Managing the Volatility Risk of Portfolios of Derivative

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MANAGING THE VOLATILITY RISK OF

PORTFOLIOS OF DERIVATIVE SECURITIES

THE LAGRANGIAN UNCERTAIN VOLATILITY MODEL

Marco Avellaneda and Antonio Paras

Abstract We present an algorithm for hedging option p ortfolios and customtailored

derivative securities which uses options to manage volatilityrisk The algorithm uses a

volatility band to mo del heteroskedasticity and a nonlinear partial dierential equation

to evaluate worstcase volatility scenarios for anygiven forward liability structure This

equation gives subadditive p ortfolio prices and hence provides a natural ordering of pref

erences in terms of hedging with options The second element of the algorithm consists of

a p ortfolio optimization taking into account the prices of options available in the market

Several examples are discussed including p ossible applications to marketmaking in equity

and foreignexchange derivatives

Courant Institute of Mathematical Sciences New York University Mercer St New York NY

Email avellane cimsnyuedu parascimsnyuedu We are grateful to Thomas Artarit Raphael

Douady Nicole ElKaroui Arnon LevyHoward SaveryJoseScheinkman Nassim Taleb and Paul Wilmott

for their useful comments and suggestions All mistakes are of course ours

Introduction

Volatility is a crucial variable in the trading and riskmanagement of derivative securi

ties The uncertain nature of forward volatility is recognized as one of the main factors

that drive marketmaking in options and customtailored derivatives

Derivative asset prices are aected by new information and changes in exp ectations as

much as they are bychanges in the value of the underlying index Largescale changes

in implied volatilities were notorious in the crash of and in the aftermath of the

Mexican Peso devaluation of Decemb er Currency markets saw signicantreversals

in the skewness of the implied volatility curve in the summer of as central banks

moved to protect the dollar and again in late Septemb er as the dollar fell against

the Deutschemark These events cannot b e understo o d forecast or mo deled using deter

ministic volatility

Incorp orating heteroskedastic b ehavior ie volatilityofvolatility is essential for the

riskmanagement of derivatives Several attempts in this direction have b een made most

notably with autoregressive mo dels and with the use of sto chastic dierential equations to

mo del volatilitychanges However random volatility mo dels are more delicate and complex

to implement than BlackScholes due to the larger numb er of parameters that need to b e

estimated More imp ortantlyintro ducing multiple parameters to dene a pricing measure

may not necessarily b e the b est way to incorp orate information ab out future market risk

Pricing mo dels with a priori statistical distributions for the sto chastic volatility will not

capture abrupt changes in volatility exp ectations and the fact that ultimately only one

volatility path will b e realized

Option prices provide concrete information ab out the markets volatility exp ectations

Therefore options are crucial for hedging in an uncertain volatilityenvironment Typi

cally hedging with options is presented in two paradigms replicationorsynthetization

of liabilities and dynamical hedging or management of p ortfolio sensitivities In the rst

approach a derivative pro duct is identied through reverseengineering as a series of

simpler optionlikepayos Whenever cashows can b e matched exactly to those of mar

ket instruments the value of a given contingent claim should b e equal to the price of

the synthetic p ortfolio which also represents a sure hedge against market movements In

practice however p erfect replication is seldom p ossible due to its high cost and to market

incompleteness and liquidity constraints Traders are therefore faced with the problem of

determining howtodiversify risk bysynthetization and how to price and manage the

residual risk to b e carried forward Ideally they would like to hedge their riskexp osure

in a secondary market whenever this leads to protable trading but it is often said that

options are to o exp ensive a thorough coverage using derivatives can price a deal out of

the market

The alternative is dynamical hedging If traders had p erfect foresight on forward volatil

ity then Deltahedging continuously in the cash market would b e essentially riskless In

practice continuous hedging is imp ossible and there is volatility risk Such risks are usually

taken into accountby hedging with derivative instruments which through their convexity

allow for adjustments in the exp osure to higherorder sensitivities of the mo del suchas

Gamma dierential change in Delta Vega dierential change in volatility Rho dier

ential change in interest rates etc From a theoretical p oint of view the main problem

with Greek matching is that i the metho d provides protection only to dierential smal

l market changes ii the metho d is parametric ie requires having a precise view on

forward volatility and iii the cost of markingtomarket the dynamic hedge cannot b e

determined in advance

There is an intrinsic inconsistency in the way pricing and hedging are viewed in clas

sical frameworks of dynamic hedging On the one hand pricing is made with parametric

mo dels Once the market price of risk has b een calculated every derivative instrument

is equivalent to a dynamic p ortfolio of basic securities On the other hand matching

the Greeks esp ecially Gamma Vega and other higherorder derivatives implicitly

recognizes that the probabilistic assumptions will not b e valid at later times We b elieve

that a key factor missing in current derivative pricing mo dels is the idea that heteroskedas

ticity gives rise to a preference ordering in terms of which trades should b e made under

particular market conditions This ordering of preferences cannot b e captured by linear

presentvalue mo dels in which derivative asset prices are additive functions of their future

cashows

From the ab ove considerations we contend that a reasonable mo del for managing the

risk of derivative securities in markets with uncertain volatility should satisfy the following

requirements

Heteroskedasticity or uncertainty in the values of forward volatilities should b e taken

into accountby assuming that more than one arbitragefreemeasurecan realize current

prices at any given time

Portfolio values should b e subadditive for the sellside and superadditive for the

buyside due to diversication of volatilityrisk

Option prices as well as the prices of other liquid optional instruments traded in

the market should b e incorp orated into the mo del since they provide the information

necessary to narrowdown volatility uncertainty

This pap er is an attempt to implement these ideas in a mo del which is b oth theoreti

cally sound and easy to implement The main elements in our approach are i mo deling

volatility uncertaintyby using volatility bands and ii optimization over the class of ad

missible probabilities according to market prices of derivative instruments The rst idea

is implemented using the Uncertain Volatility Mo del UVM intro duced recently byAvel

laneda Levy and Paras Paras This pricing metho d assumes that

forward volatility paths vary inside a band and calculates the value of assetsliabilities

It is imp ortant to note that matching the higherorder derivatives of the BlackScholes formula Greek

s is not a theoretical consequence of dynamical asset pricing theory but rather a practical device used

by traders which are wellaware of the shortcomings of statistical presentvalue mo dels and market

heteroskedasticity

In the sense of parametric statistics wherein a sp ecic probability distribution is assumed

under the worstcase volatility scenario Portfolio valuation is subadditive for the sellside

and sup eradditive for the buyside The second feature optimization is fundamental

we use options as hedging instruments to minimize the worstcase scenario value of the

contracted liabilites Our approach can b e summarized schematically by the equation

