AMS Spring Western Sectional Meeting April 4, 2004 (USC campus) Two-sided barrier problems with jump-diffusions Alan L. Lewis

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1 AMS Spring Western Sectional Meeting April 4, 2004 (USC campus) Two-sided barrier problems with jump-diffusions Alan L. Lewis No preprint yet, but overheads to be posted at (Publications) (This work is preliminary and incomplete.) Objectives: 1. Develop analytic formulas for Green function s for certain two-sided barrier/exit problems, where the underlying stochastic process: { X t } = 1D jump-diffusion (Levy process) 2. Develop simple alternative numerical methods to check the results. 1

2 Connection with Finance Suppose a stock price S t follows an exponential Levy process: St = S 0 exp( Xt), (Under a martingale pricing measure Q ) Properties of Levy class: (i) independent increments: X t X s and X s are independent ( t > s) (ii) time-homogeneous (stationary): Distribution of { X t+ s X s} does not depend on s. (iii) continuous (in probability) at any fixed time: The jumps occur at unpredictable times. Levy sub-class: the jump diffusions: B t = X = ωt+ σb + X, t t t std. Brownian motion; X t N t = ξi i= 1 (i) jump events N t follows a Poisson process: Jum p probability(t) λ t, λ = intensity (ii) jump amplitudes ξ are then drawn independently: ξ ~ F( ξ ) (Jump size density) 2

3 Figure 1 Connection with Finance (cont.) SPX Options: Implied Volatility vs. Strike on Aug. 16, 2002 H1 monthto ExpirationL ConstantVolatility +Jumps Ø SPX Example Jump-diffusion model: Merton s 1976 jump-diffusion model: Jump sizes: F( ξ) = 1 exp 2 ( ξ µ J) / 2σ J 2 2πσ J 2 Typical S&P500 Index Option Smile Fit λ 03., µ 025,. σ 010. J Plain vanilla options like these are easily priced under any Levy/jump-diffusion process. (single integral transform) Exotics: single barrier options; more difficult, but formulas are known. (Integrals using Wiener-Hopf factors φ ± ( z) ) Double barrier: No general formulas for G are known rt OptionValue( x) = e G( x, y, T ) Payoff ( y) dy 0 b J 3

4 The Two-sided Exit Problem X0 = x ; T e = first exit from I = ( 0, b). Some probability densities: : Support for y Unrestricted law: P ( X dy) = p( x, y, t) dy. (, ) x t (Conditional) Green function: P ( X dy, T > t) = G 0 ( x, y, t) dy. I = (0,b) x t e Exit densities: Px( X( Te) dy, Te dt) I c=(, 0) ( b, ) = Ge ( x, y, t) dydt = [ G( x, yt, ) + G( x, yt, )] dydt 1 2 4

5 The Markov Property + Conservation of Probability px (, yt, ) = G( x, yt, ) + dξ dsg( x, ξ, s) p( ξ, yt, s) 0 But, for a Levy process: p( x, y, t) = p( y x, t) I c p ( y xt, ) = G( x, y, t) + dξ dsg ( x, ξ, s) p( y ξ, t s) Next: 0 I c 1. Take Fourier transform of both sides dy e (...) 0 t 0 t e e 0 2. Take Laplace transform of both sides dt e (...) izy t izy izx tψ ( z) 3. Use e p( y x, t) dy = e, where ψ ( z) =characteristic exponent of Levy process t Define Q ( z) dte dy e G ( x, y, t) i izy =,. Fundamental Transform Identity: (FTI) eizx ( ) ( ) ( ) Q 1 z + Q 2 = Q z z ψ( z) + ψ( z) i i = 012,, 5

6 Kemperman, A Wiener-Hopf Type Method for A General Random Walk with a Two-sided boundary. (Ann. Math. Statistics 1963) His starting point is the FTI: e izx ( ) ( ) ( ) Q 1 z + Q 2 = Q z z 0 +, x ( 0, b) + ψ( z) + ψ( z) (The dependence upon x in Q 's is implicit/suppressed) His focus is the discrete-time Random Walk: X n = ξ1+ ξ2 + ξ n ; ξ F( ξ ). He shows how to find Q 0 when F( ξ) = ce ηξ 1{ ξ 0} + G ( ξ) 1 { ξ 0}, G arbitrary. < My goal is to generalize this to the Continuous-time case: Xt = ωt+ σbt + ξi, ξi F( ξ ) N t i= 1 6

7 Ingredients to the argument 1. Introduce the WH Factors (z horiz. strip about origin) + = φ ( z) φ ( z) + ψ( z) Properties: φ + ( z) is a characteristic function of a probability distribution with support on ( 0, ). 2. Introduce M( ab,, ) a class of functions with support in (a,b). Their Fourier transforms are, by definition, in the class Mˆ ( ab., ) For example, φ ( ) ˆ + z M( 0, ). 3. Introduce the truncation-by-a operation: ( ˆf = Fourier transform of f ) [ ˆ A ] [ ˆ A izx f = f] ( z) = e f( x) dx A 7

8 Steps in the argument ex FTI: = Q0 + ( Q1+ Q2) ; + ψ( z) + ψ( z) ex = e izx x 1 2 φ WH factors: Q = e φ φ ( Q + Q ) φ Q0 + + exφ ( Q Q ) φ φ = (1) Then, Kemperman proves, and I rely upon: A. Q φ + Mˆ ( b, ) 2 Q0 B. Mˆ (, b) φ (, b) Apply the truncation operator [ ] to both sides of (1). This knocks out Q2 φ + since its transform = 0 on (,b). Q0 + (, b) + (, b) [ exφ ] [ Qφ ] φ = 1 8

