# Remarks on American Options for Jump .Remarks on American Options for Jump Diï¬€usions...

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Remarks on American Options forJump Diffusions

Hao Xing

University of Michigan

joint work withErhan Bayraktar, University of Michigan

Department of Systems Engineering & Engineering ManagementChinese University of Hong Kong, Feb. 5, 2009

Why jump models?

Pmarket(K ,T ) = PBS(S0,K ,T , imp).

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.20

10

20

30

40

50

60

70

Moneyness = strike / spot index

Impl

ied

Vol

atili

ty%

Implied Volatility of S&P 500 index put option

Jan. 16, 09Dec. 30, 05

Crash Fear!

D. S. Bates [00] studied the index option data for the 87 crash:

After the crash out of the money (OTM) put options that provideexplicit portfolio insurance against substantial downwardmovements in the market have been trading at high prices (asmeasured by implicit volatilities) . . . even more overpriced relativeto OTM calls that will pay off only if the markets risessubstantially. Crash Fear

Crash Fear!

D. S. Bates [00] studied the index option data for the 87 crash:

After the crash out of the money (OTM) put options that provideexplicit portfolio insurance against substantial downwardmovements in the market have been trading at high prices (asmeasured by implicit volatilities) . . . even more overpriced relativeto OTM calls that will pay off only if the markets risessubstantially. Crash Fear

Comparing to models with stochastic volatilities, models withjumps can better explain this phenomenon.

Jump diffusions

Assume the stock price S follows jump diffusions (under a riskneutral measure calibrated from the market)

dSt = bStdt + (St, t)StdWt + St dNt

i=1

(eYi 1

),

in which

Nt is a Poisson process with rate independent of W ,

Yi are iid. random variables with distribution (dy),

b , r ( 1), we assume , E[eYi ] < + so that{ertSt}t0 is a martingale.

We assume (S , t) > 0 for some positive constant .

Jump diffusions

Assume the stock price S follows jump diffusions (under a riskneutral measure calibrated from the market)

dSt = bStdt + (St, t)StdWt + St dNt

i=1

(eYi 1

),

in which

Nt is a Poisson process with rate independent of W ,

Yi are iid. random variables with distribution (dy),

b , r ( 1), we assume , E[eYi ] < + so that{ertSt}t0 is a martingale.

We assume (S , t) > 0 for some positive constant .Examples:

Mertons model: (dy) is the dist. of normal r.v.

Kous model: (dy) is the dist. of double exponential r.v.

The American put optionThe value function of the American put option is given by

V (S , t) , supTt,T

E

[er(t)g(S )

St = S],

where g(S) = (K S)+.

The American put optionThe value function of the American put option is given by

V (S , t) , supTt,T

E

[er(t)g(S )

St = S],

where g(S) = (K S)+.

The domain Rn [0,T ) can be divided into the continuationregion and the stopping region:

C , {(S , t) R+ [0, T ) : V (S , t) > g(S)} andD , {(S , t) R+ [0, T ) : V (S , t) = g(S)}.

The American put optionThe value function of the American put option is given by

V (S , t) , supTt,T

E

[er(t)g(S )

St = S],

where g(S) = (K S)+.

The domain Rn [0,T ) can be divided into the continuationregion and the stopping region:

C , {(S , t) R+ [0, T ) : V (S , t) > g(S)} andD , {(S , t) R+ [0, T ) : V (S , t) = g(S)}.

There exists an optimal exercise boundary s(t), such thatC = {S > s(t)} and D = {S s(t)}.

The American put optionThe value function of the American put option is given by

V (S , t) , supTt,T

E

[er(t)g(S )

St = S],

where g(S) = (K S)+.

The domain Rn [0,T ) can be divided into the continuationregion and the stopping region:

C , {(S , t) R+ [0, T ) : V (S , t) > g(S)} andD , {(S , t) R+ [0, T ) : V (S , t) = g(S)}.

There exists an optimal exercise boundary s(t), such thatC = {S > s(t)} and D = {S s(t)}.

One of the optimal exercising time D is the hitting time of thestopping region D.

V (St, t) is a super-martingale,

V (StD , t D) is a martingale.

Parabolic integro-differential equations (PIDE)

Proposition ([Pham 95], [Yang et al. 06], [Bayraktar 08])The value function V (S , t) is the unique classical solution of

(t + LD (r + )) V +

R

V (S ey , t) (dy) = 0, S > s(t),

V (S , t) = K S , S s(t), V (S , T ) = (K S)+,

in which LD , 122S2 2SS + b S S . Moreover,

SV (s(t), t) = 1, and

(t + LD (r + )) V +

R

V (S ey , t) (dy) 0.

