Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 4 / 29K (Stochastic Financial Models)

Between times $n-1$ and $n$, the investor holds ${\theta }_n$ shares worth ${\theta }_n{S}_{n-1}$ and places the remaining wealth $X_{n-1}-{\theta }_n{S}_{n-1}$ in the risk-free asset. The self-financing condition therefore gives

Under a risk-neutral measure, the discounted stock must be a martingale. If $q$ is the probability of the factor $1+b$, this condition issoThe hypothesis $a<r<b$ ensures $0<q<1$, making this equivalent to the original probability measure. Since the $N$ returns are independent under $Q$, the number of up moves has a binomial distribution, and hence

The backward option pricing recursion iswith terminal value $V(N,s)=g(s)$. At a node with $S_{n-1}=s$, define the replicating stock holding byThis is a predictable process because it depends only on information available at time $n-1$.

Indeed, if $X_{n-1}=V(n-1,s)$, substituting the risk-neutral formula for $q$ into the self-financing update shows separately in the up and down states thatAn induction over $n$ therefore gives $X_N=V(N,S_N)=g(S_N)$. The required initial capital is

For the European call option, risk-neutral valuation givesDefine the stock-numeraire measure in a binomial market byIt is a probability measure because the risk-neutral martingale property gives ${\mathbb{E}}_Q{S}_N=S_0(1+r{)}^N$. Consequently

Tilting one up move by its stock factor changes its probability toSimilarly $1-\hat{q}=(1-q)(1+a)/(1+r)$. Thus under $\hat{Q}$ the number of up moves is binomial with parameter $\hat{q}$, and