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Written Exam
Write answers in this file or on paper.
Let (B_t) be standard Brownian motion. The Bachelier model of stock price is F_t = f(1 + sigma B_t), where f is the forward and sigma is the volatility
- What is E[F_t] and Var(F_t)?
Solution:
Consider a European option that pays (F_T)^2 at time T.
- Derive a formula for V_t = E_t[F_T^2] in terms of F_t.
Hint. Use B_t^2 - t is a martingale.
Solution:
Let Z be a standard normal random variable and N(z) = P(Z <= z) be its cumulative distribution function.
- Show E[Z 1(Z <= a)] = -n(a) where n(z) = N'(z) is the standard normal density function.
Solution:
-
Prove the forward value of a European put option with strike k and expiration T is
p = E[max{k - F_T, 0}] = (k - f) N(x) + f sigma sqrt(t) n(x)
where x = (k - f)/(sigma sqrt(t)).
Hint. B_t/sqrt(t) is standard normal.
Solution:
- Derive a closed form solution for the Bachelier delta dp/df.
Hint. Use dg(F_T)/df = g'(F_T) dF_T/df where g(x) = max{k - x, 0}.
Solution:
Coding Exam
Follow the instructions in xll_template.cpp.