Stop-loss Strategies and Derivatives Portfolios Patrick L. Leoni University of Southern Denmark Department of Business and Economics Campusvej 55 DK-5230 Odense M, Denmark E-mail : [email protected]

Abstract We carry out a Monte-Carlo simulation of the long-term behavior of a standard derivatives portfolio to analyze the performance of stop-loss strategies in terms of loss reductions. We observe that the more correlated the underlyings the earlier the stop-loss activation for every acceptable level of losses. Switching from 0-correlation across underlyings to a very mild form of correlation significantly decreases the expected time of activation, and it significantly increases the probability of activating the stop-loss. Adding more correlation does not significantly change those features. We introduce the notion of laissezfaire strategies, and we show that those strategies always lead to lower average losses than stop-losses.

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1

Introduction

Stop-loss strategies are the most commonly used trading strategies in financial practice for portfolio management. This strategy involves liquidating a position once a pre-determined level of losses is reached, typically 10% in practice. This managerial practice is thus at the heart of reducing the risk of volatile assets, and they become strikingly important for highly risky assets such as derivatives (see Pedersen [10], and also Basak et al. [1] and Demirer and Lien [3] for more standard stock portfolios). Derivatives have caused most of the largest financial disasters since their introduction in the early 70s, and controlling losses on those portfolios has become an issue of paramount importance in practice. Some of the most spectacular cases were Barings Bank in the early 90s causing a $1bn loss and the bankruptcy of the bank, and the $4bn losses suffered by Long Term Capital Management in 1998. Derivatives have thus been under severe and permanent watch both by financial regulators and financial managers. Despite a plethoric number of accounting measures to limit this downside risk, the French bank Soci´et´e G´en´erale realized a record loss of $7.1bn in January 2008 after dubious trades on vanilla products similar to those we consider in this paper. Stop-losses thus appear as a natural and easily implementable method of controlling those potentially staggering losses, explaining their popularity among traders (see also Lakshman [8] for other methods and their relative cost).

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This paper analyzes the performance of stop-losses for derivatives portfolios in terms of loss reductions and control, an issue that has been omitted in this important case despite a considerable amount of efforts for stock portfolios (see Shen and Wangb [12] among many others). In a Monte-Carlo simulation of a standard trading exercise with derivatives, our objective is two-fold: 1- we identify which factors affect the early and sub-optimal activation of those strategies, and 2- we exhibit a more effective strategy capable of reducing the risk of those portfolio without forgoing recovery possibilities. The fact that stop-losses are sub-optimal is rather intuitive, since liquidating a portfolio corresponds to a problem of optimal stopping time and this stopping time does not coincide in general with the first day a given threshold is reached (see Boyle and Mao [2]). Identifying a dominating strategy is more challenging, and the one that we give here is rather simple. We carry out a Monte-Carlo simulation of the long-run behavior of standard derivatives portfolio, similar to that in Leoni [9]. The prices of the underlyings all follow a standard geometric Brownian motion, as assumed in the now common Black-Scholes framework. The initial wealth is equally allocated across all of the derivatives, and once the maturity common to all of the derivatives is reached the proceeds are reinvested in the same manner in a new portfolio. The reinvestment takes place until the stop-loss is activated because of a recorded loss on the portfolio value, or until an arbitrary horizon of six years is reached. The scenario is simulated 2500 times to generate Monte-Carlo estimators (see Glasserman [4] for an introduction).

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The experiment shows the extreme sensitiveness of stop-loss strategies to the cross-correlation of underlyings. In particular, we observe that the more correlated the underlyings the earlier the portfolio liquidation because of stop-loss activation, for every level of possible losses that we consider. The striking result is that switching from 0-correlation across underlyings to a very mild form of correlation significantly increases the probability of activating the stop-loss before the six-year horizon, and it also significantly decreases the expected time of activation. Adding more correlation does not significantly affects those figures, clearly indicating a discontinuity at 0 for those two variables. We consider an alternative trading method called laissez-faire strategy,1 which is defined as letting the portfolio reach the end of the 6-year horizon regardless of the losses incurred in the meantime. Our point is to compare the performances in terms of risk reduction of this latter strategy to our standard stop-losses. The recovery rate is defined as the number of scenarii where stop-losses lead to higher overall losses than for laissez-faire strategies. The experiment shows that the recovery rate is however decreasing with the pre-determined loss level albeit always significantly high. In particular, laissez-faire strategies lead to lower average losses than stop-losses, for every cross-correlation and every pre-determined loss level. The intuition for this last result is similar to that of the well-known Gambler’s Ruin problem (see Grimmett and Stirzaker [5] Chapter 3). Consider a player endowed with an initial positive wealth, and tossing a fair coin. The 1

This translates from French as “let it happen strategy.”

