算术布朗运动下 ATM call 定价

The Black- Scholes formula is derived assuming the stock price process $S(t)$ follows a geometric Brownian motion: $dS(t) = \mu S(t)dt + \sigma S(t)dw(t)$ , where $w(t)$ is a standard Brownian motion. Suppose instead that a stock price process $S(t)$ follows an arithmetic Brownian motion: $dS(t) = \mu dt + \sigma_A dw(t)$ . Derive the pricing formula for a call option on $S(t)$ . Please assume that the option is at- the- money [i.e., $S(t) = \chi$ ], that the riskless interest rate $r = 0$ , and that the stock pays no dividends.