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Omega Performance Measure and Portfolio Insurance P. Bertrand∗ J.L. Prigent† March 16, 2006 Abstract We analyze the performance of portfolio insurance methods : especially the OBPI and CPPI. For this, we use the Omega performance measure, which takes the entire return distribution into account. We show in par- ticular that for the Omega performance measure, the CPPI method is better than the OBPI one for “rational” thresholds. Keywords: Portfolio insurance, Performance, Omega measure. JEL classiﬁcation: C 61, G 11. 1 Introduction The goal of portfolio insurance is to give to the investors the ability to recover, at maturity, a given percentage of their initial capital, in particular in falling mar- kets. Two standard portfolio insurance methods are the Option Based Portfolio Insurance (OBPI) and the Constant Proportion Portfolio Insurance (CPPI). The OBPI was introduced by Leland and Rubinstein (1976). This method is based on a portfolio invested in a risky asset S (usually a ﬁnancial index such as the S&P) covered by a listed put written on it. Thus, whatever the value of S at the terminal date T , the portfolio value will be always greater than the strike K of the put. The CPPI was introduced by Perold (1986) (see also Perold and Sharpe (1988)) for ﬁxed-income instruments and Black and Jones (1987) for equity in- struments. This method uses a simpliﬁed strategy to allocate assets dynamically over time. The investor chooses a ﬂoor equal to the lowest acceptable value of her portfolio. The cushion is deﬁned as the excess of the portfolio value over the ﬂoor. It determines the amount allocated to the risky asset: this amount, called the exposure, is equal to the cushion multiplied by a predetermined multiple. Both the ﬂoor and the multiple are functions of the investor’s risk tolerance and are exogenous to the model. The remaining funds are invested in the reserve asset, usually T-bills. As the cushion approaches zero, exposure approaches zero ∗ GREQAM and University Montpellier I, France. † THEMA, University Cergy-Pontoise, France. 1 too. In continuous time, this keeps portfolio value from falling below the ﬂoor. Portfolio value will fall below the ﬂoor only when there is a very sharp drop in the market before the investor has a chance to trade. The properties of portfolio insurance have been previously studied for exam- ple by Bookstaber and Langsam (2000), Black and Rouhani (1989), Black and Perold (1992),...To summarize, OBPI performs better if the market increases moderately. CPPI does better if the market drops or increases by a small or large amount. Bertand and Prigent (2002, 2005) compare CPPI with OBPI by introducing systematically the probability distributions of the two portfolio val- ues and by comparing them by means of various criteria: the four ﬁrst moments of their returns, the cumulative distribution of their ratio and in particular some of its quantiles... Hedging risk properties involved by these two strategies are also studied, when the option has to be synthesized. The “greeks” of the OBPI and the CPPI are derived and some of their probability distributions are com- pared. The Greeks’ features show the diﬀerent nature of the dynamic properties of the two strategies, in particular if the risky asset value drops suddenly. Nevertheless, when comparing the performance of portfolios, usually we have to choose a performance measure: for example, for standard asset allocation, we can use Sharpe’s ratio, Treynor’s ratio or Jensen’s Alpha. Besides, since the payoﬀs of portfolio insurance strategies are asymmetric, we must select a per- formance measure which overcomes the inadequacy of traditional performance measures when they are used to analyze return distributions which are not nor- mally distributed. In the present paper, we use the Omega performance measure, recently intro- duced by Shadwick and Keating (2002), to compare standard portfolio insurance strategies.The Omega measure is based on a a gain-loss approach since it uses the downside lower partial moment. It has been applied across a broad range of models in ﬁnancial analysis, in particular to examine hedge fund style or