hence for C x; t; K , in eachT ;T under a regularity condition

i i1 i

on .

2.7.3 The Full Mo del

The following equations summarize the feedback pricing mo del for a Europ ean call option that we

shall study in detail in the next sections:

2 3

@C 1 @C @ C

2 2

4 5

+ + r x C = 0; t

@C @ C

@t 2 @x @x

1 x

@x @x

C x; T " = C x; T ";

C 0;t = 0;

r T t

lim jC x; t x Ke j = 0;

x!1

with C x; t=C x; t for T " t T .

That is, we study in detail, in sections 3, 4 and 5, the mo del from the demand function D x; y =

U y =x with U z = z.

3 Asymptotic Results for small

In this section, we obtain results that are valid as , the prop ortion of the total volume of the

asset traded by the program traders, tends to zero. This means that we assume is small enough

that 2.32 can b e considered a small p erturbation to the classical Black-Scholes partial di erential

equation 2.4. Rubinstein [21] assesses alternative pricing formulas by calculating their implied

Black-Scholes volatilities for various strike prices and times-to-maturity and comparing with ob-

served historical implied Black-Scholes volatilities. We present the feedback e ects from our mo del

b oth directly in terms of prices and as measured in this way. 16

3.1 Regular Perturbation Series Solution

We calculate the rst-order correction to the Black-Scholes pricing formula for a Europ ean option

under the e ects of feedback when << 1. The full problem for C x; t, the price of the option

when the underlying sto ck price is x> 0 at time t

The = 0 solution is the Black-Scholes formula 2.29. Then constructing a regular p erturba-

tion series

C x; t+ O ; 3:1 C x; t= C x; t+

and de ning

2 2

L C := C + x C + r xC C ; 3:2

BS t xx x

we expand 2.24 for small to obtain

2 3 2 2

L C + x C = O ; 3:3

C x; t satis es so that, substituting from 3.1 and equating terms of magnitude O ,

@ C

2 3

L C = x :

Di erentiating the expression 2.29 for C gives

C = ; t

2 T t

C x; T " = 0;

C 0;t = 0;

C x; tj = 0: lim j

x!1

We next transform the problem for C to an inhomogeneous heat equation. Under the transformation

x = Ke ; 3.4

t = T ; 3.5

1 1

k 1y k +1

2 4

C x; t = Ke uy; ; 3.6

2 2

where k =2r= ,we obtain the following problem for uy; in 1 " =2

2 2

@u @ u 1 y 1 y

= exp k +1 k +1 ;

@ @y 2 2 4 2

= 0; 3.7 uy;

k 1y

e uy; ! 0asy !1;

and u is b ounded as y ! +1. See Wilmott et al [27] for further details.

If the right-hand side of 3.7 is denoted by f y; , the solution can b e expressed

Z Z

uy; = B ; s; y; f ; sd ds;

2 17

where

1 y

: B ; s; y; = exp

4 s

Therefore,

Z Z

1 1 y

k +1 s k +1 d ds uy; = exp

4 s 2s 4 2

2s 4 s

Z Z

k +1 s

4 4 s

= e d ds;

2s 4 s

1 1 1

where = + and = + k +1.Evaluating the inner integral as = exp =4 ,

2s 4 s 2 s 2

and using

s [y k +1 s]

= ;

4 42 s s

we obtain the solution

i h

s[ y k +1 s]

2 y

exp k +1 s +

4 s 42 s s

uy; = ds: 3:8

2 "

2s s

Now to remove the s = 0 singularity which causes the integrand to b ecome large near the lower limit,

and make the integral amenable to quadrature metho ds, we make the transformation v = s=2

to give

1=2

M y; ; vdv ; 3:9 uy; =

"=4

where

2 2 2

1 1 y v y k +1 1 2v

exp M y; ; v= k +1 v + ;

2 2 2

2 4 1 2v 4 1 v 1 2v

1 v

which hasawell-de ned limit as 1 2v ! 0

1 y 1 2

M y; ; v= exp k +1 y k +1 : lim

4 2 2

1=2 v !

C given by 3.6 Clearly, M>0 in the interval of integration, and so the rst-order correction,

and 3.9, is p ositiveinx> 0, t

mo del due to the presence of the program traders therefore has the e ect of increasing the no-

arbitrage price of the Europ ean option. Since the Black-Scholes formula 2.29 is an increasing

d =2

T t =2>0, this result gives us function of the volatility parameter ,@C =@ =xe

initial con rmation that program traders cause market volatility to increase, and moreover, from

the form of the p erturbation series constructed 3.1, it is linearly increasing in the parameter .

A more direct indication of feedbackvolatility can b e calculated from 2.22. This tells us that,

according to the program traders, the underlying asset price under feedback follows the random

walk describ ed by 2.9, where the di usion term in this case is

x 1 C @ C

x BS

= x 1+ x + O :

1 C xC @x

x xx 18

2 2 d =2

Since @ C =@ x = e =x 2 T t > 0, feedback market `sp ot' volatility is always greater

than that used in the reference Black-Scholes mo del. See also the comments at the end of section

Throughout this section, we shall illustrate results with the example of a six-month Europ ean

option T =0:5years with a strike price K = 5 when the constantspotinterest rate is r =0:04

and the reference volatilityis =0:4. Plots of C x; t, C x; t and the p erturb ed prices for xed

x and for xed t are given in Figure 3.1. In the latter, we take =0:05 with the corresp onding

smo othing parameter " =0:003.

Black−Scholes Price First−Order Correction

20 1.5

1 10 0.5 BS Price Correction 0 0 0.5 0.5 20 20 10 10 t 0 0 x t 0 0 x

Price Perturbation at x=6 Price Perturbation at t=0.25 1.4 0.4

1.3 0.3

1.2 0.2

1.1 0.1 BS and Corrected Prices BS and Corrected Prices 1 0 0 0.2 0.4 0.6 0 1 2 3 4

Time t x

Figure 3.1: First-Order Perturbation: =0:05, =0:4, K =5, T =0:5

In Figure 3.2, we show the p erturb ed dynamic hedging strategy

@C C @

x; t= + ;

@x @x

compared with the original Black-Scholes hedging strategy. Recall that x; t tell us howmuchof

the underlying asset the program traders should hold at time t to insure against the risk of the call

option, and as the graphs show, this amount can increase or decrease under feedback p erturbation,

with e ects most noticeable around the strike price x = K .

3.2 Least Squares Approximation by the Black-Scholes Formula

Wenow further exploit the view that 2.24 is a small p erturbation to 2.4 by considering the

situation where program traders wish to use the Black-Scholes formula to price a Europ ean call

option, accounting for the feedback e ects by using an adjusted volatility estimate. E ectively,

we are estimating how the presence of the program traders changes Black-Scholes volatility; that 19 Black−Scholes Hedging Strategy First−Order Correction

1 1

0.5 0

0 −1 0.5 0.5 20 20 10 10 t 0 0 x t 0 0 x

Strategy Perturbation at x=6 Strategy Perturbation at t=0.25 1 1

0.95 0.8

0.9 0.6

0.85 0.4

0.8 0.2

BS and Corrected Strategies 0.75 BS and Corrected Strategies 0 0 0.1 0.2 0.3 0.4 0 5 10 15 20

t x

Figure 3.2: First-Order Perturbation to Hedging Strategy : =0:05, =0:4, K =5, T =0:5

is, what volatility parameter should b e used in 2.29 to b est approximate the solution to 2.24?

Our motivation b ehind this is to relate feedback e ects to a commonly quoted synoptic variable,

namely implied volatility.

3.2.1 Adjusted Volatility as a function of time

We calculate the Black-Scholes volatility that should b e used in 2.29 to minimize the mean-square

approximation error in x at each time. Thus, if C x; t; is the feedback price satisfying 2.32,

where is the reference traders' given constant estimate of the underlying asset's volatility, then

at each xed t,we calculate ^ t which solves

[C x; t; C x; t; ^ t] dx: 3:10 min

^ t

Here C x; t;^ t denotes the Black-Scholes formula evaluated at asset price x, time t with

volatility parameter ^ t, and the minimization is over values of ^ t for which the integral is

well-de ned.

