Replicating European Payoffs with Vanilla Options

Summary

The document describes a static replication result for a European payoff using calls and puts across a range of strikes. For a twice continuously differentiable payoff function, the second derivative determines the weights applied to vanilla options. The construction also includes a cash-like payoff term and a position in the underlying, with a reference strike that can be chosen freely.

The idea links the curvature of a target payoff to a portfolio of standard options, providing a general way to represent exotic terminal payoffs. The document gives the replication formula and points readers to an interest-rate modeling reference, but supplies no worked example, market data, or numerical evidence. Its stated result is limited to smooth payoffs and European exercise; it does not explain practical issues such as discrete strike availability, transaction costs, or hedging before expiry.

Key ideas

• A smooth European payoff can be represented using vanilla calls and puts across strikes.
• The option weights are determined by the second derivative of the target payoff.
• The representation also includes terms involving the payoff and its first derivative at a chosen reference strike.
• The document cites an interest-rate modeling text but gives no worked example or empirical evidence.

Tags

# How to recreate a payoff of exotic options with vanillas? ( using density function)


# How to recreate a payoff of exotic options with vanillas? ( using density function)












How to replicate exotic option (maybe hybrid) using vanillas?

I have heard once that the idea is to find a derivative wrt the strike of the option value, so that to get a CDF of some distribution, and by having second derivative wrt the strike to get a PDF which in turn somehow helps...

- Can somebody give an idea/reference to the approach I vagually described above?

- Are there other standard approaches?

P.S. I do not know any book telling about it.. But I found a post here, by nicolas, that probably tells the approach I am asking about.

Thank you

## Answer by Lipton (score 1, accepted)

https://quant.stackexchange.com/a/37841

Replication theorem:

For any twice-continuously differentiable $f(x)$, the value of a European option with payoff $f(\cdot)$ and expiry $T$ is the weighted integral of call and put options with weights equal to the second derivative of $f(\cdot)$:

$E(f(S(T)) = f(K^*) + f'(K)(S(0)-K^*) + \int_{-\infty}^{K^*}p(0, S(0); T, K)f''(K) dK + \int_{K^*}^{\infty}c(0, S(0); T, K)f''(K) dK$

for any $K^*$.

You can find it in Interest Rate Model by Andersen and Piterbarg.

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.