Separating Option Pricing Models from Numerical Methods

# Preferred Option pricing model


# Preferred Option pricing model












I am at Uni studying mathematical finance and wanted to know which is most preferred /widely used model by Finance Industry Practitioners from the list below.

- Fourier Transform for option pricing

- Jump Diffusion model for option pricing

- Constant Elasticity of variance for option pricing

- Option pricing through alternating direction implicit (ADI) Method.

In essence, which model should I focus to successfully transition to a Quant Role in the future

## Answer by THATS MY QUANT MY QUANTITATIVE (score 11, accepted)

https://quant.stackexchange.com/a/79357

General Comment: In industry, you're effectively an engineer/mechanic. You choose the best tool for the job, and there is no 1 tool that works with everything because they all have different benefits and drawbacks. Tools that have high accuracy will have high computational cost (Monte-Carlo), whilst tools that are faster can be less accurate or have stability issues (PDEs and its Fourier Transform). Industry practitioners working with options will use many different methods, and if you take graduate classes, you will be introduced to these different methods.

To your specific question, the models you have listed do not make sense in that methods != models. A jump diffusion model is a model that models the diffusion of a stock price, i.e. tells you how the stock will behave across time. Whilst the ADI method is a method to solve PDEs and is not a stock diffusion model.

The general gist of option pricing is as follows:

- Decide the type of option we are pricing, lets say a European call option, and it will have some type of equation in the form $C(S,T) = \text{max}(0,S_T - K)$. (Key point here is that our price is determined on $S$ at time $T$, so we need a distribution of $S$.

- Assume some type of diffusion for the stock price, $S_t$, which will have a stochastic differential equation (SDE). This could be models like constant volatility, jump diffusion, stochastic volatility, local (deterministic) volatility, stochastic interest rate etc.

- Then for the chosen $S_t$ model and the specific option pricing equation, we use a type of solving method. I.e. Monte-Carlo, PDE, Fourier Transform etc.

For the option equation:

$$C = \text{max}(0, S_T - K) \quad\quad(1)$$

($K$ is strike), we require some type of diffusion of the stock price, $S_t$, let's say geometric brownian motion:

$$dS_t = rS_tdt + \sigma S_tdW_t \quad\quad(2)$$

But it will also have a PDE interpretation:

$$ \frac{\partial C}{\partial t}+\frac{1}{2} \sigma^2 S^2 \frac{\partial^2 C}{\partial S^2}+r S \frac{\partial C}{\partial S}-r C=0\quad\quad(3) $$

(You can arrive at this PDE without the probabilistic interpretation (using the SDE) but lets ignore that for now).

We can do 100,00 simulations simulate the stock price, plug them into (1), take the average and that is the price of the option. But we can we also instead use equation (2) and solve the PDE using ADI (or any other PDE solving method, i.e. explicit method). But we can also transform the PDE using the Fourier Transform for a computationally quicker solution. And finally, you can take it a step further and solve the PDE analytically for the analytical solution:

$$ \begin{aligned} C\left(S_t, t\right) & =N\left(d_{+}\right) S_t-N\left(d_{-}\right) K e^{-r(T-t)} \\ d_{+} & =\frac{1}{\sigma \sqrt{T-t}}\left[\ln \left(\frac{S_t}{K}\right)+\left(r+\frac{\sigma^2}{2}\right)(T-t)\right] \\ d_{-} & =d_{+}-\sigma \sqrt{T-t} \end{aligned} (4) $$

The key point: The method that is used is determined by the diffusion of the stock you choose, the tools and data you have readily available, etc. If we choose a more complicated diffusion of $S_t$, the analytical solution (4), may not exist, or the PDE may be unstable or hard to deal with which forces us to use Monte-Carlo. You don't focus on 1 model, or 1 method as they all have their place.

2 simplistic examples:

- You are working at a bank that has some SPY ETFs and they want to write (sell) some barrier options. They have plenty of time and want the most accurate price, so they can model the diffusion of the stock with more complicated features, such as the stochastic-local volatility model, which is typically used for path-dependent derivatives. Since they have a lot of time, they can implement a monte carlo simulation and do millions of simulations to come to a near-exact solution for the option given their parameters.

- You are working on an option's trading desk that is constantly evaluating the price of the options for the stock they have in inventory. You need a method that is fast, but you don’t necessarily need an exact solution, so their team opts-in to use PDE methods for their pricing tools.