Borrowing Constraint and the Effect of Option Introduction

Transcription

1 Insttut des Hautes Etudes Commercales de Carthage From the electedworks of Khaled Bennour 00 Borrowng Constrant and the Effect of Opton Introducton Khaled Amra, uffolk Unversty Khaled Bennour, Insttut des Hautes Etudes Commercales de Carthage Avalable at:

2 Borrowng Constrant and the Effect of Opton Introducton Khaled Amra awyer Busness chool uffolk Unversty Boston, MA 008 UA Tel: (00) Fax: (00) E-mal: Khaled Bennour Insttut des Hautes Etudes Commercales Unversty 7 Novembre Carthage Présdence 006 Tuns, Tunsa Tel: (6) Fax: (6) E-mal: khaled.bennour@hec.rnu.tn Abstract Ths paper studes how optons tradng, by crcumventng constrants on borrowng, permts optmstc nvestors to hold the desred portfolo. Unconstraned nvestors proceed to a portfolo rebalancng by constructng a zero-ncome portfolo that conssts of a short poston n the opton, a long poston n the stock and a short poston n the rskless asset. We show that aggregate demand for the stock s what prevals when optons do not exst and no constrants hold. Furthermore, the opton lstng causes an ncrease n the aggregate demand for the stock and consequently an ncrease n the equlbrum stock prce. JEL classfcaton: G, G3 Keywords: optons; credt constrants; stock prce; arbtrage

3 . Introducton In the tradtonal valuaton of optons the prce process of the underlyng asset s exogenous. The opton prce s derved by usng an arbtrage argument. In ths approach the opton s a redundant asset whose payoff can be replcated by portfolos of prmary assets. Introducng an opton contract has no mpact on the prce of underlyng stock and rsk-sharng possbltes are not modfed. However, when the assumpton of complete and (or) frctonless markets s relaxed, the ntroducton of an opton may affect the prce of the underlyng asset. In the presence of asymmetrc nformaton, the ntroducton of optons affects nformaton revelaton through opton prces and traders (Grossman (988), Back (993), Bas and Hllon (994)). Grossman (988) argues that for an opton that can be replcated by dynamc tradng strateges, ts absence affects the prces of underlyng assets due to the nformatonal content of a traded opton relatve to ts synthetc counterpart. If the nformaton s symmetrc, the mpact of optons tradng s generally analyzed n the context of ncomplete fnancal markets (Hart (975), Detemple (990), Detemple and elden (99)). As shown by Detemple and elden (99), ntroducng optons may expand opportuntes for rsk sharng and wll n general affect the prce of the underlyng assets. A usual justfcaton to the creaton of optons s that they allow the completon of the market (Ross (976)) and the openng of many suffcent securtes allows an effcent allocaton of resources. But ths argument cannot explan the presence of redundant assets lke mutual funds. Fnancal ntermedares create securtes to permt lower transacton costs. Another argument behnd fnancal nnovaton s the exstence of fnancal restrctons. Generally nvestors are lmted n ther ablty to short sell assets and n borrowng. Introducng an opton may be proftable even f the span of the suppled assets s unaffected by the nnovaton. The ntroducton of a redundant opton permts constraned nvestors to crcumvent fnancal restrcton and then mprove ther wealth transfer among dfferent states of nature. Ths leads to a change of ther demand functons on underlyng assets and then prces could be modfed. In ths paper we analyze the role of a redundant opton when some nvestors are not allowed to borrow at the rskless rate for the purpose of nvestng n the underlyng stock. These nvestors are optmstc on the chance of the occurrence of the hgh payoff but are not suffcently wealthy to hold the quantty of stocks as desred. The ntroducton of the opton permts the achevement of maxmum welfare by holdng a long poston n the opton. The presence of unconstraned nvestors ensures the equlbrum of the opton market and the valuaton of the opton by arbtrage. They sell the opton and modfy ther demand for the stock and for the rskless asset. We show that aggregate demand for stock s what prevals n a fnancal market wthout fnancal constrants and wthout opton tradng. One of the man assumptons n ths paper s that nvestors who provde lqudty to the opton market are not subject to wealth constrants. Gârleanu et al. (009) assume that market makers take the other sde of the net demand of prvate nvestors. They cannot hedge ther opton postons perfectly but they do not face fnancal constrants. anta-clara and aretto (009) show that, due to margn requrements and lmted access to captal, non-market makers are restrcted when

