Cautiousness and Measuring An Investor s Tendency to Buy Options


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1 Cautousness and Measurng An Investor s Tendency to Buy Optons James Huang October 18, 2005 Abstract As s well known, ArrowPratt measure of rsk averson explans a ratonal nvestor s behavor n stock markets whle Kmball s measure of prudence explans a ratonal nvestor s behavor when they make precautonary savngs. What s mssng s a measure of an nvestor s tendency to buy optons. In ths paper we show that cautousness, whch s equvalent to the rato of prudence to rsk averson, s the measure. We also dscuss some propertes of ths measure. The model n ths paper does not assume all nvestors are ratonal utltymaxmsers. Keywords: Cautousness, tendency to buy optons, prudence, rsk averson, optons. JEL codes: D81, G1. Introducton Investors pursue three mportant fnancal actvtes, namely makng savngs, buyng equty, and tradng dervatves. How do we measure the strength of ther motve n dong these actvtes? Pratt (1964) and Arrow (1965) developed the measure of rsk averson, whch s defned as the negatve rato of the second dervatve to the frst dervatve of a utlty functon. Pratt (1964) showed that the hgher an nvestor s measure of rsk averson, the more rsk premum he demands and the less nvestment he makes n equty. Leland (1968) and Kmball (1990) among some others nvestgated how to measure the strength of an nvestor s motve to make precautonary savngs. Kmball (1990) developed the measure of prudence, whch s defned as the negatve rato of the thrd dervatve to the second dervatve of a utlty functon. The hgher an nvestor s measure of prudence, the more precautonary savng he wll make respondng to a rsk n hs wealth. 1
2 An nvestor s tendency to trade dervatves s more complcated to measure. When the measures of rsk averson and prudence are developed an nvestor s actvtes of makng precautonary savngs and buyng stocks are separated; however, when decdng an nvestor s optmal poston n a dervatve t makes lttle sense f hs poston n the underlyng equty s gnored. Hence we cannot separate an nvestor s actvty n the dervatve market from that n the underlyng equty market. Ths s the reason that makes t dffcult to develop a measure of an nvestor s tendency to trade dervatves. In ths paper we show that the rato of prudence to rsk averson (mnus one), measures an nvestor s tendency to buy optons. Ths measure has long been called cautousness though t has never been explaned f ths s really a measure of cautousness. 1 We show that an nvestor wth unformly hgher cautousness has a stronger tendency to buy optons. More precsely, f nvestor has unformly hgher coeffcent of cautousness than j, then nvestor j buys an opton only f does so, and nvestor sells the opton only f j does so, regardless of ther ntal wealth, the underlyng stock prce, and the opton prce; and the reverse s also true. The dea to use cautousness to explan the demand for optons can be attrbuted to Leland (1980). He used cautousness to explan the convexty of an nvestor s optmal payoff functon. More recently, usng a smlar framework, Franke, Stapleton, and Subrahmanyam (hereafter FSS)(1998) nvestgated the mpact of background rsk on nvestors optmal payoff functons n an economy n whch nvestors have dentcal postve constant cautousness. They showed that n such an economy the nvestors wthout background rsk wll have globally concave optmal payoff functons. However, we have a few problems to use the above two models to explan the demand for optons. Frstly, Leland and FSS assume that there are a complete set of optons on the stock wth strke prces from zero to nfnty. Secondly, Leland exogenously assumes a representatve nvestor whose preference determnes the opton prces whle FSS assumes that every nvestor s an expectedutltymaxmzer wth constant cautousness. 2 These two assumptons have a smlar feature, whch s the overall ratonalty of the nvestors. Assumng all nvestors are ratonal utlty maxmzers s questonable accordng to emprcal studes. Thrdly, n both of the above models, an opton buyer wll buy all optons wth strke prces from zero to nfnty whle an opton seller wll sell all optons. Ths can hardly be used to descrbe the realty. Fourthly, above all, both of the models explan the demand for optons n specal cases where nvestors have constant cautousness; they do not, nether they can establsh cautousness as a measure of an nvestor s tendency to buy optons. The model presented n ths paper establshes cautousness as a measure of an nvestor s tendency to buy optons whle overcomng all the above problems. We do not assume a representatve nvestor, nether do we assume that all nvestors are utltymaxmzers. Indeed we do not even assume that all nvestors 1 See Wlson (1968). We have used quotaton marks on cautousness to dstngush t from ts lteral meanng. For brevty, hereafter we wll just wrte t n talc style. 2 See also Secton 6 of ths paper for the dscusson of ther framework. 2
