= Exponential claim in a driftless square-root model
{title2=$V(t,S)=\exp\!\left(\frac{vS}{1-v(T-t)/2}\right)$}
In the zero-interest <local volatility model> $dS_t=\sqrt{S_t}\,dW_t^Q$, the <risk-neutral pricing> value of the <European contingent claim> $e^{vS_T}$, for $v>0$ and $v(T-t)<2$, is
$$
V(t,S_t)=\exp\left(\frac{vS_t}{1-v(T-t)/2}\right).
$$
The <compound Poisson transition law of a driftless square-root diffusion> proves finiteness and this conditional-expectation formula. Alternatively, substituting $V(t,S)=e^{a(t)S+b(t)}$ into the <pricing equation for a local volatility model> gives $a'+a^2/2=0$ and $b'=0$, with terminal values $a(T)=v,b(T)=0$. Hence $a(t)=v/[1-v(T-t)/2]$ and $b(t)=0$. The conditional-expectation calculation verifies that this formal solution is the price of the unbounded claim. The <delta hedge> holds $a(t)V$ stocks and $(1-a(t)S_t)V/B_t$ bank units.
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