Stochastic Dominance of Portfolio Insurance Strategies

Transcription

1 Annals of Operations Researc manuscript No. (will be inserted by te editor) Stocastic Dominance of Portfolio Insurance Strategies OBPI versus CPPI Rudi Zagst, Julia Kraus 2 HVB-Institute for Matematical Finance, ecnisce Universität Müncen, Boltzmannstrasse 3, D Garcing, pone: , fax: , zagst@ma.tum.de 2 HVB-Institute for Matematical Finance, ecnisce Universität Müncen, Boltzmannstrasse 3, D Garcing, pone: , fax: , krausj@ma.tum.de e date of receipt and acceptance will be inserted by te editor Abstract e purpose of tis article is to analyze and compare two standard portfolio insurance metods: Option-based Portfolio Insurance (OBPI) and Constant Proportion Portfolio Insurance (CPPI). Various stocastic dominance criteria up to tird order are considered. We derive parameter conditions implying te second- and tird-order stocastic dominance of te CPPI strategy. In particular, restrictions on te CPPI multiplier resulting from te spread between te implied volatility and te empirical volatility are analyzed. Key words Portfolio insurance, CPPI, OBPI, stocastic dominance, volatility spread, risk-averse investor Introduction In te last years, private retirement arrangements ave become an issue of more and more importance to lots of investors. Wit tis respect, customers usually demand a guaranteed minimum performance on teir invested capital from te o ering banks and insurance companies. Suitable investment strategies to provide tis required guarantee are so-called portfolio insurance strategies. ey provide downside protection in falling markets wile keeping te potential of pro t in rising markets at te same time. e variety of portfolio insurance models is wide as any rule tat takes less risk at lower wealt levels and more risk at iger wealt levels is basically a

2 2 Rudi Zagst, Julia Kraus candidate. However, tis paper focuses on te two most prominent examples, te Constant Proportion Portfolio Insurance (CPPI) strategy and te Option-based Portfolio Insurance (OBPI) strategy. e CPPI strategy was introduced by Perold (986) (see also Perold and Sarpe (988)) for xed income instruments and Black and Jones (987) for equity instruments. It as been furter analyzed in Black and Rouani (989) and Black and Perold (992). Basically, it implements a simple strategy to allocate assets dynamically over time. e option-based portfolio insurance strategies date from 976, wen H. Leland and M. Rubinstein were te rsts to tink about put options for portfolio edging reasons. Basically, it consists of buying simultaneously a portfolio invested in a risky asset and a put option written on it. Wereas te CPPI strategy being a dynamic investment strategy requires continuous reallocation of te corresponding portfolio, te OBPI strategy represents a static investment strategy and tus no furter rebalancing of te portfolio is necessary after te initial purcase of te protecting put option. It is terefore frequently discussed weter te comfort of te static OBPI comes at a price compared to te dynamically rebalancing CPPI if we want to guarantee a minimum performance over a given time orizon. Analyses of te two portfolio insurance strategies were already conducted in Black and Rouani (989), Black and Perold (992) (for te CPPI metod) and Bookstaber and Langsam (2000). Bertrand and Prigent (2005) compare te two metods wit respect to various criteria, introducing systematically te probability distributions of te two portfolio values. ey conclude tat neiter of te two strategies dominates te oter one statewisely or stocastically in rst order. e present paper extends teir analysis in two di erent aspects. Similar to te previous analyses, we assume a standard Black-Scoles model for te underlying assets. However, we sould not miss te fact tat te two investment strategies act in di erent market environments. Wereas te CPPI strategy represents a dynamic investment strategy tat operates on te nancial market wit its empirical market volatility, te OBPI uses put options wit di erent exercise prices tat ave to be priced using te implied volatility. It is a well-known fact in te nancial market tat one usually observes a spread between te empirical and te implied volatility. As an example Figure () visualizes te intra-mont volatility estimated from daily DJ Euro Stoxx 50 index returns 2 and te corresponding implied volatilities given by te VStoxx index. See Leland and Rubinstein (988). Actually, Leland and Rubinstein didn t use put options in order to provide portfolio insurance, as tese didn t exist at tat time for entire portfolios. Instead, tey replicated te put option according to te Black-Scoles formula and no-arbitrage arguments. is investment strategy is now known as te Syntetic Put Portfolio Insurance (SPPI) strategy. 2 o estimate te intra-mont volatility of te DJ Euro Stoxx 50 index, for eac mont we calculated te (annualized) standard deviation of te corresponding daily intra-mont returns.

3 Stocastic Dominance of Portfolio Insurance Strategies 3 Fig. Empirical and implied volatility. e empirical volatility is estimated from daily DJ Euro Stoxx 50 returns witin eac mont. e implied volatility is given by te VStoxx index. e time period considered in te calculation is 0/2000- /2007. Since te implied volatility usually exceeds te empirical volatility, we actually ave to pay a iger price for te put option used in te OBPI strategy compared to te Black-Scoles price based on te empirical volatility or te corresponding edging strategy in te underlying market. Hence, te impact of te volatility spread sould be considered in te performance analysis of te two strategies and will be one of te main focuses in our analysis. Secondly, previous analyses only examined rst-order stocastic dominance criteria, wic is related to increasing utility functions. is signi es tat te corresponding investors prefer more return to less return, wereas te associated risk is not taken into account. In general, owever, we observe a certain saturation of te investor. Usually, te gain in utility from an additional unit decreases wit te income level and tese so-called riskaverse investors are described by increasing, concave utility functions. is is te reason wy we extend te analysis of Bertrand and Prigent (2005) to stocastic dominance criteria up to tird order. More precisely, we seek to deduce parameter conditions under wic te CPPI strategy stocastically dominates te OBPI strategy. In all of our considerations we comprise te e ect of te spread between te empirical and te implied volatility.

