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Replicating a Shorter-Dated Option with a Longer-Dated Option

Article Quant Q&A · Author: actinidia
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Summary

The note asks whether an option expiring sooner can be replicated using a traded option with a later expiry. It sets out a theoretical construction using the underlying asset, a money-market account, and two option values. Each option’s dynamics are first expressed through its underlying and cash account exposures. Eliminating the underlying exposure between those relationships produces a hedge in which the shorter-dated option is represented using the longer-dated option and the money-market account.

The result is a theoretical replication argument, not a demonstration that a discounted short-dated contract can be profitably arbitraged in practice. The hedge depends on option sensitivities and assumes the stated dynamics and trading relationships are valid. The note gives no worked numerical example and does not discuss transaction costs, liquidity, discrete rebalancing, model error, or whether suitable pricing and trading access are available.

Key ideas

  • The shorter-dated option can be related to a longer-dated option by eliminating exposure to the underlying.
  • The resulting replication uses the longer-dated option together with the money-market account.
  • The hedge weights depend on the options’ sensitivities to the underlying asset.
  • The argument is theoretical and does not establish a practical arbitrage after costs or model limitations.

Tags

OptionsDerivatives pricingArbitrage
Full text
# Can I replicate an option with time to expiry $t$ by trading in another with expiry $T > t$?


# Can I replicate an option with time to expiry $t$ by trading in another with expiry $T > t$?












Suppose there's a salesman who will always sell me an option expiring in two weeks. His options trade at a steep discount, but I can't directly arb it because the closest exchange-traded contract expires in 4 weeks. Is there anything I can do, other than waiting for two weeks?

Generally, is there any way to replicate an option with time to expiry $t$ by trading in another with expiry $T > t$?

## Answer by Kurt G. (score 2, accepted)

https://quant.stackexchange.com/a/72107

From a theoretical point of view there is no reason to believe that this might not be possible:

Let $B_t=e^{rt}$ be the money market account. For expiries $T_1<T_2\,,$ let $C_i(t,S_t)$ be the value of the option with expiry $T_i$ at time $t$. We know that \begin{align} dC_1(t,S_{t})&=\partial_SC_1(t,S_t)\,dS_t+\frac{C_1(t,S_t)-\partial_SC_1(t,S_t)}{B_t}\,dB_t\\ dC_2(t,S_{t})&=\partial_SC_2(t,S_t)\,dS_t+\frac{C_2(t,S_t)-\partial_SC_2(t,S_t)}{B_t}\,dB_t\\ \end{align} which says how the options $C_1,C_2$ are traditionally replicated by trading in the underlying $S_t\,.$

The above system of equations allows to eliminate $dS_t$ which gives

\begin{align} dC_1&=\frac{\partial_SC_1}{\partial_SC_2}\,dC_2-\frac{\partial_SC_1}{\partial_SC_2}\frac{C_2-\partial_SC_2}{B_t}\,dB_t+\frac{C_1-\partial_SC_1}{B_t}\,dB_t\,. \end{align} This shows how $C_1$ can be replicated by trading in $C_2\,.$

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.