= Call price in an arithmetic stock model with interest
{title2=$C=(s-Ke^{-r\tau})\Phi(d)+\nu\phi(d)$}
For the <arithmetic stock model with constant volatility>, let $\sigma>0$, $\tau=T-t>0$, $a=Ke^{-r\tau}$, and
$$
\nu^2=\sigma^2\int_0^\tau e^{-2ru}\,du,\qquad d=\frac{s-a}{\nu}.
$$
Thus $\nu=\sigma\sqrt{(1-e^{-2r\tau})/(2r)}$ for $r\ne0$, and $\nu=\sigma\sqrt\tau$ for $r=0$. With $\Phi,\phi$ the <standard normal distribution function> and <standard normal density>, the <European call option> price is
$$
C=(s-Ke^{-r\tau})\Phi(d)+\nu\phi(d).
$$
Indeed, 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 $(C-sC_s)/B$ 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.
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