3 are ratonal. All we assume s that there are some nvestors who are ratonal expectedutltymaxmzers, and whose behavors n the optons market are the subject of the research n ths paper. Moreover, we do not assume the market completeness. We assume there s one opton whch the nvestgated nvestor s consderng n hs optmal nvestment strategy. Ths way of modelng s n lne wth the approach we use when we study the demand for equty. For example, when Pratt (1964) and Arrow (1965) developed the measure of rsk averson they assume one rsky asset and then show how rsk averson affects an nvestor s demand for ths asset. After establshng cautousness as the measure of an nvestor s tendency to buy optons, we present some propertes of ths measure. We also dscuss the mpact of background rsk on an nvestor s cautousness hence on hs tendency to buy optons. We show that f an nvestor has HARA class utlty wth postve cautousness then when he has a background rsk, ether addtve or multplcatve, the cautousness of hs derved utlty functon wll be strctly hgher, hence he wll have a stronger tendency to buy optons. Although for the case of addtve background rsk the concluson on the mpact of background rsk s smlar to that n FSS (1998), unlke the result there, ths result s stll vald when there s no complete market of contngent clams on the stock and even when many other nvestors are not ratonal utlty maxmzers. Apart from the papers mentoned above, ths paper s also related to Bennnga and Blume (1985), Brennan and Cao (1996), and Carr and Madan (2001). Bennnga and Blume nvestgated the optmalty of a certan nsurance strategy n whch an nvestor buys a rsky asset and a put on that asset. Brennan and Cao (1996) nvestgated the mpact of asymmetrc nformaton on the demand for optons n an economy wth exponental utlty and normally dstrbuted returns. They concluded that well nformed nvestors tend to buy optons on good news and sell optons on bad news. Carr and Madan dscussed how nvestors preferences and belefs affect ther postons n dervatves. The structure of ths paper s as follows. In the frst secton we ntroduce a measure of nvestors preferences, whch s called cautousness. In the second secton we present the model. In Secton three we establsh an orderng of utlty functons by cautousness and show cautousness s a measure of an nvestor s tendency to buy optons. In the fourth secton we dscuss some propertes of cautousness. In the ffth secton we dscuss the mpact of background rsk on cautousness hence on an nvestor s tendency to buy optons. In Secton sx we dscuss the noton of ncreasng cautousness. The fnal secton concludes the paper. 1 Cautousness To ntroduce the concept of cautousness we frst have to explan the concepts of rsk averson and prudence. Pratt (1964) and Arrow (1965) developed the concept of rsk averson to explan nvestors behavor n the equty market. As nterpreted by Pratt (1964), gven utlty functon u(x), the functon R(x) = 3
4 u (x)/u (x) s a measure of rsk averson. 3 The hgher rsk averson an nvestor has, the larger rsk premum he demands for a small and actuarally neutral rsk. More precsely, the rsk premum demanded by an nvestor wth utlty u(x) wll be approxmately the functon R(x) tmes half the varance of the rsk. It s also shown to be a global measure of rsk averson n the sense that f the functon R(x) of an nvestor s always larger than that of the other, then the former wll demand a larger rsk premum than the latter for any rsk, large or small, at any wealth level. Kmball (1990) developed a theory regardng nvestors precautonary savngs analogous to Pratt s (1964) theory of rsk averson. Absolute prudence s defned as P (x) = u (x)/u (x). The hgher prudence an nvestor has, the more equvalent precautonary premum he demands for a rsk n hs wealth and the more precautonary savngs he makes n response to the rsk. The frst dervatve of rsk tolerance, where rsk tolerance s the nverse of absolute rsk averson, was called cautousness by Wlson (1968) 4. Gven a utlty functon, u(x), ts cautousness s C(x) (1/R(x)) =( u (x)/u (x)). Equvalently t can be defned as the rato of absolute prudence to absolute rsk averson mnus one. Ths can be shown as follows. Gven an ncreasng and concave utlty functon u(x), we have (ln R(x)) = (ln u (x)) (ln u (x)) = (P (x) R(x)) whch can be wrtten as It follows that R (x) = R(x)(P (x) R(x)). (1/R(x)) = R (x)/r 2 (x) =P (x)/r(x) 1. Thus we have C(x) =P (x)/r(x) 1. More explctly we can wrte t as Note that C(x) =u (x)u (x)/u 2 (x) 1. (R(x)) = R 2 (x)(1/r(x)) = R 2 (x)c(x). Thus decreasng absolute rsk averson (hereafter DARA) s equvalent to postve cautousness and constant absolute rsk averson (hereafter CARA) s e quvalent to zero cautousness. It s well known that exponental utlty functons have zero cautousness whle other HARA utlty functons have constant postve cautousness. For example, gven a HARA utlty functon u(x) =(x + a) 1 γ /(1 γ), we have C(x) =1/γ. Now we defne a key concept n ths paper. 3 Throughout ths paper we assume all utlty functons are strctly ncreasng, strctly concave, and three tmes dfferentable. 4 See Wlson (1968). 4
5 Defnton 1 Investor s called to have unformly hgher cautousness than nvestor j f there exsts a constant C such that for any w > 0 and v > 0, C (w) C C j (v), where C (w) and C j (v) are the coeffcents of cautousness of nvestors and j respectvely. It s straghtforward that the condton n the defnton s equvalent to nf w>0 C (w) sup v>0 C j (v). The above concept gves an orderng of utlty functons n terms of ther cautousness. Snce HARA class utlty functons have constant cautousness thus they can be ordered perfectly n ths way. 2 The Model Assume a twodate economy wth startng tme 0 and endng tme 1. Assume there s a rskfree bond traded n the market; the rskfree nterest rate s denoted by r. Assume there s a stock avalable n the market whose prces at tme 0 and 1 are denoted by S 0 and S respectvely. Assumpton 1 Assume the dstrbuton of the stock prce S s contnuous and ts support s an nterval n [0, + ). We denote the support nterval by I. The nterval I can be ether bounded or unbounded. Although we assume that the stock prce follows a contnuous dstrbuton, the result can be easly extended to the dscrete case. Assume there s a convex dervatve wrtten on the stock that s traded n the market. Here we frst clarfy the concept of convex dervatve. Defnton 2 A dervatve wrtten on a stock s called to be convex f ts payoff functon s pecewse contnuously dfferentable and everywhere convex n the stock prce S I, and the convexty s strct for at least one pont. The above defnton ensures that the convex dervatve wll not degenerate to a fracton of the stock; thus t ensures that the nvestment problem of