American Option Pricing Using A Markov Chain Approximation

American Option Pricing Using A Markov Chain Approximation

Jin-Chuan Duan

Hong Kong University of Science and Technology

Correspondence to:

Prof. Jin-Chuan Duan Department of Finance

Hong Kong University of Science & Technology Clear Water Bay, Kowloon, Hong Kong

Tel: (852) 2358 7671; Fax: (852) 2358 1749 E-mail: jcduan@ust.hk

Markov Chain OPM JC Duan (3/2000)

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A two-way classification of option valuation problems

• Path-independent vs. path-dependent payoff functions

Path-independent: European options, American options, Bermudan options, digital options Path-dependent: lookback options, Asian options,

knock-out (in) options

• Path-independent vs. path-dependent underlying stochastic processes

Path-independent: Black-Scholes model, CEV model, jump-diffusion

Path-dependent: GARCH model

• In a numerical sense, it is typically easier to deal with path- dependence (a non-Markovian feature) arising from the payoff function than path-dependence inherent in the underlying dynamic.

• The traditional numerical methods are poor in either handling early exercise or the cases involving path-

dependent payoffs (and worse for the cases involving path- dependent underlying dynamic).

Different numerical methods for option valuation

• Monte Carlo (quasi and pseudo) methods

Exceedingly flexible in dealing with path-dependent payoffs and path-dependent underlying dynamic. Poor in handling options with early exercise possibilities.

Computing time intensive even with some variance reduction technique.

• Finite difference/element methods

Good at solving the option pricing problem that can be cast as a partial differential equation. Cannot deal with the

discrete time valuation models.

• Lattice methods

Hard to ensure recombination and harder still to

accommodate path-dependent payoffs and/or underlying dynamics.

• Analytical approximation methods Highly valuation problem specific.

• Neural network methods

Don’t stand alone and are intended to be a complementary speed accelerator for real-time runs (see Hanke, 1997).

• Markov chain method (Duan & Simonato, 1999)

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Black-Scholes Model

• Asset return under the risk neutral probability measure :

t

t r dt dW

S

d σ  +σ



 

 −

= 2

2 ln 1

where

σ : volatility rate r : risk free rate

W : standard Brownian motiont

• American option price with discrete exercise points :

[ ]

{

}

t t S

V( , ) = max ( , ), − ( +1, +1)|ℑ

where

V S t( _t, ) : American option’s price K : Strike price

f S K( _t, ) : European option’s payoff

V S( _T , )T = f S( _T , )K

• Discretize the underlying asset price

Let 2) ln( )

2

(r 1 t St

pt ≡ − − σ +

Then,

dWt

St d dt t r

dp

σ

σ

=

+

−

−

= 2) ln( )

2 ( 1

• Jarrow and Rudd’s (1983, eq (13-18)) n-step binomial tree approximation

Approximation target: tp

Up and down moves: and

n d T

n

u =σ T = −σ

Probability:

2

= 1 q

An n-step binomial tree has m discrete prices where 1

2 +

= n

m .

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Vector for the logarithmic adjusted stock price:

[

]

P = !

where

nu p

m p

u p n

p

p n

p

d n

p p

nd p

p

+

=

+

= +

= +

− +

=

+

=

0 0 0 0 0

) (

) 2 (

) 1 (

) 1 (

) 2 (

) 1 (

"

"

Example: A 2-step binomial tree (m = 5)

p(1) p(2) p(3) p(4) p(5)

p(1) p(2) p(3) p(4) p(5)

p(1) p(2) p(3) p(4) p(5)

A Markov chain interpretation with the following transition probability matrix





















= Π

1 0

0 0

0

5 . 0 0 5 . 0 0 0

0 5 . 0 0 5 . 0 0

0 0

5 . 0 0 5 . 0

0 0

0 0

1

American option valuation numerically

{ }

V P t( , ) = max g P K t( , , ) , e^−rΠV P t( , +1)

where

V P t( , ) : American option’s price K : strike price

g P K t( , , ) : European option’s payoff

) , , ( )

,

(P T g P K T

V =

Example: a put option







 − − +

= ) 1 ],0

2 exp[( 1

1 max )

, ,

(P K t k r ² t P

g σ

Shortcoming of the lattice approach

1. Rigidity of geometry, i.e., the number of states is tied to the number of steps.

2. Recombined lattices are hard to construct.

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• A general Markov chain method The structure

p(1) p(2) p(3) p(4) p(5)

p(1) p(2) p(3) p(4) p(5)

p(1) p(2) p(3) p(4) p(5)

