PCA Coordinates and Dimensional Meaning in Basket Option Pricing
Summary
The document addresses questions that arise when applying a finite difference method to a basket option under the Black–Scholes framework. A principal component transformation rotates log-price coordinates using the eigenvectors of the covariance matrix, while a drift adjustment accounts for the risk-free rate and asset volatilities. Dimensional analysis resolves a disputed formula: volatility must be squared in the drift term so that it has the same units as the rate.
The response also clarifies what the solution variable represents. The unknown is the value of one option on a basket, so it is scalar-valued at each point in the multidimensional state space; it is not a vector with one component per underlying asset. Spatial discretization represents option values across possible transformed asset states. The discussion is conceptual and does not provide a complete numerical scheme, implementation, or convergence evidence for the BDF2 method.
Key ideas
- PCA can rotate log-price coordinates using covariance eigenvectors in basket option pricing.
- The drift adjustment combines the risk-free rate with half the squared volatility.
- Dimensional analysis distinguishes the squared-volatility formula from an expression using volatility directly.
- A single basket option has one scalar value at each point in its multidimensional underlying-price state space.
- Spatial discretization samples that state space to approximate the option value function.
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Full text
# Finite difference methods
# Finite difference methods
I am simulating the price of a basket option with the help of equations from the report http://www.it.uu.se/edu/course/homepage/projektTDB/vt07/Presentationer/Projekt3/Dimension_Reduction_for_the_Black-Scholes_Equation.pdf .. I am a beginner in using Numerical methods in Finance.. hence I am stuck at probably the most trivial questions.. This paper uses BDF2 method.. The method includes performing PCA and then translating coordinates according to the following equations.. \begin{equation} \bar{x}=\textbf{Q'} \ln(S) + b{\tau} \end{equation} where $\tau=T-t$ and $b_i= \sum_{j=1}^d q_{ij} (r- \frac{\sigma_j^2}{2})$. I saw two different formulas in two different places.. alternate one was $b_i= \sum_{j=1}^d q_{ij} (r- \frac{\sigma_j}{2})$ , and I can't understand which one is correct. By applying change of variables to the Black Scholes Equation we get, \begin{equation} \frac{\partial u}{\partial \tau} =\frac12 \sum\limits_{i=1}^d \lambda_i \frac{\partial^2 u}{\partial x_i^2}-ru \end{equation} where $(\bar{x},\tau) \in \mathbb{R}^d \times (0,T)$ and $\lambda_i$ is the eigenvalue number $i$ of the covariance matrix. The payoff for the basket option is, \begin{equation} u(\bar{x},0)=\max(\sum\limits_{i=1}^d \mu_i \exp(\sum_{j=1}^d q_{ij} x_j),0) \end{equation} where $\bar{x} \in \mathbb{R}^d$. Now my question is.. What is in the vector U? According to me the dimension of U should be no_of_underlying_assets, but then I don't understand how spatial discretization plays a role here. I understand that the initial condition of U is the contract function(eq. 12) but then, it appears as a 1 x 1 scalar to me. Am quite confused here.
## Answer by Luigi Ballabio (score 2, accepted)
https://quant.stackexchange.com/a/9698
$u$ is the value of the option, and is in fact a scalar (which, of course, is a function its several underlyings). You're studying a single option on a basket, not a basket of options.
As for the two different formulas: you can pick the correct one just by looking at the units of its terms. The rate $r$ is the inverse of a time; each volatility $\sigma_j$ is the inverse of the square root of a time (you can see that from their definitions, or from their role in the Black-Scholes formula). The formula that combines them in a meaningful way is the one with the $\left( r - \frac{\sigma_j^2}{2} \right)$ term.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.