Pricing a best of

PanzerMeyer · Post by **PanzerMeyer** » Fri Dec 17, 2004 10:41 am

Hello, since this end of Dec is cooler, I have some time to learn a few things... I d like to price a best of between 2 indices. Mat 1 year, European Call.

so I guess the idea is to make a MC simulation, but where does the correlation gets into the scene ?

Johnny · Post by **Johnny** » Fri Dec 17, 2004 11:57 am

A common way of generating correlated share price paths is to use Cholesky. Any good text book will tell you everything you need. Or a Google search. Or that other place.

Veegan · Post by **Veegan** » Fri Dec 17, 2004 1:08 pm

If you are simulating each index as a single variable then the procedure is the following:

1) Draw two independent N(0,1) variables: X1 and X2.

2) Construct two new correlated random variables Y1 and Y2 as:

    Y1 = p*X1          Y2 = p*X1 + X2*(1-p^2)^0.5

    Where p is the correlation.

3) Use Y1 and Y2 in your MC simulation for each index.

This gives Y1 and Y2 correlation p, mean 0 and variance 1. As Johnny says this is Cholesky decomposition (in 2 dimensions).

If you want to simulate your indices as baskets of correlated underlyings the principle is the same - you generate an (nx1) vector X of independent N(0,1) random variables and multiply the vector by the lower-triangular matrix A:

    Y = A * X

The matrix A is produced by the cholesky decomposition of the correlation matrix for your underlyings. It is easily performed in MATLAB using the 'chol()' function, or I think I have some VBA code for it somewhere if you want to go down this route.

V.

Veegan · Post by **Veegan** » Fri Dec 17, 2004 1:10 pm

BTW, some analytical formulae for checking your MC output can be found here.

dadeto · Post by **dadeto** » Fri Dec 17, 2004 2:02 pm

"Y1 = p*X1

Where p is the correlation.

This gives Y1 and Y2 correlation p, mean 0 and variance 1"

ahem ahem...

Veegan · Post by **Veegan** » Fri Dec 17, 2004 2:24 pm

Oops. Typo. It should be:

Y1 = X1.

Thanks Dadeto.

PanzerMeyer · Post by **PanzerMeyer** » Wed Dec 29, 2004 3:10 pm

I was reading the Global Derivatives tutorial on Best of option and I was wondering what was the [b]Bivariate Cumulative normal distribution[/b] ?

PanzerMeyer · Post by **PanzerMeyer** » Wed Dec 29, 2004 3:53 pm

Okay I found out in Hull, in the annex

In fact I don't remember if I can include in VBA a function in a function, or do I have to create one ?

I need to include in the main function (Cumul_biva_Norm) the functon F(X,Y)... any hints ?

---

Function Cumul_biva_Norm(a, b, Rho)

Dim A1, A2, A3, A4, B1, B2, B3, B4, Rho As Single

Dim a, b, a_1, b_1 As Double

Dim i, j As Integer

A1 = 0.325303

A2 = 0.4211071

A3 = 0.1334425

A4 = 0.006374323

B1 = 0.1337764

B2 = 0.6243247

B3 = 1.3425378

B4 = 2.2626645

a_1 = a / ((2 * (1 - Rho ^ 2)) ^ 0.5)

b_1 = b / ((2 * (1 - Rho ^ 2)) ^ 0.5)

F(x,y) = Exp(a_1 * (2 * x - a_1) + b_1 * (2 * y - b_1) + 2 * Rho * (x - a_1) * (y - b_1))

For i = 1 To 4

    For j = 1 To 4

        Cumul_biva_Norm = (((1 - Rho ^ 2) ^ 0.5) / Pi)*f(X,Y)

    Next j

Next i

End Function

baghead · Post by **baghead** » Wed Dec 29, 2004 8:52 pm

when I tested hull, nelken & white's equity vol 2 credit model I used this:

edit: I had to price a compound option I needed the bivariate normal dist for.

http://www.mathfinance.de/FF/vblib.html