Advanced Numerical Methods

1. When the asset pays continuous dividend yield at the rate q, the expected rate of return of the asset is r − q under the risk neutral measure (see Chap 3 of Kwok's text for justification). Under the continuous Geometric Brownian process model, the logarithm of the asset price ratio over ∆t interval is normally distributed with mean r − q − σ 2 2 ∆t and variance σ 2 ∆t. Accordingly, the mean and variance of S t+∆t S t are e (r−q)∆t and e 2(r−q)∆t (e σ 2 ∆t − 1). By equating the mean and variance of the discrete binomial model and the continuous Geometric Brownian process model, we obtain pu + (1 − p)d = e (r−q)∆t pu 2 + +(1 − p)d 2 = e 2(r−1)∆t e σ 2 ∆t. Also, we use the usual tree-symmetry condition: u = 1/d. Solving the equations, we obtain u = 1 d = σ 2 + 1 + (σ 2 + 1) 2 − 4R 2 2R , p = R − d u − d , where R = e (r−q)∆t and σ 2 = R 2 e σ 2 ∆t. As an analytic approximation to u and d up to order ∆t accuracy, we take u = e σ √ ∆t and d = e −σ √ ∆t. The only change occurs in the binomial parameter p, where p = e (r−q)∆t − d u − d , while u and d remain the same. The binomial pricing formula takes a similar form (discounted expectation of the terminal payoff): V = [pV ∆t + (1 − p)V ∆t ]e −r∆t. The discount factor e −r∆t remains the same while the risk neutral probability of up move p is modified. 2. (a) With the usual notation p = R − d u − d and 1 − p = u − R u − d. If R < d or R > u, then one of the two probabilities is negative. This happens when e (r−q)∆t < e −σ √ ∆t or e (r−q)∆t > e σ √ ∆t. This in turn happens when (q − r) √ ∆t > σ or (r − q) √ ∆t > σ. Hence negative probabilities occur when σ < |(r − q) √ ∆t|. This result places a restriction on the time step. More precisely, the time step cannot …