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player wins (resp. looses) one monetary unit if head (resp. tail) occurs at every toss. The player tosses the coin until either her wealth reaches a predetermined upper-bound, or ruin occurs. Standard results claim that the game will end for sure, and the average number of tosses needed to reach one of those two bounds (the upper-bound or 0) decreases exponentially as the upper-bound gets closer to the initial wealth. A ruin in our portfolio management setting corresponds to reaching a loss level, after which the stop-loss is activated. We observe a similar result in the portfolio simulation, even if the random process characterizing our portfolio return is far more complex than this tossing game and thus requires numerical simulations. However, letting the gambler’s game continue even when ruin occurs (through retaining barriers for instance) leads to a wealth distribution at a given future horizon whose mean is different from zero. It turns out that the mean of the return distribution at the 6-year horizon, and conditional on reaching a given loss level before, is strictly positive. This implies that it is always preferable to carry on the trades until the fixed horizon is reached. The lesson is that those derivatives portfolios have a high recovery potential, and stop-losses entirely ignore this important aspect because of their one-off nature. A central question is to know whether our assumption on the price process of underlyings drives our results. We conjecture that similar results would obtain with other common processes such as mean-reverting processes with or without jumps. This issue is left for further work.

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The paper is organized as follows. In Section 2 we describe the experiment; in Section 3 we give the empirical results about the sensitivity of stop-losses to correlation across underlyings and the relative performances of stop-losses with laissez-fair strategies; Section 4 contains some concluding remarks.

2

The experiment

In this section, we describe the model, assumptions and trading strategies that we use to carry out our Monte-Carlo simulation. We first describe four classes of basic options, and the way to form our portfolio with those options. We will then describe our assumptions about the law of motion of the underlyings, which are essentially Geometric Brownian Motions exhibiting some cross-correlation. We will conclude this section by defining stop-loss and laissez-faire strategies, that we will later compare in terms loss reductions.

2.1

The options

We now describe the classes of options constituting our portfolio. We consider 400 different options, which are partitioned into four classes of 100 options each. Every option has a maturity of T = 3 months, starting with the same common date. • Class 1. 100 cash-or-nothing options with strike price K = 49 and end-payment Q = 10, each of then written on a different underlying. 6

The payoff at time T of the cash-or-nothing option is Q if ST > K and 0 otherwise, where ST is the price of the underlying in 3 months. • Class 2. 100 lookback options, each of them written on a different underlying. The payoff of a lookback option is ST −min(S), where min(S) is the minimal price of the underlying between 0 and T . • Class 3. 100 Asian options, each of them written on a different under¯ where S¯ is lying. The payoff of one Asian option is max{0, ST − S}, the mean of underlying price between 0 and T . • Class 4. 100 European calls with strike price K = 49, each of them written on a different underlying. The payoff at the end of the 3 months is max{0, ST − K}.

2.2

Portfolio formation

We now describe how our portfolio is formed. We first allocate the initial wealth w0 = 1, 000, 000 equally among the four classes of options. In every class of options, the wealth allocated to this class is equally distributed across all of those options. That is, if wj is the wealth allocated to Class j, then for every option in this class we purchase at current market price a number of contracts whose value amounts to wj /100 monetary units. The risk-neutral prices of those options will be given later in this section, once the law of motion of underlying prices will be specified. Once the first time horizon (3 months) is reached and payoffs are realized,

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the proceeds are reinvested in a similar portfolio in the same manner as above. We will discuss later in this section the limit on the number of times that we allow for those financial operations, when we will compare standard stoploss strategies with some other trading strategies. The portfolio formation is summarized in Fig. 2.2. We call a quarter any of such times where options expire and proceeds are reinvested. The fact that options are kept until expiration instead of being sold before is not restrictive. Indeed, since the current reselling price of the option reflects any loss-gain incurred during the exercise, the reinvestment of the realized gain-loss into similar assets would not affect the portfolio value since the underlyings follow a L´evy process.

