# Methods for pricing options

I'm looking at doing some research drawing comparisons between various methods of approaching option pricing. I'm aware of the Monte Carlo simulation for option pricing, Black-Scholes, and that dynamic programming has been used too.

Are there any key (or classical) methods that I'm missing? Any new innovations? Recommended reading or names would be very much appreciated.

Edit: Additionally, what are the standard methods and approaches for assessing/analysing a model or pricing approach?

• See previous questions: Are there any new Option pricing models? and Option pricing before Black-Scholes. – chrisaycock Mar 11 '11 at 20:51
• Ah excellent, I spotted the latter but missed the former (somehow) thanks! Do you know of any good reads for measuring or assessing how an option is priced? (should probably add this bit to my question) – amr Mar 11 '11 at 21:18
• I don't really understand the question. To my mind, the question of which model you use (Black-Scholes / Bachelier / local-volatility / Hull-White / etc.) is orthogonal to the numeric technique you use to implement (solve) the pricing (analytic formula / Monte-Carlo / *nomial tree / finite-difference (pde) / etc.) – SetTheorist Mar 11 '11 at 22:33
• I think this is partly due to my limited/non-existent understanding of how pricing works - I was hoping to gain a greater grasp of how it all works I suppose. Your comments are very helpful though, thank you! – amr Mar 12 '11 at 0:05

There are a wide variety of models (by which I mean the theoretical / mathematical formulation of how the underlying financial variable(s) of interest behave). The most popular ones differ depending on the asset class under consideration (though some are mathematically the same and named differently). Some examples are:

• Black-Scholes / Black / Garman-Kohlhagan
• Local-volatility [aka Dupire model]
• Stochastic-volatility - a generic term for extensions of Black-Scholes where there is a second stochastic factor driving the volatility of the spot; examples are Heston, SABR
• Levy processes (usually actually log-Levy): a wide class of models with some features that make them theoretically / technically nice; examples are VG, CGMY
• jumps (often compound Poisson) of various kinds can be added to the above models, for example Merton model is Black-Scholes with jumps
• CIR, OU processes show up in fixed-income
• There are multi-factor (ie multiple driving Brownian motions) versions of the above; e.g. Libor market model, correlated log-normal models
• Models for pricing, say, credit-default swaps are often Poisson processes with random hazard rates
• Commodities such as electricity can require specialized models to handle the particular features of that market
• etc.

Implementation methodologies can include:

• analytic formulae (usually involving special functions): examples are the classic case of European-style vanilla options in Black-Scholes / CEV / VG; many exotics in black-scholes can be "solved" in this way
• approximate analytic - for example, one might price average-rate options in Black-Scholes by approximating the final distribution (by moment-matching) with a shifted-lognormal and using the closed-form for the shifted-lognormal
• Binomial / trinomial trees can be viewed as a discretization technique for approximating, say, Black-Scholes. (Note that some people might view the approximation as a model in its own right --- a conflation of model & implementation and more of a philosophical stance than a practical consideration.)
• Numerical methods for solving or approximating the PDE governing the option price; this could be solved by finite-difference methods, finite-element methods, etc.
• Monte-carlo is a nice brute-force way to handle almost any kind of model and most options (though there are complications with early-exercise style features of options), but it typically takes a lot of computing power to get any accuracy in the price
• Interpolation could be viewed as a technique --- if you know the price of a collection of options (varying in some parameters) you can price a new option by interpolating based on the parameters (volatility surfaces implemented by interpolating a grid of given options are examples of this)
• etc.
• Superb answer, and exactly what I was looking for - very much appreciate the in depth reply! Thanks :) – amr Mar 17 '11 at 23:11
• @amr: don't hesitate to mark then as an answer then... – SRKX Apr 24 '11 at 22:01
• @JSmaga I did, long ago :) – amr May 2 '11 at 10:44

I would also look into pricing models based upon models other than lognormal (Black-Scholes). Do some research on "fat tailed" or stable distributions. There can also be known by their specific distribution names as Levy, Levy-Poisson, or Cauchy.

http://en.wikipedia.org/wiki/Fat_tail

There are two more methods that i am aware of from my academic curriculum. I am not sure if they are applicable to the real world but you can read up on them. Method 1: Binomial Valuation; Method 2: Risk Neutral Valuation; Both of the methods are fairly easy to implement(in terms of writing a program for it or simply, using excel spreadsheets)

Fourier Transform seems a good method for option pricing by take advantage of Fast Fourier Transform technique, such as the following paper written by Peter Carr and Dilip B. Madan: