# 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 $$T > t$$?

• The Heston PDE or any other stoch vol PDE does exactly what you are looking for. The problem is calculating the Greeks. If you use the pure local vol model instead then in theory delta hedging an option is sufficient, you don't need another option.
– user34971
Commented Sep 1, 2022 at 12:02
• Carr and Wu (2014) have this spanning result where you can statically hedge an option with maturity $T$ with a continuum of short-dated options (maturity $u\in(0,T)$). Follows from Breeden-Litzenberger. I don't know whether such a (mostly model-free) spanning result also exists for the opposite case that you're interested in. Commented Sep 1, 2022 at 13:39
• As mentioned by @Kevin there is the Carr-Wu result, but it is for a single factor Markovian model only, eg a Local vol model. But then imo one might just delta hedge which is in theory sufficient.
– user34971
Commented Sep 1, 2022 at 15:39
• @FridoRolloos Why would delta hedging be sufficient? Because of the difference in expirations, wouldn't the gammas be different as a function of time? Or do you mean I'd need to constantly rebalance my gammas so that they net? Commented Sep 1, 2022 at 15:45
• Because in a local vol model the only risk factor is the stock price. Carr and Wu's result will outperform if there are many jumps though. However it is about hedging longer term options with short term options, which is the opposite of your situation.
– user34971
Commented Sep 1, 2022 at 16:12

Let $$B_t=e^{rt}$$ be the money market account. For expiries $$T_1 let $$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\,.$$