Estimating Covariance for Simulated Stock and Call Option Returns

Summary

The document explains a practical way to estimate a covariance matrix for a portfolio containing a stock and multiple calls on that stock, with a common maturity and different strikes. Rather than relying on historical return data, simulate the stock forward to the portfolio’s chosen horizon, value the stock and options in each simulated outcome, and calculate the corresponding returns for every instrument.

Repeat the process across many simulated paths, storing each outcome as a vector of returns. The sample covariance matrix of those vectors estimates the joint return covariance for portfolio optimization. The answer gives the procedure but no numerical example or discussion of model selection, option valuation assumptions, simulation error, or how the chosen horizon affects the estimate.

Key ideas

• Simulate the underlying stock to the portfolio’s chosen horizon under a specified model.
• Value the stock and each option for every simulated outcome.
• Calculate and store the implied return vector for each simulation.
• Estimate covariance from the collection of simulated return vectors.
• The resulting estimate depends on the model and option valuation assumptions.

Tags

# Mean-Var optimisation of Monte Carlo simulated model


# Mean-Var optimisation of Monte Carlo simulated model












I have a problem which involves optimisation of a portfolio containing one stock and multiple call options written on it, with the same maturity and different strikes. In order to use optimisation technique, I need a covariance matrix between the options and the stock. Usually to calculate covariance matrix one would use a matrix of returns, but what do you do if you use Monte Carlo simulation instead of historic data and portfolio containing derivatives rather than simple stocks?

## Answer by Mark Joshi (score 2, accepted)

https://quant.stackexchange.com/a/29702

evolve the stock to the requisite time horizon using some model. Get its value and that of the options on it. Compute the returns implied by these. Store this vector.

Do this many times.

Compute the implied covariance matrix of these vectors of returns.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.