allocatng money to the bond, the stock, and the convex dervatve wll not degenerate to the problem of allocatng money to the bond and the stock only. Note for a call or a put opton wth strke prce K nsde the nterval I, ts payoff functon s convex and the convexty s strct at K. Accordng to the above defnton, such an opton s a convex dervatve. However, f K s not nsde I, then the opton degenerates to the underlyng stock, thus t s not a convex dervatve. Denote the payoff of the dervatve at tme 1 by c(s). Note snce c(s) s a convex functon of S, c(s) s contnuous. Denote the prce of the dervatve at tme 0 by c 0. The nterest rate and the current prces of the stock and the dervatve are determned n the equlbrum of the market. For an ndvdual nvestor, he can only take them as gven from the market. We stress here that we do not assume all nvestors are ratonal utltymaxmzers. We only assume there are some nvestors who are ratonal expectedutltymaxmzers whose behavors n the opton market are the subject of ths 5
6 research. After all, an expectedutlty framework can only deal wth those who are utlty maxmzers. These nvestors are ndexed by =1, 2,...; and they are all prcetakers. Investor s preference s represented by utlty functon u (x). At tme 0 he has ntal wealth w 0. Assume nvestor buys x shares of the stock and y unts of dervatves, and nvests the rest of hs money n the bond, whch s w 0 x S 0 y c 0. Denote nvestor s wealth at tme 1 by w (S; x,y ). We have w (S; x,y )=(w 0 x S 0 y c 0 )(1 + r)+x S + y c(s). For brevty we often wrte w (S; x,y ) smply as w (S). Investor maxmzes the expected utlty of hs endng tme wealth w (S) whle requrng the endngtme wealth beng nonnegatve. That s, max x,y Eu (w (S)), s.t. w (S) 0. (1) We obtan the frst order condtons for an nteror soluton: Eu (w (S))(S (1 + r)s 0 )=0, and Eu (w (S))(c(S) (1 + r)c 0 )=0, whch can be wrtten as E[u (w (S))S] Eu (w (S)) =(1+r)S 0, and E[u (w (S))c(S)] Eu (w (S)) =(1+r)c 0. (2) The soluton, (x,y ), depends on the utlty functon, (S 0,c 0 ), and the ntal wealth of the nvestor gven the nterest rate r and the dstrbuton of the stock prce. We now make the followng assumptons. Assumpton 2 All utlty functons are strctly ncreasng, strctly concave, and three tmes dfferentable. The concavty of the utlty functons guarantees that the second order condton for the expected utlty maxmzaton problem s always satsfed. Before we proceed to the next secton, we frst ntroduce some notaton. Let R (x) denote nvestor s absolute rsk averson,.e., R (x) u (x)/u C (w) denote hs coeffcent of cautousness,.e., C (w) (1/R (x)). Let φ (S) u (w (S))/Eu (w (S)). Then (2) can be wrtten as (x). Let E[φ (w (S))S] =(1+r)S 0, and E[φ (w (S))c(S)] = (1 + r)c 0. (3) Thus φ (S) may be somewhat regarded as nvestor s prcng kernel, whch he uses to prce the stock and the dervatve although he actually has to take the prces as gven from the market. Snce the payoff of the dervatve, c(s), s pecewse contnuously dfferentable, so s w (S). Let δ (S) φ (S)/φ (S). We have δ (S) =R (w (S))w (S). (4) 6
7 3 Measurng the Tendency to Buy Optons In ths secton we wll show that cautousness s a measure of an nvestor s tendency to buy optons. We now present our man result. Theorem 1 (Suffcency) Assume there s an nteror soluton to the nvestment problem (1) for both nvestors and j. If nvestor has unformly hgher cautousness than nvestor j, then nvestor j buys the dervatve only f nvestor does so, and nvestor sells the dervatve only f nvestor j does so, regardless of ther ntal wealth, the stock prce, and the dervatve prce. Before we proceed to prove ths result, we frst explan the sgnfcance of the statements n the theorem. The above result essentally states that f nvestor has unformly hgher cautousness than nvestor j, then nvestor always has a stronger tendency to buy the dervatve regardless of ther ntal wealth, the stock prce, and the dervatve prce. Note an nvestor wth hgher coeffcent of cautousness does not necessarly have hgher rsk averson rato. In fact, gven a certan level of cautousness, we can make an nvestor arbtrarly less rskaverse. For example, suppose an nvestor has HARA class utlty functon u(x) = (x+a)1 γ 1 γ, where γ>0 and a are constant. Obvously the nvestor has constant cautousness 1/γ for any a; γ x+a however, the nvestor s rsk averson rato s equal to, whch depends on a. It s straghtforward to see that ncreasng a to nfnty we can make the nvestor extremely less rsk averse; the concluson from the theorem s that however less rsk averse the nvestor may become, t can never change hs poston n the demand and supply chan of the dervatve. Ths s really a bg surprse consderng that many people thnk that rsk averson explans an nvestor s decson to buy or sell optons. Proof: Note c(s) s globally convex n S and there exsts at least one pont, S,at whch c(s) s strctly convex. Thus nvestor j buys (sells) the dervatve f and only f hs optmal strategy s convex (concave) and the convexty (concavty) s strct for at least one pont, S. Suppose nvestor sells the dervatve but nvestor j does not do so, then w (S) s concave, and for at least one pont, S, the concavty s strct; whle w j (S) s convex. Recall from (4) we have δ (S) =R (w (S))w (S). If w (S) were twce dfferentable, dfferentatng the above equaton, we would obtan δ (S) = C (w (S))δ 2 (S) +R (w (S))w (S) However, snce w (S) s not necessarly twce dfferentable, we do not have the above result. Nevertheless, when S ncreases, the ncrement of δ (S) stll 7
