# Matching Dynamic Trading Strategies with Derivatives

The famous Black-Scholes framework is usually derived using a hedging approach where a self-financing portfolio is constructed and the resulting stochastic differential equation is being solved under some conditions.

The self-financing portfolio is basically a dynamic trading strategy where according to the actual price development and the strike of the replicated option parts of the underlying are being bought or sold. For example when you want to replicate a call you would have to buy when the price rises and sell when it develops in the direction of the strike.

My question: I want to build a framework where you could match different dynamic trading strategies with derivatives. For example I would like to find the characteristics of a moving average approach in terms of a derivative, e.g. the P/L-diagram of this would in my opinion look like an covered call writing combination.

How would you do that?

First of all I guess in this example one could take the price process (e.g., a geometric Brownian motion) and e.g. convolute that with a rectangular function to model the moving average. After that one would have to find the resulting distribution density function...

But all of that is just speculation... (also quite literally ;-)

Has any one of you some ideas on how to do that - or are there even ready-to-use approaches (on the web, in articles, books, etc...)

• 1) The usual approach is to start with the derivative and derive the dynamic trading strategy from that. You seem to want to go backwards. It seems clear that any dynamic trading strategy corresponds to some derivative, but it's not clear to me what the possible types of derivatives you can get this way are. 2) A pragmatic approach is to discretize and chase down every possible path in the binomial lattice to see what the ultimate payoff is. An continuous analogue of this analysis doesn't seem out of the question to me. – Deane Yang Oct 28 '09 at 18:51

At last I found two papers as a starting point that do exactly that:

(i) You have a time-dependent p&l(I call it pay-off) from trading a portfolio of financial assets (although this doesn't really matter)

(ii) You want to find a static portfolio of assets, replicating this p&l in every state of the world (called the replicating portfolio).

This can be done very easy if:

(a) you work in discrete time, i.e. the pay-off you are trying to match is a finite vector of random variables $G_1$, ..., $G_T$

(b) the pay-off does not depend on the path of the underlying assets you are trading in.

Under these conditions it suffices to find a replicating portfolio at a single time $t$ of pay-off. The pay-off is a real-valued function $f=f(S_t)$ within some function-space and what you really want to do is to approximate $f$ by a set of basis function (i.e. the options).

Of course there are many ways to approximate $f$ as well as many possible choices for basis functions. The best approach will depend on your specific set-up. You have to decide (A) What function space you are working in: Is piecewise linear OK or will your trading strategy produce discontinuities? (B) What are your basis instruments? If you are looking for a practical solution you will be restricted to traded instruments, i.e. no far out-of-the money stuff, no digital options and so on. (C) What is your notion of error? $L^2$ is natural but maybe you have a specific utility or risk aversion? (D) What is your method of discretizing domain and range of $f$?

Anyway if there are not too many times and underlying assets involved, this might be something simple enough to do in Excel.

On the other hand if condition (b) above is not fulfilled, you are most likely in for some trouble. Path dependency creates lots of problems ("curse of dimension") and from a practical perspective you will have trouble finding sufficiently many path-dependent instruments for replication. The method of choice in that case is a regression approach. I.e. regress the pay-off from your strategy against various instruments you deem appropriate.

• @g g: Thank you. Well, not exactly: What I try to do is to find a static portfolio of derivatives to replicate a(ny) dynamic trading strategy. What you do here is - as far as I understand it - to find a static portfolio of assets to replicate a given P&L. Did I understand this correctly and do you see the difference? – vonjd Jul 18 '10 at 15:33
• @vonjd: you see correctly (replication of a given P&L) but I do not see the difference between a P&L and a dynamic trading strategy. Please, explain thank you gg – g g Jul 18 '10 at 19:41
• @g g: OK, thank you - In my opinion the question remains: How do you get from a dynamic trading strategy to a P&L? If you have an answer for that I understand your point! (BTW: There is a freely available tool that does exactly what you describe: longvega.com/FETool/feapplet.php) – vonjd Jul 19 '10 at 8:28
• Yes, nice link and a good idea to put something like this on the net. – g g Jul 19 '10 at 21:50

How to get from trading strategy to P&L? P&L might be a bit tricky since this involves a valuation at each time step. So lets focus first on cash-flows:

You start at t=0 with an initial Wealth which you invest in a portfolio of assets. You hold for one period. At t=t_1 the prices of the assets are revealed. You sell all your assets at then current prices. You might consume (positve cash-flow) or borrow (negative cash-flow) and you invest the remaining cash according to your strategy. This repeats until some point in time (maturity) when you sell off all remaining assets and deduct your debt. This is the final cash flow.

This random vector of cash-flows defines a synthetic asset. By the law of one price the vector of cash-flows in each state determines the value (=price) of this asset uniquely. It doesn't matter how you produce cash-flows, be it a trading strategy in underlying assets, a static portfolio of derivatives or a political lobbying strategy providing you with the right amount of bail-out cash at each point in time, it always has the same vector of values.

So using those values (and discounting) you can define the P&L from your trading startegy (or option portfolio or whatever) as difference of the value process.

Does that sound reasonable?