The Center for Applied Probability Columbia University: Nov. 9, th Annual CAP Workshop on Derivative Securities and Risk Management

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1 The Center for Applied Probability Columbia University: Nov th Annual CAP Workshop on Derivative Securities and Risk Management A Simple Option Formula for General Jump-Diffusion and other Exponential Lévy Processes Alan L. Lewis Overheads and additional notes to be posted at (Publications) Topics Why jump-diffusion models? European-style options.. Solutions in Stock price space are complicated. The Fourier-space solution is simple 3. Moving integration contours around is useful. American-style options 4. Simple numerical method (method of lines) 5. The analytic T solutions

2 Why Jump-Diffusion Models for Options? I. Benchmark model (exponential Brownian motion): Attractive features: limited liability stock prices uncorrelated level independent returns simple formulas (methods) for option prices (euroamer) Weak points: Actual stock price distributions have wider tails Lacks volatility clustering (auto-corr. of absolute returns) Lacks stock price jumps Poor fit to real-world option prices (smile/skew) II. Jump-diffusion generalization (exponential Lévy processes) - Class of all stationary independent increment processes - Subclass: Brownian motion plus Poisson jumps Attractive features: all the attractive benchmark features + large flexible class of models each with a few parameters wide return tails common (exponential decay moments) Good fits to expiring options (fear of jumps/crashes?) Weak points: Lacks volatility clustering (auto-corr. of absolute returns) Brownian motion Large number of small jumps

3 Stock price Evolution and Examples St = S 0 exp( Xt) (Assumption: this is under the martingale pricing measure Q ) where Xt = ct+ σbt + Xt; X = y ( y) (a measure) τ µ Type A: Poisson sub-class µ ( ydy ) < µ ( y) = λp( y) where p( y) dy= Examples: (A.) Merton s 976 jump-diffusion model with log-normally distributed jumps: p( y) = exp ( y α) / δ πδ (A.) Kou s 000 jump-diffusion model with exponentially distributed jumps: p( y) = exp [ y κ / η] η 3

4 Stock price Evolution and Examples (cont.) S = S 0 exp( X ) t where Xt = ct+ σbt + Xt; X = y ( y) (a measure) τ µ t Type B: No Poisson intensity exists: µ ( ydy ) = Example: (B.) Carr and Wu s (000) Finite Moment Logstable Process µ ( y) = c y ± + α c c y> 0 y 0 + ±= c < < α < ( α = is Brownian motion) 4

5 European-style options. Solutions in Stock price space are complicated Example: Madan Carr and Chang s Variance Gamma Process Pure jumps: BM sampled at random times The Call Option Price: c C( S ) S d ( α s) ν τ 0 = 0Ψ + ν c ν r c Ke τ Ψ d ( α s) ν τ + ν c ν S c where d log rτ τ log = s K ν c and γ+ + / c exp( sign( a) c)( u) Ψ ( ab γ) = X πγ( γ) ( + u) + Kγ+ / ( c) Φ γ γ + γ; sign( a) c( + u) +... (4 more lines) γ 5

6 European-style options (cont.) Solutions in Fourier space are simple Ingredients:. The generalized Fourier transform of the payoff function: For the call option: iz+ izx x + K wz ˆ ( ) = e ( e K) dx = Im z > ( z iz). The characteristic function of the Lévy process: ϕ ( z) = e izx p ( x) dx = E[ exp( izx )] = exp ( TΨ( z) ) T T T where Ψ ( z) is the characteristic exponent. For the VG model example: ( ) T / ν ϕt ( z) = exp( icωt) izνθ+ σ νz α< Im z < β Then the option price is given by ( y= log( S0 )) rt iν+ ( ) e izy V S0 = e ϕt ( z) w( z) dz π ν has conditions iν The integration is along a line parallel to the real z-axis. (Closely related results: Carr & Madan Bakshi & Madan Raible) 6

7 Generalized Fourier Transforms for various Payoffs Financial Claim (Option) Payoff Function: wx ( ) Call option x ( ) e K + Put option ( K e x ) Covered call/ cash-secured put + x ( ) Payoff Transform: wz ˆ ( ) iz+ K z iz K iz + min e K iz+ K z iz Delta function δ( x ln K) iz K Money market z iz Strip of regularity S Im z > Im z < 0 w 0< Im z < Entire z-plane πδ( z) Im z = 0 7

8 European-style options (cont.) Solutions in Fourier space are simple The solution is very easy to derive and obvious : First we need the inversion formula for the payoff function: iν+ izx wx ( ) = e wzdz ˆ ( ) π x = log S T z Payoff strip iν Then by martingale pricing: rτ iν+ rτ izlog ST V( S ) e [ w(log ST )] e 0 = E = E e w( z) dz π iν rτ i e ν+ iz log S izxt e e 0 = E w( z) dz π iν rt iν+ e iz log S 0 = e ϕt ( z) w( z) dz π i ν Ok to exchange the integrations (sufficient conditions) if:. wx ( ) is Fourier integrable in some Payoff strip S w and bounded for x <. *. ϕt ( z) is regular in some strip S X : α< Im z < β 3. ν = Im z lies in the intersection of these two strips 8

9 European-style options (cont.) Moving integration contours is easy In practice: for typical financial claims: There is complete freedom to integrate anywhere in the strip of regularity of the characteristic function (Residue Theorem) Example: the call option with strike K. * Let ϕt ( z) be regular in some strip S X : where α < 0 and β >. Then α< Im z < β rt iν + izk C( S ) = Ke e ϕt ( z) w( z) dz π 0 iν z iz where < ν < β and k log( S / K) = 0 Now move contour to 0< ν <. There are simple poles at z = 0 and z = i. You will pick up a residue at z = i. rt iν + qt izk C( S ) = Se Ke e ϕt ( z) w( z) dz π 0 iν z iz where 0< ν <. Applications: Black-Scholes style formulas Asymptotics (T K other parameters ). 9

