Replicating Exotic Payoffs with Vanilla Options and Optimization

# Software for decomposing payoff diagrams into plain vanilla products


# Software for decomposing payoff diagrams into plain vanilla products












Nowadays structured products (or packages) with complex payoff diagrams are omnipresent.

Do you know of any software, add-ons, apps, code whatever, that enables you to enter a payoff diagram or a cashflow profile which gives you the basic building blocks like the underlying, zero coupon bonds and esp. all the option components with their different strikes to replicate this payoff?

EDIT: Because some people asked what the input of such a tool could be, have a look at this example - I am asking for a software that is able to do this kind of decomposition automatically: http://www.risklatte.com/Articles_new/Exotics/exotic_28.php

## Answer by SBF (score 8, accepted)

https://quant.stackexchange.com/a/853

I do not know such a software - but we can think about the code. There are tow points which you have to define properly:

- which assets (correspondently, payoffs) are you allowed to replicate the complicated option?

- as barrycarter has already asked - what should be the form of the input?

Further procedure should be quite easy. You are trying to find a linear combination $\lambda$ of basic assets $s_1,s2_,...$ (because in practice this is the only possibility for you to "combine" it) which fits the complex payoff $\gamma$. It's just a peace-wise affine optimization problem. Once you minimize the difference $|\lambda - \gamma|$ you have either zero (so you have found the replication formula) or smth greater than zero (which means that there is no replication formula which perfectly covers this complicated payoff).

Once you will determine the points I've mentioned - I believe I will be able to help you to solve this problem.

Edited: Let us call your payoff $P(S)$ and simple payoff functions are $P_1(S,\theta_1),P_2(S,\theta_2),...$, where $\theta$ are parameters, e.g. strike for Call or Put.

Then you would like to check if there exist $a_1,a_2,...$ such that $$ P(S) = \sum\limits_i a_i P_i(S,\theta_i). $$

You can solve this problem by defining $$ J(a,\theta) = ||P(\cdot) - \sum\limits_i a_i P_i(\cdot,\theta_i)|| $$ where you can use any norm - and in fact due to the structure of payoffs, this norm should be defined only on some finite interval $[0,S']$. Then you solve $$ J(a,\theta)\to \min $$ and if the extremum value is $0$ - you can cover your exotic payoff with simple ones, if non-zero - you cannot cover it perfectly, but the obtained values of $a,\theta$ will be optimal.

If you need more details about the solution of optimization problem -just tell me.

P.S. I think the paper you have refereed to is not correct - the payoff is not peace-wise affine while they plot it (and considered it) as peace-wise affine function.

## Answer by fabien (score 2)

https://quant.stackexchange.com/a/1099

There are already quite a lot of softwares that do that. Quite expensive however for most of them. Then it depends whether you're interested into a trading software (trade capture and stuff) or a pricing engine.

Trading softwares : murex, misys summit, calypso ... provide tools to structure deals and value them. Then they are processed front to back. Pricing engines : NumeriX, Pricing partners ... are able to define payoff scripts and value them.

disclaimer : I used to work for one of these vendors, but I don't think my answer is biased.

## Answer by nicolas (score 1)

https://quant.stackexchange.com/a/1206

the example uses an european payoff. every european payoff can be decomposed by call and put combinations. just hold for each strike a qtty = to the 2nd derivative of the payoff....

in the general case, that is non european payoff, there is not always a perfect hedge.

## Answer by spike_tt (score -1)

https://quant.stackexchange.com/a/868

This sounds like a perfect application for genetic algorithms to me.