Mo del Value

Min f Value of Option Hedge Max PV Residual Labilityg

where i the residual liability liabilityminus option payos is valued under the worst

case scenario according to UVM and ii the minimum is takeover all option p ortfolios

ie over p ossible hedges which use options and cash instruments For any given liability

structure this pro cedure selects a p osition in options which under the assumptions of the

mo del will cancel the p ortfolio risk at the least cost

This pap er describ es the implementation of the algorithm for derivative securities based

on a single market index We study several examples which can lead to p otential applica

tions and analyze the sensitivity of the mo del to the assumptions of volatility band and on

the numb er of input options used to hedge Other examples indicate how the mo del could

b e used for trading equity and and foreignexchange derivatives using a large number of

input options In particular we consider an example based up on the Telmex options mar

ket in the rst months of and an analysis of riskmanagementofknockout options

on DollarMark using recent market conditions Finally in an App endix we connect the

mo del with the theory of arbitragefree pricing in a market with many derivative securities

based on a single underlying asset

The uncertain volatility model uvm

Let us consider a mo del for contingentclaimvaluation based on an underlying asset

with uncertain volatilityFor this purp ose we assume that the sp ot price of the underlying

follows a sto chastic dierential equation of the typ e

P dt dZ

t t t

where and are sp ot drift and volatility parameters These functions may dep end

t t

on present andor past market information Since the underlying asset is a traded security

we can assume for valuation purp oses that

r d

t t t

where r is the sp ot domestic riskless rate and d is the dividend rate or foreign interest

t t

rate Our primary concern is with volatilityrisksowe assume for simplicity that r

and d are constant For eachvolatility pro cess f g the symbol P in represents the

t t

probability measure induced on price paths fS g

Volatility uncertainty is mo deled by assuming that the volatility pro cess whichdrives

the price in will uctuate within a band ie

t T

min t max

Here and are constants or more generally deterministic functions of the sp ot

min max

price and time and T is some terminal horizon The assumption of a forward range for

the sp ot volatility is the only hyp othesis made on heteroskedastic b ehavior In particular

we shall not sp ecify an a priori probability distribution for Instead we will consider

the set P of all p ossible measures P induced byanyvolatility pro cesses whichvary within

the band

We consider the following valuation problem calculate the present value of terminal

assetsliabilities under the worstcase scenario for a forward volatility path assuming that

it remains inside the apriori band To x ideas consider an agent that must deliver a

stream of cashows

F S F S F S F S

t t t N t

where F arepayos due at settlement dates t t t The worstcase

j N

scenario presentvalue estimate of his liabilityisgiven by

P r t t

V S t sup E e F S

t j t

P P

where E represents the exp ectation op erator asso ciated with the sto chastic dierential

equation

As shown in Avellaneda Levy and Paras V St satises the nonlinear pro

gramming equation

Probabilistic b eliefs on the b ehavior of price volatilityasastochastic pro cess such as Hull White

can b e incorp orated in the mo del by constructing a condence interval for the volatilitypath

Avellaneda Levy and Paras applied this idea to the case of a lognormal meanreverting volatility

pro cess In realworld situations a statistical analysis of the range of sp ot volatilities can b e used to

determine a suitable band for the market of interest

Under this convention the liabilities are included with a p ositive sign and the assets with a negative

sign So if the agent is short a call expiring at for example then F S maxS K but if he was

long that same call then F S maxS K

V V V V

rV S

t S S S

where

if X

max

fX g

if X

min

This equation is similar to the BlackScholes PDE with the dierence that

V St

ie by the input volatility is not constant it is determined by the sign of

the convexityof V Thus the worstcase volatility path is V S S

a function of S and which dep ends up on the stream of cashows liabilities under

consideration

The value V S t corresp onds to the cost of dynamic hedging with the underlying asset

under the worstcase volatility path In fact it was shown that a dynamic strategy for

risklessly hedging in this uncertain volatilityenvironment starting with wealth V S t

consists in maintaining a p osition of V S S contracts in the cash market

and adjusting it p erio dically as the market moves Avellaneda Levy and Paras An

agent who follows this strategy will end up with a nonnegative cashow after delivering

the payos F S Moreover if the volatility path actually followed the worstcase path

j t

this is the only nonanticipative strategy that exactly replicates the stream of cash

ows generating no excess returns Therefore this is the least costly dynamic hedging

strategy that can b e constructed using the underlying asset no options as hedging

instrument which never generates losses regardless of the path followed byvolatility

The partial derivatives V S t S and V S t S can b e viewed as riskadjusted

Delta and Gamma For example a short option p osition represents a convex liability F S

for the agent and is therefore priced with the maximum volatility Similarlya

max

long option p osition is valued using The quantity

min

V S t

can b e interpreted as a new Gamma sensitivity of the hedgeratio to price movements

adjusted for volatilityrisk For atthemoney options the relation b etween this adjust

ed Gamma and the classical BlackScholes Gamma is the following given any constant

BlackScholes volatility within the band eg the eect of

BS min max

the nonlinear PDE is to overestimate when the agent is short Gamma and thus

to reduce Gamma in absolute value in comparison with BlackScholes theory When the

agent is long Gamma is underestimated and thus the Gamma owned also increases

Akey feature of equation is its subadditivity with respect to payos The sum of

the worstcasescenario values of each cashow will b e generally greater than the value of

the stream of cashows considered as a lump ed liability This is due to the cancelation

of volatility risk that o ccurs if the overall p ortfolio has mixed convexity Hence unlike

linear mo dels the present mo del quanties the p ortfolios exp osure to volatility risk

Hedging with options

the Lagrangian uncertain volatility model uvm

Consider an agent who must deliver cashows F S j N and wishes to

j t

hedge his exp osure using a mix of options and the underlying asset Let us assume that

there are M Europ eanstyle options available for hedging Their payos and expiration

dates are denoted resp ectively by G S and with We

i i M

assume that the options are available for trading at the prices C C C Weomit

temp orarily the bidoer spreads in these prices to simplify the discussion but these will

b e incorp orated later on

Supp ose that at time t the agentpurchases a p ortfolio of options consisting of

contracts of eachstrikematurity to hedge his exp osure The market value

of this p ortfolio is

i i

The combination of this p osition and the agents previous liability mo dies the agents

overall risk prole After the trade the residual liability in presentvalue terms is

N M

X X

r t t r t

j i

e F S e G S

j t i i

j i

The total cost of hedging computed by adding the cost of the options and the worst

case cost of dynamically hedging the residual is

V S t

t M

Thus G S Max S K or G S Max K S where K is the strike price Due

i i i i i

to the putcall parity relation it is sucient to consider one typ e of option call or put p er strikeand

maturity

Liquidity plays an imp ortant role in hedging esp ecially if the trader considers a complex hedge using

a large amount of options

N M

X X

r t t r t P

j i

e F S e G S sup E

j t i i

j i

P P

j i

i i

The supremum over exp ectations E is calculated with UVM equations

A p ortfolio of options issaidtobeanoptimal hedge if it solves the

optimization problem

V S t inf

t M

To this optimization we give the name of Lagrangian Uncertain Volatility Model or

