Roger Mercken 1, Lisette Motmans 2, Ghislain Houben Call options in a nutshell

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1 No more replicaing porfolios : a simple convex combinaion o undersand he ris-neural valuaion mehod for he muli-sep binomial valuaion of a call opion Roger Mercen, Lisee Momans, Ghislain Houben 3 Hassel Universiy, Faculy of Business Economics, KIZOK, roger.mercen@uhassel.be Hassel Universiy, Faculy of Business Economics, ZW, lisee.momans@uhassel.be 3 Hassel Universiy, Faculy of Business Economics, KIZOK, ghislain.houben@uhassel.be Absrac This paper covers he valuaion, from beginning o implemenaion, of a European call opion on a soc using he muli-sep binomial model in a ris-neural world. The aim is o inroduce his model in a simple bu raher unconvenional way. The usual presenaion of he ris-neural valuaion, see Hull (9), among ohers, relies on replicaing porfolios. For mos praciioners, his echnique loos raher myserious. We presen a new ransparen analysis requiring no replicaing porfolios. The new finding o undersand why he ris-neural pricing is consisen wih invesors being ris-averse is he noion of a convex combinaion. Key words : invesmens, soc, Blac-choles, volailiy Our discussion is summarized as follows. We sar by considering a one-sep binomial valuaion model using he ris-neural principle. Nex, we explain how, involving he no-arbirage principle, ris-neural valuaion maes no assumpion of ris neuraliy. The added value of his paper is he use of a convex combinaion insead of replicaing porfolios o explain why he model is no ignoring he ris. This is he core of he ex. We hen show how he mehod can be generalized o a muli-sep binomial valuaion model. In he nex secion we describe how hisorical daa can be used o esimae he parameers in he model. Finally, we presen a real case o illusrae he model.. Call opions in a nushell A call opion is a conrac ha gives he holder he righ, wihou he obligaion, o buy an asse on or before a specified dae for a guaraneed price K. The price K in he conrac is called he exercise price, he dae in he conrac is nown as he exercise dae or expiraion dae, he purchase price of he opion is called he premium. In he financial world, he underlying asse is generally a share of soc. A European opion can be exercised only on he exercise dae, an American opion can be exercised a any ime during is life. In his paper we limi ourselves o a European call opion on a share of soc paying no dividends and ignoring ransacion coss and axes. We can describe he value of he call opion on he exercise dae in erms of he soc price a expiraion and he exercise price. Le C be he value of he call opion on he exercise dae, he soc price a expiraion and K he exercise price. Evidenly he opion will only be exercised if > K. Hence, C ( ) = max{ K,} I is much more difficul o compue he curren value of he call opion. In his paper he popular binomial valuaion model is used, bu here are alernaives, e.g. he Blac-choles model.

2 . One-sep binomial valuaion model uppose he soc's curren price is, and one period laer he price will eiher increase o u or fall o d wih < d < +r < u and r he ris-free ineres rae per period and generally d <. We do no now he probabiliies of he soc price moving up or down. The ris-free ineres rae r is he rae you can earn by leaving money in ris-free asses such as governmen bonds. Wha abou he inequiies d < +r < u? No-arbirage argumens give he answer. We defer a deailed discussion o he nex secion. Le's calculae he curren value C of a call opion on his soc. uppose an exercise price K and one period ime o exercise dae. Table : soc price and call opion value oc price Time ime Time Call opion value ime u max{u - K,} C? d max{d - K,} We use an imporan general principle nown as ris-neural opion valuaion. The ris-neural principle saes : The curren value C of a call opion is obained by discouning, a he ris-free ineres rae, he expeced opion value a expiraion, compued in a ris-neural world. Recall ha his concep is based on he absence of arbirage opporuniies and will be discussed in he nex secion. Reurn o he problem : ep : A expiraion, he expeced soc price in a ris-neural world mus equal he soc price invesed a he ris-free ineres rae r. olve for p : ( + r) = pu + ( p) d + r d p = < p < u d This probabiliy p is referred o as he ris-neural probabiliy. ep : Use his p o compue he expeced value a expiraion of a call opion on he soc. { u K,} + ( p) max{ d K,} E(C ) = p max ep 3 : The curren value C of he call opion is obained by discouning E(C ) a he ris-free ineres rae r.

