Risk-neutral pricing ($V_t=B_t{\mathbb{E}}_Q[X/B_T\mid {\mathcal{F}}_t]$)

In a finite-state discrete-time financial market, a replicable terminal financial payoff $X$ has the displayed value under any equivalent martingale measure $Q$ for the bank account numéraire. Here ${\mathcal{F}}_t$ represents current information and conditional expectation averages over future states. The self-financing portfolio's discounted portfolio wealth is a martingale, so its current value equals the discounted expectation of its terminal value. In an incomplete market, nonreplicable claims need not have a unique price; continuous-time extensions require further conditions on admissible trading strategies and integrable discounted wealth.

For the European call option in the example,Discounting the physical expected value instead gives $200/21\approx 9.52381$ when $p=1/2$, or $320/21\approx 15.23810$ when $p=4/5$. Those amounts describe discounted physical expectations, not the replicating portfolio's cost. Holding $S_0,u,d,R$ fixed is crucial: changing beliefs can influence traded prices in a richer financial market model.

For a non-dividend-paying traded asset $dS=\mu Sdt+\sigma dW$ and bank account $dB=rBdt$, delta hedging sets the share holding to $V_s$. Its bond value is $V-s{V}_s$. Itô's formula and the self-financing condition give the displayed pricing equation on the whole real line. The linear drift change of measure supplies risk-neutral dynamics $dS=rSdt+\sigma d{W}^{\mathbb{Q}}$, so the same price is the discounted conditional payoff expectation under the usual admissibility and integrability hypotheses.