# 零波動期權定價

$$dS = \ mu S dt + \ sigma S dW，$$其中$$\ mu> 0，\ sigma> 0$$是固定常數，而$$dW$$是維納處理。要為該基礎期權定價，您需要使用德爾塔套期保值或風險中性定價的整個過程來獲取BS方程，並最終獲得期權的價值$$V（S，t）$$。現在，我正在考慮如果波動，即$$\ sigma = 0$$，會發生什麼。我的問題是：在這種情況下，您如何定價期權？

$$S（t）= S_0e ^ {\ mut}$$

$$S_0e ^ {\ mut} -Ee ^ {-r（T-t）} \ leq C（t）\ leq S_0e ^ {\ mut}$$

$$C（t）\ equiv 0，\ \ forall E \ geq S_0e ^ {\ mu T}$$

$$\ text {Profit} _T = S_0e ^ {\ mu T}-E$$

The only missing point is that, by NA, if an asset has zero volatility, it is riskless and must therefore grow at the risk-free interest rate: $$\mu \equiv r$$ (Else, you buy the highest yielding asset and sell the lowest yielding).

In such situation, the valuation of an option is straightforward: it is the discounted payoff $$e^{-r\left(T - t\right)} \left[S_t e^{r\left(T - t\right)} - K \right]^+ = \left[S_t - Ke^{-r\left(T - t\right)} \right]^+$$.

That makes every OTM/ATM option worthless, and every ITM option worth exactly the PV of its known payoff, when the “money” is defined as $$K = S_t e^{r\left(T - t\right)}$$, the forward price.