4 4 Rudi Zagst, Julia Kraus e remainder of tis paper is organized as follows: In Section 2, we brie y introduce and discuss te two portfolio insurance strategies under consideration. We examine teir nal payo s and compute teir expectations and variances. Section 3 provides a teoretical comparison of te payo s wit respect to various criteria of stocastic dominance. e focus lies on second- and tird-order stocastic dominance. o conclude te analysis Section 4 summarizes te main ndings and gives some concluding remarks. 2 Basic properties of te CPPI and te OBPI strategy 2. e nancial market In order to compare te performances of te two portfolio insurance strategies, we start wit de ning te two strategies matematically. We consider a classic Black-Scoles model were two basic assets are traded continuously in time during te investment period [0; ] : Witin te context of te two portfolio insurance strategies te time orizon can, e.g., be regarded as te time orizon for te given guarantee or te time of retirement. e rst of te two assets is a risk-free asset, like a zero-coupon bond or cas-account, and is denoted by B. Its value grows wit constant continuous interest rate r > 0 according to B t = B 0 e rt ; () and positive initial value B 0 > 0. e second asset, denoted by S, is subject to systematic risk, suc as a stock, stock portfolio or market index. Now and in te following, we call S te risky asset and te stocastic dynamics of its market value are given by te geometric Brownian motion ds t = S t (dt + dw t ) ; (2) and positive initial value S 0 > 0: W = (W t ) 0t is a standard Brownian motion and > r > 0 and > 0 are constants tat represent te drift 3 and te volatility of te asset price S, respectively. en, following from Itô s lemma, te log-returns of te risky asset are normally distributed according to ln St S 0 N 2 2 t; 2 t : (3) Witin te scope of tis paper we limit our considerations to self- nancing investment strategies, i.e. strategies were money is neiter injected nor 3 Note tat te assumption > r can easily be understood by observing tat a typical risk-averse investor is caracterized by a monotonely increasing, concave utility function u. Hence if < r tere would be no reason for a rational investor to invest in stocks, since at any time t 2 [0; ] E u S 0 e rt = u S 0 e rt u S 0 e t u (E [S t]) E [u (S t)] :

5 Stocastic Dominance of Portfolio Insurance Strategies 5 witdrawn during te trading period (0; ). Furtermore, following Black and Scoles (973), we assume "ideal conditions" in te market for stocks and options. Markets are terefore frictionless and do not provide any arbitrage opportunities. Moreover, tere are no transaction costs, taxes or margin requirements. Borrowing and sort-selling as well as divisibility of sares are allowed witout restriction. As te borrowing and lending rates are bot assumed to be equal to te risk-free rate of return r, default risk is excluded. As far as options are considered, we restrict ourselves to European options tat can only be exercised on a predetermined date. Furtermore, te underlying stocks do not pay dividends during te life of te option. 4 Wenever te price of an option as to be determined, we take account of te spread between te empirical and te implied volatility by using te Black-Scoles model (),(2) wit te implied volatility i instead of te underlying empirical volatility. In te following, V IS = V IS t denotes te portfolio value process 0t associated wit te investment strategy IS. By means of simplicity we sometimes omit te index IS if one can conclude from te context wic strategy is referred to. We start wit a brief review of te CPPI strategy. 2.2 Constant Proportion Portfolio Insurance (CPPI) e basic idea of te CPPI approac consists of managing a dynamic portfolio, so tat its terminal value V CP P I at te end of te investment orizon lies above an investor-de ned level F, given as a percentage 0 of te initial investment V0 CP P I, i.e. F = V CP P I 0 : (4) Note tat in te absence of any arbitrage opportunities it is impossible to nd an investment tat returns more tan te risk-free rate of return r wit no risk, and tus te maximum guaranteed portfolio value at te end of te investment period is limited by e r : (5) Let (F t ) 0t denote te present value of te guarantee, te so-called oor. By discounting wit te risk-free rate of return r, it evolves according to F t = t V CP P I 0 ; t = e r( t) : (6) e surplus of te current portfolio value Vt CP P I over te oor F t is called cusion C t and its value at any time t 2 [0; ] is given by C t = max V CP P I t F t ; 0 : (7) 4 Note tat in te case of a stock, modeled as a geometric Brownian motion, tat pays dividends continuously over time at a constant rate d per unit time, te drift term in Equation (2) is simply replaced by te drift d and all te oter calculations remain te same (see, e.g., Sreve (2004)).

6 6 Rudi Zagst, Julia Kraus In order to ensure a minimum nal portfolio value V CP P I F ; te basic idea of te CPPI metod now consists of investing a constant proportion m of te cusion C t in te risky asset. is is te reason wy te strategy is called constant proportion portfolio insurance. e investment in te risky asset is called exposure (E t ) 0t and is determined by E t = m C t = m max V CP P I t F t ; 0 : (8) e remaining part of te portfolio B t = V CP P I t E t is invested in te riskless asset. Notice tat te payo function is convex if te so-called multiplier m satis es m. By applying Itô s lemma, te value of te CPPI portfolio Vt CP P I at any time t during te investment orizon [0; ] can be derived as 5 V CP P I t = F t + C t = t V0 CP P I + C t (9) m St ( m)(r+ e 2 m2 )t : = e ( t)r V CP P I 0 + C 0 S 0 us, te CPPI metod is parametrized by te level of insurance and te multiplier m. Note tat te value of te CPPI portfolio Vt CP P I is always above te current oor F t = t V0 CP P I as C t > 0. Hence, te oor F t represents te dynamically insured amount of te portfolio. Furtermore, from Equation (9) we can see tat te value process of te CPPI strategy is pat-independent, i.e. does not depend on te stock price evolution in te investment period (0; t). 6 We conclude te section wit te determination of te expected value as well as te variance of te value of te CPPI portfolio V CP P I at te end of te investment orizon, wic will be needed in te upcoming stocastic dominance analysis. Proposition e mean and te variance of te CPPI portfolio value at te end of te investment period are given by V CP P I = E V CP P I = V0 CP P I + C 0 e [r+m( r)] ; (0) 2 = C 2 0 e 2[r+m( r)] e m2 2 ; () V CP P I were C 0 = V CP P I 0 e r : Proof See te Appendix A. 5 Details about tis formula are provided in te Appendix (see also Bertrand and Prigent (2005)). 6 is important property of te CPPI strategy was earlier sown by Bertrand and Prigent (2005).