Wehave the regular p erturbation solution 3.1, and we also linearize ^ t= + t+O ,

assuming the adjusted volatility^ t will b e close to for small . Then we nd

C x; t; ^ t = C x; t; + t R x; t; + O ;

BS BS

where

@C T t xe

R x; t; = x; t; = : 3:11

2 20

The linearized minimization problem is now

i h

C x; t; t R x; t; dx: min

The minimizing correction term is given by

C x; t; R x; t; dx

; 3:12 t=

1 2

R x; t; dx

and the adjusted volatility at time t is ^ t= + t+O .

We next consider how t b ehaves as t ! T in order to obtain an indication of the validity

of the asymptotic estimate to ^ t. The integral in the denominator is given by

3 3=2

K T t 1

R x; t; dx = exp 3T tr ;

2 4

3=2

which b ehaves like const.T t as t ! T .We also nd

C x; t; R x; t; dx D T " t ;

for some constant D . Hence, as t ! T ",

1=2

t D T " t ;

for some constant D , which suggests the asymptotic estimate to the adjusted Black-Scholes volatil-

2 2

ityisvalid as long as T t>O , since " = O when << 1.

Finally we note that C x; t and Rx; t are p ositive, by insp ection, so that t > 0 for all

t< T which implies that the rst-order correction to the reference volatilityisalways p ositive:

Black-Scholes volatility, calculated in this manner, always increases as a result of the presence of

program traders in this << 1 setting.

In Figure 3.3, and ^ t are plotted for our standard example, using Simpson's Rule to evaluate

the integral in the numerator of 3.12 numerically. With =0:05, base volatility is seen to increase

bybetween 10 18 over time in the region of validity t 0:37.

3.2.2 Adjusted Volatility as a function of Asset Price

Similarly we can nd a p erturbation to the base volatility whichvaries with the sto ck price:

^ x= + x+O ;

where, for each xed x> 0, ^ x solves

min [C x; t; C x; t; ^ x] dt: 3:13

^ x

The rst-order correction x is given by

C x; t; R x; t; dt

; 3:14 x=

T 2

R x; t; dt

0 21 Time−dependent Volatility Correction x−dependent Correction 0.5 0.45

0.44 0.48

0.43 0.46

0.42

0.44

0.41

0.42

Base and Adjusted Volatilities Base and Adjusted Volatilities 0.4

0.4 0.39

0.38 0.38 0 0.1 0.2 0.3 0.4 0 5 10 15 20

Time t x

Figure 3.3: First-Order Corrected Black-Scholes Volatilities: =0:05, =0:4, K =5, T =0:5

and, as for the time-dep endentvolatility correction, it is strictly p ositive. A plot for the example

option app ears in Figure 3.3, and volatility app ears to increase by up to 12.

We could similarly minimize the mean-square error in b oth x and t and lo ok at the adjustment

to volatility as a function of the strike price K or the reference volatility . It turns out that the

adjusted volatility increases near-linearly with , and is relatively constantasK varies, provided

that K is not near zero.

3.3 Extension to Multiple Options

Many authors see [1,13,16,21] have attempted to construct pricing mo dels that capture the ob-

served non-constantvariation of implied Black-Scholes volatility with strike price. That is, given

observed prices for Europ ean options on the same underlying asset, but with di erent strike prices,

inverting the Black-Scholes formula 2.29 for , and plotting the resulting implied volatilities

against the strike prices reveals a non-constant graph. There has b een some success with sto chastic

volatility mo dels as prop osed by Hull and White [14] for example, and reviewed in [8, Chapter

8]. An approach to sto chastic volatility e ects based on separation of time scales and asymptotic

analysis is given in [26].