Parabolic integro-differential equations (PIDE)

Proposition ([Pham 95], [Yang et al. 06], [Bayraktar 08])The value function V (S , t) is the unique classical solution of

(t + LD (r + )) V +

R

V (S ey , t) (dy) = 0, S > s(t),

V (S , t) = K S , S s(t), V (S , T ) = (K S)+,

in which LD , 122S2 2SS + b S S . Moreover,

SV (s(t), t) = 1, and

(t + LD (r + )) V +

R

V (S ey , t) (dy) 0.

RemarkBecause of the nonlocal integral term, the classical finite difference

method cannot be applied directly to solve the PIDE numerically.

Numerical algorithms

Various numerical algorithms were studied in the literature:

[Zhang 97]

[Andersen & Andreasen 00]

[Kou & Wang 04]

[Hirsa & Madan 04]

[Cont & Voltchkova 05]

[d Halluin et al. 05]

Numerical algorithms

Various numerical algorithms were studied in the literature:

[Zhang 97]

[Andersen & Andreasen 00]

[Kou & Wang 04]

[Hirsa & Madan 04]

[Cont & Voltchkova 05]

[d Halluin et al. 05]

We propose yet another numerical algorithm which

1. A slight modification of classical finite difference schemes

2. Shows the connection between jump diffusion problems anddiffusion problems

The functional operator J

Instead of the jump diffusions, we consider the GBM:

dS0t = b S0t dt + S

0t dWt .

The functional operator J

Instead of the jump diffusions, we consider the GBM:

dS0t = b S0t dt + S

0t dWt .

We introduce a functional operator J, for any f : R+ [0, T ] R+,

J f (S , t) , supTt,T

E

[

t

e(r+)(ut) P f(S0u , u

)du +

e(r+)(t)(K S0

)+S0t = S],

in which

P f (S , u) ,

R

f (ey S , u) (dy).

The functional operator J

Instead of the jump diffusions, we consider the GBM:

dS0t = b S0t dt + S

0t dWt .

We introduce a functional operator J, for any f : R+ [0, T ] R+,

J f (S , t) , supTt,T

E

[

t

e(r+)(ut) P f(S0u , u

)du +

e(r+)(t)(K S0

)+S0t = S],

in which

P f (S , u) ,

R

f (ey S , u) (dy).

RemarkJ f is the value function for an optimal stopping problem for the

diffusion S0.

The approximating sequenceLet us iteratively define

v0(S , t) = (K S)+ ,vn+1(S , t) = J vn(S , t), n 0, for (S , t) R+ [0,T ].

The approximating sequenceLet us iteratively define

v0(S , t) = (K S)+ ,vn+1(S , t) = J vn(S , t), n 0, for (S , t) R+ [0,T ].

TheoremFor each n 1, vn is the unique classical solution of the followingfree boundary PDE:

(t + LD (r + )) vn(S , t) = (P vn1) (S , t), S > sn(t),vn(S , t) = K S , S sn(t), vn(S , t) > (K S)+, S > sn(t),Svn (sn(t), t) = 1, vn(S ,T ) = (K S)+.

The approximating sequenceLet us iteratively define

v0(S , t) = (K S)+ ,vn+1(S , t) = J vn(S , t), n 0, for (S , t) R+ [0,T ].

TheoremFor each n 1, vn is the unique classical solution of the followingfree boundary PDE:

(t + LD (r + )) vn(S , t) = (P vn1) (S , t), S > sn(t),vn(S , t) = K S , S sn(t), vn(S , t) > (K S)+, S > sn(t),Svn (sn(t), t) = 1, vn(S ,T ) = (K S)+.

Remark

vn(S , t) = supTt,T

E

[er(nt) (K Sn)+

St = S],

where n is the n-th jump time of the Poisson process Nt .

The convergence of the sequence

Proposition

{vn}n0 is an monotone increasing sequence, moreover vn K.

The convergence of the sequence

Proposition

{vn}n0 is an monotone increasing sequence, moreover vn K.

Proof.

J v0 = v1 (K S)+ = v0, J maps positive functions to positive functions.

The statement follows from the induction.

The convergence of the sequence

Proposition

{vn}n0 is an monotone increasing sequence, moreover vn K.

Proof.

J v0 = v1 (K S)+ = v0, J maps positive functions to positive functions.

The statement follows from the induction.

There exists a limit v(S , t) , limn+ vn(S , t).

The convergence of the sequence

Proposition

{vn}n0 is an monotone increasing sequence, moreover vn K.

Proof.

J v0 = v1 (K S)+ = v0, J maps positive functions to positive functions.

The statement follows from the induction.

There exists a limit v(S , t) , limn+ vn(S , t).Moreover, v is the fixed point of J = v is the uniqueclassical solution of the PIDE.

The convergence of the sequence

Proposition

{vn}n0 is an monotone increasing sequence, moreover vn K.

Proof.

J v0 = v1 (K S)+ = v0, J maps positive functions to positive functions.

The statement follows from the induction.

There exists a li

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