strat- egy indices. This measure splits the return into two sub-parts according to a threshold. The "good" returns are above this threshold and the "bad" returns below. Therefore, the Omega measure is deﬁned as the ratio of the gain with respect to the threshold and the loss with respect to the same threshold. The Omega function is deﬁned by varying the threshold. As mentioned for example in Bacmann and Scholz (2003), the main advantage of the Omega measure is that it involves all the moments of the return distribution, including skewness and kurtosis. Moreover, ranking is always possible, whatever the threshold in contrast to the Sharpe ratio. Section 2 recalls the main properties of Omega measure. In Section 3, the Omega measure is computed for OBPI and CPPI. Section 4 provides some numerical comparisons of performance of both strategies. Some proofs are rel- egated to an Appendix. 2 2 The Omega performance measure 2.1 Deﬁnition and general properties The Omega performance measure was ﬁrst introduced by Keating and Shadwick (2002, 2003). It was designed to overcome the shortcomings of performance mea- sures based only on the mean and the variance of the distribution of the returns. Omega measure takes into account the entire return distribution while requiring no parametric assumption on the distribution. Moreover, it is a function of the expected return threshold that is set according to investor preferences. Omega measure takes into account the returns below and above a given loss threshold. More precisely, Omega is deﬁned as the probability weighted ratio of gains to losses relative to a return threshold. The exact mathematical deﬁnition is given by: Rb L (1 − F (x)) dx ΩF (L) = RL , a F (x) dx where F (.) is the cumulative distribution function of the asset returns deﬁned on the interval (a, b), with respect to the probabilty distribution P and L is the return threshold selected by the investor. For any investor, returns below her loss threshold are considered as losses and returns above as gains. At a given return threshold, investor should always prefer the portfolio with the highest value of Omega. Omega function exhibits the following properties : • First, as shown for example in Kazemi et al (2004), Omega can be written as : h i EP (X − L)+ ΩFX (L) = h i. EP (L − X)+ Remark 1 It is the ratio of the expectations of gains above the threshold L to the expectations of the losses below the threshold L. As noted by Kazemi et al (2004),Omega can be considered as the ratio of the prices of a call option to a put option written on X with strike price L but both evaluated under the historical probability P. • For L = EP [X], ΩFX (L) = 1, • ΩFX (.) is a monotone decreasing function. • ΩFX (.) = ΩGX (.) if and only if F = G. • Kazemi et al (2004) deﬁne the Sharpe Omega measure as: EP [X] − L SharpeΩ (L) = h i = ΩFX (L) − 1. + EP (L − X) 3 Remark 2 If EP [X] < L, the Sharpe Omega will be negative otherwise it will be positive. Typically, consider the payoﬀ X of a stock S at time T which is modelled by a geometric Brownian motion : X = S0 exp[(µ − σ 2 /2)T + σWT ], where WT has the Gaussian distribution N (0, T ). Then, EP [X] = S0 exp[µT ] does not depend on the volatility. Thus, if S0 exp[µT ] < L then the Sharpe Omega is an increasing function of the volatility (due to the Vega of the put). If S0 exp[µT ] > L, the Sharpe Omega is a decreasing function of the volatility. The level must be speciﬁed exogeneously: it varies according to investment type, individual risk aversion. It might be for example a rate of inﬂation for pension’s incomes or the rate of a benchmark index. 