funny.md

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title: Something Funny author: Keith A. Lewis institution: KALX, LLC email: kal@kalx.net classoption: fleqn abstract: Path dependent volatility. ...

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Thanks to Bill Goff, Ioanis Karatzis, and Jesper Andreasen for giving feedback that helped improve the exposition, hopefully.

Consider a stochastic volatility model of stock price $(S_t){t\ge0}$ satisfying $dS_t/S_t = r,dt + \Sigma_t,dB_t$ where $r$ is constant, $B_t$ is standard Brownian motion, and $(\Sigma_t){t\ge0}$ is an Ito process.

A first guess at path-dependent volatility $\Sigma_t$ might be $\Sigma^2_t = (1/t)\int_0^t (dS_s/S_s)^2 = (1/t)\int_0^t \Sigma^2_s,ds$, the average realized variance.

Exercise. Show $\Sigma_t^2$ is constant.

Hint: Compute $d(t\Sigma^2)$ two ways.

Solution

Clearly $d(t\Sigma^2) = \Sigma^2\,dt$. We also have $d(t\Sigma^2) = t\,d\Sigma^2 + \Sigma^2\,dt$ so $d\Sigma^2 = 0$.

Consider the discrete time version where $\Sigma^2_n = 1/(t_n - t_0)\sum_{0\le j < n} \Sigma^2_j (t_{j+1} - t_j)$.

Since $\Sigma^2_1 = 1/(t_1 - t_0)\Sigma^2_0(t_1 - t_0)$ we have $\Sigma_1 = \Sigma_0$. Rinse and repeat until you stop laughing.