DIFFERENT PATHS: SCENARIOS FOR n-asset AND PATH-DEPENDENT OPTIONS II

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1 DIFFERENT PATHS: SCENARIOS FOR n-asset AND PATH-DEPENDENT OPTIONS II David Murphy Market and Liquidity Risk, Banque Paribas This article expresses the personal views of the author and does not necessarily represent the views of Banque Paribas or any other body. The author gratefully acknowledges the help given to him in the preparation of the paper by Gareth Anthony and his colleagues at BZW and Andrew Street and his colleagues at SFA s Risk Assessment Group. Any errors remain, of course, the author s own. Scenarios have achieved widespread use in the risk management of portfolios including vanilla options, and are sometimes permitted as a tool for the calculation of regulatory capital. Sadly, however, it is not straightforward to extend this appealing technique to exotic options because the scenarios become higher-dimensional, making them hard to display, understand and calculate. In a previous paper, we discussed the difficulties and presented some kinds of tractable scenarios for n-asset options; here we consider the case of path dependent options, treating barrier options in particular, and touching on digitals and lookbacks 1. VANILLA SCENARIOS AND THE PROBLEM OF SIZE Scenario matrices have achieved widespread use in the risk management of portfolios of securities including options in recent years. The basic idea is to take a portfolio of instruments over a given underlying, such as a single stock position, forwards on the stock, and options with that stock as underlying, and to revalue the portfolio at various levels of the underlying stock around the current level, and at various implied volatilities. The resulting table, often called simply a scenario, indicates the change in value of the portfolio if these new market conditions come to hold. In particular, the scenario indicates the effectiveness of a hedge over a range of market movements, and thus provides more risk management information that a single (point) Greek sensitivity such as delta, gamma or vega. Scenarios are designed to give a quick and easy to understand picture of the space of possible behaviours of a portfolio. Before treating specific scenarios, therefore, it is worth discussing the size of the space of total behaviours occupied by a portfolio. In the previous article we called this space the phase space of the portfolio. The value of a vanilla option depends (at least) on the level of the underlying stock, the stock volatility, dividends, risk-free interest rates and the time to expiry of the options concerned. However, the crucial variables are often stock price level and implied volatility, so we can concentrate on a two-dimensional phase space with perhaps 45 points (a point giving the value of the portfolio for each of 9 underlying stock price levels and 5 volatility levels). For path-dependent options the situation is even less tractable: using a binomial tree approximation with daily branching for an arbitrary six month path-dependent option gives rise to a scenario with over points. Of course, many of these points lead to very similar 1 For the sake of clarity, the payouts for all the options discussed in this article are given in the display below. Page 1

2 portfolio values, and the point with the worst portfolio value could be vanishingly unlikely to be visited. Clearly, a more sophisticated approach is needed. The previous paper presented some approaches for n-asset options; here we extend these techniques to path-dependent options. Knock in barrier call option: max[s maturity K, 0] if S i > H for some i Cash or nothing Digital call option: FixedCashSum if S i > K for some i Lookback put option: max[max(s i ) S maturity, 0] DISPLAY: Option Payouts CLASSES OF OPTIONS Before deciding upon a style of scenario for a class of options, it is important to understand the area of phase space the option is likely to explore: the value of barriers and digitals can be highly sensitive to the precise position in phase space, whereas asian options do not have the same sensitivityarticularly once the averaging period has been entered. One way of thinking about this is to realise that a general path dependent option has a payoff dependent on the precise path S 1, S 2, S 3, (where S i is the stock price at time i) an underlying takes. However, a given class of path-dependent options identifies certain paths: for instance, asians options identify all paths with the same average, lookback puts all those with the same maxima and end-point, and so on. The size of the equivalence classes of paths indicates roughly how path-dependent the option is. The more path-dependent an option is, the smaller the equivalence classes and the more paths a scenario must contain to explore all the option s behaviour. Our chief example of a path-dependent option will be a barrier, although exactly the same techniques can be used for digital options. A scenario-based analysis of a portfolio containing cash, futures, vanilla options and barriers over a given underlying should account both for the ordinary sensitivities of the portfolio (i.e. to underlying price and volatility) and for the particular risks posed by the barrier options exit risk and concentration risk. Exit risk is the risk that the market will move sufficiently quickly that the hedge on a barrier cannot be reversed at the barrier level; concentration risk is the risk that if a particular market level is reached a large amount of the firm s capital is at risk, and the liquidity necessary to rehedge at that level may not be available. It is usually appropriate to spread risk due discontinuities around a range of market levels rather than concentrating it at one point. This is especially true if that point is a technical level in the market. Other techniques for managing barrier options include treating them as if they were a slightly riskier option. These risks are worth considering in more detail; consider a written 3 month knock-in JPY put/usd call currency option struck at 100 knocking in at 110 with the spot rate at 108. If JPY/USD reaches 110, the holder will have a highly in the money vanilla call option. The option could be naïvely delta-hedged, but since its gamma tends towards infinity around the barrier level, an options-based strategy such as a tight vertical call spread around the barrier level may be much more appropriate. A standard (or vanilla) scenario for a portfolio short the barrier, long some call spreads (to gamma-neutralise the barrier) and futures (to delta-neutralise the portfolio) is shown in figure Page 2