4 seekng to wrte &P500 optons. Ths leads them to the concluson that nvestors do not compete wth market makers for supplyng lqudty to the opton market. The rest of the paper s organzed as follows: ecton descrbes the model. ecton 3 analyzes the mpact of ntroducng a redundant opton. The effect of modfcaton the fracton of constraned nvestors on the stock prce s studed n ecton 4. ecton 5 concludes.. The model We consder a sngle good and an exchange economy wth one perod (two dates, zero and one). The fnancal market s composed of three assets: a stock n fxed supply, a call opton wrtten on the stock whch s n zero net supply and a rskless asset n perfectly elastc supply at a prce of one and yeldng a rate of return equal to zero. The prces of the stock and the opton are respectvely p and p 0. We denote by p the equlbrum stock prce n the absence of the opton market and p n the case of opton tradng. We denote v ~ the stock payoff, g = Max( v k,0) the payoff of the opton and k the exercse prce. We normalze the supply of stocks to be one unt. Investors have pror belefs regardng the dstrbuton of the stock payoff. The formaton of expectatons s exogenous to the model. In our model, nvestors are compettve and form a contnuum wth measure. These nvestors are ether borrowng constraned or unconstraned. Investors of the frst type ( = ), n fracton N, have unlmted access to credt. Investors of the second type ( = ), n fracton N, cannot rely on borrowng to buy stocks. At t = 0 nvestors determne ther portfolo. At t = the uncertanty resolves and nvestors consume. Let x and x 0 represent respectvely the shares holdng of the stock and opton, W (0) the frst date wealth and W () the fnal date wealth of nvestor. Let U and U be the utlty of an nvestor of the frst type and second type respectvely, and whch are strctly ncreasng and strctly concave. We assume W (0) W (0), that s nvestors who are subject to the borrowng constrant do not have more ntal wealth than unconstraned nvestors. 3. The effect of opton ntroducng Investors generally use credt or margn to ncrease ther purchasng power so that they can own more stocks wthout fully payng for t. They borrow money from ther broker to buy a stock and use the nvestment as collateral. Accordng to U regulaton, nvestors may borrow up to 50 percent of the purchase prce of securtes that can be purchased on margn. Even f nvestors use the margn system frequently, they are restrcted n ther ablty to rely on ths possblty. In ths secton we analyze the possbltes of tradng created by the ntroducton of a redundant opton n the presence of a borrowng constrant. We consder an extreme stuaton, where some nvestors, who form the second type, are not allowed to borrow but nvestors from the frst type are unconstraned. Many authors have studed the case of agents havng dfferent ablty to borrow at the rskless rate for the purpose of nvestng n rsky assets. In Kyotak and Moore (997), farmers are credt constraned whereas Gathers are unconstraned. In Yuan (005), a fracton of nformed nvestors face borrowng 3

