Replicating a Call Plus an Earlier-Expiring Put in a CRR Model

Summary

The document explains how to replicate a payoff composed of a European call expiring at time T and a scaled European put expiring two time units earlier. In a Cox-Ross-Rubinstein binomial model, the suggested construction is to find the standard replicating portfolio for each option and combine the positions. The put portfolio is multiplied by its payoff coefficient, while the call portfolio retains its own weights.

Because the put expires before the call, its replicating positions are held only until that earlier expiry; for the final two periods, the put component is removed and the call replication continues. The response relies on the reader already knowing how to replicate ordinary calls and puts in the CRR model. It gives no tree parameters, numerical example, or derivation of the hedge ratios, so implementation still requires computing the separate option portfolios for the chosen model inputs.

Key ideas

• The combined payoff can be decomposed into a call and a scaled put.
• Replicate each option separately using the CRR model.
• Scale the put replicating positions by the coefficient on the put payoff.
• Set the put portfolio to zero after its earlier expiry while continuing the call replication.

Tags

# In the CRR model, describe the strategy replicating the payoff $X=(S_T-K)^{ +} +a(K-S_{T-2})^{+ }$ for $a \neq 0$


# In the CRR model, describe the strategy replicating the payoff $X=(S_T-K)^{ +} +a(K-S_{T-2})^{+ }$ for $a \neq 0$












> In the CRR model, describe the strategy replicating the payoff $X=(S_T-K)^{ +} +a(K-S_{T-2})^{+ }$ for $a \neq 0$

$X$ consists of two parts:

- European call option with strike price $K$ and expiration date $T$

- $a$ European put options with strike price $K$ and expiration date $T-2$

So I think I should replicate these two parts separately, but I don't know how to do that.

## Answer by Bob Jansen (score 1)

https://quant.stackexchange.com/a/75227

Do you know how to use the CRR model for a standard Call and a Put? If not, you can learn that from a textbook or this site. If you know how to:

- Find the replicating portfolio for the Call

- Find the replication portfolio for the Put

- Add the weights together, multiplying the weights of the put by $\alpha$. For the final two days, set the weight of the Put-replicating portfolio to 0.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.