3 one. This seems to imply that the worst situation we could expect to find ourselves in is the underlying going up. However, this disregards the path-dependent nature of the portfolio: if the underlying swings up from 108 to 111, the barrier is triggered, and we are left with a very different portfolio: short a plain vanilla call option and long the call spreads and futures. The market may then swing back down, and we are left with a mark to market loss. Here, the hedge has lost us money as we were not able to exit it before the market fell. Clearly, whiplash market movements pose a significant risk in this situation, and a scenario-based analysis of portfolios containing barriers should account for them. These are not the only kinds of movements of interest, one can easily construct more pathological paths, but they do form a usefullausible and computationally tractable subclass of the (much larger) class of all paths. SCENARIOS FOR OPTIONS WITH DISCONTINUITIES Consider a portfolio of vanilla instruments and barriers over a single underlying. Suppose that a reasonable range of price moves of this underlying is ±x, and for volatility is ±v: for JPY/USD, x = 3%, v = 4% is historically reasonable, allowing for conditions of extreme market turbulence. Take spot S today as 108, and round all FX rates to the nearest big figure. We wish to add whiplash moves to this scenario to account for the extra risk due to the barriers. Clearly we do not expect the market to move by more than x%, so this places a natural bound on the size of the moves considered. Consider just the price of the underlying of some portfolio s, and suppose there is a whiplash move up to a and back down so that S tomorrow = b. Let such a path be notated p a, b so that p 111, 107 is the path whereby the underlying moves to 111, then down to 107. If we only allow ourselves paths of total length x + 1, (which in the example is c. 4,) there is only one whiplash going from 108 to 111 within the range considered, namely the degenerate whiplash p 111, 111. There are two starting at 108 and ending up at , 110 and p 111, 110 ; similar reasoning gives the paths shown in table one for each node. S tomorrow JPY/USD Direct path Indirect whiplash paths of length no more that 4 111, p 110 p 110, 110 p 111, , p 109, 109 p 110, , p 108, 108 p 110, , , , p 97, 97 p 106, , p 106, 106 p 105, , , p TABLE ONE: Whiplash paths starting at 108 with total length less than or equal to 4. As the size of the underlying moves increases from x to x +1, each old node acquires at most two more paths, and we have two new nodes each with one path, so the total number of paths only increases as the square of the size of the scenario. This can be substantially improved upon by careful analysis of which portfolios need to be valued, making this notion of scenario computationally fairly tractable. Consider a node S tomorrow = b in the scenario. Clearly, for each path p a, b we can revalue the portfolio if it had taken this path (i.e. by recognising any barrier events that would have Page 3

4 happened had this path been taken and changing the portfolio to be valued accordingly). Let this profit/loss be V a, b. The worst of these numbers min(v a, b ) measures, admittedly somewhat crudely, the worst loss that the portfolio could experience starting off at the origin and ending up at S tomorrow = b, given whiplash moves of size no greater than x + 1. Hence our proposed scenario has the same form as a vanilla scenario, but the number at each node is simply min(v a, b ). Figure two depicts the whiplash scenario calculation for the JPY/USD position described above: notice the extra risk due to the path dependency. The worst situation considered for the holder of the portfolio is for underlying to swing up to trigger the barrier, then back down with volatility decreasing. SCENARIOS FOR LOOKBACKS Lookbacks form a relatively benign class of path-dependent options in that they do not often display the discontinuities of behavior common in barriers or digitals. To see the option behaving badly, suppose that we are short a lookback put with S today = 110 and S max = 110. A natural hedge for this is a long put position struck at 110. Clearly this hedge only fails if the underlying whiplashes up and then down before we have time to sell the put and buy one struck at the new maximum: for instance, if S leaps to 115 and then falls back to 110, we have a potential loss of 5 underlying points. Given this insight, we can design a scenario calculation for lookbacks simply by appropriating the extremal whiplashes for barriers: we need merely look at all whiplashes with length precisely x+1. Since shorter whiplashes cannot be worse than longer ones for lookbacks, we can safely include these too. Hence exactly the barrier scenarios proposed above will suffice. CONCLUSIONS We have presented some relatively simple scenario-based techniques for visualising the risk on path-dependent option portfolios with equity or foreign exchange rates as underlyings. Clearly, for fixed income products we have another dimension again (as a portfolio will, in general, be sensitive to more than one point on the yield curve), but the same techniques could be applied. These techniques, though, are only part of the solution; they must be used together with a constraining structure of limits (including exit and spread risk limits), an awareness of any special features of an underlying and, above all, a vigilant risk manager. BIBLIOGRAPHY [Murphy 1996] D. Murphy, Exotic Places: Scenarios for n-asset and Path-Dependent Options I. Page 4

5 Vanilla Scenario for a Barrier Option Value % 9% Volatility 7% 5% 3% Underlying FIGURE ONE: Representations of the vanilla scenarios for a 3 month JPY/USD ITM knock in call (ABOVE), and a hedged portfolio containing the barrier, call spreads and futures one month from expiry (BELOW). Vanilla Scenario for the Hedged Position 300 Change in Value % % 11% Volatility Underlying Page 5

6 FIGURE TWO: A representation of the whiplash scenario for the hedged JPY/USD barrier portfolio one month before expiry (BELOW). The effect of the whiplash moves is pronounced. Whiplash Scenario Volatility 3% 5% 7% 9% 11% Underlying Change in Value Page 6

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