5 constrants. In Detemple and errat (003), some agents are subject to lqudty constrants n that the value of ther portfolos must be nonnegatve at all tmes. Unconstraned agents, however, can trade the stock and the rskless asset wthout restrcton and then use ther labor ncome as collateral for an aggregate short poston. As n our framework, markets are complete from ther pont of vew. In ths secton we assume that the stock payoff can take only two possble values v H and v L where ( 0 < v L < vh ) and ( v L < k < v H ). There are several crtera to characterze a complete fnancal market. In the absence of mperfectons, these crtera are equvalent. The fnancal market s sad to be complete f the number of non-redundant assets s equal to the number of successor states. In a second defnton, a fnancal market s complete whenever nvestors can transfer as much wealth as desred among dfferent states. In our context and because of the borrowng constrant, these two crtera are not equvalent when the opton s not traded. In fact, a constraned nvestor cannot construct the frst Arrow asset, wth payoff (,0). To see ths, suppose the contrary. Then we would have x and x B verfyng x B (,) + x ( vh, vl ) = (,0), whch leads to xb = xvl and x = /( vh vl ). Hence x > 0 and x B < 0, a contradcton to the borrowng constrant. The creaton of the opton leads to the completon of the market by the two crtera. The frst Arrow asset s formed by purchasng /( v H k) optons. Also wth a long poston of / vl stocks and a short poston of vh / vl ( vh k) optons one can construct the second Arrow asset. We then adopt the second defnton and consequently the fnancal market s ncomplete when the opton s not traded. We say that t s more complete when the fracton of unconstraned nvestors ncreases. For unconstraned nvestors the opton contract s redundant. The condton of no-arbtrage opportunty requres that v L < p < vh and p0 = ω+ ωp g = ω + ωv () where ω ( v k) v /( v v ) 0, ω ( v k) /( v v ) 0. = H L H L < = H H L > 3.. Wthout the opton market The wealth W (0) allows nvestor to nvest p x n the stock and W (0) px n the rskless asset. Hs tme wealth s gven by W = W0 + x( v p). Each nvestor chooses hs portfolo at date zero so as to maxmze expected utlty of date one wealth. Let Γ ( ) ( (0) x = EU W + x( v p)). In the absence of the borrowng constrant, the demand for the stock of nvestor, denoted by X, s soluton of equaton Γ ( x ) = 0. The quantty X verfes E[( v p ) U ( W (0) + X ( v p ))] = 0 () 4

6 Investor of type s unconstraned; hs demand functon for the stock s gven by X (.). For the nvestor of type, her demand functon for the stock s the soluton of the program: MaxEU ( W ()) px W (0) For a gven stock prce, f X W (0) / p, her demand for the stock s X but f X > W (0) / p she does not hold the rskless asset and then nvests all her wealth n the stock. The borrowng would allow reachng a superor expected utlty compared to the case when all her wealth s nvested n the rsky asset. A Constraned nvestor, who s optmstc about the chance of the stock apprecaton, would nvest more n the stock f she were allowed to borrow. We are nterested n the stuaton where the borrowng constrant mposed on nvestors of type s bndng. Assumpton We restrct the set of parameters descrbng the economy such that, wth and wthout optons, X > W (0) / p n equlbrum. Ths assumpton ndcates that constraned nvestors are optmstc about the stock payoff. 3.. Wth the opton market The date- wealth of each nvestor s gven by: W = W (0) + x( v p) + x0( g p0) By () we have: W = W (0) + ( x + ωx0)( v p) (3) For the unconstraned nvestor the opton s redundant. Its ntroducton has no mpact on hs expected utlty (for the same stock prce). Hs optmal demands for the stock and opton verfy: x + ωx0 = X (4) The ntroducton of the opton nduces a change of hs demand for the stock so that x + ω x0 corresponds to hs demand for the stock when optons are not traded. We show below that n equlbrum he holds a short poston n the opton ( x < 0 0 ) and then x > X ; ths nvestor sells x 0 optons and buys a quantty of stocks superor to what happens f the opton does not exst. For nvestor the program s to maxmze her expected utlty wth the constrant that p x + p0 x0 W0. MaxE U ( W (0) + x ( v p ) + x ( g p )) 0 0 We could assume that for a fracton of constraned nvestors the borrowng constrant s not bndng n equlbrum. However, ths framework does not change the results of ths paper. 5