Transition probability matrix





















= Π

)

; 5 , 5 ( )

; 4 , 5 ( )

; 3 , 5 ( )

; 2 , 5 ( )

; 1 , 5 (

)

; 5 , 4 ( )

; 4 , 4 ( )

; 3 , 4 ( )

; 2 , 4 ( )

; 1 , 4 (

)

; 5 , 3 ( )

; 4 , 3 ( )

; 3 , 3 ( )

; 2 , 3 ( )

; 1 , 3 (

)

; 5 , 2 ( )

; 4 , 2 ( )

; 3 , 2 ( )

; 2 , 2 ( )

; 1 , 2 (

)

; 5 , 1 ( )

; 4 , 1 ( )

; 3 , 1 ( )

; 2 , 1 ( )

; 1 , 1 (

τ π

τ π

τ π

τ π

τ π

τ π

τ π

τ π

τ π

τ π

τ π

τ π

τ π

τ π

τ π

τ π

τ π

τ π

τ π

τ π

τ π

τ π

τ π

τ π

τ π

where

{ }

{ }

π τ τ

σ τε

σ τ ε

σ τ

σ τ σ τ

( , ; ) Pr ( ) ( ) | ( )

Pr ( ) ( )| ( )

Pr ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

i j Q

c j pt c j pt p i Q

t t t

Q

t

c j p c j p p i

c j p i c j p i

c j p i c j p i

= ≤ + < + =

≤ + < + =

− ≤ < + −

 

 + −

 

 −  −

 



1

1 1 1

=

=

= Φ Φ

and Φ(.) is the standard normal distribution function

Density of the transition probability matrix

Fact: For a fixed partition, the density of the transition probability matrix depends on the length of one operation time interval

Transition probability matrix using one step T=90, m=101, σ=0.2, S0=50, τ=90/365

Transition probability matrix using 90 steps T=90, m=101, σ=0.2, S0=50, τ=1/365

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• A numerical example

European put options in the Black and Scholes framework

Parameters:

1

Strike Price 48

Stock Price 50

Interest Rate 0.05 d1 0.55411

Standard Deviation 0.2 d2 0.35411

Price computed by Black and Scholes Formula: 2.024003 Binomial Tree Approach # of steps 2

Step size 0.141421

3.62918 3.62918 3.62918

3.77060 3.77060 3.77060

3.91202 3.91202 3.91202

4.05344 4.05344 4.05344

4.19487 4.19487 4.19487

Backstep 2 Backstep 1 Payoff K - S

8.72325 8.94408 9.17050 9.17050

5.13971 4.47204 3.27191 3.27191

2.18081 1.59556 0.00000 -3.52273

0.77808 0.00000 0.00000 -11.34954

0.00000 0.00000 0.00000 -20.36532

Transition probability matrix

1 0 0 0 0

0.5 0 0.5 0 0

0 0.5 0 0.5 0

0 0 0.5 0 0.5

0 0 0 0 1

Maturity

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Markov Chain Approach # of prices 5

3.62918 3.62918 3.62918

3.77060 3.77060 3.77060

3.91202 3.91202 3.91202

4.05344 4.05344 4.05344

4.19487 4.19487 4.19487

Backstep 2 Backstep 1 Payoff K - S

5.71204 6.95589 9.17050 9.17050

3.91773 3.98155 3.27191 3.27191

1.96324 1.36892 0.00000 -3.52273

0.69951 0.24891 0.00000 -11.34954

0.17864 0.02115 0.00000 -20.36532

Transition Probability Matrix

0.69146 0.24173 0.06060 0.00598 0.00023 0.30854 0.38292 0.24173 0.06060 0.00621 0.06681 0.24173 0.38292 0.24173 0.06681 0.00621 0.06060 0.24173 0.38292 0.30854 0.00023 0.00598 0.06060 0.24173 0.69146

• Numerical results for the Black-Scholes model (Duan &

Simonato, 1999)

⇒ Table 1 : European puts

⇒ Table 2 : American puts

GARCH Option Pricing Model

• The reasons for GARCH option pricing A. Option prices

1. Volatility smile/smirk

2. Term structure of volatilities

3. Black-Scholes implied volatilities are higher than historical (or realized) volatilities

B. Historical returns

1. Volatility clustering

2. Negative return skewness 3. Excess return kurtosis

• Asset return under the data generating probability measure P

1 2 1

2 1

1 0

2

1 1 1

1 1

) (

2 ln 1

θ ξ

β β

β

ξ λ

− +

+

=

+

− +

=

+ +

+ +

+ + +

+ +

t t

t t

t t t

t t

t

h h

h

h h

h S r

S

where

r : risk-free rate

λ : risk premium parameter θ : leverage parameter

ξt+1 ~ N(0,1) with respect to P

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• Asset return under the risk neutral probability measure Q (Duan 1995)

