Options should play an important role in asset allocation. They allow for kernel spanning and provide access to additional (priced) risk factors such as stochastic volatility and negative jumps. Unfortunately, the traditional methods of asset allocation (such as mean-variance optimization) are not adequate for considering options because the distribution of returns is non-normal, and the short sample of option returns available makes it difficult to estimate the distribution.

  • A mean-variance result is presented for portfolios containing vanilla options. The main result is an analytic formula for the optimal payoff function f (S) which maximises the Sharpe ratio for a position in a single stock S. The main inputs to the formula are the risk-neutral probability distribution .RN (S) which is used in option pricing, and the real-world probability distribution .RW (S) associated with Markowitz mean-variance. The result is completely general and applies to any probabilty distributions for S, not just normal or log-normal distributions.
  • An equivalent Markowitz style formula which involves options directly is also derived. This shows how to convert the optimal payoff function f (S) into a practical option position.
  • Taking the S&P 500 index as an example with data from July 2007, it is shown that the payoff which maximises the Sharpe ratio can be well approximated by a position in the index itself together with a long position in a 15% delta put option and a short position in a 15% delta call option. This provides downside protection on the index, as well as giving a higher Sharpe ratio.
  • Going beyond mean and variance, it is shown that analytic formulas can also be found when an aversion to the skewness or the kurtosis of .RW (S) is incorporated into the framework. However a consideration of the probability distribution of the optimal payoff suggests that optimising moments of .RW (S) may not be appropriate when considering options. Nonetheless, the analytic formula produced by mean-variance does behave well in the example cases considered.

INTRODUCTION


Traditional mean-variance applies to portfolios of assets where the returns are driven by separate stochastic processes which are usually correlated. Put and call options on an asset, however, are derivative assets whose values are driven by the same stochastic process that drives the underlying asset. This article extends basic Markowitz mean-variance to take account of these kinds of assets in the portfolio selection process.
The layout of this article is as follows: Section 2 is a short recap of basic Markowitz mean-variance. Section 3 discusses the properties of the two probability distributions .RN (S) and .RW (S) which are required for this work. The main result is then derived in section 4, followed by a Markowitz-style discrete approximation to the result in Section 5. Although the risk measure used here is variance, it will be seen that the result is sensitive to the entire probability distribution, not just the second central moment. The results are illustrated in section 6, .rst for log-normal distributions, and then for another couple of distributions including an example taken from the S&P500 index. Section 7 discusses issues beyond the mean-variance result of section 4. To show that an analytic formula is possible when considering an aversion to some higher moments of .RW (S), section 7.1 re-derives the main result when the investor has an aversion to the skewness of .RW (S) as well as the variance. Section 7.2, however, discusses issues raised by a consideration of the probability distribution of the optimal payoff f (S), which raises doubts about the main result of section 4. Nonetheless, the main result seems to have all the right properties, and this is discussed in the conclusion in section 8. The appendix starting on page 18 gives explicit formulas for the results of sections 4 and 5 for the cases of normal and log-normal distributions.

BASIC MARKOWITZ MEAN-VARIANCE

The mean-variance portfolio optimisation technique developed by Markowitz[1] calculates the optimal weights for assets in a portfolio by assuming that the asset returns Ri are jointly normally distributed. To be explicit, parameterise the distribution of Ri by the expected excess returns .i and the covariance matrix .ij so that


where rf is the risk-free simple interest rate. Using matrix notation, let . be the column vector of .i, and let . be the square matrix .ij . To derive meanvariance, look for w to maximise the quadratic utility function


where . is the scalar risk-tolerance parameter. In the absence of any constraints, U (w) is maximised by choosing
w =

This article will replicate this analysis for the case where the investable assets are a single stock together with all European put and call options on that stock which expire at the investment horizon.

MODELLING THE OUTCOMES FOR A SINGLE STOCK

The mean-variance result derived below depends on the relationship between two probability distributions for the possible stock prices S at the investment horizon:
  • The risk-neutral probability distribution .RN (S), which is the basis for option pricing at the start of the investment period;
  • The real-world probability distribution .RW (S), which is the basis for meanvariance asset allocation.

The risk-neutral probability distribution .RN (S)
Put and call options which expire on the investment horizon, where the underlying is the stock S, are priced using the risk-neutral probability distribution .RN (S). One of the key features of .RN (S) is that the mean of the distribution is the forward stock price F. Assuming that the stock pays no dividends during the investment period, F can be calculated directly from the stock price S0 at the start of the investment period and the risk free rate rf , so that


where . is the time from the start to the end of the investment period. Note that simple interest rates are being used here, to keep compatibility with the results of section 2.

Write the payoff for call options and put options at strike K using the the following notation

Call option payoff =  (5)
Put option payoff =  (6)

where . (x) is the Heaviside step function de.fined by
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