To address this issue, we return to the sp ecial case of section 2.6 in which the program traders

are trading to insure against n call options with strike prices K

1 2 n

we assume they all have the same expiration date T and uniform frequency of o ccurrence. Then,

the smo othing parameter " is the same as for the scalar case. Their prices C x; t satisfy 2.27-

2.28 in t< T ", with h x= x K , T T and =n, and the smo othing correction

i i i i

C x; T "= C x; T "; K .

i BS i 22

Wenow generalize the calculations of this section for the single option to obtain a least-squares

adjusted Black-Scholes volatility for the feedback price of each option and thereby the variation of

this implied volatility with the discretely distributed strike prices.

First we construct regular p erturbation series

C x; t= C x; t; K + C x; t+ O ; 3:15

i BS i i

where = =S as b efore. Expanding for small and using 2.29 gives

d K

xe 2

d K

e ; C = L

i BS

2 T t

j =1

where d K is de ned by 2.30 with K replaced by K , and b oundary and terminal conditions

1 j j

are zero. Solving for each i,

1 1

k 1y k +1

2 4

u y ;; 3:16 C x; t= K e

i i i i

where

y = log x=K ;

i i

= T t=2;

1=2

k +1

u y ; = e M y + a ;;vdv ;

i i j i ij

"=4

j =1

a = log K =K ;

ij i j

and

1 1 y

M y; ; v = exp k +1 v

2 2

2 4 1 2v 8v

1 v

a 12v

2 2

v y k +1 1 2v +

7 2

+ :

2 2

4 1 v 1 2v

Then we can calculate ^ which solves

Z Z

1 T

[C x; t; C x; t;^ ;K ] dxdt; min

i BS i i

0 0

whose linearized expansion is given by^ = +=n + O , where

R R

1 T

C x; t; R x; t; dxdt

i i

0 0

= ; 3:17

R R

T 1

R x; t; dxdt

0 0

where

x; t; ;K : R x; t; =

i i

@ 23 Multi−Option Feedback Adjusted Volatility against Strike Price 0.435

0.43

0.425

0.42

0.415

0.41 Base and Adjusted Volatilities 0.405

0.4

0.395 4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6

Strike Price K

Figure 3.4: First-Order Corrected Black-Scholes Volatilities for 21 Options with strike prices at

even intervals of 0:1between K = 4 and K = 6 and =0:05

min max

An interp olated plot of against K is shown in Figure 3.4 for evenly distributed strike prices.

We see that volatility increases by up to 8, and that the p eak volatilty rise is at the arithmetic

average of the strike prices, K = 5. This re ects qualitatively that observed volatility patterns

might reveal information ab out the distribution of strike prices of options b eing hedged. However,

this computation and others where, for example x and/or t are xed rather than integrated over

suggests that feedback cannot by itself explain observed smile patterns of implied volatility. Since

it is known see [20,26] that these patterns can b e pro duced by sto chastic volatility mo dels, it will

be interesting to build feedback from hedging strategies into these mo dels and see how the resulting

smile curves are attened.

4 General Asymptotic Results

We generalize the results of Section 3 to obtain analogous conclusions ab out the impact on market

volatility of hedging strategies for an arbitrary Europ ean derivative security with payo function

hx. Our main to ol is the Minimum Principle for the Black-Scholes partial di erential op erator,

and with this and assumptions of almost everywhere smo othness and convexityonhx, we nd

that feedback always causes derivative prices and market volatility to increase, as in the particular

case of the Europ ean call option.

Theorem 4.1 Minimum Principle Suppose ux; t is a suciently smooth function satisfying,

for some t

L ux; t 0; for x> 0;t t

BS 0

ux; T 0; for al l x> 0, 4.2 24

u0;t 0; for al l t t

ux; t U x; t 0; as x !1, 4.4

where L is de ned by 3.2. Then for t t T , ux; t 0.

BS 0

Proof. See, for example Protter & Weinb erger [19].

As in Section 3.1, wenow lo ok at the rst-order correction to the feedback price of a deriva-

h h

tive security C x; t satisfying 2.24 with non-negative terminal payo C x; T = hx 0. We

shall also require that h is twice continuously di erentiable almost everywhere. In the following,

we shall not explicitly refer to the smo othing detailed in section 2.7.1, but the results are clearly

applicable if T is replaced by T " and hxby C x; T ".