7 px + px 0 0 W (0) The exstence of the opton contract, other thngs equal, allows her to ncrease expected utlty of date- wealth. The frst order condtons are: E[( v p) U ( W ())] λ p = 0 E[ ω( v p ) U ( W ())] λp0 = 0 λ( W (0) px p0x0) = 0, λ 0 The frst two equatons mply that λω p = λ p0. From () t comes thatλ = 0. Hence E [( v p ) U ( W ())] = 0. Consequently, we obtan the followng equaton: x + ωx0 = X (5) From equatons (3) and (5) we can state the followng Proposton. Proposton The opton ntroducton permts constraned nvestors to crcumvent mperfectons n that ther expected utlty s what wll be attaned f constrants are nonexstent. The condton of clearng on the opton market s Nx 0 + ( N ) x0 = 0. From (4) and (5) we deduce that the aggregate demand for the stock when the opton s traded s NX + N) X. Then ( Proposton When the opton market s ntroduced, the aggregate demand for the stock s the same as when the opton market does not exst and there are no portfolo constrants. Ths result, consstent wth the fndng of ten (987), permts to conclude to a relaton between openng a dervatve market and the aggregate demand for the underlyng asset. The ntroducton of the opton permts constraned nvestors to choose ther holdngs of rsky assets so that ther wealth constrant s respected and expected utlty s at maxmum. The opton contract allows constraned nvestors to crcumvent market mperfectons. When optons do not exst the clearng of the stock market yelds NX ( p) + ( N) W (0)/ p = (6) In the presence of the opton market, the stock market-clearng condton s NX ( p ) + ( N) X ( p ) = (7) Proposton 3 In equlbrum and under assumpton, constraned nvestors hold a long poston n the opton. Proof: Usng () and (5) we have p x The budget constrant yelds + p 0 x 0 = p X + ω x 0 6

8 x0 ( W (0) px)/ ω (8) The result follows from the assumpton that px > W(0) n equlbrum. In equlbrum, a constraned nvestor holds a long poston n the opton such that (8) s verfed and completes her portfolo by a quantty of stocks that satsfes equaton (5). he has an nfnte number of possbltes for the composton of her optmal portfolo, whch lead to the same equlbrum prces for the stock and opton. As an example she could not hold the rskless asset so the quanttes of stocks and optons n her portfolo are such that x + ω x0 = X and px + px 0 0 = W (0). Even f nvestors of the frst type do not share the optmsm of nvestors of the second type, they sell them the opton and then demand an addtonal quantty of the stock to put a perfect hedge of ther poston on the opton market. If the stock prce was unchanged, the creaton of the opton market ncreases the expected utlty of constraned nvestors but do not modfy the expected utlty of unconstraned nvestors. As we show later, when the opton s created, the aggregate demand for the stock s modfed and then the equlbrum stock prce changes. We consder a second restrcton on the parameters of the economy. Assumpton The demand functons X (.) and X (.) are strctly decreasng n the stock prce. Ths hypothess guarantees, among others thngs, that the equlbrum stock prce s unque. Let us examne the dervatve dx / dp for the arbtrary utlty functon n order to determne suffcent condtons that make Assumpton hold. Dfferentatng () wth respect to stock prce we get: dx E U ( W ()) ( ) ( ()) + X E v p U W = dp E ( v p) U ( W ()) Let R A (.) = U (.) / U (.) denote the absolute rsk averson. The sgn of E[( v p) U ( W ())] depends on the sgn of dr A ( z) / dz (Huang and Ltzenberger (988) page ). Under a strctly decreasng absolute rsk averson, that s when dr A ( z)/ dz < 0, E[( v p) U ( W ())] has the sgn of X and hence assumpton s verfed. The same result holds n the case of a utlty functon of class CARA snce E[( v p) U ( W ())] = 0. In contrast, when dr A ( z) / dz s strctly postve then E[( v p) U ( W ())] and X have dfferent sgns and consequently the sgn of dx / dp s ambguous. Assumpton also holds for preferences of mean-varance type. We can now establsh the followng result on the effect of ntroducng an opton market on the stock prce. Proposton 4 Under assumptons and, ntroducng an opton contract ncreases the equlbrum stock prce. 7

