# 連續三角洲對沖公式

How does the formula look like in terms of $\sigma_r$? What happens if $\sigma_r =0$? $> \sigma_i$? $< \sigma_i$?

## 最佳答案

\開始{}方程 \ mathrm {d} \ Pi_t = \ mathrm {d} V_t ^ {（i）} - \ Delta_t ^ {（i）} \ mathrm {d} S_t - r \ left（V_t ^ {（i）} - \ Delta_t ^ {（i）} S_t \ right）\ mathrm {d} t。 \ {端方程}

\開始{}方程 \ mathrm {d} V_t ^ {（i）} = \ frac {\ partial V ^ {（i）}} {\ partial t} \ mathrm {d} t + \ underbrace {\ frac {\ partial V ^ {（ i）}} {\ partial S}} _ {= \ Delta ^ {（i）}} \ mathrm {d} S_t + \ frac {1} {2} \ underbrace {\ frac {\ partial ^ 2 V ^ { （i）}} {\ partial S}} _ {= \ Gamma ^ {（i）}} \ mathrm {d} \ langle S \ rangle_t， \ {端方程}

\開始{}方程 \ mathrm {d} \ Pi_t = \ left（\ frac {\ partial V ^ {（i）}} {\ partial t} + \ frac {1} {2} \ sigma _ {（r）} ^ 2 S_t ^ 2 \ frac {\ partial ^ 2 V ^ {（i）}} {\ partial S ^ 2} - r \ left（V_t ^ {（i）} - \ Delta_t ^ {（i）} S_t \ right）\ right） \ mathrm {d}噸。 \ {端方程}

\開始{}方程 \ frac {\ partial V ^ {（i）}} {\ partial t} + r S_t \ underbrace {\ frac {\ partial V ^ {（i）}} {\ partial S}} _ {= \ Delta ^ { （i）}} + \ frac {1} {2} \ sigma _ {（i）} ^ 2 S_t ^ 2 \ underbrace {\ frac {\ partial ^ 2 V ^ {（i）}} {\ partial S ^ 2 } {_ \ = Gamma ^ {（i）}} - r V ^ {（i）} = 0 \ {端方程}

\開始{}方程 \ mathrm {d} \ Pi_t = \ frac {1} {2} \ left（\ sigma _ {（r）} ^ 2 - \ sigma _ {（i）} ^ 2 \ right）S_t ^ 2 \ Gamma ^ {（i ）} \ mathrm {d} t。 \ {端方程}

<�強>參考</強>

Ahmad, Riaz and Paul Wilmott (2005) "Which Free Lunch Would You Like Today, Sir? Delta Hedging, Volatility Arbitrage and Optimal Portfolios," Wilmott Magazine, available here

Carr, Peter (2005) "FAQs in Option Pricing Theory", Working Paper, available here

Wilmott，Paul（2006） Paul Wilmott on Quantitative Finance ，Vol。 1：Wiley，第2版