8 conssts of two parts: one s from the ncrement of R (w (S)), and the other s from the ncrement of w (S). And we have δ (S + S) δ (S) s equal to R (w (S + S))w (S + S) R (w (S))w (S) =(R (w (S + S)) R (w (S)))w (S)+R (S + S)(w (S + S) w (S)) It follows that δ (S + n S) δ (S) s equal to n (R (w (S + k S)) R (w (S +(k 1) S)))w (S +(k 1) S) k=1 + n R (S + k S)(w (S + k S) w (S +(k 1) S)) k=1 Let n S = τ>0 and n. From the above equaton we obtan δ (S + τ) δ (S) S+τ S C (w (s))δ 2 (s)ds + π (S + τ,s), (5) where π (S + τ,s) = nf R (w (S + x))(w (S + τ) w (S)), 0 x τ whch s always nonpostve and s strctly negatve for S + τ>s >Ssnce w (S) s concave and the concavty s strct at S. Smlarly we have δ j (S + τ) δ j (S) S+τ S C j (w j (s))δ 2 j (s)ds + π j(s + τ,s), (6) where π j (S + τ,s) = nf R j(w j (S + x))(w j (S + τ) 0 x τ w j (S)), whch s always nonnegatve snce w (S) s convex. Frst consder any nterval n whch the payoff of the dervatve c(s) s contnuously dfferentable. Suppose at one pont n ths nterval, say S, we have δ (S) =δ j (S). If S ncreases slghtly by a small τ, snce C (w (s)) C j (w j (s)) and π (S + τ,s) π j (S + τ,s) then from (5) and (6), δ (S) decreases faster than δ j (S), and we wll have δ (S + τ) δ j (S + τ). We assert that the above nequalty s true not only for small τ > 0 but for all τ > 0 such that S + τ s n the nterval. Ths s because after δ (S) becomes smaller than δ j (S), f t somehow ncreases to the pont such that they are close to each other agan, then agan δ (S) decreases faster than δ j (S), and δ (S) stays smaller than δ j (S) n the whole nterval. Now consder at the ponts where c (S) has jumps. These jumps wll cause the jumps n δ (S) and δ j (S) smultaneously. Snce δ (S) =R (w (S))w (S), where R (w (S)) s postve and globally contnuous whle w (S) s decreasng, when δ (S) jumps, t jumps down. For the opposte reason, when δ j (S) jumps, t jumps up. 8
9 Hence combnng the above two cases, we conclude that δ (S) δ j (S) changes ts sgn at most once from postve to negatve. Because of the strct concavty of w (S) ats = S we conclude that there exsts a neghborhood of S, A, such that δ (S) δ j (S) 0 for all S A {S }. It follows that φ (S) φ j (S) can change ts sgn at most twce. But snce these two prcng kernels both prce the stock correctly, φ (S) φ j (S) must change ts sgn at least twce n the nterval I. Thus t changes ts sgn exactly twce n I. Ths mples that there exst S 1 and S 2, where S 1 <S 2 and S 1,S 2 I, such that φ (S) φ j (S) 0, when S < S 1 and S I, or S 2 < S and S I; φ (S) φ j (S) 0, when S 1 <S<S 2. Moreover, there must exst a neghborhood of S, A 1, such that φ (S) φ j (S) 0 for all S A 1 {S }. Now construct a portfolo of the bond and the stock such that ts payoff equal to the payoff of the convex dervatve at S 1 and S 2. Denote the payoff of the portfolo by L(S). It follows from the strct concavty of w (S) ats = S that there exsts a neghborhood of S, A 2, such that c(s) L(S) 0 for all S A 2 {S }. It can be verfed that φ (S) φ j (S) and c(s) L(S) always have the same sgn. Ths, together wth the concluson that nether of them wll be zero for all S A 1 A 2 {S } and the assumpton that the probablty mass of A 1 A 2 {S } s postve, mples that E(φ (S) φ j (S))c(S) =E(φ (S) φ j (S))(c(S) L(S)) > 0. Ths contradcts the condton that the two prcng kernel both prce the dervatve correctly. Thus the ntal supposton that nvestor sells the dervatve but nvestor j does not do so cannot hold. That s, f nvestor sells the dervatve, so must nvestor j. For the same reason, f nvestor j buys the dervatve, so must nvestor. Ths completes the proof. Q.E.D. The theorem tells us that f nvestor has unformly hgher cautousness than nvestor j, then nvestor always has a stronger tendency to buy the dervatve regardless of ther ntal wealth, the stock prce, and the dervatve prce. Obvously t could happen that one nvestor has unformly hgher cautousness than another nvestor whle they both buy or sell the convex dervatve. The queston s, n ths case could ther coeffcents of cautousness tell somethng about who holds more poston n the dervatve? The answer s, however, more lkely to be negatve. We beleve that cautousness can not tell ths. It s well known that rsk averson determnes the amount of nvestment n rsky assets; however, snce cautousness s not necessarly dependent on rsk averson, t wll be surprsng f t can tell how much poston an nvestor wll hold n the dervatve. Perhaps the above queston s also related to the followng queston. Is cautousness really a measure of cautousness? We understand that a more cautous nvestor may have stronger tendency to buy a protectve put. That s, a more cautous nvestor may tend to use the convex dervatve to make a portfolo nsurance. Thus the queston become, can we tell by an nvestor s 9
10 coeffcent of cautousness that whether he buys the optons to hedge aganst stock rsk? The answer s, agan, more lkely to be negatve. Cautousness may tell who buys optons but t may not tell what the optons are bought for: they may be used to hedge the rsk n the underlyng stock; they may be also used for ther leverage effect. From the theorem we can mmedately nfer the followng result. Corollary 1 Assume there s an nteror soluton to the nvestment problem (1) for both nvestors and j. If nvestors and j have the same constant coeffcent of cautousness, then they ether both buy the opton, or both sell the opton, or both hold zero poston n the opton regardless of ther ntal wealth, the stock prce, and the dervatve prce. Theorem 1 shows that havng unformly hgher cautousness s a suffcent condton for an nvestor to have stronger tendency to buy optons regardless of hs ntal wealth, the stock prce, and the dervatve prce. We now try to show ths condton s also necessary. For purely techncal reasons, we now make the followng assumpton. Assumpton 3 Assume all utlty functons have contnuous thrd dervatve and the support of the stock prce dstrbuton, [a, b], s bounded. The case wth unbounded support of the dstrbuton can be dealt wth by addng some mnor condtons. We recall that we have assumed that the expectedutltymaxmzng nvestors n the market are strctly rskaverse. In a rare case, when the current prces of