The Pricing of Asian Options on Average Spot with Average Strike

Transcription

1 The Pricing of Asian Opions on Average Spo wih Average Srike Marin Krekel Fraunhofer ITWM, Insiue for Indusrial and Financial Mahemaics, Deparmen of Finance, P 3049, Kaiserslauern, Germany, krekel@iwm.fhg.de, Phone: Inroducion Asian Opions, also known as Average Opions, are opions whose paymens a mauriy depends on a in real world discreely moniored average of sock prices. There are wo basic ypes of Asian Opions: Fixed Srike Opions syn. Average Price Opions, Average Rae Opions) and Floaing Srike Opion syn. Average Srike Opions). The firs ype pays a mauriy he difference if posiive beween some arihmeic mean of he sock and a predeermined srike price. The second ype pays a mauriy he difference if posiive beween he arihmeic mean and he sock price a mauriy. Asian Opions are normally European-syle opions. I is no possible o derive an exac) closed-form soluion in he Black-Scholes model, since he sum of log-normally disribued random variables is no log-normal. Numerical approximaions for boh ypes of opions are among ohers: Kemna & Vors 1990) uses Mone-Carlo simulaion wih geomeric mean as conrol variae. Roger & Shi 1995), Zvan, Forshy & Vezal 1997) and Benhamou & Dugue 000) solve he problem applying PDEmehods. Binomial and rinomial rees are no he appropriae mehods o price hese kind of opions, since he branches would no recombine, and hus he calculaion ime would increase exponenially wih refinemen of he ree. Analyical approximaions for Fixed Srike Opions are o name jus a few): Levy 199) uses a log-normal disribuion wih maches he firs wo momens of he disribuion of he arihmeic mean, no o confuse wih Ruiens & Vors 1990), who simply use he geomeric mean. Turnbull & Wakeman 1991) compue an edgeworh expansion o mach he firs four momens based on a log-normal disribuion. Carverhill & Clewlow 1990) and Benhamou 000) apply fourier-ransformaion. Recenly, Henderson & Wojakowski 001) showed he equivalence of Floaing and Fixed Srike Opions. So all his mehods can also be applied o Floaing Srike Opions. We inroduce here a quie new ype of Asian Opion, he so-called Asian Opion on Average Spo wih Average Srike. The payoff of his opion depends on he raio or he difference of wo arihmeic averages of he sock prices. This is going o be specified in he nex chaper. Model We are using he Black-Scholes Model wih deerminisic coefficiens. The saving accoun reads as db) = B)r)d, B0) = B 0, 80 Wilmo magazine

