Static Replication of Longer-Dated Calls with Shorter-Dated Options
Summary
The document presents a Carr and Wu option-pricing identity for a Markov model in which the underlying asset price and time describe the state. A call expiring at a later date can be represented as an integral of calls expiring at an intermediate date, weighted by the second strike derivative of the later call’s value. This weighting is related to option gamma and to the connection between call prices and risk-neutral distributions.
The proposed hedge is to offset a short longer-dated call with a discretized portfolio of shorter-dated calls, then use the portfolio’s value at the intermediate date to close the original position. The text asks whether this interpretation is correct and how volatility that changes over time affects the result, but supplies no answer or empirical test. The identity is stated as exact under the Markov assumption; its practical use therefore depends on that modeling condition and on constructing the continuum of weighted options from available contracts.
Key ideas
- Under the stated Markov assumption, a longer-dated call can be represented by weighted shorter-dated calls.
- The weights are given by the second strike derivative of the later call’s pricing function.
- A discretized weighted portfolio can serve as a static hedge for a short longer-dated call.
- The proposed strategy relies on the identity holding through the intermediate date.
- Time-dependent volatility raises a limitation because it may violate the assumed Markov state description.
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Full text
# Intuition behind the Carr and Wu (2014) static hedging for ordinary options
# Intuition behind the Carr and Wu (2014) static hedging for ordinary options
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?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.