2.2 The Omega function for the Buy-and-Hold strategy. The Omega function can be examined for standard univariate distributions: for example the Normal, Logistic, Lognormal and Gamma distributions (see Cascon et al (2003)) and the Johnson family of distributions (see Kazemi et al. (2004)). Consider for instance a portfolio manager who invests in two basic assets : a money market account, denoted by B, and a ﬁnancial index, denoted by S. The period of time considered is [0, T ]. The value of the riskless asset B evolves according to : dBt = Bt rdt, where r is the deterministic interest rate. Assume that she wants a guaranteed level equal to pV0 (with p ≤ erT ) and uses a Buy and Hold strategy. Thus, her portfolio value at maturity is given by : VT = pV0 + αST , with α equal to: V0 (1 − pe−rT ) α= . S0 Then, the Omega of her portfolio is given by : h i + EP (ST − λS0 ) ΩVT (L) = h i, (1) EP (λS0 − ST )+ where λ = (L/V0 − p)/(1 − pe−rT ) and the threshold L is chosen obviously higher than the guaranteed amount pV0 . Thus, it can be considered as the ratio of the prices of a call option to a put option written on ST with strike price λS0 (evaluated under the historical probability P). Note that the ratio λ is higher than 1 (out-of- the-money Call) if and only if the ratio L/V0 is higher than 1 + p(1 − e−rT ). Besides, the threshold L is smaller 4 £ ¤ than the expectation EP [VT ] if and only if L/V0 ≤ 1 + p(1 − e−rT ) eµT . Thus, if we want also to compare the ratio L/V0 with the riskless return erT , we can for example consider three cases: 1) The ratio L/V0 satisﬁes: p ≤ L/V0 ≤ 1 + p(1 − e−rT ), 2) The ratio L/V0 satisﬁes: 1 + p(1 − e−rT ) ≤ L/V0 ≤ erT , 3) The ratio L/V0 satisﬁes: erT ≤ L/V0 ≤ 1 + p(1 − e−rT )eµT . Proposition 3 The Omega performance measure for the previous Buy and Hold strategy is a monotonous function of the guaranteed percentage p: 1) If the ratio L/V0 satisﬁes: L/V0 < erT , then Omega is an increasing function of the percentage p. 2) If the ratio L/V0 satisﬁes: L/V0 > erT , then Omega is a decreasing function of the percentage p. 3) If the ratio L/V0 satisﬁes: L/V0 = erT , then Omega is a constant function of the percentage p. Proof. Since Omega is a decreasing function of the threshold, we deduce from Relation (1) that it is suﬃcient to analyze the ratio λ as function of p. Recall that λ(p) = (L/V0 − p)/(1 − pe−rT ). Therefore : ∂λ(p) = (L/V0 e−rT − 1)/(1 − pe−rT )2 , ∂p from which we deduce the result. The intuition behind porposition 3 is as follows : • In part 1 of proposition 3, the low level of the threshold means that the investor is essentially concerned about risk control. As a result, the Omega becomes increasing in the insured percentage, p. • As the threshold is higher (part 2 of proposition 3), the investor worries more about the performance of her fund. As a result, the Omega becomes decreasing in the insured percentage, p. • If the threshold is just set equal to the initial capitalized portfolio value, L = V0 erT , the Omega is independent of the insured percentage, p. We consider the following numerical example : The dynamics of the market value of the risky asset S are given by the classic diﬀusion process : dSt = St [µdt + σdWt ] , where Wt is a standard Brownian motion. Remark 4 ST = S0 exp[(µ − σ2 /2)T + σWT ], where WT has the Gaussian distribution N (0, T ). Then, EP [VT ] = pV0 + α.S0 exp[µT ] does not depend on the volatility. Since we choose the threshold L smaller than the expectation EP [VT ], both Sharpe Omega and Omega are decreasing functions of the volatility. 5 The parameter values are : T = 1, µ = 10%, S0 = 100, V0 = 100, r = 3%, σ = 20% 20 15 10 5 101.5 102 102.5 103 Figure 1 : Ω as a function of L for p = 1. 8 7 6 5 101.5 102 102.5 103 I Figure 2 : Ω as a function of L for p = 0.95. The two previous ﬁgures illustrate the results of Proposition 3: for levels L smaller than the riskless return, the Omega performance measure is an increas- ing function of the guaranteed percentage p. 6 3 The Omega measure of OBPI and CPPI 3.1 Deﬁnition of the two strategies The portfolio manager is yet assumed to invest in two basic assets : a money market account, denoted by B, and a portfolio of traded assets such as a com- posite index, denoted by S. The strategies are self-ﬁnancing. The OBPI method consists basically of purchasing q shares of the asset S and q shares of European put options on S with maturity T and exercise price K. Thus, the portfolio value V OBP I is given at the terminal date by : OBP I ¡ ¢ VT = q ST + (K − ST )+ , (2) OBP I which is also : VT = q (K + (ST − K)+ ), due to the Put/Call parity. This relation shows that the insured amount at maturity is the exercise price times the number of shares, qK. The value VtOBP I of this portfolio at any time t in the period [0, T ] is : ³ ´ VtOBP I = q (St + P (t, St , K)) = q K.e−r(T −t) + C(t, St , K) , where P (t, St , K) and C(t, St , K) are the no-arbitrage values calculated under a given risk-neutral probability Q (if coeﬃcient functions µ, a and b are constant, P (t, St , K) and C(t, St , K) are the usual Black-Scholes values of the European Put and Call). Note that, for all dates t before T , the portfolio value is always above the deterministic level qKe−r(T −t) . The investor is yet willing to recover a percentage p of her initial investment V0 . Then, her portfolio manager has to choose the two adequate parameters, q and K. First, since the insured amount is equal to qK, it is required that K satisﬁes the relation1 : pV0 = pq(K.e−rT + C(0, S0 , K)) = qK, which implies that : C(0, S0 , K) 1 − pe−rT = . K p Therefore, the strike K is a function K (p) of the percentage p, which is increasing. 1 This relation can also take account of the smile eﬀect. 7 Second, the number of shares q is given by : V0 q= . S0 + P (0, S0 , K (p)) Thus, for any initial investment value V0 , the number of shares q is a de- creasing function of the percentage p. The CPPI method consists of managing a dynamic portfolio so that its value is above a ﬂoor F at any time t. The value of the ﬂoor gives the dynamically insured amount. It is assumed to evolve according to : dFt = Ft rdt. Obviously, the initial ﬂoor F0 is less than the initial portfolio value V0CP P I . The diﬀerence V0CP P I − F0 is called the cushion, denoted by C0 . Its value Ct at any time t in [0, T ] is given by : Ct = VtCP P I − Ft . Denote by et the exposure, which is the total amount invested in the risky asset. The standard CPPI method consists of letting et = mCt where m is a constant called the multiple. The interesting case is when m > 1, that is, when the payoﬀ function is convex. The value of this portfolio VtCP P I at any time t in the period [0, T ] is : m VtCP P I (m, St ) = F0 .ert + αt .St , (3) ³ ´ ³ ¡ ¢ 2 ´ C0 where αt = S m exp [βt] and β = r − m r − 1 σ 2 − m2 σ . 2 2 0 Thus, the CPPI method is parametrized by F0 and m. The OBPI has just one parameter, the strike K of the put. In order to compare the two methods, ﬁrst the initial amounts V0OBP I and V0CP P I are assumed to be equal, secondly the two strategies are supposed to provide the same guarantee qK = pV0 at maturity. Hence, FT = qK and then F0 = qKe−rT . Moreover, the initial value C0 of the cushion is equal to the call price C(0, S0 , K). Note that these two conditions do not impose any constraint on the multiple, m. In what follows, this leads us to consider CPPI strategies for various values of the multiple m2 . The portfolio payoﬀs for both strategies are given in the next ﬁgure: 2 Note that the multiple must not be too high as shown for example in Prigent (2001) or in Bertrand and Prigent (2002). 8 250 OBPI 200 CPPI , m=4 CPPI , m=6 150 CPPI , m=8 100 50 0 50 100 150 200 Figure 3 : CPPI and OBPI Payoﬀs as functions of S Note that the value of the level L corresponding to the ﬁrst intersection of both graphics is about 103 which is approximately the value of the riskless return. In what follows, we compare the CPPI and OPBI methods for “rational” thresholds which are in fact around this value. 3.2 Comparison of portfolio insurance performance with Omega measure 3.2.1 Computations of Omega As we analyze portfolio insurance, the threshold of the Omega measure must be greater than the insured amount at maturity : L > q.K = p.V0 . Proposition 5 For the OBPI strategy, the Omega function is deﬁned by : h i EP (ST − L/q)+ ΩOBP I (L) = h i h i. EP (L/q − ST )+ − EP (K − ST )+ Proof. See Appendix. The following ﬁgure illustrates this risk reduction due to a Capped Put proﬁle: Remark 6 Note that the risk measure associated to the Omega performance h i + measure at a given level H is the expectation EP (H − ST ) . Thus, the Omega of the OBPI strategy can be viewed as the stock’s Omega to which one would have removed the "risk" of falling under the level K. Note that the insured amount at maturity is q.K. Therefore the reduction in the risk is clearly due to portfolio insurance. 