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

Let $\nu =\sigma \sqrt{(1-e^{-2r\tau })/(2r)}$ and $d=(s-K{e}^{-r\tau })/\nu$, where $\tau =T-t$. The displayed discounted Gaussian positive-part expectation prices a call in the arithmetic stock model with constant volatility. Its delta is $\Phi (d)$. The stock holding $C_s$ and bank-account holding $(C-s{C}_s)/B$ replicate the payoff with nonnegative wealth. A nonnegative discounted-wealth supermartingale bound proves this is the least replication capital.

The delta of the call price in an arithmetic stock model with interest is the normal distribution function at its standardized discounted moneyness. It lies strictly between zero and one before maturity. Its maturity limit is the call payoff derivative away from the strike, an exceptional event of probability zero under the Gaussian pricing law.