UVM

The vector must b e restricted to vary over a suitable range of p ortfolio

combinations A reasonable sp ecication for this range is

i i

where and are constants These constraints on the option p ortfolio are natural from

the p oint of view of trading limits or from liquidity considerations We shall assume that

the s can take arbitrary values inside the intervals in It is convenient to assume

whenever p ossible that the s can take b oth negative and p ositivevalues

The function V S t isconvex in This follows from the

t M M

value function is a supremum of linear functions in cf App endix When the vector

is equal to zero this function reduces to regular UVM valuation as in Section and the

agent hedges only in the cash market In general the minimum in is attained for

since the implied volatilities lie inside the band they are cheap er to buy and

more exp ensive to sell than at or Accordingly the cost of the ecienthedge

max min

derived from will b e less than the exp ected cost of strictly deltahedging under the

worstcase scenario

In the examples to come wewillshow that the cost of the ecient hedge can b e often

comparable to the BlackScholes fair value using midmarket volatilities even when the

As we shall p oint out later the s can b e viewed as Lagrange multipliers for a constrained optimization

problem

Prices quoted bymarketmakers are often valid only for limited amount of contracts Optimal

strategies which require trading in large volumes may b e meaningless since the price is likely to change

with the order ow

band is very wide to in the TELMEX example of x This is due to the fact

that the algorithm constructs strike and calendar option spreads which reduce volatility

risk and the need for intensive Deltahedging

So far we regarded options as essentially static instruments once the p osition is taken

the option hedgeratios need not b e readjusted However market are far from b eing

static Desert island option strategies can b e replaced by dynamic ones as the market

p ermits In fact once the hedge is in place the agent can takeadvantage of the evolution

of prices and the issuance of new options to improve the p osition by applying UVM to

the liability structure inherited from the previous trading date If a new option hedge is

then selected the new p osition generates a b o ok prot without taking additional volatility

risk

Calibration of the volatili ty band There are imp ortant consideration to b e made

with resp ect to the choice of the volatility band An incorrect calibration of the mo del

may result in spurious arbitrage opp ortunities when the band is taken to

min max

b e to o narrow Taking into account exp ected trends in forward volatility p ossibly with a

band whichvaries with time may b e useful in order to avoid overestimating volatility risk

with the UVM This issue deserves some discussion

We know from classical valuation theory that there is a precise mathematical relation

between sp ot volatilities and implied volatilities obtained via the BlackScholes formula

In fact assuming a band for sp ot volatilities is essentially equivalent to assuming that the

implied volatilities of traded options will remain within predetermined b ounds over the

p erio d of interest In other words the band assumption implies an upper bound on the

purchase price and a lower bound on the sale price of any option traded over the p erio d

of interest For instance if and are constants the simplest kind of band the

min max

b ounds in are consistent with the b elief that

min impl max

where t represents the implied volatilityattimet of an option maturing at time

impl

If the bands are timedep endent ie

t t

min min max max

the assumption on the forward sp ot volatilityistantamount to assuming that

Z Z

s ds t s ds

min impl max

t t

for any t T

The hyp erb ole is b orrowed from E Thorp

This statementisvalid as long as the assumption made on the volatility band is not violated

The calibration of the forward volatility band should b e done using market data such

as the implied volatilities of liquid options historical and seasonal information and b eliefs

ab out abrupt changes in implied volatility levels The implied volatilities of traded options

provides an imp ortant indication ab out the range of If for instance and

t min max

are chosen to b e constant the band should contain the implied volatilities of all options

considered as input instruments to avoid mo del arbitrage In addition the band

should b e strictly wider than the range of the implied volatilities taken as the total range

when the band is at or in buckets when the band if timedep endent b ecause of the

additional volatility risk that exists and transaction costs see Section

In markets where implied volatilites vary signicantly with the maturity of options a

more conservative approachwould consists in using implied forwardforwa rd volatilites to

calibrate the band Accordingly if at time t we are given the implied volatilites of two

atthemoney options with maturities T and T T T we can deriveanapproximate

implied T toT volatility T T by solving the equation

impl

T t t T T t t T T T T T

impl impl impl

The band can then b e chosen so as to satisfy the additional constraint that

min

T T

max

impl

Cho osing a band that contains all implied volatilities is a generalization of the standard pro cedure

which consists in selecting a constantspotvolatility in usually equal to the implied volatilityofa

liquid atthemoney option

Equation denes an expected but not necesarily realized forwardforward volatilityover the

period T T The bands obtained using this typ e of constraintarealways wider than the ones obtained

by using implied volatilities of options currently traded For a study of implied forwardforward volatilities

we refer the reader to Taleb and Avellaneda forthcoming

Transaction costs and bidoffer spreads

Transaction costs increase the cost of hedging and hence aect the comp osition of the

optimal hedging p ortfolio In particular pure arbitrage opp ortunities whichmight exist

in the absence of market frictions may disapp ear once transaction costs and bidoer

spreads are prop erly accounted for

Bidoer spreads for trading the underlying asset We assume that Deltahedging

will require the agent to buy at the oer and sell at the bid Although this may not always

b e the case the assumption is consistent with the worstcase scenario approach followed

so far

It is wellknown that the impact of bidoer spreads on Deltahedging costs can b e in

corp orated a priori into the value of derivative securities by adjusting the mo del volatility

see Leland Boyle and Vorst Hoggard Whalley and Wilmott Avel

laneda and Paras among others Accordingly if the price of the underlying

asset follows a geometric Brownian motion with constant volatility the cost of Delta

hedging a derivative security including exp ected future transaction costs is obtained by

solving equation with

k V

Here k is the exp ected roundtrip transaction cost expressed in p ercent of the value of

the underlying security and dt is the timelag b etween adjustments This fundamental

result can b e applied to heteroskedastic pricing mo dels If the volatility is uncertain and

varies in the band then the trasnsactioncostadjusted sp ot volatility will vary in a

wider band with upp er and lower b ounds given by

max

min max

Transaction costs arise from brokerage fees b elowmarket returns for margin dep osits taxes and

bidoer spreads for trading cash instruments and options We shall b e primarily concerned with two

latter eects The problem of receiving b elowmarket interest for dep osits held in margin accounts can b e

studied in the framework of passiveactive dep osit rates This feature is actually taken into account in the

numerical implementation of the algorithm but we shall not elab orate on it here

the lower volatility should b e set to zero Avellaneda and Paras If

and

Max min

min

min max

Hence the impact of transaction costs on Deltahedging can b e incorp orated bycho osing

the width of the band appropriately

Bidoer spreads in option prices We assume that

bid price for the i option C

and

C oer price for the i option

for i M We shall op erate under the assumption that options can b e purchased

at the oer price and sold at the bid price for the amounts sp ecied bythe constraints

in In this case the cost of acquiring a p ortfolio is computed as in

o b

if Accordingly this cost is if orC but with C replaced by C

i i i

i i

given by

b o b o

C C C C

i i i i

j j

i i

The total hedging cost for the derivative pro duct with cashows fF S g is therefore

j t

V S t

t M

N M

X X

P r t t r t

j i

sup E e F S e G S

j t i i

j i

P P

j i

b o b o

C C C C

i i i i

j j

i i

We conclude that transaction costs for trading in the cash market and in options can

b e taken into account with slight mo dications of the algorithm presented in Section

