Probabilistic Outcomes of Investment Rules
and Derivative Hedging Strategies.

Wesley Phoa, Ph.D.
President of Research

Scenario analysis is useful for analyzing performance under subjective market views, which may correspond to analysts, forecasts or worrisome scenarios identified by risk managers. However, each market view must be crystallized into a single scenario, even if the original market view is compatible with an infinite number of specific outcomes. As a consequence, the analysis does not give probabilistic information. For example, it cannot answer a question such as, "What is the portfolio value-at-risk assuming a bear market occurs?" Also, the results for path-dependent strategies such as stop-loss rules or barrier option strategies are overly dependent on the precise scenario specification. For example, it cannot give a robust answer to a question such as, "How poorly does a stop-loss strategy perform when the market whipsaws?"

Monte Carlo simulation generates, not just a single outcome, but a large number of random outcomes which can be used to estimate a probability distribution. Furthermore, one can simulate the possible paths followed by security prices, rather than just their final values. Path simulation is the only way to analyze path-dependent strategies. For example, Dybvig has used simulation methods to illustrate the fact that, under a neutral market view, option strategies always dominate path-dependent trading rules in a certain sense.

Can one combine the two techniques? In its usual form, Monte Carlo simulation draws random paths from a "neutral" probability distribution on the space of paths, ignoring the subjective views of the investor or risk manager. Is it possible to take these views into account? For example, suppose one wants to compute value-at-risk under a "bear market" assumption. One way would be to carry out a Monte Carlo simulation which drew only paths under which security prices fell by at least 20%.

Formally, this would involve drawing, not from the neutral probability distribution on the space of paths, but on a conditional probability distribution. Unfortunately, except in the very simplest cases, drawing independent samples from this kind of distribution is extremely inefficient. It is only quite recently that efficient methods for carrying out this kind of simulation have been developed. These are the so-called Markov chain Monte Carlo algorithms. The results in this article are derived using Gibbs sampling, an algorithm from this family which was initially developed for use in image processing. Gibbs sampling is an order of magnitude more efficient than naïve methods, but still quite easy to implement.

The outcomes of option strategies or investment rules, under non-neutral market assumptions, have probability distributions with shapes that are far from obvious. In some cases, the general form of the distribution is clear, but there is no easy way to make quantitative estimates. In other cases, even the form of the distribution may be surprising. In general, the only way to obtain reliable probabilistic information on outcomes is via simulation.

As an example, we present a probabilistic analysis of several investment strategies applicable to an individual stock, under a variety of scenarios. The strategies are as follows:

• Invest all funds in the stock, and hold it to the horizon date.
• Invest in a combination of the stock and a put option expiring on the horizon date.
• Invest all funds in the stock, with a once-and-for-all stop loss rule at a specified level.
• Invest using a stop-loss rule, but also buying back at a specified (higher) level.

In each case a horizon of one year is used, with monthly time steps. Stop orders are assumed to be executed at the end of the month, so that considerable slippage is possible. (However, it is simple to modify this assumption about the execution of stop orders.)

The scenarios were defined as follows: "neutral" means all possible paths are allowed, with probabilities determined by the specified volatility; "was_bear" only includes paths under which the stock price fell by 20% at some stage, though it may or may not have recovered later; and "whipsaw" only includes paths under which the stock price fell by 20% at some stage and recovered to its initial value at some later stage. Note that each scenario is imprecisely specified: the conditions are satisfied by an infinite number of paths.

In each case, Gibbs sampling was used to generate random paths drawn from the conditional distribution corresponding to each scenario. The return from each investment strategy was computed for each path, and the results used to estimate the probability distribution of returns. In order to obtain reliable estimates about the tails of the distributions (e.g. value-at-risk), a very large number of samples are required: thus, in every case, one million paths were used.

Exhibit A shows maximum loss estimates, i.e. 95% quantiles of the estimated distribution of investment returns; these may be regarded as the investment manager's analog of "value-at-risk," in this simple single-security case. Under a neutral assumption on market direction, the put and stop loss strategies are about equally effective at limiting losses the cost of the option roughly offsets the risk of overshooting inherent in the stop loss rule. However, greater losses are possible with the stop/buy-back strategy if the stop is tight, as there is a non-negligible probability that the investor will be stopped out twice.

A: Maximum loss at 95% confidence level

It is simple to repeat the analysis for exotic derivatives such as barrier options, and also for more complex trading rules. It is also fairly simple to extend these techniques to the case of multiple correlated securities, though the computational burden will be greater. However, it is not straightforward to refine the time-step significantly for example, to simulate 52 weeks instead of 12 months. This is because of a technical property of Gibbs sampling: it draws serially correlated samples and relies on the ergodic theorem for convergence, but where there are many time steps the serial correlations become too high, and "slow mixing" severely retards convergence.