the stock and the dervatve are equal to the rsk neutral prces, a strctly rsk averse nvestor wll optmally hold zero nvestment n both the s tock and the dervatve. If we use S r and c r to denote the rsk neutral prces of the stock and the dervatve respectvely, when (S 0,c 0 )=(S r,c r ), an nteror soluton to (1) s (x,y )=(0, 0). We now show that for those (S 0,c 0 ) whch are near (S r,c r ), nteror solutons to (1) exst too. Lemma 1 There exsts a neghborhood of (S r,c r ), A, such that for any (S 0,c 0 ) A, an nteror soluton to (1) exsts. Proof: We frst prove that there exsts a neghborhood of (S r,c r ), A, such that for any (S 0,c 0 ) A, a soluton to (3) exsts. Snce the support of the stock prce dstrbuton s bounded, the prce of the stock and the dervatve under the frst stochastc domnance rule s bounded. Let S and S be the lower and upper bounds of the stock prce; let c and c be the lower and upper bounds of the dervatve prce. We now defne a map f(.) on [0, + ) [0, + ) as follows. For a par of stock prce and dervatve prce (S 0,c 0 ), f there s a soluton (x,y ) to (2), then f(s 0,c 0 )=(x,y ). Obvously, f(.) s contnuous. Consder the opposte problem n whch gven a par of (x,y ), we want to solve (2) for (S 0,c 0 ). We assert that there exsts a neghborhood of (0, 0), B, 10
11 such that for any (x,y ) B, the soluton of (S 0,c 0 ) exsts. Ths can be shown as follows. Consder the map g(s 0,c 0 )= 1 1+r (E[φ (w (S))S],E[φ(w (S))c(S)]). For any par of (x,y ), f (x,y ) s close to (0, 0) enough, the map s well defned on [S, S] [c, c]. Obvously, ths s a contnuous map of a nonempty, closed, convex set [S, S] [c, c] nto tself. Accordng to Brouwer s Fxed Pont Theorem, there s always a fxed pont. Thus a soluton to (2) always exsts. Ths proves the asserton. Hence we conclude that there s a neghborhood of (0, 0), B, such that B s a set of mages under map f(.). Snce f(.) s contnuous and B s open, the nverse mage of B s also open. Thus there s a neghborhood of (S r,c r ), A, such that for any (S 0,c 0 ) A, a soluton to (2) exsts. From the above result, notng that when (S 0,c 0 ) (S r,c r ), the soluton to (3) (x,y ) reaches ts lmt (0, 0), we can easly nfer that there exsts a neghborhood of (S r,c r ), Â, such that for any (S 0,c 0 ) Â, an nteror soluton to (1) exsts. Q.E.D. Theorem 2 (Necessty) Assume nvestor j buys the dervatve only f nvestor does so, and nvestor sells the dervatve only f nvestor j does so, regardless of ther ntal wealth and current prces of the stock and the dervatve. Then nvestor has unformly hgher cautousness than nvestor j. Proof: We need only to show that f there does not exst a constant C such that for any w>0 and v>0, C (w) C C j (v) then there s a set of w 0 > 0, w j0 > 0, S 0, and c 0 such that nvestor j optmally holds a long poston n the dervatve whle does not. When y = 0, the frst equaton n (2) becomes 1 E[u (w (S))S] 1+r Eu (w (S)) = S 0 where w (S; x j, 0) = (w 0 x S 0 )(1 + r)+x S. As n the proof for Lemma 1, we can easly show that for any small x n > 0, a soluton of S 0 to the above equaton exsts. Ths mples that there s a seres: {(x n, 0) n =1, 2,...}, where xn s strctly decreasng n n, lm n x n = 0, and for all n, (x n, 0) s the soluton to (2) correspondng to (S 0,c 0 )=(S 0n,c 0n ). Obvously we have lm n S 0n = S r and lm n c 0n = c r. When n s suffcently large, x n s suffcently small; these solutons are obvously nteror solutons to 1). Wthout loss of generalty assume for all n, (x n, 0),n= 1, 2,..., are nteror solutons. 11
12 Accordng to Lemma 1, there exsts a neghborhood of (S r,c r ), A, such that for any (S 0,c 0 ) A, the soluton to (2) exsts. Wthout loss of generalty assume for all n, (S 0n,c 0n ) A. Applyng Lemma 1 we conclude that gven the seres {(S 0n,c 0n ) n =1, 2,...}, there also exst a seres of nteror solutons {(x jn,y jn ) n =1, 2,...} to (1) for nvestor j. Snce lm n (S 0n,c 0n )=(S r,c r ) from the contnuty of the solutons we have lm (x jn,y jn )=(0, 0). n Let nvestors optmal strateges and prcng kernels correspondng to (S 0n,c 0n be marked by an addtonal subscrpt n. Snce there does not exst a constant C such that for any w and v, C (w) C C j (v), that s, for some w 0 and v 0, C (w 0 ) < C j (v 0 ), then there s a neghborhood of w 0, A, a neghbor hood of v 0, B, and a constant α, such that for all w A and all v B, C (w) =<α<c j (v). Let w 0 = w 0 /(1 + r) and w j0 = v 0 /(1 + r). Then snce the support of the stock prce dstrbuton s bounded, there exsts N>0, such that for all n>n, we must have that for all S [a, b], w n (S) A and w jn (S) B. Ths mples that for all S [a, b], C (w n (S)) <α<c j (w jn (S)). Now we assert that for all n>n we must have x jn > 0. Otherwse suppose for some n>n, x jn 0. Frst suppose x jn = 0. constants. Thus from (4), we have In ths case, w (S) and w j (S) are both postve 1 1 ( δ (S) ) = C (w (S)) and ( δ j (S) ) = C j (w j (S)) Snce for all S [a, b], C (w n (S)) <α<c j (w jn (S)), usng the fact that φ n (S) and φ jn (S) both prce the stock correctly, we conclude that φ n (S) φ jn (S) changes ts sgn twce and wll not be zero except for two ponts. Obvously these two prcng kernels can never agree on the prce of the dervatve because the one wth fatter tals wll always gve strctly hgher prces for convex dervatves. Now suppose x jn < 0. Followng the same argument as n the proof for the suffcency, we conclude that the two prcng kernels cannot agree on the prce of the dervatve. Hence for n>n we must have x jn > 0. Thus we have a stuaton where nvestor j buys the dervatve, but nvestor does not do so. Ths completes the proof. Q.E.D. The above result gves an orderng of utlty functons n terms of the tendency to buy optons. Ths orderng s not complete snce not all functons can be ordered n such a way. Note t s strong that we requre one nvestor has unformly hgher cautousness than another. The reason that we need ths strong condton s because we have to deal wth the stuaton where the nvestors may have any optmal postons n the stock market and the bond market. 12