2 TECHNICAL ARTICLE 4 where r) is a deerminisic and bounded funcion of ime. The sock is given by ds) = S)[r) d))d + σ)dw)], S0) = S 0, where d) is he deerminisic and bounded dividend yield and σ) is he deerminisic, bounded and piecewise coninuous volailiy. Le us firs inroduce some noaions: T : mauriy of opion K : srike price α : srike in percen N s : oal number of Spo-Fixings N k : oal number of Srike-Fixings A s = 1 N s Ns i=1 S s i ) A k = 1 Nk S k i N ) k i=1 { 1: Call θ = 1: Pu Call/Pu Operaor) The fixing daes are ordered, i.e. 0 < 1 k <...<k < s Nk 1...< s = T. The Nk payoffs read as: i) Fixed Srike : θ{a s K}) + ii) Average Spo wih Average Srike in Equiy : θ{a s αa k }) + iii) Average spo wih Average Srike in { }) + As Performance : θ α A k Noe ha a Floaing Srike Opion is a special case of an Average Spo wih Average Srike in Equiy Opion wih N s = 1. 3 Approximae Valuaion To our knowledge, up o now, he pricing of hese opions is rarely or even no reaed in lieraure. There are four ways o price hese opions: Approximaion by geomeric means Mone Carlo wih geomeric mean as conrol variae) Approximaion by bivariae log-normal disribuion wih maching momens PDE and FM wih dimension reducion) We implemened and esed he firs hree mehods. Le us firs describe he Approximaion by Geomeric Means: If we replace he arihmeic means by geomeric means, we ge a bivariae log-normal disribuion. Then he problem is equivalen o an exchange opion whose soluion is well-known e.g. see he Collecor s book) and was already reaed in he lae 70 s by Margrabe 1978). As usual we use his approximaion wih or wihou adjused srikes o mach he expecaion of he arihmeic means. Wihou adjusing he srike his approximaion can be used as conrol variae for variance reducion in he Mone Carlo simulaion. We will concenrae here on he hird mehod, namely he Approximaion by bivariae log-normal disribuion wih maching momens. In order o ease he calculaion we inroduce some addiional noaion: The T-forward price a ime is given by: T F T) = S exp rs) ds) ds Hence we can wrie he sock price given S as: T 1 T S T) = F T) exp σ s)ds + σs)dws) Since shor raes are no quoed, we work insead wih discoun facors of he form: Df T) = exp T rs)ds To ake observed prices ino accoun, when we wan o price he opion during he lifeime, we have o disinguish he means of observed- and fuure quoes: A s ) = 1 S s i ) sum of fuure spo-quoes) N s i : s > i A s ) = 1 S s i ) sum of observed spo-quoes) N s i : s i A k ) = 1 S k i ) sum of fuure srike-quoes) N k i : k > i A k ) = 1 ) sum of observed srike-quoes) N k i : k i S k i Our idea is o approximae he arihmeic means by an appropriae bivariae log-normal disribuion wih maching momens. More precisely, we replace A s ) and A k ) by log-normal disribuions X s = expm s + V s Y s ) and X k = expm k + V k Y k ), where Y s, Y k such ha Ys Y k ) = N ) 0, 0 )) 1 ρ. ρ 1 Thus, we choose he parameers such ha he firs and second momens are maching E[A s )] = E[X s ] = expm s + 0.5V s ) E[A s ) ] = E[X s ] = expm s + V s ) E[A k )] = E[X k ] = expm k + 0.5V k ) E[A k ) ] = E[X k ] = expm k + V k ) ^ Wilmo magazine 81

3 and he correlaions coincide E[A k )A s )] = E[X k X s ] = expm s + M k + 0.5V s + 0.5V k + ρv sv k ) which yields M = log E[A)] 1 log E[A) ] V = log E[A) ] log E[A)] ) E[Ak )A s )] log E[A k )]E[A s )] ρ =. V s V k The momens are given by he A s )-momens are similar o hose of A k )): E[A k )] = 1 N k E[A k ) ] = 1 N k i: k i > F k i ) i: k > j: k i E[A s )A k )] = 1 N k N s i: k i > k F i )F k j ) exp > j j: s j > F k i )F s j ) exp min k i,k) j min k i,s) j σs)ds σs)ds Hence we achieve he appropriae disribuion, bu we are no finished ye, since in some cases he derivaion of he expecaion is no rivial. The nex secion will address o his problem. 4 Final Compuaion 4.1 Fixed Srike Opion [ { } )+] V FixS ) = Df T) E θ A s ) + A s ) K [ { } )+] Df T) E θ expm s + V s Y s ) K A s )) The Average-so-far can be pu in he srike price, hence he opion can be reaed as a vanilla opion, 1. Case: K A s )) > 0 V FixS ) = Df T)θ{expM s + 0.5V s )Nθd 1) K A s )) Nθd )}, d = M s logk A s ))/V s, d1 = d + V s,. Case: K A s )) 0 V FixS ) = Df T) θ { expm s + 0.5V s ) K A s ))}) +. In his case our mehod is equivalen o Levy 199). 4. Average Spo wih Average Srike in Equiy V AveSK ) = Df T) E[θ{A s ) + A s )) A k) + A k ))})+ ] Df T) E[θ{expM s + V s Y s ) expm k + V k Y k ) A k ) A s))}) + ] Therefore he opion valuaion depends on A k ) A s ), which we call here Average-Srike-so-far. We disinguish hree cases: 1. Case: A k ) A s )) = 0: In his case we are concerned wih an exchange opion pricing problem, which has been solved by Margrabe 1978). ˆV = Vs + V k ρv sv k d1 = M s M k + 0.5V s V k ) ˆV )/ ˆV d = d1 ˆV V AveSK = Df T)θexpM s + 0.5V s )Nθd 1 ) expm k + 0.5V k )Nθd )). Case: A k ) A s )) > 0: We are now concerned wih a pricing problem which is equivalen o ha of a spread opion. This ype of opions has been reaed in Shimko 1994) and Zhang 1997). We were able o reduce he wo-dimensional inegral o a one-dimensional one by applying condiional expecaion echniques: d = M s loga k ) A s )) Vs φx) = 1 log[1 + expm k + V k x)/a k ) A s ))] Vs 1 V AveSK ) = Df T)θ π [ e Ms+0.5Vs N θ d + ρx + V s + φx + ρ ) V s ) 1 p e Mk+0.5Vk N θ d + ρx + V k ) + φx + ) V k ) 1 p )] A k ) A s ))N d + ρx + φx) θ e x dx 1 p The derivaion of his formula can be found in Krekel 003). 3. Case: A k ) A s )) < 0: Due o he payoff srucure and our approximaion he value of he opion is a funcion of M s, V s, M k, V k, A k ), A s ), ρ and we have he following symmeric relaionship: V AveSK θ, M s, V s, M k, V k, A k ), A s ), ρ) = V AveSK θ,m k, V k, M s, V s, A s ), A k ), ρ) So we if have a CallPu) Opion and he Average-Srike-so-far is negaive we can shuffle he parameers and price i as PuCall) Opion. In his 8 Wilmo magazine