9 Payoﬀ 6 L/q − K ST - K L/q Figure 1: Capped Put Proposition 7 For the CPPI strategy, the Omega function is deﬁned by : h i m + EP (p.V0 + αT .ST − L) ΩCP P I (L) = h i. m + EP (L − p.V0 − αT .ST ) Proof. By using relation (3). Remark 8 The expectation of the CPPI portfolio value is given by : £ CP ¤ EP VT P I = p.V0 + C0 e[r+m(µ−r)]T . Note that this expression does not depend on the volatility. Thus, as for the stock S (see Remark (2)), the Sharpe ratio SharpeCP P I (L) of the CPPI depends on Ω the volatility, only through its denominator which is equal to a Put option. Thus it is a decreasing function of the volatility and so does the Omega ratio for the CPPI. 10 3.2.2 Comparisons of Omega General case Here, we compare numerically the Omega of the OBPI and of the CPPI without any constraint on their expectations. The parameters values are the same as previously. In what follows, the threshold level is chosen lower than the lowest expectations values of the insured portfolios analyzed (CPPI and OBPI). Omega as function of the volatility σ : We ﬁrst analyze the eﬀect of volatility on OBPI Omega and on CPPI Omega for diﬀerent values of the multiple, m. First, note that the CPPI curves intersect each other behind the volatility level of 40 %. 8 6 4 2 0.1 0.2 0.3 0.4 Figure 4 : Ω as function of sigma for p = 1 and L = 102. (OBPI : solid line, line with large dashed : CPPI with m = 3, line with medium dashed CPPI with m = 5, line with small dashed : CPPI with m = 7) For a small value of the threshold, L = 102, and an insured percentage equal to 100%, the CPPI strategies dominate the OBPI strategy for all values of the volatility as can be seen on ﬁgure3 4. 3 In all the ﬁgures, it is the logratithm of the Omega function that is computed. 11 3 2.5 2 1.5 1 0.5 0.1 0.2 0.3 0.4 Figure 5 : Ω as function of sigma for p = 1 and L = 103.5. The ﬁgure 5 shows that as the threshold rises, the OBPI can dominates some CPPI strategies. This is particularly true for small values of m. We analyze now the eﬀect of the insured percentage on the comparison between OBPI and CPPI. 6 5 4 3 2 1 0.1 0.2 0.3 0.4 Figure 6 : Ω as function sigma percentage p = 0.95 for L = 102. 12 2 1.5 1 0.5 0.1 0.2 0.3 0.4 Figure 7 : p = 0.95 and L = 104.5 As the percentage p decreases, the OBPI tends to dominates some CPPI strategies: • for high volatility levels for L = 102, • for low volatility levels for L = 104.5, Omega as function of the threshold L The following ﬁgure corresponds to p = 1 and σ = 20 %. 2.5 2 1.5 1 0.5 102.2 102.4 102.6 102.8 103 103.2 103.4 Figure 8 : Ω as function of the threshold L. (OBPI : solid line, line with large dashed : CPPI with m = 3, line with medium dashed : CPPI with m = 5, line with medium dashed : CPPI with m = 7, line with small dashed : CPPI with m = 9) The following ﬁgure corresponds to p = 0.95 and σ = 20 %. 13 1.5 1.25 1 0.75 0.5 0.25 102.5 103 103.5 104 104.5 105 Figure 9 : Ω as function of the threshold L. The eﬀect of L becomes sensitive for high values of L as already seen in ﬁgure 3. It is only for L greater than 103.5 that the ranking between OBPI and CPPI’s are inverted. As soon as the threshold is small, the CPPI strategies dominates the OBPI strategy. Special case We consider now the case where both OBPI and CPPI port- folios values have the same expectation. Recall that the value of the multiple such that the expectations of the two portfolio values are equal is given by4 : µ ¶ µ ¶ ∗ 1 C(0, S0 , K, µ) m (K) = 1 + ln . µ−r C(0, S0 , K, r) 5 4 3 2 1 0.1 0.2 0.3 0.4 Figure 10 : p = 1 and L = 102 4 See Bertrand and Prigent (2005). 14 2.5 2 1.5 1 0.5 0.1 0.2 0.3 0.4 Figure 11 : p = 1 and L = 104 4 3 2 1 0.1 0.2 0.3 0.4 Figure 12 : p = 0.95 and L = 102 2.5 2 1.5 1 0.5 0.1 0.2 0.3 0.4 Figure 13 : Compensation of p et L : p = 0.95 and L = 105. Then, we analyze the eﬀect of the threshold. 