When choosing risk management strategies, an institution should consider the trade-off between downside and upside potential. The extent to which this trade-off is quantified can influence the choice of hedging strategy.
A common approach is to hedge on the basis of a well-informed view of the direction of future market rates without attempting to estimate the magnitude of those movements. This method could be described as a 'seat-of-the-pants' analysis, not in a pejorative sense but merely to indicate the level of quantification. A more quantitative approach is to use scenario analysis, perhaps identifying best and worst cases. However, this method does not provide any estimate of the probability that a particular scenario will occur. This would have to be added by the institution and its estimation may be based on another seat-of-the-pants analysis. To this extent, scenarios may not be consistent with market forward rates and volatilities, i.e. the values of the underlying variables assumed in the pricing of risk management instruments may conflict with those values used in the analysis. It is therefore of interest to consider analysis techniques which fully quantify the trade-off between downside and upside potential, automatically generate scenario probabilities and which are consistent with the pricing of relevant hedging instruments. A Monte Carlo simulation provides one such technique.
The basics of Monte Carlo simulations can be found in many textbooks but in outline, it is assumed that the underlying variable (e.g. an interest rate, fx rate, equity price etc.) can be represented by a random movement superimposed on a non-random drift. The drift is described by forward prices and the randomness is related to volatility. A simulation generates values for the underlying variable consistent with this process.
For example, assume that an institution is concerned about the fluctuations in the dollar value of JPY5 billion of income arising in one year. The institution wishes to estimate for three different hedging strategies the probability that a specified reduction in the dollar value of the income is exceeded. The three strategies considered are; (i) no hedge, (ii) a hedge using a one-year forward fx contract, and (iii) a hedge through the purchase of a one-year European-style dollar call/yen put option with the additional feature that the option knocks out if the spot rate at expiration is higher than a pre-determined trigger level. This up-and-out feature reduces the initial option cost.
The simulation is set up using the current spot and forward fx rates and the market volatility appropriate to a one-year term. Each simulation run generates a single future spot fx rate occurring in one year which can then be applied to each of the strategies to determine the present dollar value of the future yen income. Note that strategy (ii) above is unaffected by the future fx rate as the dollar value of this strategy is locked in at the forward rate. For strategy (iii), the optionality and knockout feature permit an appropriate fx rate to be determined for each simulation run. Thus, the present dollar value is unambiguously defined for all three strategies for all simulated future fx rates. Note that in strategy (iii) the up-front premium is included in the final present dollar value.
In order to see how the simulation fills in the gaps left by scenario analysis, the results (i.e. each strategy's distribution of present dollar values) should be displayed in the form of a cumulative probability distribution. For any chosen present dollar value on the horizontal axis, the corresponding value on the vertical axis is the probability of realizing any present value less than or equal to the value on the horizontal axis. Three such probabilities are printed on the graph.
Examining the graph, the forward hedged strategy appears as a vertical line since it does not vary with the future fx rate. The unhedged strategy clearly displays the potential upside (to the right of the vertical line) and the potential downside (to the left of the vertical line). The peculiar profile for the option hedged strategy demonstrates that if the knockout is not triggered, there is no downside exposure to future fx rates between the strike and the knockout (the horizontal part of the curve) but there is unlimited upside. The inclusion of option premium means that the cumulative probability curve for strategy (iii) will lie above strategy (i) if the option is not exercised or if it knocks out. Note also that allowing for statistically insignificant fluctuations, the average present dollar value is the same for all strategies. This confirms the consistency between the simulation and the pricing of the relevant risk management instruments. As a further check, the currency option price using the simulation was shown to be in good agreement with the price from a conventional option pricing model.
If an institution has a view of future rates which is markedly different from the forward market, the consistency requirement above can be relaxed and the simulation can be used to determine the new outcome of different strategies. It turns out that the resulting cumulative distributions have the same form but the probabilities that were calculated above are now altered. Also, the average present dollar values for the three strategies are significantly different. This is because the underlying hedging instruments are always priced using forward rates but the simulation (which corresponds to the institution's view) is not. Depending on the relationship of the institution's view relative to the forward market, the hedging instruments behave as though they were either overpriced or underpriced.
The previous example can easily be extended to multiple payments or indeed to other variables such as interest rates where the simulated internal rate of return is considered.
In summary, simulation analysis provides an efficient method to analyze current and implied future market information in order to arrive at sensible risk management decisions.
This week's Learning Curve was written byPeter Fink, s.v.p. and head of marketing at Sumitomo Bank Capital Markets, Inc. in New York.
