In honour of the Nobel laureates, Robert C Merton and Myron Scholes: A partial differential equation that changed the world

Transcription

1 In honour o he Nobel laureae, Rober C Meron and Myron Schole: A parial dierenial equaion ha changed he wld By Rober A Jarrow, Journal o Economic Perpecive, Fall 999 The igin o modern opion pricing hey goe back o Bachelier (900) dieraion on he hey o peculaion. He developed he neceary mahemaic relaed o diuion procee and Brownian moion. Bachelier derived an opion pricing mula ha wa he erunner o he BLACK SCHOLES MERTON PRICING MODEL. Depie he law in Bachelier dieraion, i inpired Samuelon paper on opion pricing and Io wk on ochaic procee. The conribuion o Samuelon and Io in urn helped in he developmen o he BLACK SCHOLES MERTON PRICING MODEL. Sprenkle (96), Samuelon (965) and Meron and Samuelon (969) deermined he opion price uing he maximizing condiion obained rom an inve opimal polio poiion. Their valuaion depended on knowing he expeced reurn rom he ock. Bu expeced reurn can change, accding o he changing ae o inve and changing economic undamenal. BLACK SCHOLES MERTON PRICING MODEL go around hi problem by wking wih a rik ree polio and conequenly wih he rik ree rae o reurn. Fiher Black applied he CAPM model o Sharp and Linner o value he opion in a coninuou ime eing. He wa able o mulae a parial dierenial equaion bu could no olve i o ind ou he opion value. Laer, wih he help o Schole, he wa able o olve he equaion. Subequenly, Meron howed Fiher and Schole how o derive he parial dierenial equaion in a dieren way. Meron approach wa o conruc a perecly hedged polio involving he ock and he call opion uing he noion ha no arbirage oppuniie exi. The inuiion o BLACK SCHOLES MERTON PRICING MODEL coni o hree par: The value o a call opion increae a he price o he underlying ock increae. A h poiion on he underlying ock can be ued o hedge a long poiion on he call opion. We can ind he exac number o hare o h a call opion hereby creaing a rik ree polio which hould earn he rik ree rae o reurn.

2 BLACK SCHOLES MERTON PRICING MODEL doe no need he ock expeced reurn. Bu here are wo key aumpion- conan rik ree inere rae and conan volailiy. Thee aumpion are me valid opion o her mauriy. They are alo me applicable o equiie and equiy indice han o inere rae and eign currency inrumen. Me advanced verion o BLACK SCHOLES MERTON PRICING MODEL do away wih hee aumpion. Two key concep in BLACK SCHOLES MERTON PRICING MODEL are no arbirage and complee marke. No arbirage mean rik ree rading oppuniie do no exi. A complee marke mean ynheic conrucion o any derivaive ecuriy i poible. A combinaion o ock and rik ree bond can example replicae a call opion. Thee concep can be characerized by he Maringale probabiliy diribuion. A maringale i ochaic proce whoe expeced uure value equal he curren value. No arbirage i equivalen o aying ha he ock expeced uure value i he ame a i curren price, appropriaely dicouned. In oher wd, he ock i crecly priced. Change in ock price over ime are caued by random, unanicipaed even. The concep o marke compleene i equivalen o aying ha here i only one Maringale probabiliy diribuion. There i only one value a derivaive ecuriy ha i conien wih he price o he underlying ae. Thi i becaue a derivaive can be conruced by uing he underlying ae, which i crecly priced. BLACK SCHOLES MERTON PRICING MODEL ha been generalized by relaxing variou aumpion: Volailiy can be random, raher han conan. Sock price can jump, inead o moving gradually. Inere rae need no remain conan. In hee arbirage ree erm rucure model, here i an evoluion o muliple erm rucure o uure price inere rae upon which derivaive are

3 wrien. I he no arbirage condiion hold, he opion price depend only on he erm rucure iniial value and volailiie. Implici implied volailiy i anoher impan concep in BLACK SCHOLES MERTON PRICING MODEL. Implied volailiy i ha value o he ock volailiy ha equae he marke price o a call o he BLACK SCHOLES MERTON PRICING MODEL value. Thi volailiy i inerred rom he marke price o he opion, no rom he hiical price. Implici volailiy eem o be a beer predic o uure volailiy han hiical volailiy. The irm equiy can be conidered o be a European call opion on he irm ae wih a rike price equal o he ace value o he deb and a mauriy dae coinciding wih he mauriy o he deb. Wha hi implie i ha deb holder are acually he owner o he company ae. I he irm value i below he value o he deb, he opion canno be exercied. The deb holder can eize all he ae and ell hem o ge back par all o heir due. BLACK SCHOLES MERTON PRICING MODEL alo provide good inigh ino agency hey. The value o equiy i an increaing uncion o volailiy rik. Greaer volailiy increae he probabiliy o an exremely good oucome equiy holder, while heir loe are bounded a zero. By increaing rik, he value o equiy goe up while ha o deb come down. Thi wha he agency co o deb i all abou. Maximizing irm value i no he ame a maximizing he value o equiy. Thi alo lead o he concep o an opimal deb o equiy raio. Ne Preen Value (NPV) ha been he mo popular echnique ued in capial budgeing. Bu NPV doe no conider he opion embedded in mo invemen projec. BLACK SCHOLES MERTON PRICING MODEL can be ued o ind he value o hee opion. Called real opion, hee embedded opion are applicable in a variey o raegic deciion making iuaion like oil explaion, mining and expanion o plan capaciy. Opion pricing ha been ued o model FDIC premium a he value o a pu opion iued by he governmen o he individual bank. Thi pu opion proec he depoi accoun balance rom deaul by he bank. Financial inrumen like lie inurance and penion und guaranee conrac can be priced by and hedged uing BLACK SCHOLES MERTON PRICING MODEL.