Prop osition 2 For << 1, the rst-order correction to the Black-Scholes price for the derivative

is non-negative.

Proof. We construct a regular p erturbation series

2 h h

; C x; t+ O C x; t= C x; t+

x; t satis es the Black-Scholes equation and the terminal, b oundary and far- eld con- where C

ditions. Expanding for small and comparing O terms, C x; t satis es

2 h

@ C

2 3

L C = x 0;

C x; T = 0;

C 0;t = 0;

C x; tj = 0: lim j

x!1

C x; t 0. Then by the Minimum Principle,

Thus we know that feedback causes derivative prices to increase. Next we consider how b est

to approximate the feedback price with the solution of the Black-Scholes equation for this security

in the least-squares sense of Section 3.2. We shall require the following results.

Lemma 1 If h x 0,

2 h

@ C

in x> 0;t T .

2 h 2 h

=0 twice with resp ect to x gives =@ x . Then di erentiating L C Proof. Let W x; t= @ C

BS BS

2 2 2 2

W + x W +r +2 W + r W = 0;

t xx x

W x; T = h x:

That C is twice di erentiable follows from the Green's function solution and the smo othness

assumptions on h. Then we can use the Minimum Principle, since only the constant co ecients

are di erent from the Black-Scholes op erator, to deduce that W x; t 0: 25

Lemma 2 The Black-Scholes price is a non-decreasing function of volatility.

h h h

Proof. Let R x; t; = @C =@ . Di erentiating the Black-Scholes equation for C with resp ect

BS BS

to gives

2 h

@ C

h 2 BS

0; L R = x

W x; T = 0:

By the Minimum Principle, R 0in x> 0 and t

Prop osition 3 To rst-order in , the adjusted Black-Scholes volatilities ^ t which solves 3.10,

h h

and ^ x which solves 3.13 with C and C replacedby C and C are not less than the

reference volatility ,provided the terminal payo function is convex.

t is given by^ t= + t+ O where Proof. Following Section 3.2.1, ^

C x; t; R x; t; dx

t= : 4:5

1 2

R x; t; dx

Since C 0, it is sucient to prove R 0, for then t 0 and ^ t . This follows from

the previous Lemma. The same is true for ^ x from equation 3.14.

Furthermore, all these results generalize to the multi-option case b ecause the small expansion

e ectively decouples the system:

! !

2 h

@ C

iBS

2 3 h

= x L C 0:

2 2

@x @x

In general, the increased volatility due to program trading can b e observed b oth in the sense

of implied volatilities and realised `sp ot' volatility, which refers to the co ecient in 2.10 times .

From 2.22, this can b e written

; 1+

V 1 U V 1 x

so that, assuming the denominator stays p ositive, this factor is greater than the no-feedbackvolatil-

ity whenever the C is p ositive. We are grateful to an anonymous referee for p ointing out

that volatility only increases when the of the replicated p ortfolio is p ositive which is true for the

convex payo s of call and put options. The destabilising feedback e ect of p ositive replicating

strategies is discussed further bySchonbucher [22].

5 Numerical Solutions and Data Simulation

Wenow return to the setting where , the market share of the program traders, is not necessarily

small, and equation 2.32 for the feedback price of a Europ ean option must b e solved numerically.

A nite-di erence scheme to do so for b oth scalar and multi-option cases is outlined in [25].

Using these solutions, we simulate market data for the underlying and consider typical discrep-

ancies b etween historical estimates of volatility from tting the data to a geometric Brownian 26

Motion and Black-Scholes implied volatility from feedback options prices given by our mo del.

To construct a numerical solution to 2.32, we rst transform the call option price into P x; t

where

r T t

P x; t= C x; t x Ke 5:1

to make the b ehavior of the unknown function zero as x b ecomes large . Then P x; t satis es

1 P 1

2 2

x P + r xP P = 0; t

xx x t

2 1 P + xP

x xx

P x; T " = P x; T ";

r T t

P 0;t = Ke ;

lim jP x; tj = 0;

x!1

r T t

and P x; t= P x; t:=C x; t x Ke for T " t T .

