Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 41 / 2 / ii / Solution

Suppose under the physical measure $P$ the up indicators are independent with probability $p\in (0,1)$. If $N_k$ is the number of up moves through date $k$, a particular history has probability $p^{N_k}(1-p{)}^{k-N_k}$ under $P$ and $q^{N_k}(1-q{)}^{k-N_k}$ under $Q$. Thus the binomial-market probability density, the Radon-Nikodym derivative on ${\mathcal{F}}_k$, isIt is strictly positive. Its next multiplier has conditional expectation $p(q/p)+(1-p)((1-q)/(1-p))=1$, so $L_k$ is a mean-one martingale and $L_k={\mathbb{E}}_P[L_n\mid {\mathcal{F}}_k]$. These facts verify normalization and equivalence of the two probability measures.

The state-price density for terminal cash is ${\zeta }_n=L_n/B_n=b^{-n}{L}_n$. In particular $V_0={\mathbb{E}}_P[{\zeta }_{n}H]$. At an intermediate date the conditional change-of-measure identity givesIf physical up probabilities depend on the past, replace the constant-probability expression by the product over periods of $q/p_k$ on an up move and $(1-q)/(1-p_k)$ on a down move. Each multiplier still has conditional mean one, provided $0<p_k<1$, and the pricing measure retains conditional up probability $q$.