4 TECHNICAL ARTICLE 4 new noaion our Average-Srike-so-far will be posiive, and we can price i as in he second case. 4.3 Average Spo wih Average Srike in Performance V Per ) = Df T) E [ θ Df T) E [ θ { As ) + A s ) }) ] + A k ) + A k ) α 1. Case: A k ) = 0 = A s ) Noe ha: expm s + V s Y s ) d = exp M s M k + Vs expm k + V k Y k ) { expms + V s Y s ) + A s ) }) ] + expm k + V k Y k ) + A k ) α ) + V k ρv sv k Y, where Y is sandard normally disribued. So we have again a vanilla opion, since he bivariae log-normal disribuion collapse o an onedimensional normal disribuion.. Case: A k ) >0 We ge a new ype of opion which we again have o price by numerical inegraion. Since he srike fixing daes are before he spo fixing daes, we know ha he wo evens A s ) 0 and A k ) = 0 canno occur simulaneously. momen maching mehod wih MM, Geo he approximaion by geomeric mean used as MC-conrol variae), Geo Adj geomeric mean wih adjused srikes, and MC Mone Carlo. The numerical resuls are shown in Table 1 o 4. The number in brackes is he Sd Dev of Mone Carlo. The deviaion in he las column is calculaed as Dev = 1 0 MC i V i ), 0 and is used as a performance index for he differen mehods. i=1...and he winner is: Overall MM seems o be he mos efficien and accurae approximaion mehod. In paricular, if he valuaion dae is afer firs fixing see able and able 4) i clearly ouperforms he oher wo approximaion mehods. Wih a sligh excepion in able 3 and here only for high volailiies) i is exremely close o Mone Carlo. I seems o be exremely promissing o apply his mehod o relaed opion ype such as e.g. baskes. TABLE 1. COMPARISON OF AVERAGE SPOT WITH AVERAGE STRIKE IN EQUITY OPTIONS, VALUATION AT STARTING DATE: = 0, N k = N s = 6, i k = i/5, s i = i/5, r = 5%, d = 1%, S = 100. φx) = A k ) + expm k + V k x) { logαφx) A s )) M s d 1 x) = V s : αφx) A s )) > 0 : αφx) A s ) )) 0 [ Ax) = N θ ρx d 1x) α A s ) ] 1 ρ φx) ) Bx) = N θ ρx + V s1 ρ ) d 1 x) 1 ρ 1 φx) exp M s + V s 1 ) ρ ) + ρxv s where N.) is he sandard normal disribuion wih N ) = 0 and N ) = 1. Then he opion value is given by he following one-dimensional inegral which has o be evalued by numerical inegraion: 1 V Per ) = Df T)θ π Bx) Ax))e 1 x dx The proof of his formula can also be found in Krekel 003). 5 Numerical Resuls and Conclusions Since he numerical mehods for Fixed Srike Opions were already compared in many oher papers e.g. Levy & Turnbull 199)), we will concenrae on he Average Spo on Average Srike Opions. We denoe he σ α MC Sd Dev MM Geo Geo Adj ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) Deviaion: 0.018) ^ Wilmo magazine 83

