BS 波动率错配的对冲误差

用交易波动率 $\sigma_t$ 做 delta 对冲，但真实实现波动率为 $\sigma_r$。

经典 tracking error（忽略融资/离散误差）主导项来自 gamma：

$$\boxed{\text{PnL 误差}\ \approx\ \frac12\int_0^T \Gamma_t\,S_t^2\bigl(\sigma_r^2-\sigma_t^2\bigr)\,dt}.$$

其中 $\Gamma_t=\partial_{SS}V(S_t,t)$。

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英文解析

This classic result was first discussed by El Karoui, Jeanblanc and Shreve (1996) and Carr and Madan (1997) and is known as the PnL tracking error formula or Black Scholes robustness formula. Let $V$ be the price of the option. It is designed at inception to verify the Black Scholes equation with $\sigma_{t}$

$$V_{t} + \frac{1}{2} V_{SS}\sigma_{t}^{2}S^{2} + rSV_{S} - rV = 0$$

On another hand, we apply Itô's lemma to $V$ , which evolves during the life of the trade at the realized volatility $\sigma_{r}$

$$dV = V_{t}dt + V_{sdS} + \frac{1}{2} V_{ss\sigma_{r}^{2}S^{2}dt}$$

We isolate $V_{t}$ in (14), substitute it in (15) and integrate from 0 to $T$

$$V(T) - V(0) = \int_{0}^{T}rVdt - \int_{0}^{T}rSV_{Sdt} + \int_{0}^{T}V_{Sds} + \int_{0}^{T}\frac{1}{2} V_{SB}(\sigma_{r} - \sigma_{t})^{2}S^{2}dt$$

We see that the 2 first integrals correspond to the cost of holding a position in cash equal to $(V - SV_{S})$ , which is the position in cash after selling the option at $V$ and hedging it with a $V_{S}$ position in stock

$$I_{1} = \int_{0}^{T}rVdt - rSV_{S}dt$$

The third integral is the change in value of a $V_{S}$ position in stock

$$I_{2} = \int_{0}^{T}V_{S}dS$$

And we find that the difference in value of the european claim is equal to its expected hedging cost $(I_{1} + I_{2})$ plus an error term

$$\mathrm{PnL_{error}} = \int_{0}^{T}\frac{1}{2} V_{SS}(\sigma_{r} - \sigma_{t})^{2}S^{2}dt$$