Delta-Hedged Option P&L from Implied and Realized Volatility

Summary

The document derives the instantaneous profit and loss of an option position that is delta hedged while valued using implied volatility. It applies Itô’s formula to the option value using the underlying’s actual volatility, then combines that expression with the Black–Scholes pricing relation, which uses the implied volatility. The resulting P&L is proportional to gamma and the difference between actual and implied variance.

The accompanying question asks why equations based on different volatility assumptions can be combined. The derivation separates the option’s pricing model from the process used to describe actual underlying price changes: the Black–Scholes relation defines the model Greeks and price, while Itô’s formula describes how that price changes under realized dynamics. The result is a local continuous-time relation under the stated setup. It does not account for discrete hedge timing, transaction costs, jumps, or model misspecification, so it is not a complete forecast of realized trading returns.

Key ideas

• Delta hedging removes the first-order exposure to small underlying price moves in the stated setup.
• The option’s gamma links hedged P&L to the difference between realized and implied variance.
• Itô’s formula uses actual underlying dynamics to describe changes in the option’s model value.
• The Black–Scholes equation uses implied volatility to relate the model Greeks, price, and interest rate.
• The derivation is a local continuous-time result and omits trading frictions and jumps.

Tags

# hedge with implied volatility, PnL formula


# hedge with implied volatility, PnL formula












Notations are consistent with this answer.

Selling and delta hedging the option $V^i$ using the implied volatility $\sigma_i$ while the actual volatility of the underlying asset is $\sigma_r$. Then the portfolio pnl has $$d\Pi=dV^i-\Delta^i dS-r(V^i-\Delta^i S)dt\tag{*}$$

the underlying has $$\frac{dS}{S}=\mu dt+\sigma_rdW_t$$ apply Ito formula: $$dV^i=\Theta^idt+\Delta^i dS+\frac{1}{2}\Gamma^i\sigma_r^2S^2dt\tag{1}$$

by BS equation: $$\Theta^i+\frac{1}{2}\Gamma^i\sigma^2_iS^2=r(V^i-\Delta^i S)\tag{2}$$

then plug (1)(2) into (*), then we can get $$d\Pi= \frac{1}{2} \left( \sigma_r^2 - \sigma_i^2 \right) S^2 \Gamma^i dt$$

Question:

But I find it quite confusing, since (1) holds under the assumpsion that $$\frac{dS}{S}=\mu dt+\sigma_rdW_t$$

but (2) holds under the assumption that $$\frac{dS}{S}=\mu dt+\sigma_idW_t$$

why can we plug (1)(2) into (*) simutaneously when they have different assumption?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.