10 II. American-style options (Put option example) The problem: determine the optimal exercise strategy (a stopping time strategy) rτ + V( S T) = max τ( ω) T e ( K Sτ) 0 0 E For simplicity: St = S 0 exp( Xt) where X t is a jump-diffusion (a Poisson intensity λ exists). By homogeneity the solution is V( S0 T) = K f( x T) where x = log( S0 / K) and The reduced problem: find bt ( ) 0 and f( x t ) where (i) for bt ( ) x< using k = ( ey ) p( y) dy t xx x f = σ f + ( r σ λk) f ( r+ λ) f + λ f( x+ y) p( y) dy x (ii) for < x b( t) f( x t) = e + Subject to: (i) f( x 0) = ( e ) bt () (ii) f( x = b( t) t) = e bt () (iii) fx ( x = b( t) t) = e (smooth pasting). (iv) f( x t) 0 as x A good numerical method for this problem (G.H. Meyer): The Method of Lines. Write ft = ( fn fn )/ t. Take t = T/ N and just keep solving ODEs for n=... N. (Actually there is a sub-iteration at each n for the jump term). 0 R x R

11 II. American-style options (Put option example) Numerical example: K=00 r = 008. σ = 040. T = year. Jumps: frequency λ = ; Two possible jumps: e y.5 (+5%) prob=/ = (-50%) prob=/ Results from Method of Lines computation: Critical Price Critical 00 Price No Jumps Years Jumps Time to Expiration

12 II. American-style options (Put options) Numerical example continued: Let T in the Method of Lines program (you can!) Numerical Results for the critical boundary No Jumps: S * = 50 With Jumps: S* 3. 6 Both results confirmed by exact analytic formulas: The T Perpetual Put Is Completely Solved Analytically Case Difficulty By whom: Brownian motion + I. Up jumps Easy (? Wald) II. Down jumps Moderate Gerber Landry & Shiu III. Up & Down jumps Hard Boyarchenko & Levendorskii Formula case III: (easy integration once you know answer!) : iω+ log [ r+ Ψ( z) ] S* = Kϕ ( i) = Kexp dz. π = zz ( i) iω where 0 < ω < Zero of [ r+ψ( iy) ] y = real. Will post on web site: Very direct derivation of case II.

13 American-style Put options: perpetual case boundary Moderately hard case: Brownian motion + negative jumps. 3 steps: much faster than Gerber Landry & Shiu (following suggestion by David Dickson a discussant) Step. Translate x' = x b ( now relabel x' x) and introduce G( x) x+ b f( x) ( e ) x 0 = 0 x 0 which satisfies on x 0 with c = r σ λk hy ( ) = p( y) (*) r = σ G'' + cg' ( r+ λ) G+ λ G( x y) h( y) dy B.C.: (i) G'( 0) = G( 0) = 0 (ii) ( ) x b G x e + + vanishing terms as x Step. Solve (*) with Laplace transform ˆ sx Gs ( ) = e Gxdx ( ) 0 r = ψ( sgs ) ˆ ( ) where the Laplace exponent s ψ( s) = σ s + cs ( r+ λ) + λhˆ ( s) invert χ+ i sx G( x) = r e ds πi χ > (right-most pole) sψ( s) γ i Step. 3. Move the contour to χ ' < 0. The martingale condition b causes ψ ( s = ) = 0. The residue at s = must be e to satisfy b* B.C. (ii). This determines e = r/ ψ'( s = ) or S* = Kr/ ψ'( ). 0 x 3

14 References Bakshi Gurdip and Dilip Madan (000): Spanning and Derivative-security Valuation J. of Financial Economics Boyarchenko S.I. and S.Z. Levendorskii: Perpetual American Options Under Levy Processes manuscipt. Econ. Dept. Univ. of Penn. Carr Peter and Dilip B. Madan (999): Option Valuation using the Fast Fourier Transform J. Computational Finance No.4 Summer Carr Peter and Liuren Wu (000): The Finite Moment Logstable Process and Option Pricing manuscript Feb Gerber Hans U. and Elias S.W. Shiu (998): On the Tine Value of Ruin North American Actuarial Journal Vol. No. (Jan) Gerber H.U. and B. Landry (998).: On the discounted penalty at ruin in a jump diffusion and the perpetual put option. Insurance: Mathematics and Economics Kou S.G. (000): A Jump Diffusion Model for Option Pricing with Three Properties: Leptokurtic Feature Volatility Smile and Analytical Tractability preprint Columbia University Feb Madan Dilip B. Peter Carr and Eric C. Chang (998): The Variance Gamma Process and Option Pricing manuscript University of Maryland June 998 (forthcoming European Finance Review). Merton R.C.(976): Option Pricing When Underlying Stock Returns are Discontinuous J. of Financial Economics 3 Jan-Mar Reprinted as Ch. 9 in Continuous-Time Finance. Basil Blackwell Cambridge Mass (990). Meyer G.H. (998): The Numerical Valuation of Options with Underlying Jumps Acta Math. Univ. Comenianae Vol. LXVII Raible Sebastian (000): Lévy Processes in Finance: Theory Numerics and Empirical Facts Dissertation Faculty of Mathematics Univ. of Freiburg Germany. 4

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