9 Proof: Let us suppose that p p. It follows from assumpton that X ( p ) X ( p ). nce p and p verfy respectvely (7) and (8), then X ( p ) W (0)/ p. Hence X ( ) (0)/ p W p, s a contradcton to assumpton. The same result s derved by Detemple and elden (99) who study the case of an opton contract that does not complete the fnancal market and nvestors have dverse belefs about the rsk of the stock payoff. The emprcal fndngs of Conrad (989) confrm the prce effect of the opton ntroducton and support our analyss of nvestors' behavor. He analyzes 96 optons lsted between 974 and 980 and shows that the prce effect begns three to four days before the opton ntroducton. Also the prce ncrease s postvely related to openng day tradng volume n the opton. These two facts lead Conrad to conclude that some traders are buyng securtes for hedgng purposes n antcpaton of the tradng volume n the opton. Grossman (988) has shown that the ntroducton of an opton that can be syntheszed by exstng assets can have an mpact on the prce of the underlyng asset due to the nformatonal content of the traded opton. In our framework no asymmetrc nformaton holds, however, as Proposton 4 states, the ntroducton of a redundant opton may affect the stock prce because of the mpossblty of borrowng mposed on some optmstc agents. Example Let U ( z) = exp( β z) wth β > 0 for =,. The utlty functons U and U are strctly ncreasng and are strctly concave. nce they are of class CARA then the demand functons X (.) and X (.) are strctly decreasng n the stock prce. We deduce that opton lstng nduces an ncrease n the prce of the underlyng asset for famles of preferences n the CARA class. The opton lstng also modfes the holdngs of the rskless asset. When optons are not traded, only unconstraned nvestors hold the rskless asset. When optons are traded, each unconstraned nvestor holds two portfolos; the portfolo held n the absence of the opton market and a zero-ncome portfolo. The latter portfolo conssts of a short poston n the opton wth a quantty of x 0, a long poston n the stock n quantty wx 0 and a short poston n the rskless asset, n quantty wx 0. We verfy easly that ths portfolo s a zero-ncome portfolo snce wxp 0 + xp 0 0 xw 0 = 0 and wxv 0 + xg 0 xw 0 = 0. nce the supply of the stock s unchanged and p > p, t follows that the optons tradng nduces a decrease n the aggregate holdngs of the rskless asset. 4. Modfcaton of the fracton of constraned nvestors Ths secton consders the case n whch there are only a stock and a rskless asset and we assume that the stock payoff takes an arbtrary dstrbuton functon. We analyze the mpact of a change of the fracton of unconstraned nvestors on the equlbrum stock prce. When N changes, the aggregate demand functon for the stock s modfed and then the equlbrum stock prce vares. Let us assume that, n 8