Barrier options

We discuss how to apply the mo del to barrier options Knockout options b ecome auto

matically worthless when the price of the underlying asset reaches a preestablished level

and similarly knockin options are activated when a particular price level is attained

The sensitivities of the option premium near the barriers are of particular interest from the

p oint of view of hedging In fact the payo of reverse kno ckouts and reverse kno ckins

is a discontinuous function of the index This results in large values of Delta and inversed

Gamma exp osures that make deltahedging near expiration dicult if not outright imp os

sible The p ossibility of constructing costeective hedges using vanilla options seems

particularly app ealing in the world of barrier options

We consider the problem of hedging a kno ckout option bydiversication into the vanilla

options market Supp ose that the agent decides to use an option p ortfolio with hedgeratios

The principal dierence with the case of Europ eanstyle cashows without

barriers is that pricing a p ortfolio which combines barrier options and vanilla options

requires studying the p osition after the option kno cks out and the agent is left with a

vanilla option p osition that must b e managed or unwound

For simplicitywe consider the case of a single kno ckout option with a constant bar

rier The worstcase liability along the barrier corresp onds to the UVM value of the

outstanding vanilla options p osition after the kno ckout It is given by

bar r P r t

V S t sup E e G S

M i i

P P

for S t along the kno ckout barrier This function is calculated using the nonlinear PDE

To calculate the value of the residual liability for the agent prior to hitting the

barrier this includes the exp osure to the stillactive kno ckout option wemust solve

a b oundaryvalue problem for equation with b oundary data The mo del val

ue denoted again by V S t is obtained byaddingthevalue of the residual

liability to the cost of the option p ortfolio

According to the theory of x the residual liability solution of the b oundaryvalue

problem corresp onds to the p ortion of the p ortfolio which requires active Deltahedging

in the cash market The Delta is equal to V S t S

The algorithm can b e used to hedge in an aggregate mo de a p ortfolio of barrier options

with dierent barriers strikes and expirations

Substantial losses have resulted by the imp ossibilityofunwinding kno ckout p ositions requiring a

disprop ortionate oset in the cash market An interesting account of techniques for trading barrier options

is given in Taleb

Numerical implementation

In general the nonlinear equation do es not have closedform solutions and must

b e solved numerically A simple approach describ ed in Avellaneda Levy and Paras

and Paras relies on an explicit nitedierence solver for the PDE

It is useful to regard a nitedierence scheme as an exact valuation algorithm for a dis

crete mo del approximating the sto chastic dierential equation Accordingly let us

consider a trinomial mo del in which the underlying asset can change after each trading

p erio d to one of three dierent levels

S U

S S M

S D

After we imp ose a riskadjusted drift eq the trinomial tree has one degree of

freedom at each no de since the choice of riskadjusted probabilities fP P P g is not

U M D

unique This degree of freedom is used to mo del heteroskedasticity probabilities that

assign more weight to the extreme no des will yield a larger sp ot volatility than those

assigning more weight to the center Thus byxingUM and D and allowing the risk

adjusted probabilities to vary over a onedimensional set we can accommo date a range of

variances according to the band that we wish to mo del A simple choice of parameters

p p

dt dt dt dt dt

max max

M e and D e U e

dt b eing the timemesh The oneparameter family of riskadjusted pricing probabilities

is given by

P p dt

U max

P p

This interpretation is not essential to derive the nitedierence approximation to It provides

a concrete interpretation of the uncertain volatility mo del and the nonlinear PDE in the language of

probability trees

For stability reasons the volatility assigned to the tree has to b e greater or equal to This

max

restriction is equivalent to imp osing on the pricing probabilities to b e p ositive

and

dt P p

D max

Here p is a variable parameter that satises

min

max

to reect dierentchoices for riskadjusted probabilities or equivalently for sp ot volatilities

at each no de

The numerical implementation of equation along these lines takes the form

i h

j j

rdt j j

pL e V F V

n n

where

p p

j j j

L V dt V dt V

max max

n n n

j th

The term F app earing in represents the cashow due at the n trading date The

parameter p is chosen according to the rule

if L

min max

For p the extreme branches U and D carry of the probability and the lo cal

variance achieves the maximum value dtFor p the lo cal variance

max min max

can b e interpreted as a discrete dt The expression L achieves the minimum value

min

approximation to the secondderivativeof V

Equations and are the discrete analogues of and The reader is referred

to Paras for the pro of of convergence of the algorithm as dt

The minimization of the function V S t inUVM is done with the nite

t M

dierence solver for equation in the formulation coupled to a minimiza

tion routine For our computations we used a quasiNewton routine provided in the NAG

library

We implemented UVM on a SU N workstation Our algorithm incorp orates a term

structure of volatility boundsthus allowing the band to change with time It also allows

for a termstructure of dep osit rates taking into account dierences b etween b orrowing

and lending rates and thus for instance interest rates on margin accounts Using the

formulation of xwe are also able to takeinto account transaction costs in the cash and

options markets Finally the algorithm also oers the p ossibility of calculating hedging

strategies that imp ose limits on Delta

The running time of the computations grows linearly with M thenumb er of input op

tions and is essentially indep endent of the numb er of target cashows A nonoptimized

version of the algorithm used in the preparation of this pap er solved the optimization

problem in approximately userseconds for M and userseconds for M

Running times for hedging barrier options are longer due to meshrenement near the bar

rier but still scale linearly with M The longest calculations for barrier options involving

input instruments to ok userminutes

Example Hedging OTC options with different strikes

with atthemoney options

This example illustrates the theory in the simplest case option pricing in a market with

uncertain volatility in which there is a single traded option considered as input We assume

that the latter option is atthemoney and that it trades at some known implied volatility

Using this information in conjunction with a volatility band we price call options with

dierentstrikes having the same maturity

Exp eriment We assume that the riskless interest rate is The underlying asset

is assumed to pay no dividends over the timep erio d under consideration The maturity

of all options considered is days The atthemoney sixmonth option with strike

equal to the price of the underlying asset is assumed to trade at an implied volatility

impl

We assumed a volatility band with and in this rst exp er

min max

iment We applied the optimality algorithm to compute the prices and hedgeratios for

options with strikes at and of the sp ot The results are

illustrated in the following Table

Table Option pricing with band and

atthemoney options with volatili ty M

All these are nonlinear constraints on the dynamic programming equation which is mo died

accordingly

BS value

impl

The second and third rows indicate the BlackScholes values and Deltas of the dierent

options computed with volatility All option values are expressed as a p ercentage of

the sp ot price The fourth rowgives the value V which results from solving the optimality

algorithm assuming a band and a single hedging instrument atthemoney

options with implied volatility The next tworows show the optimal hedgeratios

with resp ect to the underlying security and the atthemoney option The last row

expresses the optimal value for eachstrike in terms of implied volatility

Notice that the mix of options and shares is dierent according to the strike a larger