13 4 Propertes of Cautousness We have gven an orderng of utlty functons n terms of ther cautousness. Utlty functons can be ordered n such a way are of specal nterest when we compare nvestors tendency to buy optons. Note snce HARA class utlty functons have constant cautousness thus they are deal canddates for ths purpose. Indeed we wll see that ths orderng of utlty functons s closely related to HARA utlty functons. We have the followng result. Proposton 1 The followng two statements are equvalent. 1. There exsts a constant C>0 such that for any x and y, C (x) C C j (y). 2. We have u (x) =t(x) 1/C, where t(x) s concave, and u j (x) =s(x) 1/C, where s(x) s convex. Proof: Let v(x) =x 1 1/C /(1 1/C). Let u (x) =v (t(x)). Then we have C (x) = v (t(x))(v (t(x))t 2 (x)+v (t(x))t (x)) v 2 (t(x))t 2 (x) 1. Ths can be rewrtten as C (x) =C 1 t (x) Cx t 2 (x). Hence C (x) C s equvalent to t (x) < 0. The result about u j (x) can be proved n the same way. Q.E.D. Surely HARA utlty functons are not the only utlty functons can be ordered n such a way. Assume there are a set of ordered utlty functons; the queston s: do basc operatons on utlty functons preserve the orderng? We have the followng result. Proposton 2 The operaton u(x) : u(ax+b) preserves the orderng of utlty functons. Proof: Let u 1 (x) and u 2 (x) are two of a set of ordered utlty functons such that C 1 (x) C C 2 (x), where C (x) s the cautousness of u (x), =1, 2. We have C (ax + b) = (au (ax + b))(a3 u (ax + b)) a 4 u 2 1= u (ax + b)u (ax + b) (ax + b) u 2 1. (ax + b) It follows that C 1 (ax + b) C C 2 (ax + b). Hence the orderng s preserved. Q.E.D. Whle the above operaton completely preserve the orderng, some operatons may partally preserve t. For example, we have the followng result. 13
14 Proposton 3 If u 1 ((x), u 2 (x),..., u n (x) all have cautousness hgher than a constant then the cautousness of n 1 a u (x) s also hgher than the constant. Proof: The general statement follows from the case u(x) =u 1 (x)+u 2 (x). For ths case, C(x) = (u 1(x)+u 2(x))(u 1 (x)+u 2 (x)) (u 1. 1 (x)+u 2 (x))2 It follows that C(x) = (u 1(x)+u 2(x))((C 1 (x)+1) u 2 1 (x) u (x) Suppose C (x) C, =1, 2, then +(C 2 (x)+1) u 2 2 (x) u (x) ) (u 1 (x)+u 2 (x))2 1. Q.E.D. C(x) (C +1) (u 1(x)+u 2(x))( u 2 1 (x) u (x) + u 2 2 (x) u (x) ) (u 1 (x)+u 2 (x))2 1 C. We also have the followng result. Proposton 4 Gven utlty functon u 1 ((x) and u 2 (x), f they both have cautousness hgher than constant C 0.5, then u(x) u 1 (u 2 (x)) also has cautousness hgher than C; f they both have cautousness lower than constant C 0.5, then u(x) u 1 (u 2 (x)) also has cautousness lower than C. Proof: Let C(x) be the cautousness of u(x). Then C(x) = u 1(u 2 )u 2(u 1 (u 2 )u u 1(u 2 )u 2u 2 + u 1(u 2 )u 2 ) (u 1 (u 2)u u 1 (u 2)u 1, 2 )2 where for brevty the argument of u 2 (x) s omtted. It can be wrtten as C(x) = (C 1(u 2 )+1)u 2 1 (u 2)u u 1 (u 2)u 1 (u 2)u 2 2 u 2 +(C 2 +1)u 2 1 (u 2)u 2 2 (u 1 (u 2)u u 1 (u 2)u 1 2 )2 If C (y) C, =1, 2, where C 0.5, then C(x) (C +1) u 2 1 (u 2 )u u 1(u 2 )u 1(u 2 )u 2 2 u 2 + u 2 1 (u 2 )u 2 2 (u 1 (u 2)u u 1 (u 2)u 1=C 2 )2 If C (y) C, =1, 2, where C 0.5, then C(x) (C +1) u 2 1 (u 2 )u u 1(u 2 )u 1(u 2 )u 2 2 u 2 + u 2 1 (u 2 )u 2 2 (u 1 (u 2)u u 1 (u 2)u 1=C 2 )2 Q.E.D. 14
15 5 Impact of Background Rsk We have shown that cautousness measures an nvestor s tendency to buy optons. In ths secton we nvestgate the mpact of background rsk on an nvestor s cautousness hence on hs tendency to buy optons. We consder both addtve and multplcatve background rsks. Gven a utlty functon, u(x), when there s an addtve (multplcatve) background rsk ɛ, as usual, we call û(x) =Eu(x + ɛ) (û(x) =Eu(xɛ)) the derved utlty functon. We have the followng result. Proposton 5 Assume u(x) has postve thrd dervatve and ts cautousness s hgher than a constant. Then gven a background rsk, ether addtve or multplcatve, the cautousness of the derved utlty functon wll also be hgher than the constant. Proof: We denote the background rsk as ɛ, a random varable. Assume the cautousness of u(x) s hgher than constant C. Let R(x) and P (x) be the rsk averson and prudence of the utlty functon. Then we have P (x)/r(x) C +1. We frst assume the background rsk s addtve. Note for postve a and b we have a + b 2 ab. Thus for any e 1 and e 2, P (x + e 1 ) R(x + e 2 ) + P (x + e 2) R(x + e 1 ) 2 Rearrangng the terms n (7), we have, for any e 1 and e 2, P (x + e 1 )P (x + e 2 ) 2(C + 1) (7) R(x + e 1 )R(x + e 2 ) u (x+e 1 )u (x+e 2 )+u (x+e 2 )u (x+e 1 ) 2(C +1)u (x+e 1 )u (x+e 2 ) (8) Assumng e 1 and e 2 are ndependent and have dentcal dstrbutons as ɛ and takng the expectaton of (8) wth respect to e 1 and e 2, we obtan 2E(u (x + ɛ))e(u (x + ɛ)) 2(C + 1)(E(u (x + ɛ))) 2 (9) Rearrangng the terms n (9), we have ˆP (x)/ ˆR(x) C +1, where ˆR(x) and ˆP(x) are the rsk averson and prudence of the derved utlty. Hence the cautousness of the derved utlty, Ĉ(x) = ˆP (x)/ ˆR(x) 1 C. The proof for the case where the background rsk s multplcatve s vrtually the same. Q.E.D. Corollary 2 Assume a utlty functon s HARA class wth postve cautousness. Gven a background rsk, ether addtve or multplcatve, the cautousness of the derved utlty functon wll be strctly hgher. Proof: Note that a HARA utlty functon has constant cautousness, say C. It follows from Proposton 5 that the cautousness of the derved utlty functon s hgher than C. Note n the proof of Proposton 5 the nequaltes are strct unless the utlty functon has constant R(x) and P (x), that s, t s power utlty whch has zero cautousness. Hence f a utlty functon s HARA class 15