9 Steps in the argument (cont.) Q0 + (, b) + (, b) [ exφ ] [ Qφ ] φ = 1 Example truncation calculations for the right-hand-side Let U( z) = e ( z) φ ( z) = e u( y) dy. x + izy A izy Recall [ U] ( z) = e u( y) dy A (, ) b b izy dξ [ U] ( z) dye U( ξ) e π = 2 iξ y or = dξ U ( ξ) e ib z 2π i Imξ> Im z ( ξ ) [ i ( x b) (, b) ibz d ex ] ξ ξ + ( z) e e + φ = φ ( ξ 2πi z ξ ) Imξ> Im z 9

10 Steps in the argument (cont.) Recall that Q 1( z) is the Fourier transform of the exit density for the lower barrier at 0. There are two ways to exit at the lower barrier: 1. A continuous touch at X = 0 due to the Brownian motion component, or 2. A negative jump from some point above the barrier, ηξ with jump amplitude drawn from ce 1 { ξ<0}. Experience calculating with purely exponential jump models, leads to overshoot (exit) densities which are also exponential with the same upside/downside parameters as the jump amplitude distribution. Even though our upside jump distribution is not exponential, experience suggests the downside solution ansatz: y G1 ( x, y, t) = A( x, t) δ ( y) + B( x, t) e η ; ( y 0) ˆ ˆ Bx (, ) Q1 ( z) = A( x, ) + z iη, Im z < η. This has to be proved a self-consistent choice! (Gap #1) 10

11 Final steps in the argument At this point, we have (suppressing some dependencies on x and ) ibz Q0 ( z) = φ ( z) e H( z), where dξ φ ( ξ) H( z) = e Ae e 2πi z ξ ξ iη Imξ> Im z + i x b i b i b { } ξ( ) ξ B ξ It can be shown (Mordecki, 2002), for a mixed exponential + arbitrary jump density, that: z iη iα iα φ ( z) 1 2 = iη z iα z iα 1 2, Where ( α1, α 2) = positive reals ( iα1, iα 2 ) = purely imag. roots of + ψ( z) in upper z-plane. But Q 0( z) is the transform of a prob. density with support in ( 0, b). This transform is analytic in Im z > 0. So, the two poles in φ ( z) must be cancelled zeros in the integral H( z ) at z = ( iα1, iα2 ). That is Hiα ( 1) = 0 and Hiα ( 2 ) = 0 Two conditions that determine the unknowns ( AB, ) 11

12 Final Formulas (The Answer) Define: I 1 dξ iξ ( x b) ( x, z) e + = φ ( ξ) 2πi z ξ Imξ> Im z I 2 dξ iξb ( ) e + z = φ ( ξ) 2πi ( z ξ)( ξ iη) Imξ> Im z Determine AB, from the two linear relations: AxI ( ) (, iα ) + BxI ( ) ( iα ) = I( xi, α ), j = 1,2 1 0 j 2 j 1 j Then, ˆ (,, ) dz ( ) izy G0 x y = Q0 z e 2π dz e iz( b y) = 1 φ ( z ) ( ) H z, 2π where H( z) = + dξ φ ( ξ) e Ae e 2πi z ξ ξ iη Imξ> Im z { } ξ( ) ξ B ξ i x b i b i b There is an ambiguity in the defn. of the I 2 contour. Is Imξ < η or > η? I force it to always lie below, because this achieves numerical agreement with my lattice method. (This needs to be clarified: Gap #2). 12

13 Computations (Alternative Numerical Method). I compare results for Gˆ ( x,, ) t y = e G ( x, y, t ) dt vs. a simple explicit lattice algorithm. (Lewis, 2004). Briefly: Divide the interval (0,b) into N points of size x. Choose t = ( x/ σ ) 2 / 3, which corresponds to a well-known trinomial lattice method in finance. To compute G0( x, y, t), start with v ( x ) = G ( x, y, ) = δ =Point mass at y. old i 0 i 0 x(), i y Iterate recursively: v new = Pvold, where P is an N N (banded/toeplitz) transition matrix: P p p p... pn p 1 p0 p1... pn 2 =... p... p N p 0 bm Where pi ( 1 λ t) pi 1 i = 1 + λ tg( xi. 5 x, xi +. 5 x) 1i 1 x Now G = F( y) dy is the jump-size distribution, (not bm density), and p 0 = 2/ 3, p± 1 = 1/ 6 ± ( ω/ σ) t/ 12 bm 13

14 Interval I = ( 02,, ) Computations (Results) y = 1, various x. λ = 3 η = 3. Parameters: = 05, σ = 2,, Jump size density: (up point jump + down exponential): x F( x ) =. 5 δ( x 025. ) η 1 { x< 0} e η Results: G ˆ 0 ( x, y = 1, = 0. 5 ) : Explicit Lattice: (T max = 5, no changes if T max = 10). N +1 x = Analytic Formula: ξ max = zmax, Double Integration; Mathematica AccuracyGoal =AG, AG+3. AG, z max x = , , Summary: The numerical results encourage me to believe the final formulas are correct. Further work is needed to fill in the gaps in their derivation. 14

15 References J.H.B. Kemperman, A Wiener-Hopf Type Method for A General Random Walk with a Two-sided boundary. Annals of Math. Statistics 34, 1963, Alan L. Lewis, Double Barrier Options with Jumps a Simple Universal Algorithm, Wilmott magazine, Mar., 2004, Ernesto Mordecki, Optimal stopping and perpetual options for Levy processes, Finance & Stochastics, VI, 2002, No. 4, Note added after the talk: the full analytic solution, with the gaps filled in, will be given in: Alan L. Lewis, Option Valuation under Stochastic Volatility, Vol. II: Jumps and Exotics, Finance Press (Newport Beach), 2004 forthcoming. 15