BS BS

5.1 Data Simulation

In this section, we illustrate typical pricing and volatility discrepancies that might arise if feedback

e ects from dynamic hedging strategies are not accounted for and the classical Black-Scholes mo del

is used instead. We simulate typical price tra jectories of the underlying asset using the feedback

sto chastic di erential equation 2.9 in 0 t T =0:5 with our numerical solution for the

feedback option pricing equation 2.32 and =0:1, and we use the parameter values =0:15,

and =0:3 for the drift and di usion co ecients in the geometric Brownian motion mo del 2.21

for the reference traders' income which are needed for the drift term X ;Y ;t in 2.9. The

t t

paths are started at x =4:5, just lower than the strike price K = 5, and , the b est estimate of

the reference traders' volatility parameter using historical data up to t =0,istaken to b e 0:4as

in our previous examples. The simulation is done by forward Euler: let X b e the numerically

generated representation of X along a particular path, and Y the same for Y . Then,

it it

i+1 i i i i i i i

^ ^ ^ ^ ^ ^ ^

t; X = X + X ; Y ;itt + v X ; Y ;it Y ;it

h io n

i i i i

^ ^ ^

1 C X ;it X where are indep endent N 0; 1 random variables, and Y = ;

from 2.7, and i =0; 1; 2; .

est

Then we calculate the constant historical volatility parameter that would b e inferred by trying

to t X here regarded as historical data to a lognormal distribution:

" ! " !

N N

i i

^ ^

X X

1 X X 1 1

est

= log log ;

i1 i1

^ ^

N 1 N N 1

X X

i=1 i=1

using the asset prices for 0 t 0:2 t, so that N =0:2=t 1. This is the volatility parameter

the program traders would use if they were to continually calibrate the Black-Scholes formula with

up-to-the-minute volatility estimates from the latest data. Nowwelookathow this metho d of

Although this lo oks like the put-call parity relationship, the nonlinear PDE is not invariant under this transfor-

mation, so wehave not assumed that put-call parity holds; this transformation is only a to ol to make the equations

more tractable to numerical metho ds.

Note that and are not needed in the feedback option pricing mo del.

1 1 27 Feedback Asset Price 7

4 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4

Feedback & Black−Scholes Option Prices 2

0 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4

Estimated & Implied BS Volatilities

0.5

0.4 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4

Time

Figure 5.1: Data Simulation: =0:1, =0:4, K =5, T =0:5. The top graph shows the feedback

asset price X in 0:2 t 0:4. In the middle graph, the solid line is the feedback price and the

est

dotted line the Black-Scholes price using from 0 t<0:2. Here e ects are shown for an

est

asset price path rising ab ove the strike price of the option, and =0:446. The average relative

mispricing by the Black-Scholes formula is =6:6.

pricing compares with using the numerical solution of the feedback equation in the time interval

0:2 t 0:4. The top graph of Figures 5.1-5.3 shows the simulated X in this p erio d.

i i est

^ ^

Next we compute C X ;it using the numerical solution and C X ;it; from the

Black-Scholes formula, and these are plotted in the middle graphs. Frey and Stremme [11] asked

the question whether the Black-Scholes mo del was still valid given the market inelasticity induced

by the program traders: these pictures show that the answer is yes, as exp ected since the feedback

mo del was constructed to b e in the neighb orho o d of the classical theory, but also that signi cant

mispricings can o ccur. The average p ercentage price discrepancy over each path,

i i est

^ ^

C X ;it C X ;it;

C X ;it

iN 28 Feedback Asset Price 4.5

3.5

3 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4

Feedback & Black−Scholes Option Prices 0.2

0.1

0 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4

Estimated & Implied BS Volatilities 0.52

0.5

0.48

0.46 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4

Time

Figure 5.2: Data Simulation: Here e ects are shown for an asset price path falling b elow the strike

est

price of the option, and =0:464, =22:0.

is given with each of the pictures, and has b een observed to vary b etween 1 and 28 for various

paths. An average value of over 100 simulated paths was found to b e =8:2.