16 wth postve cautousness then gven a background rsk, the cautousness of the derved utlty functon wll be strctly hgher. Q.E.D. The above result shows that f an nvestor has HARA class utlty wth postve cautousness then when he has a background rsk, the cautousness of hs derved utlty functon wll be strctly hgher, hence he wll have a stronger tendency to buy optons. FSS (1998) also studed the mpact of an addtve background rsk on an nvestor s demand for optons. They showed that n an economy n whch nvestors have dentcal constant postve cautousness the nvestors wthout background rsk wll have globally concave optmal payoff functons, whch they nterpreted that an addtve background rsk makes an nvestor more lkely to buy optons. The dfference between Corollary 2 and FSS s man result s worth notng although they gve the smlar concluson. FSS s model reles on the assumpton that there s a complete market of contngent clams on the stock and the assumpton that all nvestors have dentcal cautousness whle Corollary 2 does not need these two assumptons at all. Note Corollary 2 s even vald when many other nvestors are not ratonal utlty maxmzers. Corollary 2 can also be used to gve a smple proof of FSS (1998) man result that n an economy n whch nvestors have dentcal constant postve cautousness the nvestors wthout background rsk wll have globally concave optmal payoff functons. The proof s shown n the appendx. 6 Increasng Cautousness Huang (2000) showed that f the margnal utlty of zero wealth s nfnty then ncreasng (decreasng) cautousness mples decreasng (ncreasng) relatve rsk averson. Snce decreasng relatve rsk averson s slghtly more popular, t may be reasonable to favor ncreasng cautousness. In the followng we gve another argument for ncreasng cautousness. We show that when all nvestors have constant cautousness then the prcng representatve nvestor wll have ncreasng cautousness. The framework used s from Leland (1980) and FSS (1998). Assume n a oneperod economy there are N nvestors and every nvestor s wealth conssts of a portfolo of statecontngent clams on the market portfolo. Let X be the payoff of the market portfolo at the end of the perod. Assumpton 1 Assume that there s a complete market for statecontngent clams on X. Ths assumpton ensures that all nvestors can buy and sell statecontngent clams on X so that, as dscussed n Leland (1980), any nvestor can choose a payoff functon x (X). Assumpton 2 Assume all nvestors are ratonal expectedutltymaxmzers. 16
17 Ths assumpton wll ensure that we can derve a prcng representatve nvestor whose preference determnes the prce of contngent clams. As s well known, the prcng representatve nvestor s preference wll be reflected n the unque prcng kernel, φ(x), whch s determned n the equlbrum of the economy. Note the dfference between ths model and the model used n Secton 2 to develop the measure of an nvestor s tendency to buy optons. There we do not assume the above two assumptons. Let u (x) denote nvestor s utlty functon. Let w 0 be nvestor s ntal endowment, expressed as the fracton of the spot value of the total wealth n the economy. Let x be hs optmal payoff functon respectvely. Then the nvestor has the followng utlty maxmzaton problem: max x Eu (x ). (10) Subject to E(φx )=w 0 E(φX). (11) where E(.) denotes the expectaton operator. In equlbrum, the market s cleared, thus we have x (X) =X. (12) We have the frst order condton u (x )=λ φ(x). (13) Dfferentatng both sdes of (13) wll lead to the followng result: x (X) =R e (X)/R (x ), (14) where R (x) = u /u (x) s nvestor s absolute rsk averson and R e(x) = φ (X)/φ(X) s the prcng representatve nvestor s absolute rsk averson. Dfferentatng both sdes of (14), we obtan x (X) =R 2 e(x)[c (x ) C e (X)]/R (x ), (15) where C (x) s nvestor s cautousness and C e (X) =(1/R e (X)) s the prcng representatve nvestor s cautousness. From (12) and (14) we obtan R e (X) =( R 1 (x )) 1 (16) From (12) and (15), we obtan C e (X) = s C (x ). (17) where s = R 1 (x )/ R 1 (x ). We have the followng result. 17
18 Proposton 6 Assume all nvestors have ncreasng cautousness. Then the prcng representatve nvestor also has ncreasng cautousness. Moreover, the cautousness of the prcng representatve nvestor s strctly ncreasng unless all nvestors have dentcal constant cautousness. Proof: Dfferentatng both sdes of (15), we have x x (X) (X) = e (X) 2R R e (X) R (x ) R (x ) x (X)+C (x )x (X) C e (X). C (x ) C e (X) It can be rewrtten as: x /x =2(P e R e )+(P R )x +(C x C e)/(c C e ), where we have omtted the arguments of the functons. Applyng (14) and (15) to the above equaton and rearrangng the terms, we obtan x (X) = R2 e (C C e )(2(P e R e )+(P R ) R e ) R2 e C e + C R R R x Snce x R e =0wehave R 1 (C C e )(2(P e R e )+(P R ) R e )+ R R 2 e R. C x 2 C e =0. Snce C 0, we have C e 2R e(p e R e ) (C C e )/R + R 2 e (C C e )C /R. From (17) we obtan (C C e )/R = 0. Thus we have C e R 2 e C (C C e )/R. Applyng (16) and (17) we can rewrte t as C e R 2 e( C 2 /R R e ( C /R ) 2 ). Rearrangng the terms, we obtan C e R 3 e( R 1 C 2 /R ( C /R ) 2 ). Applyng the Cauchy nequalty, we obtan C e 0. C e = 0 f and only f C (x) =C j (x) =C s a constant for any and j. Q.E.D. Ths shows f every nvestor has ncreasng cautousness then so does the prcng representatve nvestor. As a specal case, when every nvestor has constant cautousness we have the followng corollary. 18