prop ortion of options is used when the strike is close to the money Consistently with the

theory the mo del value of the atthemoney option is identical to the market price

and the hedge is purely synthetic with and As a rule the amount

of atthemoney options held long diminishes as the strikegoesaway from the money

Exp eriment We use the same mo del parameters as b efore except for the volatility

band which is taken to b e and The results given in Table

min max

give an approximate idea of how sensitive the pricing is to the width of the volatility band

compare the last rows of TableandTable

Table Option pricing with band and

atthemoney options with volatili ty M

impl

The option hedgeratios obtained with the band are generally larger than the

ones with the narrower band Note also that the implied volatilities are larger in Table

and increasingly so as the strikes moveaway from the money

Exp eriment A further assessment of the eect of changing the width of the band can

b e made by considering the limiting case In this case the optimality

min max

algorithm for hedging one option with M admits an explicit solution namely

K K

V KS

S S

V K S

Here V represents the value of a call with strike K expressed in p ercentage of the price

of the underlying In particular the hedgeratios are

K S

and

K S

These hedgeratios corresp ond to trivial cheap est static strategies using sto cks and at

themoney options In the example considered ab ove wehave V Using the

ab ove formulas we generate the corresp onding table for

min max

Table Option pricing with band and

atthemoney options with volatili ty M

impl

The interest of the table resides in the last row which provides absolute upp er b ounds on

the implied volatilities conditionally on the fact that the implied volatility of atthemoney

options is

Example Constructing volatility term structures

from market datafor atthemoney options

The values of options on a particular asset with dierent expiration dates are often

quoted using a term structure of implied volatilities for liquidly traded contracts which are

usually the atthemoney options When trading overthecounter OTC options with o dd

expiration dates agents must estimate the option values using market data In this exam

ple we use the optimality algorithm to estimate the prices of options with o ddexpiration

dates We derive in this wayaworstcase volatility term structure that interp olates b e

tween the implied volatilities corresp onding to maturities which are traded Both the

sellside and the buyside are considered

We shall assume in the example that there exist four liquidly traded atthemoney option

contracts maturing in and days The input data is as follows

Maturity Implied volatili ty

days

As b efore we assume an interest rate of and neglect transaction costs The limits for

trading options were taken to b e contracts for each maturity as we shall see these

limits were not reached in the calculations made b elow The volatilitybandwas chosen to

be We considered the problem of pricing atthemoney options

min max

calls with maturities of and days

Table Atthemoney call options held short

Matdays

impl

In this table represents the hedgeratio for the option maturing in x days The last

row gives the implied volatilities for each options calculated using V An examination of

the option hedgeratios shows that the next closest maturity usually carries the largest

part of the hedge

For example the implied volatility for the day option is and the amount

of day options used to hedge volatilityriskisvery small The p ointis

that holding the day option do es not oer much protection against an increase in the

dayvolatility Given the wide exp ected range of volatilities given by the band and

the comparativelow cost of day options the dayvolatility is only the

algorithm yields ie an option hedge using predominantly the day option

Consider instead the option maturing in days which is ten days b efore the expiration

of the day trading at volatility The algorithm values this option on the oer side

at volatility which is reasonably close to the esp ecially given the width of

the UVM band In this case the option hedgeratios are essentially negligible with the

exception of the one corresp onding to the nearestnext maturity option

Similar analyses can b e made for the other maturities

Wenow study the pricing and hedging of long atthemoney options with intermediate

maturities The buyside problem is equivalent to asking what is the least possible

arbitragefree value of a derivative security conditionally on the prices of input instruments

Table Atthemoney call options held long

Matdays

impl

As exp ected the implied volatilities are lower in comparison with the values for short

p ositions The buyside implied volatilities approach the market values when there is an

option expiring near and before the option under consideration For instance the day

option has a buyside volatilityof which is essentially in the middle of the band

and p ercentage p oints b elow the dayvolatility The corresp onding hedge ratio is

which represents essentially a oneforone hedge consisting in shorting

one day option In contrast the day option has a volatility which is practically near

the lower b ound implied by the band of The corresp onding option p osition consists

in shorting the and day options and implies substantial Deltahedging Notice that

the option which expires after all maturities considered here is not selected by the

algorithm on the buyside

Telmex options

This examples involves a larger set of input options We consider the Telmex TMX

Advanced Dep ository Receipts and options on this security traded in the NYSE in the

months following the December Peso devaluation During this p erio d the market

exhibited very large implied volatilities in comparison with levels Variations in im

plied volatilities on a given trading date were also large according to b oth strike levels and

maturities Overall the volatility termstructure was inverted decreasing as the maturity

increased A complex volatility structure such as this one is wellsuited for applying the

optimality algorithm

The following table describ es the Telmex market on March

Prices are p er ADR and the contract size is ADRs Source The New York Times March

Closing prices do not representmarket data at a particular instant in time We do not takeinto

consideration bidoer spreads volume traded op en interest etc Our sole aim is to illustrate how the

algorithm could b e applied to a market with many options and to analyze the hedges that result

Table Telmex March

March closing TMX Assumed interest rate

Mat Strike Call Ivol Put

Mar

Apr

May

Aug

The options are Americanstyle The implied volatilities of the calls computed using

the BlackScholes formula with an interest rate of app ear in the column Ivol

Puts app ear only for reference purp oses and are not used as inputs Call options can b e

treated as b eing essentially Europ eanstyle due to a a wellknown prop erty of options on

sto cks that pay no dividends In contrast American puts can b e exercised early and hence

cannot b e used in the optimality algorithm

The maximum range for implied volatilities is in the front month from Mar

to Mar These volatilities reect most likely the substantial transaction costs

and riskpremia asso ciated with farfromthemoney options expiring in a few days We

For instance an agent that sells a put to hedge an OTC derivative p osition runs the risk that the

put will b e exercised early and hence that the hedge will b e lifted

discard these two options from the set of input instruments reagarding them as illiquid

We see that the next two extreme volatilities are Aug and Mar

To implement the uncertain volatility mo del wecho ose a at band from Marchto

Octob er with

min max

Notice that the range of implied volatilities of the remaining input instruments is

whichiswellcontained within the chosen band Wehaveeven assumed a comfort

zone on each side to account for transaction costs in the cash market and p ossible volatility

uctuations outside the range of the implied volatilities of the input options The limits

Transaction and on the s of the hedge p ortfolio are taken to b e

i i

costs for trading options are not taken into account

Exp eriment Writing a Europeanstyle digital option Consider a contingent claim with

the following terms

Expiration date third FridayofMay

Payo if TMX or if TMX at the expiration date

We used only the calls which expire in March April and May excluding the Mar

and Mar calls as input instruments

The optimality algorithm yields the following output

Value V

ADRs sold short p er notional

Call hedgeratios

Mat Strike Quantity

Apr

May

We use the standard mathematical convention that represents the numb er of options to buy one

share Therefore the numb er of contracts is actually Given that the notional amount of the digital

is only options with s greater than are rep orted

Figure

2 2 1.5 1.5 1 1 0.5 0.5 0 0 −0.5 −0.5 −1 −1 −1.5 −1.5 −2 −2 15 20 25 30 15 20 25 30 May: Short Digital May: Short Digital Long 0.34 Puts(22.5)

