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STOCHASTIC CALCULUS 627 HW 6 PROBLEMS (PRELIMINARY) – FALL 2024 JOHN C. MILLER 1. Exercise 5.5 in Shreve II Please do Exercise 5.5 in Shreve II (i.e. prove Corollary 5.3.2). 2. Find α and β for the transformation of the B-S-M PDE to the heat equation Show why α was chosen to be 12(k 1) and β was chosen to be 14(k + 1)2. 3. Example of risk neutral and PDE valuation Suppose r is the risk free rate and suppose S(t) follows geometric Brownian motion with constant volatility σ = 0. Let V (T ) = { S(T ) if S(T ) ≥ K, 0 if S(T ) < K. (i) Find V (0) using risk-neutral valuation. V (0) should be a function of S(0), K, T , r, σ, and the standard normal cumulative distribution function N(·). (ii) Also find V (0) using PDE valuation (following example given in class). 4. American digital call Suppose r is the risk free rate and suppose S(t) follows geometric Brownian motion with constant volatility σ = 0. Let V (T ) = { 1 if S(t) ≥ B for some 0 ≤ t ≤ T, 0 otherwise. Find V (0) using the reflection principles for solutions of the Black-Scholes-Merton PDE. V (0) should be a function of S(0), B, T , r, σ, and the standard normal cumulative distribution function N(·). 5. Expectation of exit time For constants a, b > 0, let τ be the first time that Brownian motion W (t) exits the interval [ a, b] (see example from class). What is E[τ ] 1 2 JOHN C. MILLER 6. American exercise put In the notes, we did an example of a three-period model with u = 2, d = 1/2, r = 1/4 and S0 = 4, with a payout of a put struck at $5. Using the same parameters, calculate the value of a put with strike $5, using a four period model. 7. Arbitrage Consider the multidimensional market model with one Brownian motion (d = 1) and two assets (m = 2): dS1 = α1S1 dt+ σ1S1 dW, dS2 = α2S2 dt+ σ2S2 dW with constant risk free rate r. (i) What are the market price of risk equations for this market model (ii) What are the conditions (expressed in terms of α1, α2, σ1, σ2, r) for a risk neutral measure to exist (iii) If those conditions do not hold, please describe an arbitrage strategy. 8. Induction proof Please complete the induction proof (see Tuesday’s lecture) to show that vT k(x) = log (x k )k (1 + ρ)k(k 1)/2. Email address: john.miller@jhu.edu