MFIN7003FG (2022) Assignment 2
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MFIN7003FG (2022)
Assignment 2
Due by the end of November 15
Q1. Find the stochastic differential equation dZ when
1. Zt = l0(t) g(s)dWs ,
2. X has the stochastic differential equation
dXt = µdt + σdWt
(µ and σ are constants).
3. Zt = (Xt)2 , where X has the stochastic differential equation
dXt = αXtdt + σXtdWt
(α and σ are constants).
Q2. Find the expectation and variance of the stochastic integral
\0 2 t3 dWt
Q3. Show that process Xt = Et[e −r(T −t)YT] for all t < T is not a martingale.
Q4. Stochastic integrals have many properties similar to the properties of Riemann integrals.
Using It’s Lemma with
Y (t) = (Wt)n+1
and Wiener process Wt, establish the reduction of the order formula
\0t (Ws)n dWs = (Wt)n+1 − \0t (Ws)n − 1 ds.
Q5. Solve the following PDE:
(t,x) + κ(θ − x) (t,x) + σ 2 (t,x) − rF = 0
F(T,x) = xeκT ,
where r, κ , θ, and σ are constants.
Q6. A 3-month American call option on a stock has a strike price of $20. The stock price is $20, the risk-free rate is 3% per annum, and the volatility is 25% per annum. A dividend of $2 is expected in 1.5 months. Use a three-step CRR binomial tree to calculate the option price. (You need to write down the tree for the stock and the tree for the option.)
2022-11-14