2.3

Price evolution of underlyings

In this section, we describe the underlying assets on which the options are written. Before describing the laws of motion of the underlyings, we define κ ∈ {e, a, l, c} to be an index denoting the class of options the underlying is assigned to as described earlier, and j = 1, ..., 100 to uniquely describe the option within the class of options κ. The simulation involves a set of 400 different underlyings whose price processes in a risk-neutral world are described by the following standard stochastic differential equation dStκ,j = µStκ,j dt + σStκ,j dWtκ,j ,

(1)

where St is the price of the underlying at time t, µ > 0 is the drift of process, σ > 0 is the variance of the jumps assumed to be constant over time, and 8

Wtκ,j is a Brownian motion with law N (0, t) for every time t. Using standard arguments in Stochastic Calculus, the solution to the stochastic differential equation in (1) also satisfies for every t ∈ [0, T ] dln(Stκ,j ) = (µ −

σ2 )dt + σdWtκ,j . 2

(2)

Applying Itˆo’s Lemma to the previous stochastic differential equation gives the closed-form solution

Stκ,j

σ2 = S0 exp (µ − )t + σWtκ,j 2 

 for every t ∈ [0, T ].

(3)

We need a discretized version of the continuous-time process described in Eq. (3) to carry out our numerical simulations. We use the common √ κ,j approximation Wt+∆t −Wtκ,j ≈ κ,j ∆t for every small enough time variation ∆t, where κ,j is a random variable with law N (0, 1) generating the jumps (see Karatzas and Shreve [7] for a justification). Using the independence of time increments in Brownian motions, and together with the previous approximation, the price process (3) above can be effectively approximated by Stκ,j i +∆ti

=

Stκ,j i

 p  σ2 exp (µ − )∆ti + σκ,j ∆ti 2

(4)

for every sequence of times 0 < t0 < ... < tn . We will consider our standard time horizon of T = 3 months, and we will assume that there are 15 jumps of equal length within those 3 months for every underlying. We assume that there exists a riskless asset whose return is r = 5% per annum. Since we deal with risk-neutral valuations there must be no arbitrage

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opportunity between the risk-free asset and any other asset described so far, which implies that µ = r. We assume that St0 = 50 for every underlying, with same volatility σ = 45% per annum. The underlyings thus differ by the nature of the realized jumps κ,j , and the way those jumps are correlated is central to our analysis. We now assume that the pairwise correlations are described by   1   ρ  cov(e,j , a,j , l,j , c,j ) =   ρ   ρ

variance-covariance matrices  ρ ρ ρ   1 0 0    0 1 0    0 0 1

for every j = 1, ..., 100 and for some ρ ∈ (0, 1); all the other underlyings exhibiting 0-pairwise correlation. In words, for every j the underlying with index j has a pairwise correlation of ρ with the underlying of the same j where the European call is written on, and it exhibits 0-pairwise correlation with all of the other 397 underlyings. The coefficient ρ will be called the internal correlation of the portfolio. This assumption on the correlation across assets captures the idea of a low correlation degree in the underlying structures while remaining tractable; it is meant to emphasize the importance of switching from 0-correlation to a positive albeit low correlation. Other correlation structures could have been implemented using copulae methods for instance, we leave this issue for further work. Given the previous assumptions, it is now possible to pin down the riskneutral prices of the options described above by using numerical methods. 10

Table 1: Monte-Carlo estimations of the risk-neutral prices of the options. Codes are written in R (see R project [11]). Figures between brackets are the variances of the estimators. Estimators are calculated with N=100,000 simulations. cash-or-nothing

lookback call

Asian call

European call

5.0668

7.3431

3.1966

5.2745

(0.01560)

(0.02470)

(0.01425)

(0.02542)

Table 1 gives the theoretical prices of those options, obtained with standard Monte-Carlo simulations independent of the other simulations used for evaluating our trading strategies (see Glasserman [4] Chapters 4-5 or Hull [6] Chapter 22 for an introduction to the methods used here).

2.4

Trading strategies

We now describe the trading strategies that we will use in our analysis. The way the portfolio is formed is not affected by those strategies, but instead they come down to choosing a stopping time of trading activities when losses in the overall value of the portfolio are recorded. We call a stop-loss strategy the action of liquidating the portfolio in a given quarter (or not reforming the portfolio in a similar way) once a predetermined loss level has occurred, where the loss level is measured as a negative fraction of the initial wealth. The most commonly used loss level

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in practice is 10% of the initial wealth, although we will consider the range 5-30% for possible losses. We say that a stop-loss strategy is activated when trades are stopped because the loss level is reached. We call a laissez-faire strategy the action of letting the portfolio reach a pre-determined time horizon, without interrupting trades before this time regardless of any losses incurred in the meantime. We will consider a time horizon of 6 years, which corresponds to 24 possible portfolio quarterly formations as described earlier. One of our point will be to compare the relative performances of those two strategies in terms of reducing the losses of the portfolio. We will consider N = 2500 simulations of the investment scenario above, and we define the activation rate (or AR forthwith) to be the number of simulations where stop-loss strategies are activated, divided by N . Similarly, we define the natural failure rate (or NFR forthwith) to be the number of simulations where the final wealth is below a pre-determined level at the end of the d = 24 quarters, divided by N . Those two notions allow us to define the recovery rate (or RR forthwith)as RR =