2 1

1 2 1

1 0

2

1 1 1 1

) (

2 ln 1

λ θ ε

β β

β

ε

−

− +

+

=

+

−

=

+ +

+ +

+ + + +

t t

t t

t t t

t t

h h

h

h h

S r S

where

εt+1 ~ N(0,1) with respect to Q

• American option valuation in the GARCH framework

[ ]

{ }

V S h( ,_{t t+1}, )t = max f S K( , ) ,_t e^−rE^Q V S( _t+1,h_t+2,t + ℑ1)| _t

where

V S h( _t, _t+1, )t : American option’s price

K : Strike price

f S K( _t, ) : European option’s payoff

V S( _T,h_T+1, )T = f S( _T, )K

• Discretize the underlying asset prices

1) 1 ln(

) ln(

*) 2 ( 1

= + +

+

−

−

=

ht qt

St t

h t r

p

where

] ) (

1 [ 1

*

2 2

1

0

λ θ β

β

β

+ +

−

= − h

Example: m=3, n=3

(p(1), q(1)) (p(2), q(1)) (p(3), q(1))

(p(1), q(1)) (p(2), q(1)) (p(3), q(1))

(p(1), q(2)) (p(2), q(2)) (p(3), q(2))

(p(1), q(2)) (p(2), q(2)) (p(3), q(2))

(p(1), q(3)) (p(2), q(3)) (p(3), q(3))

(p(1), q(3)) (p(2), q(3)) (p(3), q(3))

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• Transition probability matrix

Π =

π π π π π

π π π π π

π π π π

( , ; , ) ( , ; , ) ( , ; , ) ( , ; , ) ( , ; , ) ( , ; , ) ( , ; , ) ( , ; , ) ( , ; , ) ( , ; , )

( , ; , ) ( , ; , ) ( , ; , ) ( , ; ,

1 11 1 1 1 2 1 1 1 1 1 11 2 1 1

2 11 1 2 1 2 1 2 1 1 2 11 2 2 1

11 1 1 2 1 1 1 11 2

! !

! !

" " # " " # "

!

m m n

m m n

m m m m m ) ! ( , ; , )

" " # " " # "

π 1 2 m n





















Since

2 1

1 2 1

1 0

2

1 1 1 1

) (

2 ln 1

λ θ ε

β β

β

ε

−

− +

+

=

+

−

=

+ +

+ +

+ + + +

t t

t t

t t t

t t

h h

h

h h

S r S

q_t+2 is a deterministic function of q_t+1, p_t+1, p_t; that is, q_t+2 = Φ( q_t+1, p_t+1, p_t )

{ } ( )



 ∈ = = Φ ∈

=

+ +

otherwise 0

) ( )

( ), ( ), ( if

) ( ),

(

| ) ( Pr

) ,

; , (

1

1 C k p p i q q j q j p k p i D l

p l k j i

t t

t Q

π

• American option prices in the GARCH framework

Price vector

[

]

' p p p m p p p m

P = ! ! !

American option price computation

{

}

max )

,

(P t = g P K t e⁻ ΠV P t +

V ^r

where

V P t( , ) : American option’s price K : Strike price

g P K t( , , ) : European option’s payoff function V P T( , ) = g P K T( , , )

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• Density of the transition probability matrix (GARCH)

m=101, n=11,

S₀ = 50,r = 0 05. ,β₀ = 0 00001. ,β₁ = 0 8. ,β₂ = 0 1. ,λ = 0 2.

• Numerical results for the GARCH model (Duan &

Simonato, 1999)

⇒ Table 3 : European put

⇒ Table 4 : American put

References

Duan, J.-C. and J.-G. Simonato, 1999, “American Option Pricing under

GARCH by a Markov Chain Approximation,” Journal of Economic Dynamics and Control, forthcoming.

Hanke, M., 1997, “Neural Network Approximation of Option Pricing Formulas for Analytically Intractable Option Pricing Models,” Journal of Computational Intelligence in Finance 5, 20-27.

Jarrow, R. and A. Rudd, 1983, Option Pricing, Richard D. Irwin, Inc.