Remarks on American Options for Jump .Remarks on American Options for Jump Diffusions...

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Transcript of Remarks on American Options for Jump .Remarks on American Options for Jump Diffusions...

  • 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