Pricing Derivative Financial Products: Linear Programming (LP) Formulation

Donald C. WilliamsDoctoral Candidate

Department of Computational and Applied Mathematics, Rice University

Thesis AdvisorsDr. Richard A. Tapia, Department of Computational and Applied Mathematics

Dr. Jeff Fleming, Jesse H. Jones Graduate School of Management

26 November 2003

Computational Finance Seminar

Outline

• Motivation– Nature of Derivative Financial Products– European-style option

• Modeling: American-style option• Linear Complementarity Problem• Optimization Framework

– LP– Constraints

• Concluding Remarks

Option Contract Specification

An American-style option is a financial contract that provides the holder with the right, without obligation, to buy or sell an underlying asset, S, for a strike price K, at any exercise time where T denotes the contract maturity date.

T,0

Basic Financial Contracts:

An European-style option is similarly defined with exercise restricted to the maturity date, T .

Option Types

A call option gives the holder the right to buy the underlying asset.

A put option gives the holder the right to sell the underlying asset.

Two Basic Option Types:

Payoff: Fundamental Constructs

Payoff Functions

ST STKK

ST STKK

Call Option)0,max(),( KStS

)0,max(),( SKtS Put Option

Long position Short position

Modeling Assumptions

• The market is frictionless

– e.g., no transaction cost, all market participants have access to any information, borrow and lending rate are equal

• No arbitrage opportunities

• Asset price follows a geometric Brownian motion

• Riskless rate, r, and volatility, , are constant

• Option is European-style

Classic Black-Scholes Economy:

Modeling Building Framework

• Define State Variables. Specify a set of state variables (e.g., asset price, volatility) that are assumed to effect the value of the option contract.

• Define Underlying Asset Price Process. Make assumptions regarding the evolution of the state variables.

• Enforce No-Arbitrage. Mathematically, the economic argument of no-arbitrage leads to a deterministic partial differential equation (PDE) that can be solved to determine the value of the option.

Asset Price Evolution

Given a constant-variance diffusion approach to asset price changes (i.e., one-factor model of asset price evolution)

where • dW is a standard Browian motion, • μ is the expected return (or drift), and • σ denotes the volatility of asset price returns.

S dWdS=µS dt+

The value function V, for an option on an underlying asset that evolves according to dS, satisfies the well-known and celebrated Black-Scholes (1973) parabolic PDE. (cf. Hull (2000))

Black-Scholes PDE

02

12

222

rVS

VrS

S

VS

t

V

In the case of European-style options, the value function solves the Black-Scholes equation with appropriate boundary conditions.

Initial & Boundary Conditions: (Put Option)

IC:

BC:

),( tSV

)0,max(),( SKTSV

00),(

,),0( )(

SastSV

KetV tTr STK

Payoff Functions

Computational Domain

S

= T

0

S0

SM = Sm ax

Time marchingdirection

Contract Expiration Boudary

Contract Instantiation Boudary

V(S i N )

MV(S nV(S0 n

V(S i 0

In order to solve theparabolic Black-Scholes PDE for a given option valuation problem using anexplicit …nite volume method, wede…nea rectangular computational domain

­ = [0 max]£ [0 ]

where is contract expiration timeand max is selected su¢ ciently largeso that asymptotic bound-ary conditions (e.g., max = 10¢ or max =

p ) can beemployed. Spatial discretization is given

by:

f : = 012 g

= ¢ = 012

T heoption value function at nodepoint ( ) within the discretegrid is denoted by

´ ( )

Remark 6: In the spatial discretization, unequal node spacing should employed to be used to min-imize computational time complexity. In general, this implies dense node spacing near the strikeprice, , and sparse node spacing as ! 0 and as ! max .

Consider the illustration of the computational domain given in Figure (8.6). T he initial andboundary conditions aregiven below.