3 C = p max { u K,} + ( p) max{ d K,} + r 3. A simple convex combinaion o undersand why he ris-neural principle is no ignoring he ris Because he binomial valuaion model uses he ris-neural probabiliy p and he ris-free ineres rae r, his approach suggess ha we are ignoring he ris. I is worh exploring why he resuling formulas are no jus correc in a ris-neural world, bu in oher worlds as well. In conras o he usual approach, we presen an analysis requiring no replicaing porfolios. For a clear discussion of how a replicaing porfolio can be used o explain why he ris-neural principle is no ignoring he ris, we refer o he famous boo by Hull (9). The general approach adoped by Hull is similar o ha in he imporan seminal paper by Cox, Ross and Rubinsein (979). Numerous oher auhors have aemped o describe his finding, such as ampfli and Goodman (), Capinsy and Zasawnia (3) and McDonald (3). For a simple, exended illusraion of he noion of replicaion, we refer o mar, Megginson and Giman (4) and Bodie, Kane and Marcus (5). For a mahemaically horough discussion, see Eheridge () and Ross (3), among ohers. Now, suppose an asse's curren price is A and he price can increase one period laer o ua wih probabiliy q or fall o da wih probabiliy -q. The expeced price of he asse one period laer is E(A ) = qua + ( q) da A ris-averse invesor requires compensaion for ris aing. o he wans a larger expeced price as ris increases. Consequenly, in a risy world, he asse's curren price is A qua = + ( q)da + R wih R = r + r' he ris-adjused ineres rae, r he ris-free ineres rae and r' he ris premium on he risy asse. An increase in an asse's ris decreases is curren price. In he absence of arbirage opporuniies here is a value p ( < p < ) ha can be subsiued for q o modify R ino he ris-free ineres rae r. Here is a way o illusrae his proposiion. The assumpion of no arbirage requires d < +r < u. We verify hese condiions. uppose ha u + r. An invesor who invesed in governmen bonds would be cerain o mae more profi han invesors holding soc. No one would wan o buy soc. uppose + r d. An invesmen in he soc financed by deb would lead o a cerain profi. The soc would be a grea buy. No one would wan o buy he deb. A real mare would no suppor such soc behaviors. Consequenly, a complee mare wih no arbirage requires d < +r < u. Using elemenary mahemaics, we can find values p ( < p < ) and - p such ha + r = pu + ( p)d In words, + r is a convex combinaion of u and d. 3

4 Muliplying his expression by A and dividing by + r yields he formula A = pua + ( p)da + r Clearly, an asse can be priced using he ris-neural probabiliy p and discouning a he ris-free ineres rae r. Generally, he arificial ris-neural probabiliy p is no equal o he probabiliy of an up movemen. The probabiliy p only yields an asse reurn equivalen o he risless reurn. Hence, i is easy o undersand ha he relaionship beween he curren opion value and he underlying soc in a risy world is he same as i would be in a ris-neural world. Ris-neural valuaion and no-arbirage argumens are equivalen and lead o he same opion values. Risneural valuaion uses he assumpion of no arbirage, bu maes no assumpion of ris neuraliy. I urns ou ha for he purpose of valuaion of call opions he relevan probabiliy is he absrac risneural probabiliy p. The procedure described in his secion, requiring no replicaing porfolios, is unconvenional. The ey is a convex combinaion. 4. Muli-sep binomial valuaion model A soc ha can ae one of only wo possible prices a expiraion is no realisic. We can however generalize he one-sep model o incorporae more realisic assumpions. uppose ha he life of an opion on a soc is divided ino n subinervals. We assume he soc price sars a, and in each period he soc price can increase o u or decrease o d wih < d < +r < u and r he risfree ineres rae per period (subinerval) and generally d <. Le's calculae he curren value C of a call opion on his soc. Table : soc price ree Time ime ime ime 3 ime n u 3 u u u d ud u d n- d ud d d 3 A any node he srucure is idenical. Firs, solve for p : + r = pu + ( p) d o find he ris-neural probabiliy p. The price of he soc a expiraion for up movemens and n - down movemens is u d every such oucome having probabiliy p ( p). There are n n! = pahs hrough he ree leading o u d. Therefore he probabiliy ha he!(n )! n soc price is u d a expiraion is p ( p). This analysis says ha if he random variable x denoes he number of up movemens, hen x is a binomial random variable wih parameers n and p. I follows ha he expeced value a expiraion of a call opion on he soc is : 4