15 1.75 1.5 1.25 1 0.75 0.5 0.25 102.5 103 103.5 104 104.5 105 Figure 14 : Ω as a function of L with equality of expectations : p = 1, σ = 20%. When the expectations of the two strategies are set equal, the six ﬁgures above show that, according to the Omega performance criterion, most of the time the CPPI dominates the OBPI. 4 Conclusion The Omega performance measure takes potentially into account all the moments of the returns distribution. Thus, it can be used to study asset with non- normally distributed returns, such as hedge funds, equity in illiquid markets.... However, as for performance measures based on downside deviations, we have to assume that the return level of the omega function is exogenously deﬁned but the loss threshold may be deﬁned by the investor’s preferences. The evaluation of an investment with the Omega function should be consid- ered for thresholds between 0% (above the guarantee in this paper) and the risk free rate. Intuitively, this type of threshold corresponds to the notion of capital protection. In this paper, we have shown that, for this criteria, the CPPI method seems better than the OBPI ’one, when assuming Lognormality of the stock price. Further studies can extend this analysis when jumps may occur or may be based on generalized downside risk-adjusted performance measures such as the Kappa. References [1] Bacmann, J.-F. and Scholz, S., (2003): Alternative performance measures for hedge funds, AIMA Journal, June. [2] Bertrand P. and Prigent J-L., (2002), “Portfolio insurance: the extreme value to the CPPI method”, Finance, 23, p. 69-86. 16 [3] Bertrand P. and Prigent J-L., (2005), “Portfolio insurance strategies: OBPI versus CPPI”, Finance, 26, p. 5-32. [4] Black, F. and Jones, R. (1987). Simplifying portfolio insurance. The Journal of Portfolio Management, 48-51. [5] Black, F., and Rouhani, R. (1989). Constant proportion portfolio insurance and the synthetic put option : a comparison, in Institutional Investor focus on Investment Management, edited by Frank J. Fabozzi. Cambridge, Mass. : Ballinger, pp 695-708. [6] Black, F. and Perold, A.R. (1992). Theory of constant proportion portfolio insurance. The Journal of Economics, Dynamics and Control, 16, 403-426. [7] Bookstaber, R. and Langsam, J.A. (2000). Portfolio insurance trading rules. The Journal of Futures Markets, 8, 15-31. [8] Cascon, A., Keating, C. and Shadwick, W.F., (2003), The Omega Function, The Finance Development Centre London. [9] Favre-Bulle, A. and Pache, S., (2003). The Omega Measure: Hedge Fund Portfolio Optimization, MBF Master’s Thesis, University of Lausanne. [10] Kaplan, P. and Knowles, J. A.: Kappa, (2004): A Generalized Downside Risk-Adjusted Performance Measure. The Journal Performance Measure- ment, 8 (3), 42-54. [11] Kazemi, H., Schneeweis, T. and R. Gupta, (2004), Omega as performance measure, Journal of performance measurement, Spring. [12] Keating, C. and Shadwick, W.F., (2002), A universal Performance measure, The Journal of Performance Measurement, Spring, 59-84. [13] Leland, H.E. & Rubinstein, M. (1976). The evolution of portfolio insurance, in: D.L. Luskin, ed., Portfolio insurance: a guide to dynamic hedging, Wiley. [14] Perold, A. (1986). Constant portfolio insurance. Harvard Business School. Unpublished manuscript. [15] Perold, A. & Sharpe, W. (1988). Dynamic strategies for asset allocation. Financial Analyst Journal, January-February, 16-27. [16] Prigent, J-L. (2001). Assurance du portefeuille: analyse et extension de la méthode du coussin. Banque et Marchés, 51: 33-39. 17 5 Appendix Proof of Proposition (1): recall that for a given random variable X, the Omega performance measure at the threshold L is given by: h i + EP (X − L) ΩFX (L) = h i. EP (L − X)+ For the OBPI, the value of X is given by: X = qK + q (ST − K)+ . Thus: ³ ´+ + (X − L)+ = q (ST − K) − (L/q − K) . + Then, (X − L)+ 6= 0 is equivalent to (ST − K) > (L/q − K). Therefore, since we must assume that L > qK, we deduce that (X − L)+ 6= 0 is equivalent to ST > L/q and, in that case, (X − L)+ = q(ST − L/q). Consequently, we have: h i h i + + EP (X − L) = q EP (ST − L/q) . Using the same arguments, we deduce that: ³ ´+ + (L − X)+ = q (L/q − K) − (ST − K) . Therefore, we have two cases: 1) If ST ≤ K, then: (L − X)+ = L − qK. 2) If ST > K, then: + + (L − X)+ = q ((L/q − K) − (ST − K)) = q ((L/q − ST ) . Thus, for all cases: £ ¤ (L − X)+ = q (L/q − K)I[ST ≤K] + (L/q − ST )+ I[K<ST ] . h i = q (L/q − ST )+ − (K − ST )+ . Consequently, we have: h i h £ ¤ h ii EP (L − X)+ = q EP (L/q − ST )+ − EP (K − ST )+ . Finally, we deduce: h i EP (ST − L/q)+ ΩOBP I (L) = h i h i. + + EP (L/q − ST ) − EP (K − ST ) 18