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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2018

Paper 2

Let $\tau=T-t$. Conditional on $\mathcal F_t$ under the risk-neutral measure,
$$
S_T=S_t\exp\left(\sigma(W_T^Q-W_t^Q)
+\left(r-\frac12\sigma^2\right)\tau\right).
$$
Using the <moment-generating function of a normal distribution> in the risk-neutral expectation gives the <Power payoff in the Black-Scholes model> value
$$
\boxed{
V_t=S_t^\gamma
\exp\left[
\left((\gamma-1)r+\frac12\gamma(\gamma-1)\sigma^2\right)(T-t)
\right].}
$$
In particular,
$$
\boxed{
V_0=S_0^\gamma
\exp\left[
\left((\gamma-1)r+\frac12\gamma(\gamma-1)\sigma^2\right)T
\right].}
$$
The <delta hedge> holds
$$
\boxed{
\Delta_t=\frac{\partial V}{\partial S}(t,S_t)
=\gamma S_t^{\gamma-1}
\exp\left[
\left((\gamma-1)r+\frac12\gamma(\gamma-1)\sigma^2\right)(T-t)
\right]}
$$
units of the risky asset.

Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 2 / 29K / b / Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 2 / 29K / b / Solution

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Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 2 / 29K / b / Solution

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