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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 }.