19 Corollary 3 Assume all nvestors have constant cautousness. Then the prcng representatve nvestor has strctly ncreasng cautousness unless all nvestors have dentcal constant cautousness. Proof: It drectly follows from Proposton 6. The above result shows clearly that f nvestors have dfferent postve constant cautousness, then the prcng representatve nvestor wll not have constant cautousness. Instead he wll have strctly ncreasng cautousness. 7 Conclusons In ths paper we have shown that cautousness, whch s equvalent to the rato of prudence to rsk averson, s a measure of an nvestor s tendency to buy optons. We have shown that f nvestor has unformly hgher coeffcent of cautousness than j, then nvestor j buys an opton only f does so, and nvestor sells the opton only f j does so, regardless of ther ntal wealth, the underlyng stock prce, and the opton prce; and the reverse s also true. It s nterestng to see that the measure of an nvestor s tendency to buy optons s closely related to both measures of rsk averson and prudence whch explan nvestors actvtes n the bond market and stock market. Regardng the latter two actvtes, t s now wdely accepted that nvestors should have decreasng absolute rsk averson whle Kmball (1993) proposed decreasng absolute prudence. It s also sad that nvestors are more lkely to have decreasng relatve rsk averson. As Huang (2000) showed that ncreasng (decreasng) cautousness mples decreasng (ncreasng) relatve rsk averson gven that margnal utlty of zero wealth s nfnty, then ncreasng cautousness may be more lkely. Accordng to the model n ths paper, an nvestor s cautousness can tell f he has a stronger tendency to buy optons than others. However, f two nvestors both buy optons, ther coeffcents of cautousness cannot tell f one buys more optons than the other. Moreover, although the model n ths paper shows cautousness tells who buys optons but t does not tell what the optons are bought for: they may be used to hedge the rsk n the underlyng stock; they may be also used for ther leverage effect. Thus cautousness s not necessarly a measure of cautousness. We have also showed that background rsk wll ncrease the cautousness of HARA class utlty. Further research wll be nterestng to show the mpact of background rsk on the cautousness of a general utlty functon. 19
20 Appendx A Proof of Theorem 3 n FSS (1998) In the economy s descrbed n Secton 6, we have the followng result. [Theorem 3, FSS (1998)] Assume all nvestors have dentcal postve constant cautousness and some nvestors have unnsurable background rsk. Then the nvestors wthout background rsk have concave optmal payoff functons. Proof: Assume all nvestors have dentcal postve constant cautousness C. When nvestor s exposed to background rsk ɛ, the utlty functon u (x )n the utlty maxmzaton problem (10) s replaced by the ndrect utlty functon û (x )=E ɛ (u (x + ɛ )). Thus on the rght hand sde of (17) C (x ) s replaced by the cautousness of nvestor s derved utlty functon, Ĉ (x ), f nvestor has background rsk. For the nvestors wthout background rsk, C (x )=C s a postve constant. From Corollary 2, we know that for every nvestor who has background rsk, Ĉ (x ) > C. Thus from Equaton (17), we easly verfy that the prcng representatve nvestor s cautousness s strctly hgher than those of the nvestors wthout background rsk. From (15) the optmal payoff functons of those wthout background rsk are strctly concave. Q.E.D. 20
21 REFERENCES 1. Amersh, A. H. and J. H. W. Stoeckenus, The Theory of Syndcates and Lnear Sharng Rules, Econometrca 51 (1983), Arrow, Kenneth J. (1965), Aspects of a Theory of Rsk Bearng, Yrjo Jahnsson Lectures, Helsnk. Reprnted n Essays n the theory of Rsk Bearng (1971). Chcago: Markham Publshng Co. 3. Bennnga S. and M. Blume (1985), On the Optmalty of Portfolo Insurance, Journal of Fnance 40, No. 5, Bennnga, S. and J. Mayshar, (2000), Heterogenety and Opton Prcng, Revew of Dervatves Research 4, Borch, K. (1962), Equlbrum n a Rensurance Market, Econometrca 30, Brennan, M. J. and H. H. Cao (1996), Informaton, trade, and dervatve securtes, Revew of Fnancal Studes 9, No.1, Brennan, M. J. and R. Solank (1981), Optmal Portfolo Insurance, Journal of Fnancal and Quanttatve Analyss 16, Carr P. and D. Madan, 2001, Optmal Postonng n Dervatve Securtes, Quanttatve Fnance 1, Eeckhoudt L. and H. Schlesnger (1994), A Precautonary Tale of Rsk Averson and Prudence. B. Muner and M. J. Machna (eds.), Models and Experments n Rsk and Ratonalty, 7590, Kluwer Academc Publshers. Prnted n the Netherlands. 10. Franke, G., R. C. Stapleton, and M. G. Subrahmanyam (1999), When are Optons Overprced: The BlackScholes Model and Alternatve Characterzatons of the Prcng Kernel, European Fnance Revew 3, Franke, G., R. C. Stapleton and M. G. Subrahmanyam (1998), Who Buys and Who Sells Optons, Journal of Economc Theory 82, Goller, C. and C. J. W. Pratt (1996), Rsk Vulnerablty and the temperng Effect of Background Rsk, Econometrca 49, Huang C. and R. Ltzenberger (1988), Foundatons for Fnancal Economcs. PrentceHall Canada, Incorporated. 14. Huang, James (2000). Relatonshps between Rsk Averson, Prudence, and cautousness. Lancaster Unversty Management School workng paper. 15. Huang, James (2004). Rsk Neutral Probabltes and Opton Bounds: A Geometrc Approach. Lancaster Unversty Management School workng paper. 21
22 16. Huang, James (2004a). Opton Bounds and Second Order Arbtrage Opportuntes. Lancaster Unversty Management School workng paper. 17. Khlstrome, Rchard E., Davd Rome, and Steve Wllams (1981): Rsk Averson wth Random Intal Wealth, Econometrca 49, Kmball, Mles S. (1990), Precautonary Savng n the Small and n the Large, Econometrca 58, Kmball, Mles S. (1993), Standard Rsk Averson, Econometrca 61, Leland, H. E. (1980), Who Should Buy Portfolo Insurance? Journal of Fnance 35, MasColell, A. (1985), The Theory of General Economc Equlbrum: A Dfferentable Approach. Econometrc Socety Monograph, Cambrdge, Cambrdge Unversty Press. 22. Pratt, J. W. (1964), Rsk Averson n the Small and n the Large, Econometrca 32, Pratt, J. W., and R. Zeckhauser (1987), Proper Rsk Averson, Econometrca 55, Rubnsten, M. E. (1976), An Aggregaton Theorem for Securtes Markets, Journal of Fnancal Economcs 1, Wlson, R. (1968), The Theory of Syndcates, Econometrca 36,
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