Call price in an arithmetic stock model with interest ($C=(s-K{e}^{-r\tau })\Phi (d)+\nu \varphi (d)$)

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.

For positive volatility $\sigma >0$ and positive remaining maturity, the call price in an arithmetic stock model with interest has option delta $\Delta =\Phi (d)$, where $d$ is the standardized discounted moneyness. Since a finite argument of the standard normal distribution function gives a value strictly between zero and one,For a fixed spot $s\ne K$, the maturity limit is $1_{\{s>K\}}$, the derivative of the call financial payoff away from its kink. The payoff derivative is undefined at $s=K$. Under the nondegenerate Gaussian terminal law, the event $S_T=K$ has probability zero; it is this strike event that is exceptional. At zero interest use the same formula with normal scale $\sigma \sqrt{T-t}$.