Option Portfolios

A difficult situation arises when options are written on two separate but correlated assets.
Let us assume a bank writes call options for two stocks that have a correlation of -0.7.
Due to the negative correlation it is unlikely that the call options will be exercised at the same time.
An adjustment will have to be made otherwise a zero correlation will be assumed, leading to an over-estimated VAR.

To overcome this problem of correlation between assets a procedure called Cholesky Decomposition is used
1) If correlation of +1 is required, the figures are adjusted so they both rise and fall together
2) If correlation of -1 is required, the figures are adjusted so when one goes up the other goes down
3) If correlation of 0 is required, no adjustment is made
This involves using matrix algebra

VAR - Two Asset Portfolio
Let's consider a portfolio containing the following:
100 call options on Microsoft shares
Stock price is $62
Option delta is 0.62
Stock price Volatility is 40%

200 call options on Sony shares
Stock price is $53
Option delta is 0.43
Stock price volatility is 47%

The correlation between the two stocks is 0.2

VAR = "portfolio value" x "no of standard deviations" x ( "volatility" / ("days" ^ 0.5) )

VAR - N Asset Portfolio

Calculate the value of the portfolio for each scenario.
Suppose we have the following amounts in a portfolio: E[t+1], E[t+2],,,,E[t+N]
Let F[t+1], F[t+2],,,,F[t+N] be the corresponding value of these assets
For example if E is an equity then the value would be E x share price
Then for each simulation, the value of the portfolio would be:

P = SUM(from I=1 to N) w[i] x F[i]
Where w[i] is the proportion of the asset relative to the value of the portfolio

Generate a large number of scenarios and calculate the portfolio value of each scenario
The results are reordered by the magnitude of change in the value of the portfolio for each scenario
The relevant VAR is then selected from the reordered list according to the required confidence level

If 10,000 iterations are run and the VAR at the 95% confidence level is needed, then we would expect the actual loss to exceed the VAR in 5% of cases (500), so the 501st worst value

Partial versus Full Valuation

For an options portfolio, depending on the size of the portfolio, it may be more efficient to use an approximation rather than a full option pricing model (like Black Scholes) for ease of calculation
Using a partial Monet Carlo VAR approach, the portfolio can be revalued using the delta approximation

Delta approx >

This says that the change in the value of an option is the product of the delta of the option and the change in the price of the underlying asset
There are also other approximations that use delta, gamma and theta in valuing the portfolio

Delta-gamma approx >

Delta-gamma-theta approx >

By using summary statistics such as delta and gamma the computational difficulties associated with a full valuation can be reduced

The full valuation / black schools valuation is the most precise but tends to be slower and more costly.