7 Stocastic Dominance of Portfolio Insurance Strategies 7 Note tat te expected terminal value of te CPPI strategy is independent on te stock price volatility. In contrast, its volatility grows exponentially wit te market volatility, wic can be intensi ed by a ig value of te multiplier m. An increase in te desired level of insurance obviously reduces te investment risk 2 V CP P I. However, te expected portfolio value V CP P I is decreased at te same time. Opposite e ects can be observed wit respect to te coice of te multiplier m, wic determines te portfolio s participation in te stock market. Next, we will give a sort description of te protective put strategy as an example for an option-based portfolio insurance strategy. 2.3 Option-based Portfolio Insurance (OBPI) In contrast to te CPPI strategy, te OBPI strategy is a static investment strategy. It basically guarantees a minimum terminal portfolio value of V OBP I = V0 OBP I for a portfolio consisting of q sares of te risky asset S, by purcasing European put options wit maturity and strike price X on te same number of sares. o simplify our presentation, we assume tat q is normalized and set equal to one and tat te put option is leverage- nanced at te risk-free interest rate r at inception t = 0. e corresponding loan will be refunded at maturity. At inception t = 0, te total portfolio value is ten given by V OBP I 0 = S 0 + P ut S 0 ; X; r; i ; 0; P ut S 0 ; X; r; i ; 0; = S 0 ; were P ut S t ; X; r; i ; t; denotes te Black-Scoles value of a European put option (value of te underlying asset S t ; strike price X; risk-free rate of return r; implied volatility i ; valuation time t ; maturity ). Since te OBPI strategy is a static investment strategy, no trading takes place during te investment period (0; ). Hence, te nal portfolio value V OBP I at maturity is given by V OBP I = S + P ut S 0 ; X; r; i ; ; P ut S 0 ; X; r; i ; 0; e r = max fx; S g P ut S 0 ; X; r; i ; 0; e r : (2) In order to guarantee a minimum terminal portfolio value of V OBP I = V0 OBP I, te strike X of te edging European put option must equal 7 X = P ut S 0 ; X; r; i ; 0; e r + V 0 ; S 0 = V OBP I 0 : (3) Notice, tat similar to te restriction of te insurance level in te case of te CPPI strategy by (5), relation (3) also caps te maximum guaranteed portfolio value for an OBPI strategy. Generally, in contrast to te CPPI approac, te strike price X of te edging put option (wic depends 7 e corresponding strike price X can be determined from tis equation by a zero searc metod, e.g. te Newton metod.

8 8 Rudi Zagst, Julia Kraus on te desired level of insurance ) represents te only parameter of te OBPI strategy. For simplicity, we presume in te following analysis tat te required European put option P ut S 0 ; X; r; i ; 0; is available in te (OC) market. 8 o simplify te notation, now and in te following, we often use te notation P ut S 0 ; r; i : = P ut S 0 ; X; r; i ; 0; ; Call S 0 ; r; i : = Call S 0 ; X; r; i ; 0; ; wen te underlying strike price X, and te inception and terminal date, 0 and, respectively, are clear from te context. Wit respect to te edging put option P ut S 0 ; X; r; i ; 0; we also use te abbreviation P ut 0 := P ut S 0 ; X; r; i ; 0; : Similar to te CPPI strategy, we nally determine te expected value as well as te variance of te terminal value of te OBPI portfolio V OBP I at maturity. For tis purpose, we recall te de nition of lower and upper partial moments. De nition Given te bencmark X and a random variable S, te Lower Partial Moment LP M z and te Upper Partial Moment UP M z of S wit respect to X and z2 N 0 is de ned as LP M z (S; X) = E [max fx S; 0g z ] ; (4) UP M z (S; X) = E [max fs X; 0g z ] : (5) In terms of an asset price S and a corresponding bencmark X te lower partial moment LP M 0 represents te sortfall probability and LP M te expected value of te loss, wen te asset price falls below te bencmark. Vice versa, UP M 0 denotes te probability of outperformance and UP M te expected value of te pro t in te case wen te asset price beats te bencmark X. Based on tese de nitions, te mean and te variance of te terminal portfolio value of an OBPI strategy can be determined as follows. Proposition 2 e mean and te variance of te value of te OBPI portfolio at maturity are given by V OBP I = E V OBP I = UP M (S ; X) + V0 OBP I ; (6) 2 = UP M2 (S ; X) UP M (S ; X) 2 : (7) V OBP I Proof See te Appendix B. 8 Note tat tis is not a very restrictive assumption, since te investment orizon is typically very long and te underlying OC market o ers European put options of virtually any maturity.

9 Stocastic Dominance of Portfolio Insurance Strategies 9 Notice tat an increase in te desired level of insurance, or correspondingly in te strike X, results in a lower call premium of te call option wit payo max fs X; 0g tat corresponds to te upper partial moment UP M (S ; X). is reduces te exercise probability of te call option and tus te value of te upper partial moment in Equation (6). Correspondingly, te expected terminal value V OBP I of te OBPI strategy decreases wit an increase in te level of insurance and, at te same time, te variance of te terminal value 2 V OBP I decreases. Based on te deduced payo s and distribution caracteristics of te two investment strategies under consideration, we can now proceed wit te comparison of te two strategies using stocastic dominance criteria. 3 CPPI versus OBPI In order to compare te two metods, te initial portfolio values V0 CP P I and V0 OBP I are assumed to equal te current value of te risky asset S 0, i.e. V 0 := V CP P I 0 = V OBP I 0 = S 0 : Also, te two strategies are supposed to provide te same guarantee at te end of te ( nite) investment period 9 expressed as proportion of te initial investment V 0. 0 Hence, t = e r( t) ; F t = t V 0 ; in te case of te CPPI strategy and te strike price of te European put option for te OBPI strategy satis es X = P ut 0 e r + V 0 : Note tat tese two conditions do not impose any constraint on te multiplier m. In wat follows, tis leads us to consider CPPI strategies for various values of te multiplier m. 9 Note tat if! te oor of te CPPI strategy converges to zero t = e r( t)! 0; F t = 0; and tus results in a constant mix strategy wit leverage factor m E t = m C t = m V t; B t = ( m) V t: Hence, no minimum portfolio value is guaranteed anymore and no puts are needed to insure te portfolio. 0 It is our understanding, tat te insurance level satis es te Constraints (5) and (3). e multiple, owever, must not be too ig as sown for example in Bertrand and Prigent (2002 or 2005).

10 0 Rudi Zagst, Julia Kraus Let us start wit looking at te payo functions of bot strategies. In te simplest case, one of te payo functions of te two strategies would statewisely dominate te oter one, wic would imply tat one of te strategies results for all S values in a iger terminal value tan te oter one. However, Bertrand and Prigent (2005) argue tat, since V CP P I V OBP I 0 = 0 and due to te absence of arbitrage, tis is not possible, wic leads to te following proposition. Proposition 3 Neiter of te two payo s is greater tan te oter for all terminal values S of te risky asset. e two payo functions intersect one anoter. is nding can be illustrated using a simple numerical example wit typical values for te nancial market: = 7:50%; = 5%; i = 8% and r = 3:5%. In tis market, te two portfolio insurance strategies are set up assuming = 5 (years), V 0 = S 0 = 00 and = 03:5%. If not mentioned oterwise, now and in te following, we consider tis setting as our reference model scenario for numerical calculations. e value of te CPPI strategy is calculated for di erent values of te multiplier m = ; 2; 3; 4; 5. Figure (2) visualizes te obtained terminal values of te two strategies dependent on te terminal value S of te risky asset. Notice tat a more teoretical motivation of Proposition 3 will be given in te proof of eorem 3. For eac value of te multiplier m te payo s of te CPPI and te OBPI strategy intersect at least once. Since we could not observe a simple dominance of one of te two strategies, we will consider more sopisticated criteria of stocastic dominance starting from rst- up to tird-order in te sequel. 3. First-order stocastic dominance In general, stocastic dominance criteria try to rank two random variables V and V according to special classes of utility functions U. 2 It is said, tat te random variable V stocastically dominates te random variable V wit respect to U, i.e. V U V, if and only if E [u (V )] E [u (V )] ; for all u 2 U for wic te two expected values exist. In te case of rst-order stocastic dominance U is te class of all real, measurable and increasing functions denoted by U := fu : R! R : u measurable, u 0 0g : 2 For furter details concerning te concept of stocastic dominance, see, e.g., Mosler (982).

11 Stocastic Dominance of Portfolio Insurance Strategies Fig. 2 CPPI and OBPI payo s as functions of S, were m = ; 2; 3; 4; 5 and = 5 (years); = 03:5%, V 0 = S 0 = 00; = 7:50%, = 5%; i = 8% and r = 3:50%. is can be interpreted tat investors like more money rater tan less money and are non-satiated. Recall tat a common criterion to test for te rst-order stocastic dominance of te random variable V is to compare te cumulative distribution functions F V and F V of te two random variables. e random variable V stocastically dominates te random variable V in rst order (V U V or brie y V V ), if and only if for any outcome x te random variable V gives a iger probability of receiving an outcome equal to or better tan x compared to V. Hence, V V, F V (x) F V (x) ; 8x 2 R: (8) Wit respect to te CPPI and te OBPI strategy Bertrand and Prigent (2005) sow tat neiter of te two strategies stocastically dominates te oter one at rst order. However, rst-order stocastic dominance represents te strongest criterion, i.e. it implies second- and tird-order stocastic dominance. We terefore extend teir analysis to te weaker principles of second- and tird-order stocastic dominance. In particular, we try to nd special conditions for wic te CPPI strategy stocastically dominates te OBPI strategy. Wit tis respect, te multiplier m; tat determines te risk exposure, will be our most important parameter.

12 2 Rudi Zagst, Julia Kraus 3.2 Second-order stocastic dominance In comparison to rst-order stocastic dominance, te second-order stocastic dominance criterion restricts to risk-averse investors. As mentioned earlier, investors described by utility functions u 2 U are non-satiated, wic means tat teir utility is strictly monotone increasing in te income level witout taking account of te associated risk. However, in te nancial market we traditionally observe a di erent beavior of te investor. Again, more money is preferred to less money. Neverteless, te gain in utility from an additional unit decreases wit te income level. is beavior is represented by te class of increasing, concave utility functions, denoted by U 2 := fu : R! R : u 2 U and u 00 0g ; and we say tat te random variable V stocastically dominates te random variable V in second order if V U2 V or brie y V 2 V. Our goal in tis section is to deduce conditions for te parameters of te two portfolio insurance strategies suc tat te CPPI strategy stocastically dominates te OBPI strategy in second order at te due-date for te given guarantee. For tis purpose, Mosler (982) provides a useful criteria using intersection conditions, tat is independent from any speci c utility function u 2 U 2. eorem (Mosler (982)) Let V; V be two random variables wit - nite expectation. Furtermore, let H (x) := F V (x) F V (x) for all x 2 R. en, H 2 S ; E [V ] E [V ] ) V 2 V : Proof See Mosler (982). S k describes te set of all real functions H wit k canges of sign, i.e. H : R! R : 9s ; :::; s k 2 R; s 0 := ; s k+ := +; S k := were ( ) j : H (s) 0; 8s 2 (s j ; s j+ ) ; j = 0; :::; k; H 6= 0 Example 8 9 < H : R! R : 9s 2 R; = S = 0; s 2 ( ; s ) : were H (s) ; H 6= 0 ; ; 0; s 2 (s ; ) 8 H : R! R : >< 89s ; s 2 2 R; < 0; s 2 ( ; s S 2 = ) were H (s) 0; s 2 (s ; s 2 ) ; H 6= 0 >: : 0; s 2 (s 2 ; ) 9 >= : >;

13 Stocastic Dominance of Portfolio Insurance Strategies 3 In terms of te cumulative distribution functions F V (x) and F V (x), te condition H (x) := F V (x) F V (x) 2 S k implies tat te two functions intersect exactly k-times. In order to derive conditions for te second-order stocastic dominance of te CPPI strategy, we analyze te two conditions postulated in eorem. eorem 2 e following statements are equivalent:. E V OBP I E V CP P I : 2. Call S 0 ; r; i e (m )( r) Call (S 0 ; ; ), i.e. Call m + ( r) ln (S0 ; ; ) Call (S 0 ; r; i =: m ) min: (9) Proof See te Appendix C.. eorem 3 Let m > and H (x) := F V OBP I en, (x) F V CP P I (x) ; 8x 2 R: (S 2 ) : m e r m )2 If m =, H 2 S is true. e 2 (m Proof See te Appendix C.2.! m m < C 0 X e r ) H 2 S 2: Remark Note tat using standard algebraic calculus one can easily sow, tat for m large te left and side of Condition (S 2 ) is smaller tan te rigt and side and tus Condition (S 2 ) is satis ed. Based on te relationsip H 2 S, if m =, following from eorem 3 and te constraint on te call prices resulting from eorem 2 to provide E V CP P I E V OBP I, we can directly conclude from eorem te second-order stocastic dominance of te CPPI strategy for m =. eorem 4 Let m = and Call S 0 ; r; i Call (S 0 ; ; ) : en, V OBP I 2 V CP P I :

14 4 Rudi Zagst, Julia Kraus Fig. 3 Payo functions of an OBPI and a CPPI strategy for di erent values of te implied volatility i = 8%; i = 24% and = 5 (years); = 03:5%; m = ; V 0 = S 0 = 00; = 7:50%, = 5%; r = 3:50%. able Call prices based on = 5(years); = 03:5%; m = ; V 0 = S 0 = 00; = 7:50%; = 5%; r = 3:50%. m = X Call S 0; r; i Call (S 0; ; ) i = 8% 27:87 3:2 < 9:48 i = 24% 49:65 3:2 > 2:3 Remark 2 From eorem 4 we conclude tat in times of low expected return forecasts and ig implied volatility in comparison to te empirical volatility te CPPI strategy will stocastically dominate te OBPI strategy in second order. Figure (3) visualizes te statement of eorem 4. e grap illustrates te payo functions of te OBPI as well as te standard CPPI strategy for di erent values of te implied volatility i. able () provides te corresponding call prices according to eorem 2 and 4, respectively. From able () and eorem 4 we conclude tat for m = and i = 24% te CPPI strategy stocastically dominates te OBPI investment strategy in second order. If i = 8%, te strike of te edging put option used in te realization of te OBPI strategy is smaller. Consequently, te put option is not as expensive as in te case of a iger implied volatility of i = 24%: eorem 2 tells us, tat in tis case te expected OBPI return exceeds tat of te CPPI strategy (i.e. E V CP P I E V OBP I ).

15 Stocastic Dominance of Portfolio Insurance Strategies 5 Remark 3 Call S 0 ; X; r; i ; 0; is independent of te coice of te parameter i : From put-call-parity follows Call S 0 ; X; r; i ; 0; = P ut S 0 ; X; r; i ; 0; + S 0 X e r : Furtermore, since X = V 0 + P ut 0 e r we obtain Call S 0 ; X; r; i ; 0; = S 0 V 0 e r : eorem 4 provides a stocastic dominance criterion in te special case m =. In order to derive an analogue criterion for te more general case m > we analyze tird-order stocastic dominance. As already mentioned, tird-order stocastic dominance follows from second-order stocastic dominance and furter cuts down te class of utility functions U 3 under consideration. 3.3 ird-order stocastic dominance ird-order stocastic dominance adds ruin aversion to te risk aversion involved in second-order stocastic dominance. Investors prefer positive to negative skewness. Notice tat portfolio insurance strategies, like te CPPI or te OBPI strategy, are caracterized by providing downside protection wile still participating in upside markets. Matematically, te additional ruin aversion is expressed by requiring u Hence, te corresponding class of utility functions U 3 is given by U 3 := fu : R! R : u 2 U 2 and u 000 0g ; and we say tat te random variable V stocastically dominates te random variable V in tird order if V U3 V or brie y V 3 V. In particular, U 3 includes te class of utility functions U DARA and U HARA providing Decreasing Absolute Risk Aversion (DARA) and Hyperbolic Absolute Risk Aversion (HARA), respectively. Here, absolute risk aversion is measured by te Arrow-Pratt measure of absolute risk aversion ARA (v) := u00 (v) u 0 (v) ; and describes te investor s willingness to cover, based on er current wealt v, risks by paying an insurance premium. en, te subsets U DARA,U HARA U 3 are de ned as u 2 U3 : u U DARA := 0 (v) = u 0 (a) e R v a r(z)dz ; u 0 (a) 0; a 2 R; r 0; r 0 ; 0 i.e. ARA (v) = r (v) ; (20)

16 6 Rudi Zagst, Julia Kraus ( ) (v c) u 2 U U HARA := 3 : u (v) = a + b ; v c; < ; b > 0; ; a; c 2 R i.e. ARA (v) = v c : (2) Here, te absolute risk aversion is a decreasing function in wealt v. Hence, te iger er wealt v te less willing is te investor to edge iger risks. 3 According to Elton and Gruber (995) common investors are usually described by te class of HARA utility functions. Similar to te previously analyzed second-order stocastic dominance, our goal is to deduce conditions for te parameters of te two portfolio insurance strategies suc tat te CPPI strategy stocastically dominates te OBPI strategy in tird order at te end of te investment period. Again, Mosler (982) and Karlin and Novikov (963), respectively, provide useful criteria using an intersection condition S 2, tat is independent from any utility function u 2 U 3. eorem 5 (Karlin, Novikov (963), Mosler(982)) Let V; V be nonnegative wit nite second moment. Furtermore, let H (x) := F V (x) F V (x) for all x 2 R: en, H 2 S 2 ; E [V ] E [V ] ; E (V ) 2i E V 2 ) V 3 V : Proof See, e.g., Mosler (982). Su cient parameter restrictions to assure te outperformance of te expected terminal value of te CPPI strategy (i.e. E V OBP I E V CP P I ) and H 2 S 2 are already provided by eorem 2 and 3. Hence, in order to derive furter parameter conditions implying te tird-order stocastic dominance of te CPPI strategy according to te Karlin and Novikov eorem (963), we still ave to analyze te condition E i 2 E. V OBP I V CP P I i 2 eorem 6 Let f max (m) := e Call 2 S 0 ; r; i e 2(m )( r) +m S 0 Call S 0 ; r; i e (m )( r) ; b := Call (S 0 ; ; ) S e ( + ) := (S 0 ; ; ) := Call and m max := fmax (b) : en, te following statements are equivalent: S 0e 2 ;; Call(S 0;;) Call(S 0;;) ; X 3 See Arrow (965).

17 Stocastic Dominance of Portfolio Insurance Strategies 7 able 2 Analysis of te conditions for tird-order stocastic dominance of te CPPI strategy resulting from eorem 7 for di erent values of te multiplier m. m = m = 2 m = 3 m = 4 m = 5 m min 2:98 m min m X X X Condition (S 2) X X X X m max 3:05 m m max X X 3 rd order dominance X. E 2. m m max : i V CP P I 2 E Proof See te Appendix D. i V OBP I 2 : By combining eorem 2, 3 and 6 we ave now everyting we need to conclude te tird-order stocastic dominance of te CPPI strategy from eorem 5. eorem 7 Let m min and m max be de ned as in eorem 2 and eorem 6 and m min := max ; m min : Furtermore, let Condition (S 2 ) of eorem 3 be satis ed. en, m 2 [m min ; m max ] ) V OBP I 3 V CP P I : o get a better understanding of te statement of eorem 7, we analyze te payo s of te OBPI and te CPPI strategy for te parameterization visualized in Figure (2). More precisely, te underlying market parameters are tose of te reference model 4 and te CPPI multiplier takes te values m = ; :::; 5. able (2) analyzes for eac value of te multiplier te conditions for second-order (if m = ) or tird-order (if m > ) stocastic dominance, resulting from eorem 4 and eorem 7, respectively. Since m min = 2:98, we conclude from eorem 4 tat te CPPI strategy does not dominate te OBPI strategy in second order for m =. Furtermore, following from eorem 7, we only observe tird-order stocastic dominance in te case wen m = 3. o get a better understanding of wat appens in te di erent parameterizations, Figure (4) visualizes te di erence V OBP I V CP P I in te nal payo s of te two strategies for di erent terminal stock values S and di erent values of te multiplier m. 4 = 7:50%; = 5%; i = 8%, r = 3:5%, = 5 (years), V 0 = S 0 = 00, = 03:5%; X = 27:87:

18 8 Rudi Zagst, Julia Kraus Fig. 4 Di erence V OBP I V CP P I for te nal payo s of te two strategies, depending on te terminal stock price S as well as te multiplier m. If m = ; 2, te CPPI strategy is more likely to underperform te OBPI strategy, wic results in E V OBP I > E V CP P I. In contrast, if m = 3; 4; 5 te expected terminal value of te CPPI strategy exceeds tat of te OBPI strategy. However, te risk associated wit te iger probability of i outperformance exceeds tat of te OBPI strategy, i.e. E V CP P I 2 > i 2 E for m = 4; 5. V OBP I Finally, Figure (5), (6) and (7) more generally visualize te lower bound m min and te upper bound m max on te multiplier m resulting from eorem 7 in dependence on te drift and te implied volatility i of te underlying market, as well as te interval between te two bounds m max m min. e remaining model parameters are given by te reference scenario. From Figure (5) we conclude tat te minimum multiplier m min is te iger te lower te implied volatility i. Since for low values of te implied volatility i te edging put option for te OBPI strategy is ceaper, te CPPI strategy must be allocated in a riskier fasion to outperform te protective put strategy. Wit respect to te drift, no de nite dependence of te value of m min can be observed. Analogously, te upper bound m max on te multiplier m decreases wit an increase in te implied volatility i (see Figure (6)). Since an increase in te implied volatility results in a iger premium for te put option used in te OBPI strategy at maturity, te strike price X increases as well.