4 Addiional noe which may be ueul Io lemma Black and Schole mulaed a parial dierenial equaion which hey laer olved, wih he help o Meron by eing up boundary condiion. To underand heir dierenial equaion, we need o appreciae Io lemma. Conider G, a uncion o x. The change in G a mall change i x can be wrien a: ΔG = dg x dx We can underand hi inuiively by aing ha he change in G i nohing bu he rae o change wih repec o x muliplied by he change in x. I we wan a me precie eimae, we can ue he Tayl erie: ΔG = dg x + dx d G ( dx x) 3 d G ( x) 3 6 dx 3... Now uppoe G i a uncion o wo variable, x and. We will have o wk wih parial derivaive. Thi mean we mu diereniae wih repec o one variable a a ime, keeping he oher variable conan. We could wrie: ΔG = G G x dx d Again, i we wan o ge a me accurae eimae, we could ue he Tayl erie: ΔG = G G G x ( x) x x G G ( x)( ) ( ) xd.. Noe: G ( x)( ) x G G ( x)( ) ( x)( ) x x Suppoe we have a variable x ha ollow he Io proce. Thi ecion and he ollowing draw heavily rom John C Hull, Opion, Fuure and Oher Derivaive, Prenice Hall, 006.

5 dx = a (x,) d+ b(x,) dz Δx = a(x, ) Δ + b(x, ) Δ Δx = a Δ + b Δ ollow a andard nmal diribuion, wih mean = 0 and d deviaion =. We can wrie (Δx) = b Δ + oher erm where he power o Δ i higher han. I we igne hee erm (a Δ approache zero) auming hey are oo mall, we can wrie: Δx = b Δ All he oher erm have Δ wih greaer power. They can be igned. Bu Δx iel i big enough and canno be igned. Le u now go back o G and wrie: G x G G x ΔG = x ( x) Bu Δx = b Δ a we ju aw a lile earlier. I can be hown ha he expeced value o Δ i Δ, a Δ become very mall. Thu (Δx) = b Δ Since we are approaching he limiing cae, we replace ΔG by dg, Δx by dx and Δ by d. So we can wrie he equaion change in G a: dg = G G G dx d b d x x Bu dx = a(x, ) d + b(x,) dz So we can rewrie: dg = G G ( ad bdz) d b x x G d

6 G G G G = ( a b ) d b dz x x x Thi i called Io lemma. The Black Schole dierenial equaion The Io lemma i very ueul when i come o raming he Black Schole dierenial equaion. Le u aume ha he ock price ollow Geomeric Brownian moion, i.e. Or Δ = μδ + Δz Le be he price o a call opion wrien on he ock whoe price i modeled a S. i a uncion o S and. The change in ock price i given by: - ΔS = a (, ) d +b (, ) d. Applying Io lemma, we can relae he change in o he change in S. Comparing wih he general expreion Io Lemma, we ge: G =, a = μ and b =, x =, z Our aim i o creae a rik ree polio whoe value doe no depend on S, he ochaic variable. Suppoe we creae a polio wih a long poiion o hare and a h poiion o one call opion. The value o he polio will be: = - + Value reer o he ne poiive invemen made. So a purchae ge a plu ign and a h ale ge a negaive ign. We will ee laer ha i nohing bu dela

7 and he echnique ued o creae a rik ree polio i called dela hedging. Aume dela i conan over a h period o ime. Change in he value o he polio will be: Δ = - Δ + Δ = - z S Bu Δ = μsδ+sδz Δ = - ) ( z z = - z z S S Δ = - = - Thi equaion doe no have a Δ erm. I i a rikle polio, wih he ochaic riky componen having been eliminaed. The oal reurn depend only on he ime. Tha mean he reurn on he polio i he ame a ha on oher h erm rik ree ecuriie. Oherwie, arbirage would be poible. So we could wrie he change in value o he polio a: Δ = r Δ where r i he rik ree rae. (Becaue hi i a rik ree polio) Bu = - + r ) (

8 Alo ( ) So - = r r r r r Thi i he Black Schole dierenial equaion. I mu be remembered ha he polio ued in deriving he Black Schole dierenial equaion i rikle only a very h period o ime when conan. Wih change in ock price and paage o ime, i can change. So he polio will have o be coninuouly rebalanced o achieve wha i called a perecly hedged zero dela poiion. Thi i alo called dynamic hedging. The Black Schole Fmula C = S0 N(d) Ke -rt N(d) d = [ln (S0/k) + (r+ /) T] / T d = d - T C p = S0 Ke -rt p = C S0 + Ke -rt = S0N (d) Ke -rt N (d) S0 + Ke -rt = Ke -rt [ N (d)] + S0 [N (d) ] = Ke -rt N (-d) S0 N (-d)