Finally,we plot in the b ottom graphs of Figures 5.1-5.3 the implied Black-Scholes volatility

i est

from the feedback option price C X ;it in comparison with the constant . Exp eriments

reveal that in nearly all cases simulated, this calibration to the Black-Scholes formula underesti-

est

mates the implied Black-Scholes volatility. This means that, although >, the full feedback

e ect on market volatility and option prices is not captured by the classical mo del with a constant

est

volatility parameter. In practice, users of the Black-Scholes formula would up date more often

than once in the lifetime of the six-month option, but similar discrepancies will result.

5.2 Numerical Solution for many options

We brie y outline the pro cedure to obtain the numerical solution to the system 2.27, initially

when all the strike times are equal: T T , but the strike prices vary: h x= x K . This

i i i

was considered for small in section 3.3. For simplicity,we also assume a uniform distribution

, for all i =1; ;n.

i 29 Feedback Asset Price 6

5.5

4.5 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4

Feedback & Black−Scholes Option Prices 1.5

0.5

0 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4

Estimated & Implied BS Volatilities

0.5

0.4 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4

Time

Figure 5.3: Data Simulation: Asset price path b oth ab ove and b elow the strike price, and

est

=0:442, =9:6.

Analogous to 5.1, we transform to put option prices

r T t

P x; t= C x; t x K e ; 5:3

i i i

and obtain a system of n equations analogous to 5.2:

1 1 ~ P~ @ P @P @P

i i i

2 2

+ x + r x P = 0; t

@t 2 1 ~ ~ P + xP @x @x

x xx

P x; T " = P x; T "; K ;

i BS i

r T t

P 0;t = K e ;

i i

lim jP x; tj = 0;

x!1

n 1

where P x; t:= P x; t, ~ := S =n = =n, and " is the smo othing parameter corresp ond-

i=1

ing to ~ from section 2.7.1. Now a nite-di erence scheme is readily applicable. 30

5.2.1 Volatility Implications

We calculate the average implied Black-Scholes volatility under feedback e ects from many call

options spread around the strike price K = 5 that we used as our example in the scalar case. That

is, we solve for C x; 0; ;C x; 0 in 2.27 at time t = 0, when the strike prices are evenly

1 n

distributed b etween K = 4 and K = 6 at intervals of 0:1. Then we calculate the implied

min max

av e

Black-Scholes volatilities x; ; xatt = 0 and average these to obtain x. This is

1 n

shown in gure 5.4 with =0:1.

The graph shows that the biggest increase in Black-Scholes volatility is in the neighb orho o d of the

Averaged Implied Black−Scholes Volatility 0.46

0.455

0.45

0.445

0.44

0.435

0.43 2 4 6 8 10 12 14 16 18

av e

Figure 5.4: Averaged Black-Scholes implied volatility xatt = 0 from options with strike

prices 4; 4:1; ; 6, and =0:1, =0:4.

strike prices, and there is an averaging of the p eaks over this price range. This is consistent with

our results from the scalar case. In addition, there is an overall volatility increase at all prices as

a consequence of the program trading.

We also compare the e ect of the spreading of strike prices on the actual volatility increase. Frey

and Stremme [11] found that increasing heterogeneity of the payo function distribution reduced

the level of the volatility increase in their mo del. This is qualitatively con rmed for our mo del in

gure 5.5 in whichwe plot the volatility co ecient

1 C

V x; t:=

1 C xC

x xx

at t = 0 when there is only one strike price K = 5, and the equivalent

1 =nC

V x; t:=

=nxC 1 =nC

xx x 31

for the n = 21 option typ es with strike prices b etween K = 3 and K = 7 at intervals of 0:2

min max

apart. The interpretation is that the sensitivity of program trading to price uctuations around

one particular strike price is spread over a larger range of prices: as x increases from $4 to $5, for

example, the program traders buy more sto ck to insure against likely losses on the options with

strikes less than $5, but the ones with strikes close to $7 are still relatively `safe' and insurance for

these options do es not, at this stage, induce as signi cantavolatility rise as if they to o had b een

struck at $5.