19 Stocastic Dominance of Portfolio Insurance Strategies 9 Fig. 5 Value of te tresold m min as de ned in eorem 2 depending on te drift as well as te implied volatility i of te underlying market. Fig. 6 Value of te tresold m max as de ned in eorem 6 depending on te drift as well as te implied volatility i of te underlying market.

20 20 Rudi Zagst, Julia Kraus Hence, at maturity te put option is more likely to be exercised and tus te variance of te terminal value of te OBPI strategy 2 V OBP I decreases wit an increase in te implied volatility. In order for te risk associated wit te CPPI strategy to be smaller tan tat of te OBPI strategy (wic is exactly te interpretation of Statement of eorem 6), we now ave to allocate te CPPI strategy in a more conservative fasion by using a smaller multiplier m. Fig. 7 Di erence of te upper and te lower bound m max m min on te multiplier as de ned in eorem 7 depending on te drift as well as te implied volatility i of te underlying market. Finally, from Figure (7) we conclude, tat tere exist parameterizations of te nancial market so tat te interval [m min ; m max ], derived in eorem 7, actually includes admissible values for te multiplier m. Additionally, Figure (8) visualizes te di erence C 0 X e r m e r e 2 (m )2 m! m m corresponding to Condition (S 2 ) of eorem 3 in dependence on te value of te multiplier m as well as te implied volatlity i. e underlying empirical volatility is assumed to be = 5%. Condition (S 2 ) is satis ed wenever we observe a positive value of te function. As we can see from te gure, for common parameterizations of te underlying nancial market Condition

21 Stocastic Dominance of Portfolio Insurance Strategies 2 (S 2 ) is always satis ed. Altogeter, we conclude tat te CPPI strategy stocastically dominates te OBPI strategy in tird-order in times of ig implied volatilities (compared to te empirical volatility). Fig. 8 Di erence corresponding to Condition (S 2) of eorem 3 in dependence on te value of te multiplier m as well as te implied volatility i. e condition is satis ed, wenever we observe a positive value of te function. o conclude our analysis of te CPPI and te OBPI strategy we will summarize te main results and give some concluding remarks. 4 Conclusion In tis paper, we ave compared te two main portfolio insurance metods - te CPPI and te OBPI strategy - wit respect to various criteria of stocastic dominance. Wit tis respect, we ave taken into account te impact of te spread between te (usually iger) implied volatility and te empirical volatility. Furtermore, we extended te work of previous papers by focussing our analysis on te relevant group of risk-averse investors tat are described by increasing, concave utility functions. Altoug riskaverse investors prefer more money to less money, te gain in utility from an additional unit decreases wit te income level. In te past, neiter statewise nor rst-order stocastic dominance wit respect to te terminal payo s of te two strategies and increasing utility

22 22 Rudi Zagst, Julia Kraus functions, respectively, could be con rmed. However, by considering riskaversion in our stocastic dominance analysis we were able to derive speci c conditions for te market parameters as well as te CPPI multiplier m implying te second- and tird-order stocastic dominance of te CPPI strategy. More precisely, second-order stocastic dominance was based on te value m =, wereas we were able to derive an interval for te value of te multiplier m inducing tird-order stocastic dominance. e resulting admissible multipliers signi cantly depend on te parameterization of te underlying nancial market. More precisely, te CPPI strategy is more likely to stocastically dominate te OBPI strategy in tird-order te iger te implied volatility i. So far we excluded te default risk of stocks and bonds in our analysis. e inclusion of default risk would result in a pat-dependency of te CPPI strategy and will be subject of furter researc. A Calculation of te CPPI value, mean and variance Wit respect to te derivation of te value of te CPPI portfolio V t we basically follow te proof of Bertrand and Prigent (2005). However, since for te derivation of te expected value and te variance we especially need te probability distribution of te cusion C t, we brie y present te corresponding proof. Recall tat V t = C t + F t ; E t = mc t ; F t = t V 0 and d t = t rdt: e value of te self- nancing CPPI portfolio at time t 2 [0; ] is given by dv t = (V t mc t ) db t B t + mc t ds t S t = [V t m (V t t V 0 )] B trdt B t + mc t ds t S t = [V t ( m) + m t V 0 ] rdt + mc t ds t S t : Hence, te stocastic dynamics of te cusion C t satisfy dc t = d (V t t V 0 ) = dv t V 0 d t = [V t ( m) + m t V 0 ] rdt + mc t S t ds t V 0 t rdt = [(V t V 0 t ) ( m)] rdt + mc t ds t S t = C t ( m) rdt + mc t S t ds t : Substituting te geometric Brownian motion for te dynamics of te risky asset leads to dc t C t = [m + r ( m)] dt + mdw t :

23 Stocastic Dominance of Portfolio Insurance Strategies 23 By applying Itô s lemma, it can be deduced tat ln C t ln C 0 = m (ln S t ln S 0 ) + ( m) r + 2 m2 t: (22) us, C t = C 0 St S 0 m ( m)(r+ e 2 m2 )t : Substituting te lognormal distribution ln S t N(lnS 0 + ( 2 2 )t; 2 t) for te risky asset S t, we can deduce from (22) tat te cusion C t is lognormally distributed wit ln C t N ln C 0 + r + m( r) 2 m2 2 t; m 2 2 t : Wit respect to te derivation of te mean and variance of te value of te CPPI portfolio V CP P I at te end of te investment orizon, we recall tat te mean and te variance of a lognormally distributed random variable lnx N(; 2 ) are given by (X) = E[X] = e 2 (23) 2 (X) = V ar[x] = e 2+2 e 2 : (24) us, following from te lognormal distribution of te value of te cusion C and te nal portfolio value of te CPPI strategy (9) we obtain and V CP P I V CP P I = V CP P I 0 + C ; = E V CP P I = V0 CP P I + E [C ] = V0 CP P I + e ln C0+[r+m( r) 2 m2 2 ] + m2 2 2 = V0 CP P I + C 0 e [r+m( r)] ; 2 V CP P I = V ar V0 CP P I + C = V ar [C ] = e 2fln C0+[r+m( r) = C 2 0 e 2[r+m( r)] 2 m2 2 ]g+m 2 2 e m2 2 : e m2 2 5 See, e.g., Farmeir (2003), p.299.