10 equlbrum, demand X of unconstraned nvestors s not smaller than the unconstraned demand X of constraned nvestors. Assumpton 3 X X n equlbrum. Next, we derve suffcent condtons for assumpton 3 to hold. In the specal case where the utlty functon of nvestors s exponental and the stock payoff can take only two possble values v H and v L ( 0 < v L < vh ), we have γ ( vh p) X = log (9) β( vh vl) ( γ)( vl p) where β denotes the rsk averson and γ the belef about the probablty of the hgh payoff v H ( 0 < γ < ) of agent. We say that agent becomes more optmstc f γ ncreases. If β = β we show easly that the demand functon of unconstraned nvestor X (.) s hgher than the demand functon of constraned nvestor X (.) when γ > γ and the two functons are equal whenγ = γ. In the case where nvestors have the dentcal percepton of the probablty of states of nature ( γ = γ ), they share the same expected return on the stock. By Assumpton, the nvestment n the stock by the constraned nvestor s strctly postve. Consequently hs expectaton on expected return s strctly postve and t s the same for nvestor of type. We deduce from (9) that assumpton 3 holds when unconstraned nvestors are not more rsk averse ( β β). Let us fnally examne the case where the two types of nvestors have arbtrary but dentcal utlty and they agree on the probabltes they assgn to stock payoffs. The unconstraned demand functons for the stock X (.) and X (.) may be dfferent only f W (0) and 9 W (0) are dfferent. Recall that f the nvestment on the stock s postve, then t s an ncreasng functon of ntal wealth when absolute rsk averson s strctly decreasng n wealth (Huang and Ltzenberger (988) page ). By assumpton and snce expectaton on expected return are common to both types of nvestors, then demands X and X are postve n equlbrum. Consequently, assumpton 3 holds n the case of decreasng rsk averson. The effect of varyng the fracton of constraned nvestors s summarzed as follows. Proposton 5 Under assumptons, and 3, the equlbrum stock prce ncreases wth the fracton of unconstraned nvestors. Proof: Usng (6) and dfferentate wth respect to N we get: dp dx W (0) W (0) N ( N) X ( ) = dn dp p p It follows from Assumpton that dx / < 0. Assumptons and 3 mply that dp W (0) / p < X X. Hence 0 dp / dn >.

11 That s because ncreasng the fracton of frst type nvestors act as f some constraned nvestors, who nvest ntally all ther endowment n the stock, ncrease ther holdng of the stock as they now belong to the frst type. nce we assume that the demand of unconstraned nvestor s superor to the demand of constraned nvestor, then the aggregate demand for the stock ncreases whch nduces an ncrease n the equlbrum stock prce. Let us consder the case where the only heterogenety between the two types of nvestors concerns the possblty to borrow. In ths case, ther wealth, utlty functon and subjectve probabltes of successor states are dentcal. It then follows that unconstraned demand functons X (.) and X (.) are dentcal. We denote these demands by X (.). From (6) and (7), the followng proposton s mmedate. Proposton 6 The equlbrum stock prce n the presence of the opton s equal to the stock prce when optons do not exst and N converges to. From the results of Propostons 5 and 6, t follows that the prce effect of ntroducng an opton depends on the mportance of the two types and t s relatvely small when most nvestors are unconstraned ( N s close to one). The fracton N could be seen as a determnant of completon degree of the market. It follows that the prce effect of the opton lstng decreases when the market s becomng more complete. The theoretcal results of Detemple and Joron (988) are smlar. They consder two rsky assets and two nvestors wth dfferent but constant relatve rsk averson. For certan values of parameters, the prce of the frst rsky asset ncreases when an opton on that asset s traded. Ths prce effect lasts but becomes relatvely small when an opton on the second asset s ntroduced. Detemple and Joron (990) examne the mpact of opton lstng n the perod They remark that prce ncrease and volatlty decrease are dsspated after 98. A possble nterpretaton of our prevous result and the emprcal fndng of Detemple and Joron (990) suggest that, as the market becomes nearly complete, the prce effect becomes nsgnfcant n the perod Concluson In ths paper we demonstrate that n a fnancal market wth borrowng constrants the ntroducton of an opton that leaves the span unaltered nduces changes n the demands for the underlyng stock and hence the equlbrum stock prce may change. We consdered two types of nvestors who dffer n ther ablty to borrow at the rskless rate. The ntroducton of the opton permts each nvestor to transfer wealth as desred among dfferent states. Investors of the second type, who are optmstc about the chance of the stock apprecaton but are not allowed to borrow, hold a long poston n the opton. Unconstraned nvestors, who form the frst type of nvestors, supply the opton and then modfy ther demand for the stock. We show that the aggregate demand for the stock s what prevals n a fnancal market wthout optons and wthout constrants. The opton ntroducton has the same mpact on equlbrum allocatons and stock prce as abandonng fnancal mperfecton. Under the condton that demands for the stock are decreasng n the stock prce, the ntroducton of the opton ncreases the equlbrum stock prce. 0

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