A Robust Option Pricing Problem

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IMA 2003 Workshop, March 12-19, 2003

A Robust Option Pricing Problem

Laurent El Ghaoui

Department of EECS, UC Berkeley

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Robust optimization

standard form:

minx

supuU

f0(x, u) : u U, fi(x, u) 0, i= 1, . . . , m

x Rn is the decision variable

u is a parameter vector affecting the problem data

set Udescribes uncertainty on u

asemi-infiniteoptimization problem

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Agenda

option basics option pricing problem

worst-case approach

joint work with: A. dAspremont

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Options

Options are time bombs. Warren Buffett, 2003

let x(T) denote the price of an asset at time T

anEuropean call option with maturity Tand strike price K is a

contract with payoff

(x(T) K)+ = sup(x(T) K, 0)

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Options

Options are time bombs. Warren Buffett, 2003

let x(T) denote the price of an asset at time T

anEuropean call option with maturity Tand strike price K is a

contract with payoff

(x(T) K)+ = sup(x(T) K, 0)

what is the price of the option,

assuming no arbitrage (free lunch) is possible?

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Price of option

fundamental result of finance: assuming a zero risk-free interest rate,

the no-arbitrage price of the option is

E(x(T) K)+

for some distribution on the asset price at time T

such a is calledrisk-neutral

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Forwards

under the risk-neutral distribution , thecurrentasset price is

Ex(T)

i.e., is a martingale

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Basket options

let x(T) be a n-vector of prices of assets at time T

abasketoption with weight vector w and strike K is the contract with

payoff(wTx(T) K)+

denote the basket option by (w, K) and its price by

C(w, K) := E(wTx(T) K)+

(maturity date is implicit here)

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Option pricing problem

given

w0, w1, . . . , wm in Rn+ (basket weights)

K0, K1, . . . , K m in R+ (strike prices)

p1, . . . , pm in R+ (observed option prices)

determinethe price of the basket option with weight w0 and strike

price K0

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Option pricing problem

given

w0, w1, . . . , wm in Rn+ (basket weights)

K0, K1, . . . , K m in R+ (strike prices)

p1, . . . , pm in R+ (observed option prices)

determinethe price of the basket option with weight w0 and strike

price K0

in practice, we are also given the currentasset prices themselves,

q Rn+ (forwards)

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Challenges & methods

challenges:

the risk-neutral measure is notthe empirical distribution of assets

hence, we may have to rely on option & forward prices only

two approaches:

model-based approach

arbitrage-based approach

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Model-based approach

assume a log-normaldiffusion modelfor the asset prices

ds= Asdt+Bsdw,

where s= log x, and w is a multidimensional Brownian motion

fit the model to observed option prices

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ds= Asdt+Bsdw,



with some approximations, this problem is an SDP (Aspremont, 2000)

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ds= Asdt+Bsdw,



pros and cons:

very versatile, and easy to solve (albeit only recently)

makes astructural assumptionabout the risk-neutral measure

provides a point estimate for the price of basket (w0, K0)

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No-arbitrage approach

findboundson the price of basket (w0, K0) by solving semi-infinite LP

sup/ inf E(wT0x K0)+

s.t. E(wTi x Ki)+ =pi, i= 1, . . . , m

the optimization variable is the risk-neutral probability measure

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No-arbitrage approach

findboundson the price of basket (w0, K0) by solving semi-infinite LP

sup/ inf E(wT0x K0)+

s.t. E(wTi x Ki)+ =pi, i= 1, . . . , m

the optimization variable is the risk-neutral probability measure

pros and cons:

problem may bedifficult

approach provides only bounds, but . . .

...makesno assumptionsabout market dynamics

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Upper bound problem

upper bound problem:

psup := supP

0(x)(x)dx :

(x)dx= 1,

i(x)(x)dx= pi, i= 1, . . . , m ,

where

= Rn+

Pis the set of densities with support in

i(x) := (wTi x Ki)+, i= 0, 1, . . . , m

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Upper bound problem


psup := supfF

0(x)f(x)dx :

f(x)dx= 1,

i(x)f(x)dx= pi, i= 1, . . . , m ,

Lagrangian:

L(, ,0) =

0(x)(x)dx+ 0

1

(x)dx

+T

p

(x)(x)dx

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Upper bound problem


psup := supfF

0(x)f(x)dx :

f(x)dx= 1,

i(x)f(x)dx= pi, i= 1, . . . , m ,

the dual is therobust linear programmingproblem

dsup := inf Rm

Tp+ 0

s.t. x , T(x) + 0 0(x)