5 E(C ) = n = n p ( p) n The value of he call opion oday is : max { u d K,} n n C = p ( p) max n ( + r) = { u d K,} m n m The summaion sars wih he smalles m such ha u d K. > Consequenly, he formula for he curren value C of a call opion on a soc can be simplified o : n n C = p ( p) n ( + r) = m ( u d K) The firs sudy o clearly show his finding is he seminal paper by Cox, Ross and Rubinsein (979). 5. Adjusing he model o real soc daa How do we choose he parameers u and d in he binomial valuaion model? We illusrae a convenional mehod for esimaing hese values from hisorical soc prices h h h,,...,. We assume Δ for he ree (from one node o anoher node) equals he Δ for he daa N se. If Δ for he ree does no equal he Δ for he daa se, see ampfli and Goodman (). If we consider he raio in he binomial model as a random variable x hen x is a Bernoulli random variable wih values u and d. Table 3 : Bernoulli random variable x x x probabiliy p -p x u d probabiliy p -p In his secion, he probabiliies p and - p are he probabiliies of he soc price moving up or down. The mean for a Bernoulli random variable x is E(x) = px and he variance is E((x + ( p) x μ) ) = p( p)(x x ) Hence, for x = is μ = pu + ( p)d 5

6 σ = p( p)(u d) In a simple model, we se p = / : u + d μ = u d σ = and deermine u and d : u = μ + σ d = μ σ Reasonable poin esimaors of μ and σ are he sample mean and sample variance compued from h h h he hisorical real-world daa,,..., : N x = N h h = N s = N h ( h = N ) N x N The value μ - is nown as he drif parameer, he parameer σ as he volailiy. 6. Applying he model : he Ficion soc case To illusrae he model jus discussed, we presen a real-world example. We wan o use he binomial valuaion model o calculae he value of a 4-monhs call opion on a Ficion soc, lised on New-Money oc Exchange. We assume Δ for he ree equals one monh. The exercise price is assumed o be he soc's curren price of 87.8 euro (he price on June 3, 9). The monhly ris-free ineres rae is.466% (3% per annum). Using he sequence of monhly hisorical soc prices we firs esimae he parameers μ and σ o compue u and d and hen calculae he curren value of a call opion on his soc. A spreadshee program is useful. In his illusraion we have only daa. Evidenly in pracice we wan much more monhly hisorical soc prices. Table 4 : Ficion soc 6

7 dae (8-9) closing price (euro) h July Augus ep Oc Nov Dec Jan mean : Feb March Apr sandard deviaion : May h h μ x =.397 σ s =.433 u = μ + σ =.6649 d = μ σ = Nex, we fill in he ree. Table 5 : Ficion : soc price ree and call opion values Time ime ime ime 3 ime 4 probabiliy call opion value ime p p 3 (-p) p (-p) p(-p) (-p) 4 This ree allows us o compue he curren value of a call opion on he soc. ep : From one node o he following period, he expeced soc price in a ris-neural world mus equal he soc price invesed a he ris-free ineres rae r. olve for p : + r = pu + ( p)d + r d p = =.4664 u d 7

8 This probabiliy p is he ris-neural probabiliy. ep : Use his p o compue he expeced value a expiraion of a call opion on he soc. E(C ) = p 4 3 (5.68) + 4p ( p) (6.67) + 6p ( p) (8.38) + 4p( p) 3 (.74) + ( p) E(C ) =.9 ep 3 : The curren value C of he call opion is obained by discouning E(C ) a he ris-free ineres rae r. E(C) C = =.87 4 (.466) 4 The curren value of a call opion on a Ficion soc is.87 euro. 7. Conclusion In his paper, we have presened a framewor, from beginning o implemenaion, how he valuaion of a European call opion on a soc wors using he muli-sep binomial model in a risneural world. The ey o undersand why his ris-neural principle is no ignoring he ris is he noion of a convex combinaion. Therefore, our approach, requiring no replicaing porfolios, is unconvenional. References Bodie, Z., Kane, A., Marcus, A.J. (5). Invesmens. New-Yor : McGraw-Hill. Capinsy, M., Zasawnia, T. (3). Mahemaics for finance : an inroducion o financial engineering. London : pringer-verlag. Cox, J.C., Ross,.A., Rubinsein, M. (979). Opion pricing : a simplified approach. Journal of financial economics, 7 (epember) : Eheridge, A. (). A course in financial calculus. Cambridge : Cambridge Universiy Press. Hull, J.C. (9). Opions, fuures, & oher derivaives. London : Pearson Prenice Hall. McDonald, R.L. (3). Derivaives mares. Boson : Pearson Educaion. Ross,.M. (3). An elemenary inroducion o mahemaical finance : opions and oher opics. Cambridge : Cambridge Universiy Press. mar,.b., Megginson, W.L., Giman, L.J. (4). Corporae finance. Mason : Thomson ouh-wesern. ampfli, J., Goodman, V. (). The mahemaics of finance : modeling and hedging. Pacific Grove : Broos/Cole. 8