2 2 1.5 1.5 1 1 0.5 0.5 0 0 −0.5 −0.5 −1 −1 −1.5 −1.5 −2 −2 15 20 25 30 15 20 25 30

April: Rollback April: Rollback Short 0.15 Puts(22.5)

Long 0.05 Puts(25)

Figure Residual liabilityvalues corresp onding to the short digital Telmex option and

the p ortfolio of synthetic puts shown at dierent dates p er dollar notional The upp erleft

box represents the liability corresp onding to the digital option with May expiration The

upp errightboxshows the liability including the option hedge consisting of long May

synthetic puts The lowerleft b ox represents the value of the liability after the April

expiration date The lowerrightbox represents the liability in April including the p osition

in the two options expiring in April Notice that the hedge selected by the algorithm gives

rise to at residual liabilities Thus the algorithm reduces market risk signicantly Present

values are calculated with the volatility band

We analyze this output from the p oint of view of pricing rst The value V

represents the total hedging costs options plus residual The worstcase scenario value

with a band without using options turns out to b e or ab out three times

higher This gives a measure of the eciency of using options to reduce hedging costs

An assessment of the comp etitiveness of V as an oer price for the digital

option can b e made by calculating the nominal implied volatilitywhichwould make V

equal to the BlackScholes fair value of the digital This volatility turns out ot b e

Given that the implied volatilities of the May options nearesttothe

impl

money are and to the nearest decimal we conclude that an implied volatility

of is reasonable for this exotic pro duct and even more so when the full range of

implied volatilities is taken into account

The result is also interesting from the p oint of view of hedging What kind of protection

was achieved using options Using the putcall parity relation we see that the option

comp onent of the hedge p ortfolio is equivalentto

Long May synthetic Europ ean put contracts

Apr synthetic Europ ean put contracts

Short Apr synthetic Europ ean put contracts

The hedge consists essentially in buying synthetic May puts to dominate the liabilityofthe

digital payo selling April synthetic puts with higher volatility and covering the exp osure

by shortselling TMX ADRs Figure represents the liability proles for this hedge at

dierent expiration dates The p ortfolio of synthetic puts eliminates almost completely

the volatility risk of the digital option

Example DollarMark foreignexchange barrier options

We consider the problem of pricing and hedging kno ckout options with inthemoney

barrier reverse kno ckouts using vanilla options and the sp ot market The data used in

this example corresp onds to the actual DEM market on August and incorp orates

bidoer spreads It consisted of

sp ot DEMUSD exchange rate on that date DEM

USD dep osit rates for and months bidoer

DEM dep osit rates for the same p erio d bidoer

volatility curve for options with these maturities bidoer

riskreversals curve for these maturities bidoer Maturity Type Strike bid offer Call 1.5421 0.0064 0.0076 Call 1.5310 0.0086 0.0100 30 days Call 1.4872 0.0230 0.0238 Put 1.4479 0.0085 0.0098 Put 1.4371 0.0063 0.0074 Call 1.5621 0.0086 0.0102 Call 1.5469 0.0116 0.0135 60 days Call 1.4866 0.0313 0.0325 Put 1.4312 0.0118 0.0137 Put 1.4178 0.0087 0.0113 Call 1.5764 0.0101 0.0122 Call 1.5580 0.0137 0.0160 90 days Call 1.4856 0.0370 0.0385 Put 1.4197 0.0141 0.0164 Put 1.4038 0.0104 0.0124 Call 1.6025 0.0129 0.0152 Call 1.5779 0.0175 0.0207 180 days Call 1.4823 0.0494 0.0515 Put 1.3902 0.0200 0.0232 Put 1.3682 0.0147 0.0176 Call 1.6297 0.0156 0.0190 Call 1.5988 0.0211 0.0250 270 days Call 1.4793 0.0586 0.0609 Put 1.3710 0.0234 0.0273

Put 1.3455 0.0173 0.0206

Table Options prices used as input in the UVM algorithm for hedging barrier

options The prices are in DEM p er dollar notional b ecause we p erformed the calculations

using DEM as the domestic currency and USD as the underlying asset Subsequently all

prices and Deltas were converted back to dollars p er dollar notional

Option prices and bidoer spreads in Table were calculated using the data and

standard conventions for DollarMark options The range of implied volatilities of the

options considered was approximately taking into account bidoer spreads

and the riskreversal curve To calibrate the UVM weused

avolatility band

min max

a riskadjusted average DMUSD spread

a DEM carry r lendb orrow

The riskadjusted drift and discounting rates were computed using average dep osit rates

for and bidoer spreads for r

Exp eriment Barrier option with months to expiration Consider a reversekno ckout

dollar put with strike K DEM and kno ckout at H DEM with six

months days to expiration Assume that this option is written on a notional amount

of US dollars one bp

In the optimization pro cedure we considered input vanilla options with maturities

ofandmonths and with strikes at each maturity the dollar puts

and calls the dollar puts and dollar calls and the calls The results were as

follows

USD p er USD notional Price oer

Hedge in Options

typ e strike maturity quantity notional

put days

New Delta sp ot USD to b e held in the hedge p ortfolio p er dollar notional

Figures and give a graphical analysis of the residual liability ie the presentvalue

of the combined p osition short kno ckout option long option short and month

options The graphs represent the residual liabilityatseveral dates Essentially the s

trategy consists in buying sixmonth Gamma by buying deepinofthemoney dollar

puts and simultaneously selling threemonth Gamma by selling days atthemoney and

inthemoney dollar puts

The eect of buying the day puts is to reduce the size of the liability at the kno ckout

barrier USD DEM by ab out from the onset if the option do esnt kno ckout

but ends up near the barrier the agentwillown the put and hence oset part of the risk

Notice that the discontinuityattheknockout barrier is pfennig or p er dollar

Outofthe money calls were converted into inthemoney puts to clarify the analysis of the hedge USD USD 0.03

0.02 0.05

0.01

0 0 −0.01

−0.02 −0.05 −0.03 1.3 1.4 1.5 1.6 DM1.3 1.4 1.5 1.6 DM

Day 180: Short USD Put KO Day 150: Rollback Long 0.1544 Put(1.6025)

USD USD 0.03 0.03

0.02 0.02

0.01 0.01

0 0

−0.01 −0.01

−0.02 −0.02

−0.03 −0.03 1.3 1.4 1.5 1.6 DM1.3 1.4 1.5 1.6 DM

Day 90: Rollback Day 90: Rollback Short 0.0538 Put(1.5580)

Short 0.0948 Put(1.4856)

