AMS Computational Finance

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1 AMS 54 - Compuaional Finance European Opions Rober J. Frey Research Professor Sony Brook Universiy, Applied Mahemaics and Saisics frey@ams.sunysb.edu Feb Pu-Call Pariy for European Opions A ime T for opions on he same underlying S a a given srike price K and expiry T, if you buy one call, sell one pu and lend an amoun e -rht L K, hen his resuls in a pay-off a expiry of Max[S(T) K, 0] Max[K S(T), 0] + K. Bu his is simply S(T) iself. Pu-call pariy for European opions is, herefore, a ime SHL = CHL - PHL + e -rht L K () The individual componens a expiry CHTL@redD, -PHTL@blueD and K@blueD and heir oal. -00

2 2 ams-54-lec-04-p.nb 200 CHTL-PHTL+K ö S Soluions o SDEs Unlike he deerminisic case, here is no separae heory of differeniaion for sochasic processes. Differeniaion, as such, is acually defined in erms of inegraion. SDEs Are Inegral Equaions I s imporan o remember d S = mhs, L d + shs, L d W, for 0 (2) is simply anoher way of wriing he inegral equaion +D +D +D d S = mhs u, ul d u + shs u, ul d WHuL, for D Ø 0 and 0 (3) Alhough Iô s lemma provides a powerful ool for applying he ools of differenial calculus o sochasic calculus, someimes he soluion of a problem in sochasic calculus demands ha we hink in erms of he acual inegral equaions. This is in sharp conras o deerminisic calculus, in which differeniaion and inegraion are disinc conceps ha are hen linked by he Fundamenal Theorem. Tesing Soluions via Iô s Lemma In a srong soluion, we assume ha we know he drif, volailiy and Wiener process and mus find a sochasic process S ha saisfies he inegral equaion (7). In pracice, given a candidae soluion, we can check i by applying Iô s lemma o o see if i resuls he arge SDE. Consider he SDE, consan coefficien geomeric Brownian moion, and he candidae soluion d SHL = m SHL d + s SHL d WHL, for 0 (4) SHL = SH0L e Im- ÅÅÅÅ 2 s2 M +s W HL (5) If we apply Iô s lemma o compue d SHL

3 ams-54-lec-04-p.nb 3 d S = S 0 e Im- ÅÅÅÅ 2 s2 M +s WHL KKm - ÅÅÅÅÅ 2 s2 O d + s d WHL + ÅÅÅÅÅ 2 s2 d O (6) Subsiuing (5) ino (6) and cancelling erms gives he desired soluion (4). 3. Risk Free Porfolios and Dela Hedging Forming Risk Free Porfolios Consider he simples realisic model of price dynamics, a geomeric, consan coefficien SDE, d SHL = m SHL d + s SHL d WHL (7) If we have a derivaive whose price F(S(), ) is a funcion of he underlying and ime, hen he dynamics of he derivaive are driven by d FHL = i k j ÅÅÅÅÅÅÅÅ S Now we wish o consider a porfolio m SHL + ÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅ S 2 { z d + ÅÅÅÅÅÅÅÅ s SHL d WHL S PHL = FHL - J S HL SHL (8) (9) which is composed of buying one uni of he derivaive and selling J S HLunis of he underlying so ha d PHL = d FHL - J S HL d SHL (0) Nex subsiue (7) ino (8) Ä d PHL = ÇÅ K ÅÅÅÅÅÅÅÅÅ S - J SHLO m SHL + ÅÅÅÅÅÅÅÅÅ É 2 s2 SHL 2 2 F ÅÅÅÅÅÅÅÅÅÅÅÅ S 2 ÖÑ d + K ÅÅÅÅÅÅÅÅÅ S - J SHLO s SHL d WHL If we now hold J S HL = ê S consan over an infiniesimal inerval d, hen he erm associaed wih dw() disappears leaving d PHL = i k j ÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅ z d S 2 { and we have eliminaed randomness from he porfolio. Noe ha because we can hink of he funcion J S HL as being held consan of a small inerval we do no have o include a erm for dj S HL. The Porfolio P() is ofen called he replicaing porfolio. Inerpreing he Replicaing Porfolio The funcion J S HLis called he hedge raio and he echnique above is called dela hedging. If we assume ha we can mainain he hedge from insan o insan and ha here are no liquidiy limiaions or ransacion coss, hen a arbirage free marke demands ha he riskless porfolio PHL reurns he risk-free rae. This suggess anoher condiion on PHL d PHL = r PHL d () (2) (3) where r is he risk free rae and (3) is he differenial equaion represening a coninuously compounded reurn. If we consider he case of geomeric consan coefficien SDE and he condiion represened by (3) r PHL d = i k j ÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅ z d S 2 { (4)

4 4 ams-54-lec-04-p.nb Subsiuing (9) wih J S HL = ê S ino (4) gives r KFHL - ÅÅÅÅÅÅÅÅÅ S SHLO d = i k j ÅÅÅÅÅÅÅÅÅ Dividing hrough by d and rearranging erms yields ÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅ S 2 z d { 2 s2 SHL 2 2 F ÅÅÅÅÅÅÅÅÅÅÅÅ + r SHL ÅÅÅÅÅÅÅÅÅ S 2 S - r FHL = 0 The is he Black-Scholes PDE (a second order PDE) and i plays a major role in he pricing of derivaives. (5) (6) 4. Black-Scholes Formula The Black-Scholes PDE is a parabolic PDE ha is a form of he well known hea equaion. For he consan coefficien case i has an analyical soluion. In is more general forms i usually mus be solved numerically using eiher Mone Carlo simulaion or finie difference mehods. European Call The analyical soluion is CHSHL, L = SHL FHd L - K e -rht -L FHd 2 L (7) where d = lnhs ê KL + Ir + ÅÅÅÅ 2 s2 M HT - L ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅ s è!!!!!!!!! T - d 2 = d - s è!!!!!!!!! T - (8) (9) and F(d) is he cumulaive sandard Normal disribuion FHdL = d - ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ è!!!!!! 2 p e- ÅÅÅÅ 2 x2 d x (20) European Pu PHL = K e -HT-L FH-d 2 L - S FH-d L (2) Example Consider a call opion a ime = 0 wih a srike price K = 00 and expiry T = 0.25 years. The riskfree rae is r = 6.5%. The underying asse is currenly a he money a S 0 = 00 and has an annual volailiy of s = 30%. Then he curren price of he opion using he Black-Scholes formula is C 0 = If we plo his opion s price over 0 T and 80 S 20, hen we realize he surface

