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
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.
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