AR − N F R , AR

with the same loss level being used in the two strategies. The recovery rate represents the percentage of simulations that have been stopped before the end of the planned exercise by the stop-loss strategy, and where it would have been optimal to keep on trading. An interesting alternative to the stop-loss strategy above is to re-invest 12

at the risk-free rate the proceeds of a liquidation after a stop-loss activation, until the last quarter d = 24. Similar qualitative results obtain with this alternative strategy, and this issue is thus omitted.

3

Statistical results

We now give the results of our Monte-Carlo simulation. In a first step, we establish the responsiveness of stop-loss strategies to internal correlation. In particular, we will see how a change in this form of correlation affects the probability of activating the stop-loss before a given horizon and the average date of occurrence of this activation. In a second step, we will see that laissez-faire strategies dominate stop-losses by showing that the recovery is high enough to privilege the first strategy.

3.1

Activation of stop-loss strategies and internal correlation

We now turn to describing how the internal correlation of the portfolio affects the activation time of stop-loss strategies. The investment scenario has been simulated N = 2500 times, and Monte-Carlo estimators of activation times are reported for three levels of internal correlation. Table 3 below gives the results for three levels of internal correlation, and for various levels of pre-determined losses. The point is to compare the activation times for the case of uncorrelated underlyings (ρ = 0) with some

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Table 2: Estimations of the average period where activation of stop-loss strategies occurs, for various levels of internal correlation. Figures between brackets are the variances of the estimators. Estimators are calculated with N=2500 simulations. Loss level ρ

.05

.1

.15

.2

.25

.3

0

13.2548

18.2428

20.5252

22.3592

22.3592

23.5388

(0.2054) (0.1727)

(0.1386)

(0.0944)

(0.0679)

(0.0470)

13.116

15.6196

17.4732

19.1684

(0.1926)

(0.1836)

(0.1662)

(0.1493)

14.0112

16.4144

18.4368

19.504

(0.1906)

(0.1777)

(0.1590)

(0.1434)

.5

8.9084

11.2352

(0.1879) (0.1950) .7

8.7936

12.146

(0.1863) (0.1967)

other cases of strictly positive correlation. For every internal correlation, the average activation time is before the end of our 6-year horizon. Moreover, the activation time is an increasing function of the loss level. This aspect is intuitive since reaching a given loss once is more likely than reaching once a higher loss, and in turn the occurrence of this event should be earlier for lower loss level. The striking fact to notice is that the activation time sharply decreases between the 0-correlation case and any other cases, regardless of the loss level. For every loss level, the activation time occurs at 40% earlier with 14

some form of internal correlation. This mild change in correlation structure shows how sensitive stop-loss strategies are to this aspect. Adding more internal correlation to the portfolio (switching from ρ = .5 to ρ = .7) does not statistically decrease the activation time, for every level loss. This fact shows that there is a discontinuity in terms of activation time between 0-correlation to some form of correlation, thus the 0-correlation case is a threshold after which some form of continuity emerges. Those findings are graphically represented in Fig. 2. We can also determine the probability that a stop-loss strategy is activated, as a function of the internal correlation. We consider the same simulation method to calculate those probabilities. The results are given in Table 3. The likelihood of activation is increasing with the loss level for every internal correlation, for the reasons explained earlier. The striking point to notice is the sharp increase in the likelihood of activating the stop-loss strategy when switching from 0-correlation to any other form of correlation. Switching from an internal correlation of 0.5 to 0.7 leads to an increase of the odds of statistically significance, since the values remain outside of the confidence intervals of our estimators. However, the difference is mild and it is about 20% higher than for the 0-correlation case for nearly every loss level. We thus observe a significant discontinuity at 0-correlation and a relative smoothness afterward for higher forms of internal correlation. The situation is similar to the activation time, which displayed a similar pattern of discontinuity. 15

Table 3: Probability that stop-loss strategies are activated before 24 periods. Figures between brackets are the variances of the estimators. Estimators are calculated with N=2500 simulations. Loss level ρ

.05

.1

.15

.2

.25

.3

0

0.5556

0.346

0.2316

0.1308

0.0792

0.0472

(0.0084)

