By Michael Mastro PhD
Content material:
Chapter 1 monetary types (pages 1–34):
Chapter 2 bounce types (pages 35–64):
Chapter three thoughts (pages 65–104):
Chapter four Binomial timber (pages 105–129):
Chapter five Trinomial timber (pages 131–165):
Chapter 6 Finite distinction equipment (pages 167–230):
Chapter 7 Kalman clear out (pages 231–244):
Chapter eight Futures and Forwards (pages 245–294):
Chapter nine Nonlinear and Non?Gaussian Kalman filter out (pages 295–347):
Chapter 10 Short?Term Deviation/Long?Term Equilibrium version (pages 349–358):
Chapter eleven Futures and Forwards innovations (pages 359–396):
Chapter 12 Fourier remodel (pages 397–457):
Chapter thirteen basics of attribute capabilities (pages 459–466):
Chapter 14 program of attribute services (pages 467–504):
Chapter 15 Levy techniques (pages 505–546):
Chapter sixteen Fourier?Based alternative research (pages 547–584):
Chapter 17 basics of Stochastic Finance (pages 585–604):
Chapter 18 Affine Jump?Diffusion strategies (pages 605–644):
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Additional info for Financial Derivative and Energy Market Valuation: Theory and Implementation in Matlab®
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8. ALTERNATE JUMP MODELS A few alternate jump distributions have been suggested in the literature to be better for certain data sets. A subset of other important jump distributions is outlined in this section. The underlying motivation has to do with matching the shape of the distribution tail to the jumps in the data as well as the ease of translating the jumpdiffusion process into models of asset, futures, and option prices. The best choice should always be judged on a case-by-case basis. 1. Normal Model The normal model generates Q with a normal density given by ϕQ (q) = ϕ(x; μj , σj2 ) = 1 − e (x−μj )2 2σj2 , 2π σj2 with mean μj and variance σj2 .
Briefly, the log-likelihood function is n L= ln f (Si |Si−1 , μ, σˆ i , λ) i=1 n L = − ln(2π ) − 2 n n 1 Si − Si−1 e−λδt − μ 1 − e−λδt 2σˆ i2 [ln(σˆ i )] − i=1 i=1 2 and the optimal parameters are n (Si −Si−1 e−λδt )/σˆ 2 i μ= i=1 n n(1 − e−λδt ) 1/σˆ 2 i i=1 σˆ 2 = 1 n n i=1 (Si − Si−1 e−λδt + μ(1 − e−λδt ))2 ⎛ n ⎜ 1 ⎜ ⎜ i=1 λ = − ln ⎜ δt ⎜ ⎝ Si − μ (Si−1 − μ) σˆ i2 n i=1 Si−1 − μ σˆ i2 2 ⎞ ⎟ ⎟ ⎟ ⎟. ⎟ ⎠ The equation just derived for σˆ is dependent on both μ and λ. Fortunately, the two coupled equations for μ and λ are only dependent on each other.
Rather, they are dependent on M1 , M2 , λ, and Qa , Qb . The last two, Qa , Qb , are interrelated to the mean jump by μj = (Qb + Qa ) 2 44 JUMP MODELS and variance by σj2 = (Qb − Qa )2 . 12 The function ModelJumpDiffusion(S ) takes in a set of asset closing prices S, or simulates a set of price data for a null input, and then converts the price data to a vector of log-returns. The heart of the program is a call to the Matlab function fminsearch that employs the LikeEval function to fit a set of parameters {λ, Qa , Qb }.