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Dual gives a hedging strategy

let 0, be feasible for the dual:

dsup := inf Rm

Tp+ 0

s.t. x , T(x) + 0 0(x) ()

strategy: invest i in basket (wi, Ki), 0 in cash

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Dual gives a hedging strategy

let 0, be feasible for the dual:

dsup := inf Rm

Tp+ 0

s.t. x , T(x) + 0 0(x) ()

strategy: invest i in basket (wi, Ki), 0 in cash

price of strategy: Tp+ 0

taking expectations in (), get

Tp+ 0 E0(x) =psup

(proves weak duality)

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A special case

we are given option prices on the m= n individual assets, as well asforward prices

min/ sup E(wT0x K0)+

s.t. E(xi Ki)+ =pi, i= 1, . . . , nEx= q

we may further relax the problem by ignoring the forward price

information

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Special case : upper bound

no-arbitrage: problem is feasible iff0 p qp+K

upper bound is given by

dsup = max0jn+1

wT0p+i

w0,imin(qipi,jKi) jK0,

where 0 = 0 j := (qjpj)/Kj 1 =n+1

upper bound is attained, hence dsup =psup

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Ignoring forward prices

ignore the constraintsEx= q

obtain formula by maximizing dsup wrt q:

dsup =psup =wT0p+ (wT0K K0)+

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Ignoring forward prices

ignore the constraintsEx= q

obtain formula by maximizing dsup wrt q:

dsup =psup =wT0p+ (wT0K K0)+

makessense:

concave in p (the RHS of our primal LP)

convex in(w0, K0)

interpolates pi at w0 =i-th unit vector ofRn, and K0 =Ki

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Sketch of proof

weak dualityfollows from homogeneity and convexity ofx x+:

E(wT0x K0)+ = E(w

T0(x K) + (w

T0K K0))+

wT0 E(x K)++ (wT0K K0)+ (w0 0)

= wT0p+ (wT0K K0)+

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Sketch of proof

weak dualityfollows from homogeneity and convexity ofx x+:

E(wT0x K0)+ = E(w

T0(x K) + (w

T0K K0))+

wT0 E(x K)++ (wT0K K0)+ (w0 0)

= wT0p+ (wT0K K0)+

strong duality: choose x= p+Kwith pty 1 ifwT0KK0, otherwise

take a limit of feasible distributions

x=

1p+K with probability ,

0 with probability 1 .

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Lower bound

dual problem reduces to a finite LP(withO(n) variables and constraints)

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Lower bound

dual problem reduces to a finite LP(withO(n) variables and constraints)

if we ignore forward prices

pinf =dinf =

i : KiwiK0

piwi

+ maxj : Kjwj


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General case: integral transform approach

the option price function

C(w, K) = E(wTx K)+

is anintegral transformof measure , with kernel the payofffunction

(wTx K)+

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General case: integral transform approach

the option price function

C(w, K) = E(wTx K)+ (1)

is anintegral transformof measure , with kernel the payofffunction

(wTx K)+

make Cthe variable, and solve interpolation problem

supC

/ infC

C(w0, K0) : C(wi, Ki) =pi, i= 1, . . . , m ,

Cof the form (1)

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Finite LP formulation

optimalCof semi-infinite LP ispiecewise affine

hence the semi-infinite LP can be reduced exactly to a finite LP

sup / inf p0

subject to gi, (wj , Kj) (wi, Ki) pjpi, i, j = 0,...,m+n+ 1

gi,j 0, 1 gi,n+1 0, i= 0,...,m+n+ 1, j = 1,...,n

gi, (wi, Ki)= pi, i= 0,...,m+n+ 1,

where the variables gi are subgradients ofCopt

(for upper bound, in special case, LP is exact)

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Robustness

in practice we haveuncertainty:

basket weights may change, or not exactly known at present time

price information may be noisy (e.g., bid-ask)

strike prices also may vary

hedging strategies may be implemented with errors

we address the upper bound problem, in the special case

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Bid-ask spread

bid-ask spread corresponds to abox uncertaintyfor the vector ofobserved option prices

p p p

the worst-case value of upper (lower) bound is attained at pworst =p

(orpworst =p)

we may want to introduce correlation in the model . . . ellipsoidal

uncertaintyon p is as easy

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Example

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Price bounds on a basket call option

Strike K0

Price

Simulated priceUpper bound (explicit)Lower bound (explicit LP)Upper bound (LP relax.)Lower bound (LP relax.)

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Robustness for the general case

i have no idea about this

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Further research

imposing smoothness constraints

multi-period problem (Bertsimas, 2003)

link with model-based approaches

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Further research

imposing smoothness constraints

multi-period problem (Bertsimas, 2003)

link with model-based approaches

for more info: dAspremont & El Ghaoui, Static arbitrage bounds for

basket option prices, submitted to Oper. Res., 2003

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A Robust Option Pricing Problem