24 24 Rudi Zagst, Julia Kraus B Calculation of te mean and variance of te OBPI value Recall te terminal portfolio value of an OBPI strategy at maturity us, V OBP I V OBP I = max fs ; Xg P ut 0 e r = max fs X; 0g + X P ut 0 e r : = E V OBP I = E [max fs X; 0g] + X P ut 0 e r : Substituting te de nition of te upper partial moment (5) and Equation (3) for te strike price X, we obtain X = P ut 0 e r + V OBP I 0 E V OBP I = UP M (S ; X) + V0 OBP I = e Call (S 0 ; X; ; ; 0; ) + V0 OBP I : In order to calculate te variance of te terminal portfolio value 2 we use te common formula 2 V OBP I = E V OBP I 2 i E V OBP I 2 : V OBP I is leads to 2 were V OBP I = E max fs = E (max fs X; 0g + V0 OBP I 2 i E V OBP I 2 X; 0g) 2i + 2 V0 OBP I UP M (S ; X) 2 UP M (S ; X) 2 V0 OBP I 2 + V0 OBP I 2 V0 OBP I UP M (S ; X) = UP M 2 (S ; X) UP M (S ; X) 2 = S 2 0e 2 +2 d + p e 2 Call (S 0 ; X; ; ; 0; ) 2 ; d = S0 ln X p ; d 2 = d p : 2XS 0 e (d ) + X 2 (d 2)

25 Stocastic Dominance of Portfolio Insurance Strategies 25 C Second-order stocastic dominance C. Proof of eorem 2 Recall te expected values of te two portfolio insurance strategies at maturity (0) and (6) E V CP P I = V0 CP P I + V0 CP P I e r e [r+m( r)] ; E V OBP I = V0 OBP I + E [max fs X; 0g] = V0 OBP I + e Call (S 0 ; X; ; ; 0; ) ; were S 0 = V CP P I 0 = V OBP I 0 and Hence, X = P ut 0 e r + S 0 ; P ut 0 = P ut 0 S 0 ; X; r; i ; 0; : E V OBP I E V CP P I, e Call (S 0 ; X; ; ; 0; ) V0 CP P I e r e e, 0 Call (S 0 ; X; ; ; 0; ) CP P I 0 V CP P I put call parity, 0 e r C A e {z } Xe r P ut 0 Call (S 0 ; X; ; ; 0; ) Call S 0 ; X; r; i ; 0; e (m )( r) : (m )( r) (m )( r) C.2 Proof of eorem 3 Recall te set of real functions wit exactly two canges of sign 8 9 H : R! R : >< 89s ; s 2 2 R; >= < 0; s 2 ( ; s S 2 = ) : were H (s) 0; s 2 (s ; s 2 ) ; H 6= 0 >: : >; 0; s 2 (s 2 ; ) e cumulative distribution functions of te two portfolio insurance strategies under consideration are de ned as follows, were V 0 := S 0 = V0 CP P I = V0 OBP I, x 2 R 0 F V OBP I (x) = V OBP I 0 {z } =:a>0 = Q a + (S + max fs X; 0g xa X) + x ;

26 26 Rudi Zagst, Julia Kraus wit and F V CP P I (x) = Q V CP P I x 0 (s X) + 0; s X = s x; s > X B = V0 CP P I + C 0 S0 m e ( m)(r+ 2 m2 ) C {z } {z } Sm xa =a = Q (a + b S m x) : =:b>0 Our goal is to prove tat H (x) = F V OBP I (x) F V CP P I (x) 2 S 2 : erefore, we ave to nd te two points x ; x 2 were te sign of te function H canges, i.e. te intersection points of te cumulative distribution functions F V OBP I and F V CP P I. Notice, tat te asset price S is always positive under te assumption of a geometric Brownian motion as underlying stocastic dynamics. Let x a 0. Since a; b; S > 0, we conclude tat and F V OBP I Hence, (x) = Q (S X) + x a F V CP P I H (x) = F V OBP I = (x) = Q (b S m x a) = 0: (x) F V CP P I 0; x a < 0 Q (S X) ; x a = 0 ; (x) 0; if x a 0; wic implies tat te function H does not cange its sign in ( ; a]. In order for H to be in S 2, it remains to sow, tat te function canges exactly twice its sign in (a; +). Consequently, we are looking for te zeros of H, i.e. te intersection points of te cumulative distribution functions F V OBP I, respectively. is leads for x > a to te condition and F V CP P I F V OBP I (x) = Q (S X) +! x a = Q (b S m x a) = F V CP P I (x) : (25) For our furter calculations, we de ne for s > 0 f V OBP I (s) := (s X) + and f V CP P I (s) := b s m ; m : ese are te payo functions of te two portfolio insurance strategies reduced by te minimum guaranteed terminal portfolio value V 0. Notice, tat te inverse functions of f V OBP I and f V CP P I bot exist in R +. Now, let x > a. Substituting f V OBP I and f V CP P I in Condition (25) leads to F V OBP I, Q f V OBP I (x)! = F V CP P I (x) (S ) x a = Q f V CP P I (S ) x a

27 Stocastic Dominance of Portfolio Insurance Strategies 27 wic is equivalent to f V OBP I us, using s := f (x a) ; V OBP I is equivalent to s = f V CP P I (x a) = f (x a) : V CP P I F V OBP I (x) = F V CP P I (x) f V OBP I (s), f V CP P I (s) = f V OBP I (s) : Hence, te cumulative distribution functions F V OBP I and F V CP P I of te two investment strategies intersect eac oter, i te corresponding payo functions intersect. 6 In order to sow H 2 S 2, we terefore ave to determine te interception points of f V OBP I and f V CP P I. is leads to f V OBP I wic is equivalent to (s)! = f V CP P I (s), (s X) + = b s m ; 0 = b s m ; if 0 < s X s X = b s m ; if X < s: Hence, te two payo functions do not intersect for 0 < s X. In fact, f V OBP I (s) < f V CP P I (s) ; if 0 < s X: o conclude for te case s > X, we de ne te function (s) = f V CP P I (s) = bs m s + X; f V OBP I and searc for zeros of tis function in order to nd te intersection points of te two payo s f V OBP I and f V CP P I. Notice, tat (s) > 0, if 0 < s X: erefore, we try to nd parameter restrictions suc tat te polynomial function possesses exactly one strictly negative minimum. en, as is continuous and diverges to + for s! +, we could conclude tat tere exist exactly two nulls of and tus, exactly two intersection points of te payo functions f V OBP I and f V CP P I, respectively for s > X. (s) 6 Since f V OBP I and f V CP P I furtermore olds tat are bot strictly monotone increasing for x > a, it F V OBP I (x) Q F V CP P I (x), f V CP P I (s) Q f V OBP I (s) :