5 ams-54-lec-04-p.nb 5 European Call 20 C S Dividend Paying Socks Coninuous Dividend Rae If a sock reurns dividends wih a consan rae D, hen he Black-Scholes soluion above mus be adjused o reflec he fac ha he opion holder does no receive he dividend paymens. In he replicaing porfolio we mus hedge he opion wih he sock wihou is dividend income. Thus, we mus replace he curren price discouned for he expeced fuure opions, i.e., e -DHT-L SHL, ino he original Black-Scholes formula. This will correcly accoun for his effec. Example Consider he example in 4 above. If he same sock paid a divdend a a consan annual rae of 4%, hen he revised price would be and he value of he call would be C = Fixed Dividend Paymens Similarly, if a sock reurns a fixed dividend d a ime 0 T, hen he Black-Scholes soluion again mus be adjused o reflec he fac ha he opion holder does no receive he dividend paymens. We mus subrac he presen value of he dividend from he curren price, i.e., S - e -dh-l. If here are muliple dividend paymens, hen each is discouned a he riskfree rae and subraced from he curren price. Example Consider he example in 4 above. If he same sock paid a divdend of $2.50 midway hrough he period a = 0.25, hen he revised price would be and he value of he call would be C =

6 6 ams-54-lec-04-p.nb 6. Risk-Neural Measure If we selec a measure P è such ha hen d SHL ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ = m d + s d W P HL SHL d SHL ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ SHL E P C d SHL ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ SHL G = m d = r d + sj ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ m - r s d + d WP HLN m - r ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ s d + d W P HL = d W P è HL d SHL ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ = r d + s d W Pè HL SHL E Pè C d SHL ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ SHL G = r d SHL = e -rht-l E (22) (23) (24) (25) (26) (27) (28) Once we have he risk neural measure, we can also compue he value of any derivaive ha depends upon he securiy, e.g., FHSHL, L = e -rht-l E TLD (29) 7. Mone Carlo Inegraion Noe, however, ha we don necessarily need o know wha P è is, only ha under i he price dynamics saisfies he risk neural measure. Also, noe ha under ha measure d W Pè HL is a Weiner process. We can, herefore, simply assume (29) and proceed from here. In oher words, we replace he acual mean wih he risk free mean and keep he same volailiy. Consider he original example: a call opion a ime = 0 wih a srike price K = 00 and expiry T = 0.25 years. The risk free rae is r = 6.5%. The underying asse is currenly a he money a S(0) = 00 and has an annual volailiy of s = 30%. Recall ha he curren price of he opion using he Black-Scholes formula is C 0 = Under he risk-neural measure he price dynamics are where WHL~ NA0, è!! E. SHL = SH0L e s2 Jr - ÅÅÅÅÅÅÅÅÅ 2 N + s W HL (30) To price a call opion we can generae n random samples of equaion 30 and hen apply equaion 4. Le S HkL ITM designae he kh such experimen, hen he Mone Carlo esimae for he inegral represened by (30) is e-rht -L n CHSH0L, 0L = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ maxas HkL HTL - 00, 0E n k= (3)

7 ams-54-lec-04-p.nb 7 Applying sandard saisical mehodology we can compue he sandard deviaion of he esimae by dividing he sandard deviaion of he sample by square roo of he sample size. A simulaion wih n = 000 yielded a mean of and a sandard deviaion of We repeaed his process for prices of 90 and 0. The resuls wih ±.96 sandard deviaion error bars are ploed agains he Black-Scholes resuls below. Long Call Mone Carlo Simulaions sample size = 000, sample mean.96 sdev CH0L * - *- * SH0L Alhough we had an analyical soluion for SHTL, we could have proceeded direcly from he SDE under he risk neural measure and simulaed a number of rajecories o consruc a suiable Mone Carlo sample. In cases involving complex condiions, paricularly hose which are pah dependen, an analyical soluion is no an alernaive. The approach of replacing he acual drif wih he riskfree rae bu preserving he original volailiy srucure is valid under a fairly large number of circumsances. For example, if he riskfree rae or volailiy are hemselves sochasic processes wih heir own SDEs, hen such siuaions can be handled wihou undue complicaion. We have explored only he mos simple sampling approach. There are a number of echniques available o increase he efficiency of Mone Carlo inegraion. Timing esimaes running in Mahemaica for a number of,000 sample analyses ake abou 5 imes longer han a corresponding Black-Scholes evaluaion. One problem of Mone Carlo echinques is he sandard error of he esimae ends o be proporional o ë è!!! n, for a sample size of n. 8. Homework Consider a pu opion a = 0 wih expiry T = year and 6 monhs, ineres rae r = 0.06, srike K = $20, and volailiy s = Use Black-Scholes o plo he curren premium for values of he curren price S(0) from $0 o $60. Use Mone Carlo inegraion o esimae he premium a S(0) = $90, $20 and $50. Plo hese Mone Carlo esimaes wih heir 95% confidence limis ino he firs plo.