FINM345A09Lecture8corrected - FinM 345/Stat 390 Stochastic...

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Unformatted text preview: FinM 345/Stat 390 Stochastic Calculus, Autumn 2009 Floyd B. Hanson , Visiting Professor Email: [email protected] Master of Science in Financial Mathematics Program University of Chicago Lecture 8 (from Chicago) More Merton BS + Option Pricing and Jump-diffusion Financial Applications 6:30-9:30 pm, 16 November 2009 at Kent 120 in Chicago 7:30-10:30 pm, 16 November 2009 at UBS Stamford 7:30-10:30 am, 17 November 2009 at Spring in Singapore Copyright c 2009 by the Society for Industrial and Applied Mathematics, and Floyd B. Hanson. FINM 345/Stat 390 Stochastic Calculus — Lecture8–page1 — Floyd B. Hanson FinM 345 Stochastic Calculus: 8. Merton BS + Option Pricing Continued and Jump-diffusion Financial Applications: • 8.1. Merton BS + Option Pricing Continued: • 8.1.1 Merton PDE of Option Pricing: To derive the PDE of Black–Scholes–Merton option pricing , with definition of the option expected return μ y in [(7.28) on L70p47 or (10.24) in textbook, p. 296], is viewed as a PDE for the option price function with the option trajectory Y ( t ) replaced by the composite function equivalent F ( s,b,t ; T,K ) as a function of three independent variables ( s,b,t ) , the triplet ( s,b,t ) having replaced the two underlying state trajectories ( S ( t ) ,B ( t )) . FINM 345/Stat 390 Stochastic Calculus — Lecture8–page2 — Floyd B. Hanson This yields the PDE, μ y F ≡ F t + μ s sF s + μ b bF b + 0 . 5 ( σ 2 s s 2 F ss +2 ρσ s σ b sbF sb + σ 2 b b 2 F bb ) . (8.1) It is conceptually important to separate the view of s , b and t as three deterministic, independent PDE variables and the view of S ( t ) and B ( t ) as the two random SDE state trajectories in time and to use each view in the appropriate place. Next, μ y is eliminated using the Black–Scholes fraction [(7.44) on L7-p57 or (10.41) in textbook, p. 298] with μ y = μ b + ( μ s- μ b ) σ ys /σ s and the option-stock induced volatility σ ys is eliminated using its definition in [(7.29), L70-p47 or (10.25) in textbook, p. 296], i.e., σ ys = σ s sF s /F . FINM 345/Stat 390 Stochastic Calculus — Lecture8–page3 — Floyd B. Hanson The option price F can be eliminated by Merton’s homogeneous condition [(7.42), L7-p55 or (10.38) in textbook, p. 298] with y replaced by F , F = sF s + bF b , incidentally eliminating both first partials F s and F b , and so, 0= F t +0 . 5 ( σ 2 s s 2 F ss +2 ρσ s σ b sbF sb + σ 2 b b 2 F bb ) . (8.2) This Merton PDE of option pricing needs side conditions, such as a final condition at the expiration time and boundary conditions in the asset variables. The PDE and conditions forming a final value problem (FVP) . For the FVP, the natural time variable is the time-to-maturity or time-to-exercise or time-to-go τ = T- t , and F t =- F τ ....
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FINM345A09Lecture8corrected - FinM 345/Stat 390 Stochastic...

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