Can I Reverse Engineer a Partial Differential Equation to get at the Model and Contract?

Short answer

Very often you can. You just need to understand what all the terms in a financial partial differential equation represent, all those 85u^ terms and their coefficients. And the final condition for the PDE defines the contract's payoff.

Example

In a typical equation you will see a V, representing 'value.' What is the coefficient in front of it? If it's r + p where r is a risk-free interest rate then the p probably represents a risk of default, so you are dealing with some contract that has the possibility of default.

Long answer

The first thing to look out for is how many independent variables and how many dimensions are there? And what are those variables?

There'll usually be a variable t representing time. The other variables can represent financial quantities that are either traded or not. Usually the symbol will give a clue. If one variable is S that might mean stock price, so you've got an equity option, or it might be an r, so you've got a fixed-income contract, or you might have both, so you've got a hybrid. (I could imagine an interviewer giving an interviewee a differential equation for deconstruction, but one in which the symbols are thoroughly mixed up to cause confusion. Don't worry, the tricks below will help you spot this!)

There are several different types of terms that you see in the financial partial differential equations used for valuing derivatives. There'll usually be a V for value, there'll be first derivatives of V with respect to various variables and there'll be second derivatives of V. There may be a term independent of V. Let's deal with each one of these.

Term independent of V

Also called the 'source term' or the 'right-hand side.' This represents a cashflow. The contract will be earning, or paying, money throughout its life. If the term is a function of S then the cashflow is S dependent, if it's a function of t then it's time dependent. Or, if the contract has default risk, see next example, the term may represent a recovery amount.

The V term

This is the present-valuing term, so its coefficient is usually r, the risk-free interest rate. If it's r + p then the p is the probability of default, i.e. in a time step dt there is a probability pd to f default.

First-derivative terms

These can represent all sorts of things. There's always a |j (unless the contract is perpetual and time homogeneous, so no expiration).

Suppose there's a term like |p but there's no |^ term. In this case the H quantity is not random, it may represent a history-dependent term in a path-dependent option. Its coefficient will represent the quantity that is being sampled over the option's life and the payoff will probably be a function of H.

Suppose there's a term like |^ and also a ^ term (although not necessarily using the symbol 'S,' of course). That means that the quantity S is stochastic. The question then becomes is this stochastic quantity traded, like an equity, or not, like volatility. We can tell this by looking at the coefficient of the first derivative. If it's rS (or (r + p)S for a contract with credit risk) then the 'S' is traded. If it's zero then the S may be a futures contract. If it's neither of these then it's not traded, it's just some other financial quantity that is being modelled, such as volatility or an interest rate. (Note that there's a subtlety here, technically the interest rate r is not traded, fixed-income products are functions of r, and it's those functions that are traded.) Take a look at the FAQ on the Market Price of Risk (page 208) to see how the coefficient of the first derivative change depending on the nature of the independent variable.

Second-derivative terms

These terms are associated with randomness, variables that are random. If you see a term then you can figure out the amount of randomness (i.e. the coefficient of the dX in its stochastic differential equation) by taking what's in front of the second derivative, multiplying it by two and then taking the square root. That's why if it's an equity option you'll see <r2S2.

If there is more than one stochastic variable then you should see a cross-derivative term, for example This is because there will typically be a correlation between two random variables. The correlation is backed out by taking the coefficient of this cross-derivative term and dividing by both the amounts of randomness (the coefficients of the two Wiener process terms) in the stochastic differential equations that you've calculated just above.

Other terms?

If you see any non-linear terms (e.g. something depending on the value V that is squared, or its absolute value) then you are dealing with a very sophisticated model! (And if you are in a bank then it's a cutting-edge bank!)

Now you've seen how easy it is to reverse engineer the partial differential equation you'll realize that it is equally easy to write down a partial differential for many contracts without having to go through all the hoops of a 'proper derivation.'

Remember, approached correctly quantitative finance is the easiest real-world application of mathematics!

References and Further Reading

Wilmott, P 2006 Paul Wilmott on Quantitative Finance, second edition. John Wiley & Sons Ltd

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