Stochastic Differential Equations - A Few General Questions

Question

I just have a few questions about stochastic differential equations. I generally did a lot of pure math but signed up for a course on probability models and stochastic differential equations because I wanted to try something different. I have really enjoyed it and am actually seriously considering going to graduate school to study this stuff. The main issue is that we didn't do a measure theory approach to it. The professor used a lot of ideas directly from Wiener and we always took a random walk approach.

Anyway, I just wanted to ask:

a) What is the point of simulating SDEs if the solution is always different due to the randomness of the Wiener process? I have been simulating Geometric Brownian Motion and read that it is used in the Black-Scholes model in finance, so how do they actually price stocks based on the SDE?

b) What are the methods used in determining the coefficients in an SDE to calibrate it to data?

c) What is a good textbook for a very applied and computational approach to stochastic calculus with a crash course on measure theory?

Answer 1

For a) Simulating the SDE of an underlying stock price is not used to price the stock; that would yield nothing more than a self-fulfilling prophecy.

Rather, practitioners use simulation methodologies to price financial derivatives of an underlying asset. For example, the "payoff" function for a simple European call option is $\max(0,P_T - K)$, where $K$ is the "strike" price and $P_T$ is the price of the underlying asset at expiry time $T$.

An analyst would simulate the underlying price, and evaluate the payoff function at time $T$. Finally, the call option value (i.e., price) is simply the present value of the expected value of the payoff function.

For b) There are several statistical and market-based methods that practitioners use to determine the parameters of the SDE model. For market makers, the most common methodology involves the use of the term structure of the market implied volatility. Other methods include non-parametric models of the PDF of the underlying. Simpler methods include statistical estimations using historical time series data.

For c)

Darrel Duffie's book Dynamic Asset Pricing Theory might be a good one.

Answer 2

I do not think you can get a solid technical understanding of SDE's without knowing some measure theory - it is worth learning the basics of measure theory (for example, you should be able to comfortably talk about dominated convergence ).

Additionally a touch of functional analysis will not hurt - the Lebesgue spaces are quite crucial (even if not apparent at first sight) in the theory of SDE's and for financial applications.

To add to Dr. Mv's (excellent) answer:

A. It depends on what you mean by 'different'. A reasonable approximation to a SDE, such as the Euler approximation, has desired convergent properties: it is basically the discretised version of an SDE. In a sense, it is not so different. Naturally, some caveats arise: but these are very deep issues that rarely come to play in practice.

B. Least squares optimisation is one method. Another method is to estimate parameters by using historical market data. Another method is to randomise the initial parameters and then use a Monte Carlo scheme to get the parameters such that an implied surface is satisfied.

C. Stefano Iacus - Simulation and Inference for Stochastic Differential Equations. It has explicit examples in the R programming language, which is free to download.