Exponential claim in a driftless square-root model ($V(t,S)=\exp \left(\frac{vS}{1-v(T-t)/2}\right)$)

In the zero-interest local volatility model $d{S}_t=\sqrt{S_t}d{W}_t^Q$, the risk-neutral pricing value of the European contingent claim $e^{v{S}_T}$, for $v>0$ and $v(T-t)<2$, isThe 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 $a^{\prime }+a^2/2=0$ and $b^{\prime }=0$, 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.