 Home

Codex Wiki

List of universities

University of Cambridge

Course of the University of Cambridge

Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2022

Paper 3

Set
$$
\theta=\frac{\mu+\frac12\sigma^2-r}{\sigma}
$$
and define $Q$ on $\mathcal F_T$ by
$$
\frac{dQ}{dP}
=\exp\left(-\theta W_T-\frac12\theta^2T\right).
$$
The <Cameron-Martin theorem for a linear drift> says that
$$
W_t^Q=W_t+\theta t
$$
is a <Brownian motion> under $Q$. Therefore
$$
S_t=S_0
\exp\left(\left(r-\frac12\sigma^2\right)t+\sigma W_t^Q\right).
$$
The discounted stock is an exponential Brownian martingale, so $e^{-rt}S_t$ is a $Q$-martingale. Thus $Q$ is a <Risk-neutral measure for the Black-Scholes model>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 3 / 29K / b / Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 3 / 29K / b / Solution

 Ancestors (11)

 Discussion (0)

 Articles by others on the same topic (0)

Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 3 / 29K / b / Solution

 Ancestors (11)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)