(0.0067)

(0.0054)

(0.0042)

0.5824

0.4816

0.3856

0.3448

(0.0098)

(0.0099)

(0.0097)

(0.0095)

0.6192

0.5136

0.4508

0.3632

(0.0097)

(0.0099)

(0.0099)

(0.0096)

(0.0099) (0.0095) .5

0.7668

0.6472

(0.0084) (0.0095) .7

0.764

0.6872

(0.0084) (0.0092)

3.2

Comparison with Laissez-faire strategies

We now present the results of our numerical simulation where trades are allowed to reach the end of the 6-year horizon, allowing us to compare the relative effectiveness of stop-loss and laissez-faire strategies. The scenarii are again simulated 2500 times, for our usual loss levels. Figure 3 gives the recovery rates for ρ = 0, ρ = .5 and ρ = .7. Recovery rates are (approximately) decreasing with loss levels, although they remain strikingly high with low loss levels. For instance, for a very common loss level of 10% and for a 0-correlation portfolio, roughly 65% of terminated trading

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exercises would have yielded a lower loss if stop-losses were avoided. That is, instead of stopping trades and thus permanently accepting a 10% loss, in 75% of the cases carrying on trades would have led to a loss (or possibly a profit) strictly less than 10%. Our derivative portfolio thus have a strong recovery potential. Another important point to notice is that the addition of mild internal correlation has reduced the recovery rate by slightly more than 50% (depending on the experiment). In other words, the addition of correlation across underlyings (even if mild) leads to a reduction of the recovery rate by roughly 25 %, whereas it remains roughly stationary for values greater than ρ = .5. The experiment shows that the loss level is much higher when stop-loss strategies are used, regardless of the accepted loss level. The benefits from avoiding stop-loss strategies depends on the loss level though, but laissez-faire strategies always dominate stop-losses.

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Conclusions

We have compared the relative performance of stop-loss strategies for derivatives portfolios in a standard Monte-Carlo simulation. The experiment shows that stop-loss strategies are increasingly likely to be activated when the internal correlation of the portfolios increases. Moreover, there is a sharp discontinuity at 0 internal correlation, and leaving this threshold leads to much higher odds of early stopping. Reducing the internal correlation thus appears as a simple and reliable way of reducing the downside risk of those 17

portfolios. It turns out that stop-losses are not an effective method of reducing this downside risk. We have seen that using laissez-faire strategies dominate stoploss in terms of risk reduction. In other words, letting the portfolio reach the end of a pre-determined horizon leads to lower losses ex-ante than liquidating the portfolio once the first loss level is reached. Laissez-faire strategies are clearly not the optimal trading strategies, but they provide a very simple example of a trading strategy that beats the most commonly used one in practice.

References [1] Basak, S., Shapiro, A. and L. Tepla (2006) “Risk management with benchmarking.” Management Science 52, 542–557. [2] Boyle, P. and J. Mao (1983) “An exact solution for the optimal stop loss limit.” The Journal of Risk and Insurance, 719–726. [3] Demirer, R. and D. Lienb (2003) Downside risk for short and long hedgers. International Review of Economics & Finance 12, 25–44. [4] Glasserman, P. (2004) Monte-Carlo Methods in Financial Engineering. New-York: Springer Science. [5] Grimmett, G. and D. Stirzaker (2006) Probability and Random Processes. Oxford: Oxford University Press.

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[6] Hull, J. (2006) Options, Futures and Other Derivatives. (6th ed.) Upper Saddle River: Prentice Hall. [7] Karatzas, I. and S. Shreve (2001) Brownian Motion and Stochastic Calculus. New-York: Springer Science. [8] Lakshman, A. (2008) An option pricing approach to the estimation of downside risk: A European cross-country study. Journal of Derivatives & Hedge Funds 14, 31–41. [9] Leoni, P. (2008) “Monte-Carlo estimations of the downside risk of derivatives portfolios.” Forthcoming in IEEE Transactions on Wireless Communications. [10] Pedersen, C. (2001) “Derivatives and downside risk.” Derivatives Use, Trades and Regulations 7, 251–268. [11] R

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[Online].

Available:

http://www.r-project.org [12] Shen, S. and A. Wangb (2001) “On stop-loss strategies for stock investments.” Applied Mathematics and Computation 119, 317–337.

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Figure 1: Portfolio formation in every period.

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Figure 2: Date of activation of stop-loss strategies as a function of the internal correlation. 21

Figure 3: Recovery rate as a function of the internal correlation.

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