 Call price in an arithmetic stock model with interest

For the arithmetic stock model with constant volatility, let $\sigma >0$, $\tau =T-t>0$, $a=K{e}^{-r\tau }$, andThus $\nu =\sigma \sqrt{(1-e^{-2r\tau })/(2r)}$ for $r\ne 0$, and $\nu =\sigma \sqrt{\tau }$ for $r=0$. With $\Phi ,\varphi$ the standard normal distribution function and standard normal density, the European call option price isIndeed, the discounted terminal price is conditionally Gaussian with mean $s$ and variance ${\nu }^2$. Integrating its positive part above the discounted strike price gives this expression. At maturity it reduces continuously to $(s-K{)}^{+}$.

Differentiation gives option delta $C_s=\Phi (d)$. Holding $C_s$ stock units and $(C-s{C}_s)/B$ bank-account units gives a self-financing portfolio replicating the financial payoff, under the usual stochastic-integrability conditions. Its nonnegative wealth is admissible, and the discounted-wealth supermartingale bound shows that this is the least replication capital among nonnegative-wealth strategies.