5 TABLE. COMPARISON OF AVERAGE SPOT WITH AVERAGE STRIKE IN EQUITY OPTIONS, VALUATION AFTER STARTING DATE WITH OBSERVED PRICES EQUAL TO FORWARDS: = 0.5, N k = N s = 6, k i = i/5, s i = i/5, r = 5%, d = 1%, S = 100. TABLE 3. COMPARISON OF AVERAGE SPOT WITH AVERAGE STRIKE IN PERFORMANCE OPTIONS, VALUATION AT STARTING DATE: = 0, N k = N s = 6, k i = i/5, s i = i/5, r = 5%, d = 1%, S = 100, Noional = σ α MC Sd Dev MM Geo Geo Adj ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) Deviaion: ) σ α MC Sd Dev MM Geo Geo Adj ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) Deviaion: 0.11) Wilmo magazine

6 TECHNICAL ARTICLE 4 TABLE 4. COMPARISON OF AVERAGE SPOT WITH AVERAGE STRIKE IN PERFORMANCE OPTIONS, VALUATION AFTER STARTING DATE WITH OBSERVED PRICES EQUAL TO FORWARDS: = 0.5, N k = N s = 6, k i = i/5, s i = i/5, r = 5%, d = 1%, S = 100, Noional = σ α MC Sd Dev MM Geo Geo Adj ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) Deviaion: ) REFERENCES BENHAMOU, E. AND A. DUGUET 000): Small Dimension PDE for Discree Asian opions, working paper. BENHAMOU, E. 000): Fas Fourier Transform for Discree Asian Opions, London school of Economics, working paper. CAVERHILL, A. P. AND L. J. CLEWLOW 1990): Flexible Convoluion, RISK, Vol. 3, No. 4, April, 5 9. HENDERSON, V. AND R. WOJAKOWSKI 001): On he equivalence of Floaing and Fixed- Srike Asian Opions, working paper. KEMNA, A. G. Z AND A. C. F. VORST 1990): A pricing mehod for opions based on average asse values, Journal of Banking and Finance, M arz, KREKEL, M. 003): The Pricing of Asian Opions on Average Spo wih Average Srike, working paper, ITWM. LEVY, EDMOND AND STUART TURNBULL 199): Average Inelligence, RISK, Vol. 6, No., February, 5 9. LEVY, EDMOND 199): Pricing European average rae currency opions, Journal of Inernaional Money and Finance, Vol. 11, MARGRABE, WILLIAM 1978): The Valuaion of Opions o Exchange One Asse o Anoher, Journal of Finance, Vol. 33, SHIMKO, D. 1997): Opions on Fuure Spreads: Hedging, Speculaing, and Valuaion, Journal of Inernaional Fuure Markes, Vol. 14, No., TURNBULL, STUART M. AND LEE MACDONALD WAKEMAN 1991): A Quick Algorihm for Pricing European Average Opions, Journal of Financial and Quaniaive Analysis, Vol. 6, No. 3, Sepember, ROGERS L. C. G. AND Z. SHI 1995): The Value of an Asian Opion, Journal of Applied Probabiliy, Vol. 3, RUTTIENS, ALAIN 1990): Classical replica, RISK, May, 7 8. ZHANG, PETER G. 1997): Exoic Opions: A Guide o Second Generaion Opions, World Scienific Publishing Co. Pe. Ld., London. ZVAN, R., P. A. FORSYTH AND K. R. VETZAL 1997): Robus numerical mehods for PDE models of Asian opions, The Journal of Compuaional Finance, Vol. 1, No. Winer 1997/98). W Wilmo magazine 85