Let $(S_t)_{t \geq 0}$ be the price of an underlying asset, $r$ be the risk-free rate of return, $q$ the dividend yield, $C_t(K,T)$ is the price of a call option written on $S_t$ at time $t$ with strike $K$ and maturity $T$. We're also going to follow their notation and consider $C(S,T;K,T;\Theta)$ is the some pricing function for the call.

Under the assumption that the underlying model is Markovian in $S$ and $t$, they show that the following holds exactly: \begin{align} C(S,t;K,T;\Theta) = \int_0^\infty w(k) C(S,t;k,u;\Theta)dk \\ w(k) := \frac{\partial^2}{\partial k^2} C(k,u;K,T;\Theta) := \Gamma(k,u;K,T;\Theta) \end{align} where $u \in [t,T]$ is some future point in time. They essentially build on the well know Breeden and Litzenberger (1978) result which ties the risk neutral density to the discounted second derivative of $C(.)$ wrt its strike.

Now, I see that this essentially says the value of a call option at time $t$ maturing in $T-t$ periods is given by a sort of gamma-weighted portfolio of calls of shorter maturities with a variety of strikes. I also gather that the weights are going to be bigger for portfolios with strikes $k$ closer to the strike $K$ of the LHS call because the the gamma is going to peak around this point.

How would I use this to hedge an option? From what I gather from the paper, if I go short on $C_t(K,T)$, I would go long with a discretized version of the RHS. If we have $T-t = 60$ days and $u-t = 30$ days, I can hedge my short position for the first 30 days and, at time $u$, the proceeds from my portfolio of shorter maturity options could be used to close my short position on the call. In essence, this equation says the prices must be exactly the same at time $u$. Am I getting this right?

How does time dependance in volatility hurt this strategy? Why, exactly? Obviously, it violates the Markovian assumption they made on $S$ and $t$, but what is the intuition here?

  • $\begingroup$ I am not sure what your question is really, I think the secod question is important: future volatility uncertainty is the most important risk here as you are hedging longer term options using short term options. $\endgroup$ – ilovevolatility Mar 21 at 12:20
  • $\begingroup$ I see. That's a good point. $\endgroup$ – Stéphane Mar 21 at 16:42

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