Quantitative Finance Stack Exchange is a question and answer site for finance professionals and academics. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|improve this question
up vote 2 down vote accepted

$u$ 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.

share|improve this answer
thank you! got it! – Rads Dec 13 '13 at 19:22
@rads: please accept this answer if it solved your problem. – Joshua Ulrich Jan 2 '14 at 17:42

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.