Feedback Volatility Coefficient for concentrated and distributed strike prices 1.14

1.12

1.1

1.08

1.06

1.04

1.02

0.98 0 2 4 6 8 10 12 14 16 18 20

Figure 5.5: The dashed curve shows the feedbackvolatility co ecient V x; 0 for a single option

typ e K = 5, and the solid curve shows the equivalent V x; 0 for strike prices evenly spread

between 3 and 7.

5.2.2 Varying strike times

To complete the picture, the numerical solution algorithm can b e adapted to the system of many

options with various strike times T

1 2 M

which options are still active unexpired at each time and contribute to the feedback through C .

That is, the numb er of options de ning C is a function of time: n = nt.

Since we are interested in the volatility co ecient V x; t, we can add up the equations for the

option prices C in the correct prop ortion if the strike prices are not uniformly distributed for each

strike time and solve a scalar equation for C x; tin t

are three strike times T =0:25;T =0:5 and T =0:75, corresp onding to 3,6 and 9month

1 2 3

contracts, and 21 options of each maturity with strike prices evenly spread b etween K = 4 and

min

K = 6 and as b efore.

max i

The volatility co ecient V x; t is shown in gure 5.6, and a cross-section at x = 5 in the 32

top graph of gure 5.7. The volatility rises to a sharp p eak as each strike time is approached and

then falls as some of the options expire, creating less feedback. We note that the height of these

spikes is controlled by our smo othing requirements: hedging very close to the strike times of the

options is not accurately mo delled by a Black-Scholes-typ e dynamic hedging strategy it is likely

to b e very individual , and it is doubtful that the program traders follow a particular program at

such times, and the so on-to-expire options are assumed not to contribute to V x; t once they

are within " of their strike time.

Volatility Coefficient for multiple strike times and prices

0.55

0.5

0.45

0.4

0.35 0.8 0.6 20 0.4 15 10 0.2 5 0 0

t x

Figure 5.6: Volatility Co ecient V x; t for strike times 0:25; 0:5; 0:75 and strike prices 4; 4:1; ; 6.

The b ottom graph of gure 5.7 shows typical volatility b ehavior V X ;t where X isatypical

t t

feedback tra jectory with constant drift ~ =0:15 for simplicity:

dX =~X dt + V X ;tX dW :

t t t t t

Again, the presence of p eaks close to the strike times is apparent.

6 Summary and conclusions

Wehave presented and analyzed an options pricing mo del that accounts self-consistently for the

e ect of dynamic hedging on the volatility of the underlying asset. Following Frey and Stremme

[11], we consider two classes of traders, reference and program traders. We then incorp orate the

in uence of the program traders into the pricing of the option. This leads to a new family of non-

linear partial di erential equations that reduce to the classical Black-Scholes mo del in the absence

of p ortfolio insurance trading. The solutions give the feedback adjusted price of the option and the

program traders' hedging strategy. Moreover, our mo del provides a quantitative estimate of the

observed increase in volatility due to program trading. 33 Variation at fixed x=5 0.52

0.5

0.48

0.46 Volatility Coefficient

0.44 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Time t Variation along a price trajectory 0.5

0.45 Volatility Coefficient

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Time t

Figure 5.7: Volatility Co ecient for strike times 0:25; 0:5; 0:75 and strike prices 4; 4:1; ; 6at

x = 5, and along a price path X .

One way in which the self-consistent option pricing mo dels can b e used is for estimating the frac-

tion of trading that is done for p ortfolio insurance. We are studying this at present. Another issue

that requires attention is incorp oration of feedback e ect into sto chastic volatility mo dels that are

b ecoming more and more p opular in the industry.

Acknowledgements

We are extremely grateful to Darrell Due for conversations and advice on the sub ject, and to

Hans Follmer for bringing the problem to our attention.

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