Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 4 / 28J (Stochastic Financial Models)

Use discrete dates $j=0,\dots ,N$, a bank account $B_j=R^j$ with $R=1+r>0$, and a positive stock with each one-period gross return either $u$ or $d$, where $0<d<u$. On the natural tree assume both moves have positive physical probability. The Cox--Ross--Rubinstein model is arbitrage-free precisely whenOtherwise a stock/bank position gives nonnegative profit in both states and positive profit in at least one. Under the condition, the unique equivalent martingale measure has independent upward moves with conditional probabilityIndeed $qu+(1-q)d=R$, making the discounted stock a martingale. Both branches have positive probability under this measure, ensuring equivalence. The uniqueness/completeness claim is for the stock-generated binomial tree; an enlarged filtration carrying additional untraded risk would change that claim.

For any finite contingent claim $C$ measurable at the final date in the natural binomial tree, backward induction givesAt a node with stock $S_j$, let $V_{j+1}^u,V_{j+1}^d$ be its two successor values. Hold, during the next period,The stock and bank holdings reproduce both successor values and cost ${\Delta }_j{S}_j+{\psi }_j{B}_j=R^{-1}[q{V}_{j+1}^u+(1-q)V_{j+1}^d]=V_j$. Rebalancing at every node is self-financing, and terminal value is $C$. Thus every such claim is replicated and the no-arbitrage price is unique.

Interpret $t=m$ and $T=N$ as trading dates in the discrete model; a payoff at nondates would require an interpolation convention absent from the question. Put $n=N-m$. Under $Q$, the independent stock returns after $m$ giveThe ratio payoff is therefore independent of ${\mathcal{F}}_m$ under $Q$, and its time-zero price isAt any date $s\le m$, its price is the same expectation discounted only for the remaining $N-s$ periods. The formula uses the ratio payoff actually specified; multiplying it by $S_m$ would define a different, conventional monetary forward-start call.