Figure a Liability diagram for day USD Put with Kno ckOut barrier at

six month Put b Rollback with UVM of the liability viewed at day

c Rollbackatday after exercise of the day put in the hedge d Liabilityatday

b efore put exercise The diagrams takeinto account the premia received for the options

sold KO premium not included Dotted lines represent the liability if the option kno cks out

notional which represents a discontinuity The reduction of the jump

due to the long put option computed conservatively at intrinsic value is of approximately

The eect of selling month Gamma is to gain some value and at the same time to

An advantage in case the sp ot exchange rate is slightly higher than DEMUSD near expiration

in which case dynamical hedging is very costly USD USD 0.03 0.03

0.02 0.02

0.01 0.01

0 0

−0.01 −0.01

−0.02 −0.02

−0.03 −0.03 1.3 1.4 1.5 1.6 DM1.3 1.4 1.5 1.6 DM

Day 60: Rollback Day 60: Rollback Short 0.0182 Put(1.5469) Short 0.0612 Put(1.4866) Short 0.0110 Put(1.4312)

USD USD 0.03 0.03

0.02 0.02

0.01 0.01

0 0

−0.01 −0.01

−0.02 −0.02

−0.03 −0.03 1.3 1.4 1.5 1.6 DM1.3 1.4 1.5 1.6 DM

Day 0: Rollback Day 0: Reiner−Rubinstein Short USD Put KO

Figure a Liability diagram viewed at day b efore exercise of short day puts

in the hedge b Rollbacktoday b efore exercise c Rollbacktoday d Reiner

Rubinstein premium for the dayKO Notice comparing c and d the dif

ference in the DeltaGamma proles b etween the UVM solution and the ReinerRubinstein

solution

atten out the residual risk The graphs clearly show that the months liabilityisquite

at Moreover if the exotic option kno cks out early the short p osition in frontmonth

options is comp ensated by the longdeepinthe money put

We observe nally that the fair value of the kno ckout option using a sixmonth

volatility computed assuming a lognormal mo del cf Rainier Rubinstein is

p er dollar notional or ab out cheap er than the oer price calculated from the

algorithm Nevertheless the mo del price is comparable to oer prices quoted by market

makers in exotic options due to the additional risk premium charged for these instruments

Exp eriment months reverseknockout dol lar put We consider the case of a reverse

kno ckout option which is inthemoney and has months b efore expiration The charac

teristics of the option are

DEM Strike

Barrier DEM

days Maturity

vol USD p er USD notional BlackScholes Fair Value

The result of applying the UVM to this option is

Price oer USD p er USD notional

Hedge in Options

typ e strike maturity quantity notional

put days

New Delta sp ot USD to b e held in the hedge p ortfolio p er dollar notional

Figure presents a graphical analysis of the residual liability for this hedge at dierent

times to maturity

The UVM solution consists of sel ling options with one and two months to maturity

This solution which seems surprising at rst can b e analyzed as follows At a level of

sp ot of approximately DEMUSD the option is well in the money Consequently the

agent which is short the kno ckout is long Gamma and will b e sub jected to large Deltas

as the dollar falls towards the barrier By selling and dayvolatility in the way

sp ecied by the output the agent b ecomes instead practically Gammaneutral and the

Delta is no longer moving against the market but instead practically static p ositioned

short dollarlong Mark Thus the hedge of the residual resembles more that of a capp ed

dollar put than that of a reverse kno ckout The solution minimizes volatility risk by

attening the residual prole An early kno ckout will leave the agentshortvanilla options

but this has already b een priced into the value of the barrier option USD USD 0.1 0.03

0.08 0.02

0.06 0.01

0.04 0

0.02 −0.01

0 −0.02

−0.02 −0.03 1.35 1.4 1.45 1.5 1.55 1.6 DEM1.35 1.4 1.45 1.5 1.55 1.6 DEM

Day 90: Short USD Put KO Day 60: Rollback

(a) (b)

USD USD 0.03 0.03

0.02 0.02

0.01 0.01

0 0

−0.01 −0.01

−0.02 −0.02

−0.03 −0.03 1.35 1.4 1.45 1.5 1.55 1.6 DEM1.35 1.4 1.45 1.5 1.55 1.6 DEM

Day 60: Rollback Day 30: Rollback Short 0.0284 Puts(1.5469)

Figure a Liability diagram for day USD Put with Kno ckOut barrier at

b Rollback with UVM of the liability viewed at day after exercise of the day

put in the hedge c Liabilityatday b efore put exercise d Rollbacktoday of short

KO short puts p osition The diagrams takeinto account the premia received for the

options sold KO premium not included Dotted lines represent the liability if the option

kno cks out

Conclusion

We prop osed an algorithm for calculating optimal hedging strategies for managing

volatility risk of option p ortfolios and OTC derivatives using vanilla options and cash

instruments The algorithm is based on a theoretical mo del the UVM which takes into

account uncertaintyofvolatility or heterorskedasticityby assuming that volatility can USD USD 0.03 0.03

0.02 0.02

0.01 0.01

0 0

−0.01 −0.01

−0.02 −0.02

−0.03 −0.03 1.35 1.4 1.45 1.5 1.55 1.6 DEM1.35 1.4 1.45 1.5 1.55 1.6 DEM

Day 30: Rollback Day 15: Rollback Short 0.1629 Puts(1.5310) Short 0.1501 Puts(1.4872) (a) (b)

USD USD 0.03 0.03

0.02 0.02

0.01 0.01

0 0

−0.01 −0.01

−0.02 −0.02

−0.03 −0.03 1.35 1.4 1.45 1.5 1.55 1.6 DEM1.35 1.4 1.45 1.5 1.55 1.6 DEM

Day 0: Rollback Day 0: Reiner−Rubinstein Short USD Put KO

Figure a Liability diagram viewed at day b efore exercise of short day puts in

the hedge b Rollbacktoday of short KO short and day puts c Rollbackto

day d ReinerRubinstein premium for the dayKO Notice comparing c

and d the dierence in the DeltaGamma proles b etween the UVM solution and the

ReinerRubinstein solution

vary inside a band This range is easily determined from the

min t max

implied volatilities of input options and the traders exp ectations and riskaversion to ex

treme volatilitymoves The UVM combined with an optimization algorithm that consists

of a regression using option prices gives rise to the UVM Lagrangian Uncertain Volatil

ity Mo del The algorithm selects costecient option hedges whichtakeinto account

the worstcase scenarios for the unhedged cashows Through the examples studied we

learned that the UVM can pro duce comp etitive bidoer prices for derivative securi

ties in heteroskedastic markets and simultaneously reduce volatility risk and the need for

intensive marktomarket of volatility

References

Avellaneda M LevyAandParas A Pricing and hedging derivative securities

in markets with uncertain volatilities Appl Math Finance

Avellaneda M and Paras A Dynamical hedging strategies for derivative securities

in the presence of large transaction costs Appl Math Finance

Avellaneda M and Paras A Dynamic hedging with transaction costs from lat

tice mo dels to nonlinear volatility and freeb oundary problems Courant Institute working

pap er submitted to Stochastics Finance

Black F Scholes M The Pricing of Options and Corp orate Liabilities J of

Political Economy

Boyle PP and Vorst T Option replication in discrete time with transaction costs

J Finance

Due D Dynamic Asset Pricing Theory Princeton Univ Press New Jersey

Harrison JM Kreps D Martingales and Arbitrage in Multip erio d Securities

Markets Journal of Economic Theory

Hoggard T Whalley E and Wilmott P Hedging option p ortfolios in the presence

of transaction costs Advances in Futures and Options Research

Hull J and White A The pricing of options on assets with sto chastic volatilities

