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I am trying to figure out how to calculate the one day expected return given I have the event volatility. In his book Trading Volatility, Correlation, Term Structure and Skew, Collin Bennet (link) explains how to calculate the one day expected return given we already calculated the event volatility. However, I am not too sure how to use the $N(\sigma)$ in the formula on Page 247:

$$\operatorname{Expected daily return} = e^{\frac{\sigma^2}{2}} \left[ 2 \cdot N(\sigma) -1 \right]$$

Do I have to run simulations and calculate the expected value of the straddle?

Assuming our event volatility is 100% would someone be able to help me make sense of this formula?

The event volatility was calculated by solving for non-event vol and subtracting it from current ivol. Below is the formula from Page 180.

$$\sigma_{\operatorname{Jump}} = \sqrt{\sigma_{\operatorname{Expiry after Jump}}^2 \cdot T - \sigma_{\operatorname{Diffusive}}^2 \cdot (T-1)}$$

where $\sigma_{\operatorname{Expiry after Jump}}$ denotes the implied volatility of an option whose expiry is after the jump, $T$ is the time to the expiry after jump ($=T_1$), $\sigma_{\operatorname{Diffusive}}$ is the diffusive volatility and $\sigma_{\operatorname{Jump}}$ is the implied volatility due to the jump.

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