Initial Condition,

0 = ( = 0) =

8<

:

max ( ¡ 0) call option

max ( ¡ 0) put option.(3)

Boundary Condition,

at = max, ( = ):

= ( ) =

8<

:

max call option

0 put option.(4)

15

Example: European-Style Put Option

Problem Data:S0 = 100; K = 100; T = 0.50; r = 0.05; sigma = 0.25;

V(S0,0) = 5.5776VBS = 5.7910

50 60 70 80 90 100 110 120 130 140 1500

5

10

15

20

25

30

35

40

45

50

Asset Price

Opt

ion

Val

ue

European Put Option Value

NumericalAnalytic

Modeling

What changes?

Basic Question:

European-style contract

American-style contract

American Option Valuation

• Early work focused on discrete dividends and analytic solutions– (1977) Roll

– (1979) Geske

– (1981) Whaley

• When closed-form solutions cannot be derived– (1977) Brennan-Schwartz: Finite-Difference-Method (FDM)

– (1978) Brennan-Schwartz: Equivalence of explicit FDM and jump model

– (1979) Cox-Ross-Rubinstein: Binomial Pricing Model

American Option Valuation

• Relaxations of underlying assumption

– Stochastic volatility: Heston (1993), Stein-Stein

– Deterministic Volatility Function (DVF): Derman-Kani (1994), Dupire (1994), Rubinstein (1994)

– Empirical test of DVF: Dumas-Fleming-Whaley (1998)

– Jump diffusion process: d’Halluin-Forsyth-Labahn (2003)

Modeling

)0,max(),( SKtSV

No Arbitrage: (put option)

02

12

222

rVS

VrS

S

VS

t

V

Optimal to Exercise Early

02

1

,

2

222

rVS

VrS

S

VS

t

V

SKV

02

1

,

2

222

rVS

VrS

S

VS

t

V

SKV

Not Optimal to Exercise Early

Modeling

)0,max()( SKS

Let:

rS

rSS

SLBS

2

222

2

1

Exercise Early Region

0

,0

VLt

V

V

BS 0

,0

VLt

V

V

BS

Continuation Region

and

Then,

Linear Complementarity Problem (LCP)

The American put value function can be expressed as the unique solution to the following LCP: (cf., Dempster-Hutton (1999))

0

0

0

),(

Vt

VVL

t

VVL

V

TV

LCP

BS

BS

Discretized LCP and Equivalent LP

Discretized sequence of LCPs:

MmBVAV

Vts

Vc

VAVBV

AVBV

V

mm

m

m

mmm

mm

m

,...,1

..

min

0

0

0

1

1

1

Equivalent sequence of LPs: (cf., Dempster-Hutton-Richards (1998))

Observations

• The discretized sequence of LCPs can be solved in an iterative manner without using the equivalent formulation as an LP. (ref., Wilmott-Howison-Dewynne (1995))

• However, our desire is to move beyond vanilla option pricing and establish a framework that allows more general economic constraints to be considered.

Example: American-Style Put Option

Problem Data:S0 = 50.00K=50.00T=0.42r=0.10sigma=0.40

Grid nodes: 201Time steps: 100Time step size: 0.00416667Discretization: Implicit

V(S0,0) = 4.2698

V = 4.24 (control variate, Hull, 4ed, p.418)

25 30 35 40 45 50 55 60 65 70 750

5

10

15

20

25

Asset Price

Opt

ion

Val

ue

American Put Option Value

Idea

Consider a 2-factor (or 2-state variable problem)

22222

11111

dWdtSdS

dWdtSdS

where is the correlation between the Wiener processes.

Employing the 2D version of Ito’s Lemma and no-arbitrage arguments a more general governing B-S PDE is obtained.

Ongoing Work

Recall the equivalent sequence of LPs:

MmBVAV

Vts

Vc

mm

m

m

,...,1

..

min

1

In the context of spread options, consider the constraint

max21min spreadSSspread

Concluding Remarks

• Transitioned from American-style option pricing under stochastic volatility to pricing spread option with economic constraints.

• Built PDE solver using finite difference.• Presently working to solidify proper numerical

implementation of model using LIPSOL to solve the associated LP.