Transcript A New Global Optimization Technique for Zero

Pricing Derivative Financial Products:
Linear Programming (LP) Formulation
Donald C. Williams
Doctoral Candidate
Department of Computational and Applied Mathematics, Rice University
Thesis Advisors
Dr. 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
Basic Financial Contracts:
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   0, T  where T denotes the
contract maturity date.
An European-style option is similarly defined with
exercise restricted to the maturity date, T .
Option Types
Two Basic 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.
Payoff: Fundamental Constructs
Payoff Functions
Short position
Long position
Call Option
(S , t )  max(S  K ,0)
K
K
ST
K
Put Option
(S , t )  max(K  S ,0)
ST
K
ST
ST
Modeling Assumptions
Classic Black-Scholes Economy:
• 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
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)
dS=µS dt+ S dW
where
• dW is a standard Browian motion,
• μ is the expected return (or drift), and
• σ denotes the volatility of asset price returns.
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
V 1 2 2  2V
V
  S
 rS
 rV  0
2
t 2
S
S
In the case of European-style options, the value function V ( S , t ) solves
the Black-Scholes equation with appropriate boundary conditions.
Initial & Boundary Conditions: (Put Option)
IC:
V (S , T )  max(K  S ,0)
BC:
V (0, t )  Ke  r (T t ) ,
V ( S , t )  0 as S  0
Payoff Functions
K
ST
Computational Domain

Contract Instantiation Boudary
V(S
i N)
 = T
V(S Mn
V(S 0n
Time marching
direction

0
S0
V(S i0
Contract Expiration Boudary
S
SM = S max
Example: European-Style Put Option
V(S0,0) = 5.5776
VBS = 5.7910
European Put Option Value
50
Numerical
Analytic
45
40
35
Option Value
Problem Data:
S0 = 100;
K = 100;
T = 0.50;
r = 0.05;
sigma = 0.25;
30
25
20
15
10
5
0
50
60
70
80
90
100
110
Asset Price
120
130
140
150
Modeling
Basic Question:
What changes?
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
No Arbitrage: (put option)
V ( S , t )  max(K  S ,0)
V 1 2 2  2V
V
  S

rS
 rV  0
2
t 2
S
S
Optimal to Exercise Early
Not Optimal to Exercise Early
V  K  S,
V  K  S,
V 1 2 2  2V
V
  S
 rS
 rV  0
t 2
S 2
S
V 1 2 2  2V
V
  S

rS
 rV  0
2
t 2
S
S
Modeling
Let:
LBS
1 2 2 2

  S

rS
r
2
2
S
S
and
( S )  max(K  S ,0)
Then,
Exercise Early Region
Continuation Region
V    0,
V    0,
V
 LBS V  0
t
V
 LBS V  0
t
Linear Complementarity Problem (LCP)
The American put value function can be expressed as the unique solution
to the following LCP: (cf., Dempster-Hutton (1999))
LCP
V (, T )  
V    0


V
0
 LBS V 
t


V 
V     0
 LBS V 
t 

Discretized LCP and Equivalent LP
Discretized sequence of LCPs:
Vm  0
BV m 1  AV m    0
BV
m 1


 AV m   V m    0
Equivalent sequence of LPs: (cf., Dempster-Hutton-Richards (1998))
min c V m
s.t. V m  
AV m    BV m 1
m  1,...,M
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
Grid nodes: 201
Time steps: 100
Time step size: 0.00416667
Discretization: Implicit
American Put Option Value
25
20
Option Value
Problem Data:
S0 = 50.00
K=50.00
T=0.42
r=0.10
sigma=0.40
V(S0,0) = 4.2698
V = 4.24
(control variate, Hull, 4ed, p.418)
15
10
5
0
25
30
35
40
45
50
55
Asset Price
60
65
70
75
Idea
Consider a 2-factor (or 2-state variable problem)
dS1  S1 1 dt   1dW1 
dS2  S 2 2 dt   2 dW2 
where r 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:
min c V m
s.t. V m  
AV m    BV m 1
m  1,...,M
In the context of spread options, consider the constraint
spreadmin  S1  S2  spreadmax
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.