28 28 Rudi Zagst, Julia Kraus It olds for s > X 0 (s) = m bs m! = 0, s = m ; bm 00 (s) = m (m ) {z} b s m 2 > 0; 8m > : >0 us, for all m > te only extremum s is a minimum ( 00 > 0) wit value m (s m m ) = b + X = (bm) m ( m) bm bm m + X: In order to force te function value at te minimum to be negative, te following restriction must be satis ed: (s )! < 0 m>, (bm) m > m m X: Hence, if (bm) m > m m s ; s 2 and te two payo functions f V OBP I times. We terefore set n s : = min s 2 : = max X; m >, te function as exactly two zeros and f V CP P I intersect exactly two s 2 R + : f V OBP I (s) = f V CP P I o (s) ; o : n s 2 R + : f V OBP I (s) = f V CP P I (s) If m = and s > X tere only exists one interception point, i.e. (s) = bs s + X! = 0, s = X b ; s > X: Notice, tat tis point s actually exists, as e constraint to i.e. b = C 0 S m 0 e ( m)(r+ 2 m2 ) m= = e r < : m m X < (bm) m, m >, can be equivalently transformed m m X < m m were C 0 = S 0 is equivalent to i C 0 S0 m e ( m)(r+ 2 m2 m ) ; m m m m X < C 0 e r m m e (r+ 2 m2 ) ; e r. Hence, m m X < (bm) m ; m > m m e r m m X < C 0 e r e 2 m2

29 Stocastic Dominance of Portfolio Insurance Strategies 29 and tus to m m e r m m < or equivalently C 0 X e r e 2 m m (m )2 m m e r )2 e 2 (m! m m < C 0 X e r : Altogeter, we ave proved for m = or m > and m m e r )2 e 2 (m! m m < C 0 X e r ; tat f V OBP I f V OBP I f V OBP I (s) < f V CP P I (s) > f V CP P I (s) < f V CP P I (s) 8 s < s (s) 8 s < s < s 2 ; (s) 8 s 2 < s; were ( s = X b s : = n ; o if m = min s 2 R + : f V OBP I (s) = f V CP P I (s) ; if m > ( s2 = n+; o if m = s 2 : = max s 2 R + : f V OBP I (s) = f V CP P I (s) ; if m > : As mentioned earlier, te zeros of te function exactly represent te zeros of te function H. us for m = or m m( e r m m ) < C0 ; )2 Xe r if m > e 2 (m H (x) 0, F V OBP I H (x) 0, F V OBP I H (x) 0, F V OBP I (x) F V CP P I (x) F V CP P I (x) F V CP P I Hence, H 2 S, if m = and H 2 S 2, if m > : (x) ; 8x s (z) ; 8s x s 2 (x) ; 8x s 2 :

30 30 Rudi Zagst, Julia Kraus D Proof of eorem 6 Recall te means and variances deduced for te CPPI and te OBPI strategy, were V0 CP P I = V0 OBP I = V 0 = S 0 and C 0 = V 0 e r : E V CP P I = V 0 + V 0 e r e [r+m( r)] ; V ar V CP P I = (V0 ) 2 e r 2 e 2[r+m( r)] e m2 2 ; and E V OBP I V ar V OBP I = V 0 + E [max fs X; 0g] = V 0 + e Call (S 0 ; ; ) ; = UP M2 (S ; X) UP M 2 (S ; X) 2 = S0 2 e 2 +2 N d + p 2XS 0 e N (d ) +X 2 N (d 2) e 2 Call 2 (S 0 ; ; ) ; were d = ln( S 0 p ; d 2 = d p : From te translation teorem for te variance we directly conclude tat E V CP P I 2 i = V ar V CP P I + E V CP P I 2 X )+(+ 2 2 ) = (S 0 ) 2 e r 2 e 2[r+m( r)] e m2 2 + ( S 0 ) S 2 0 e r e [r+m( r)] : Recall, tat X = P ut 0 e r + S 0. From put-call-parity follows Hence, wic leads to E Call S 0 ; r; i + X e r = P ut S 0 ; r; i {z } + S 0 : (26) P ut 0 Call S 0 ; r; i = S 0 e r ; V CP P I 2 i = Call S 0 ; r; i n 2 2 [r+m( e o r)] + m S 0 Call S 0 ; r; i e [r+m( r)] + ( S 0 ) 2 : Furtermore, E V OBP I 2 i = V ar V OBP I = S 2 0 e 2 +2 N + E V OBP I 2 d + p 2 X S 0 e N (d ) +X 2 N (d 2) + (S 0 ) S 0 e Call (S 0 ; ; ) :

31 Stocastic Dominance of Portfolio Insurance Strategies 3 From te Black-Scoles formula for te value of a call option Call (S 0 ; X; ; ; 0; ) we know Hence, E en, E Setting Call (S 0 ; X; ; ; 0; ) = S 0 N (d ) X e N (d 2) : V OBP I 2 i = S0 2 e 2 +2 N d + p i V CP P I 2 E X e S 0 N (d ) + (S 0 ) 2 + e Call (S 0 ; ; ) (2 S 0 X) : i V OBP I 2 is true if and only if Call 2 S 0 ; r; i e 2[r+m( r)] +m S 0 Call S 0 ; r; i [r+m( r)] e S 0 e 2 S 0 e 2 N d + p +e Call (S 0 ; ; ) (2S 0 X) : Call S 0 e 2 ; ; tis is equivalent to i X e N (d ) : = Call S 0 e 2 ; X; ; ; 0; = S 0 e 2 N d + p Xe N (d ) ; e Call 2 S 0 ; r; i e 2(m )( r) +m S 0 Call S 0 ; r; i i e (m )( r) Call (S 0 ; ; ) i S 0 e Call S 0 e 2 ; ; Call (S 0 ; ; ) +Call (S 0 ; ; ) e X e S 0 0: Setting Call S 0 e 2 ; ; Call (S 0 ; ; ) := (S 0 ; ; ) := ; Call (S 0 ; ; ) we conclude tat E V CP P I 2 i E V OBP I 2 i