Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 3 / 27J (Stochastic Financial Models)

For a finite-state, finite-horizon discrete-time frictionless market with a strictly positive numéraire, the fundamental theorem of asset pricing states no arbitrage if and only if there exists an equivalent martingale measure under which all discounted asset prices are martingales. Equivalence means the same null events as the physical measure. In this setting market completeness is equivalent to uniqueness of the equivalent martingale measure, and a replicable payoff $H$ has price $S_t^0{\mathbb{E}}_Q[H/S_T^0\mid {\mathcal{F}}_t]$. Without finite-state hypotheses, stronger integrability and closure conditions are needed; this is the standard multi-period course version.

The difference of the terminal European call option and European put option payoffs is $S_T^1-K$. Replicating that difference by one share and a borrowing position gives conditional put-call parityThus the call is more valuable exactly when the right side is positive. With a put selected at equality, the chooser option payoff isAt equality both choices have the same price, even though their eventual payoffs can differ. Put-call parity assumes no dividends or other intermediate stock cash flows, as in the stated asset-price model.

Put $K^{\prime }=K(1+r{)}^{t_0-T}$. At the choice time,Buy a European call option of strike $K$ and maturity $T$, together with a European put option of strike $K^{\prime }$ and maturity $t_0$. If the call is selected, the short-maturity put expires worthless. If the put is selected, sell the call and combine its value with the short-maturity put's cash payment; put-call parity makes this exactly enough to buy the maturity-$T$ put. This self-financing replication provesIt requires tradability of the priced options at the choice time, but not completeness of every other market claim. The identity also covers $t_0=0$ and $t_0=T$.