J of Finance XLI I

Leland H E Option pricing and replication with transaction costs J Finance

Levy A Avellaneda M and Paras A new approach for pricing and hedging

derivative securities in markets with uncertain volatility a case study on the trinomial

tree Working pap er Courant Institute New York University

Paras A NonLinear partial dierential equations in nance a study of volatility

risk and transaction costs Ph D dissertation New York University

Reiner E and Rubinstein M Breaking down the barriers RISK

Taleb N Introduction to Dynamic Hedgingmanuscript Taleb Research Larch

mont NY

Taleb N and Avellaneda M Thetatime and estimation of forwardforward volatil

ity in preparation

Thorp E O Optimal gambling systems for favorable games Rev Intl Statistical

Institute p

Appendix Arbitragefree pricing and uvm

A pricing measure P is said to b e arbitragefree if security prices are equal to the

exp ected values of their discounted cashows under P Harrison and Kreps Due

In the present framework a measure of typ e is arbitragefree if an only if

n o

P r t

C E e G S i M

i i

The UVM b ears a strong relation with the problem of constructing arbitragefree

measures In fact we will show that from the p oint of view of pricing UVM is equivalent

to determining the worstcase arbitragefree value for the contracted liability conditional ly

on the prices of input options

More preciselywe shall establish the following statements

Prop osition Suppose that the minimization problem admits a solution

with for al l i f MgLet P represent the probability measure

i i

that realizes the worstcase scenario for the residual liability

N M

X X

r t r t t

i j

e G S e F S

i j t

i i j

j i

Then P solves the program

r t t P

e F S M aximiz e E

j t

P P

n o

r t P

e G S C i M A subject to E

i i

Prop osition If

i there exists at least one arbitragefreeprobability measure of type and

ii thereare no option portfolios such that

M M

X X

P r t

C Inf E e G S A

i i i

P P

i i

then the cost of the ecient hedging portfolio V S t obtained via the op

timization problem with and is the value of the program

i i

To prove these statements we shall make use of the following facts

i the function V S t is convex in and

t M M

ii for each the probability P P whichachieves the supremum in is

unique

The rst prop ertyfollows from the fact that V S t is a supremum of linear

t M

functions in the variables More preciselywehave

V S t sup a P bP A

t M i i

P P

where

n o

P r t

a P C E e G S A

i i i

and

r t t P

A e F S bP E

The uniqueness of the probability P realizing the worstcase scenario follows from the

formula

if V S

max

S t A

if V S

min

which expresses the extremal volatility in terms of the second derivative of the solution of

the UVM see Section The latter is determined by the cashows fF S g and bythe

j t

p ortfolio The probability P is determined by A In particular the

graph of the function

V S t

M t M

has a unique supp orting hyp erplane passing through each point Therefore

V S t iseverywhere dierentiable and its gradientatthepoint

t M M

is given by

n o

V S t

r t P

e G S A a P C E

i i i

Pro of of Prop osition Consider rst the case ie when there are no

constraints on the s Since V S t is convex and dierentiable the rstorder

condition V i M is b oth necessary and sucient for a minimum to

o ccur In view of A this rstorder condition is equivalentto

n o

P r t

E e G S C i M A

i i

We conclude that

If the UVM algorithm has a solution then the probability P associated with the optimal

is arbitragefree portfolio

provided that the optimal p ort Clearly the argument extends to the case of nite

folio lies in the interior of the set of constraints ie

i M

i i

Next we establish that the value obtained with UVM coincides with the supremum

of the discounted exp ected cashows fF S g as P ranges over all arbitragefree proba

j t

bilities satisfying the band constraint

In fact using A A and A we obtain

We neglect degenerate cases in which the secondderivativeofthevaluefunction vanishes on a set of

is not uniquely dened These cases are uninteresting since they p ositive measure In the latter case

basically corresp ond to forward transactions whichhave no optionality

inf V S t inf sup a P bP

M i i

P P

inf sup a P bP

i i

Pa P

sup bP

Pa P

P r t t

sup E e F S A

P arb free

This gives a lower b ound on the b est worstcase scenario price in terms of the supremum of

exp ectations over arbitragefree probabilities But this b ound is in fact an equality since

wehave shown that the extremal measure P corresp onding to the optimal p ortfolio is

arbitragefree

Pro of of Prop osition Supp ose that the class of noarbitrage probabilities P P is

nonempty and that the supremum of the discounted cashows over all such probabilities

is nite According to A the UVM value is Moreover if an optimal p ortfolio of the

UVM exists ie the minimum is attained at nite values of weknow

from Prop osition that the value of the UVM is equal to the the supremum of the

discounted cashows

P r t t

E e F S

as P ranges over all arbitragefree probabilities in the class P Therefore to prove Prop o

sition we need to show that the optimal p ortfolio has nite Lamb das To see this notice

that for j j wehave

M M

X X

P r t

V S t C Inf E e G S O A

M i i i

P P

i i

which is a statement that the liability fF g is irrelevantas j j In fact the p ortfolio

consists predominantly of options

There are altogether three p ossibilities according to the b ehavior of

M M

X X

r t P

e G S A C Inf E

i i i M

P P

i i

for large j j

First if converges to as j j then the function V S t must

with all Lamb das nite achieve its minimum at some vector

Second if converges to along some direction as j j the mo del cannot b e

arbitragefree in the sense that equation A cannot hold of any P In fact in fact if

A were true for some P then clearly

M M

X X

r t P

e G S C Inf E

i i i

P P

i i

M M

X X

P r t

C E e G S

i i i

i i

for all f g

It remains to analyze the case in which remains b ounded as j j Notice that

isahomogeneous function of degree oneie

M M

Therefore the b oundedness of as j j along any sequence implies that there

must exist at least one p ortfolio such that A holds This contradicts

the second assumption in the statement of Prop osition Hence only the rst of the three

cases can actually o curr and the pro of is complete

Remark The existence of a p ortfolio satisfying A would imply that there exists a

nontrivial combination of options has model value zero under the worstcase volatility

scenario This situation corresp onds to a quasiarbitrage in the sense that

i the sum of the cost of taking the p osition and of Deltahedging the residual is zero and

hence the agent is riskless

ii the agent stands to make a prot by Deltahedging using the UVM solution if the

volatility path do es not followtheworstcase scenario

Of course the marginal case A can always b e avoided by making the volatility band

slightly wider In the latter case this quasi arbitrage will dissap ear