= Pricing equation for a local volatility model
{title2=$V_t+rSV_S+\frac12\sigma(S)^2S^2V_{SS}=rV$}
For a <local volatility model> $dS_t=S_t(rdt+\sigma(S_t)dW_t^Q)$ with <bank account> $B_t=B_0e^{rt}$, a nonnegative classical solution $V$ of
$$
V_t+rSV_S+\tfrac12\sigma(S)^2S^2V_{SS}=rV,\qquad V(T,S)=g(S),
$$
defines a possible price of a <European contingent claim>. Indeed, <Itô formula> gives $d(V(t,S_t)/B_t)=B_t^{-1}\sigma(S_t)S_tV_S(t,S_t)dW_t^Q$, a <local martingale>. Thus the same <equivalent local martingale measure> works for the market augmented by this price. For an <admissible trading strategy> whose discounted wealth is bounded below, the discounted gains are a <local martingale> bounded below, hence a <supermartingale> by localization and the <Fatou lemma>. A zero-cost terminal gain that is nonnegative and strictly positive with positive probability would then contradict the <supermartingale> expectation inequality. This proves the absence of <arbitrage> for such strategies. If $V$ is bounded on the finite time interval, its discounted value is a bounded <martingale> and consequently $V(t,S_t)=B_t\mathbb E_Q[g(S_T)/B_T\mid\mathcal F_t]$. Nonnegativity alone guarantees the <local martingale> calculation, not this latter equality for every possible solution.
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