c Applied Mathematics & Decision Sciences, 31, 63 73 1999 Reprints Available directly from the Editor. Printed in New Zealand. SOME APPLICAIONS OF OCCUPAION IMES OF BROWNIAN MOION WIH DRIF IN MAHEMAICAL FINANCE ANDREAS PECHL Center of Asset Pricing and Financial Products Development Deutsche Genossenschaftsbank Frankfurt am Main Am Platz der Republik, D 60325 Frankfurt am Main, Germany Abstract. In the last few years new types of path-dependent options called corridor options or range options have become well-known derivative instruments in European options markets. Since the payout profiles of those options are based on occupation times of the underlying security the purpose of this paper is to provide closed form pricing formulae of Black & Scholes type for some significant representatives. Alternatively we demonstrate in this paper a relatively simple derivation of the Black & Scholes price for a single corridor option based on a static portfolio representation which does not make use of the distribution of occupation times of Brownian motion. However, knowledge of occupation times distributions is a more powerful tool. eywords: options. Brownian motion with drift, occupation times, Black & Scholes model, corridor 1. INRODUCION AND DESCRIPION OF HE PROBLEM Since 1973 when Black and Scholes presented their seminal paper concerning the pricing of financial securities [see Black & Scholes 1973], where the no- arbitrage price-process X = X t ; t 0 of the underlying security is described by geometric Brownian motion, questions of option pricing have stimulated both, economics and stochastic calculus [see Harrison & Pliska 1981, aratzas & Shreve 1988, Duffie 1988, et al.]. Recent results have been inspired by so-called Asian options or average options and induced a rigorous study of random variables obtained by application of average functionals to the price-process. A famous problem in mathematical finance has arisen from the application of arithmetic averages to X. o discover an analytic representation of the corresponding distribution has turned out to be a resistant problem for years and has been denoted jestingly a minor holy grail in mathematical finance [see Hart & Ross 1994] till Yor 1992 found a remarkable result using Laplace transformation. Another type of average options are quantile options first introduced by Miura 1992. he α-quantile M α, z of a measurable function z on [0; ] is defined
64 A. PECHL by M α, z = inf k gr 1 [zt k] dt > α Questions concerning the distribution of M α, X are closely related to questions of distribution concerning occupation times of Brownian motion with drift Z = Z t, t 0, i.e. the random variable Γ +, k Z = }. 1 [Z t > k] dt. he distribution of M α, Z was studied successfully by Akahori 1995 and Dassios 1995. hough Lévy s arc- sine law for occupation times of standard Brownian motion is a classical probabilistic result [see Billingsley 1968] it seems that explicit formulae for the distributions of occupation times of Brownian motion with drift have not been available since both, Akahori and Dassios, only provide formulae for densities respectively integral versions for the options prices. However, there is a practical need and interest in such explicit solutions if available, especially when dealing with path-dependent options, where established approximative methods, e.g. numerical integration, Monte Carlo simulation or lattice approximation, often fail or at least consume enormous resources of computation time. Motivated by those practical needs we provide the explicit representation for the distribution of Γ +, k Z [see Pechtl 1997, heorem 1.1 and heorem 3.1] which can be used not only for pricing quantile options, but also for some types of corridor options. While quantile options seem to be highly sophisticated financial products nearly unknown to investors corridor options or range options were introduced to European options markets in 1994 and have become familiar derivative instruments. he payout profile of a traditional single corridor option is given by Ψ 1 λ,, 1, 2 ; z = λ t= 0 1 [ 1 < zt 2 ] dt 1 with, 0 0, λ > 0 and 1 < 2, whereas the more complex structure of a dual corridor option is given by Ψ 2 λ 1, λ 2,, 1, 2 ; z = λ 1 1 [zt 1 ; 2 ]] dt 2 t= 0 λ 2 1 [zt / 1 ; 2 ]] dt + with λ 1, λ 2 gr. he aim of this paper is to provide a general pricing formula for dual switch options, i.e. dual corridor options with 2 =, and related derivative securities in the well-known Black & Scholes model. he paper is structured as follows. In Section 2 the Black & Scholes assumptions and the notations used in the paper are introduced briefly. he main results are
SOME APPLICAIONS OF OCCUPAION IMES OF BROWNIAN MOION 65 presented in Section 3, in particular the explicit version for the Black & Scholes price of a dual switch option and its derivation as an application of the distribution of occupation times of Brownian motion with drift. An alternative and in the considered special case of single corridor options rather elegant method of evaluation of the option s price based on static replication techniques is provided in Section 4. 2. PRELIMINARY REMARS AND NOAIONS hroughout this paper we consider pricing of derivative securities in the Black & Scholes model, where the risk-neutral price-process X = X t, t 0, i.e. the process which has to be used for the calculation of the option s price, can be represented by X t = exp W t + µ BS t}, 0 t, 3 with a one-dimensional standard Brownian motion W = W t, t 0, the initial price of the underlying security, its volatility, r> 1 defined by one plus the risk-free interest rate and the drift µ BS = r 1 2 2 determined by the model. As well known the payout profile of an option is described by a real-valued functional Φ on the measurable space C [0; ] ; C [0; ], the linear space of all continuous functions on the time-interval [0; ] endowed with the -algebra of Borel sets. hen the Black & Scholes price π BS Φ is calculated by π BS Φ = r EΦX. 4 For the following explicit calculations we introduce the univariate and the bivariate standard normal distribution functions by N x = 1 x } exp v2 dv 2π 2 for all x gr and by 1 N x, y; ρ = 2π 1 ρ 2 x v= y for all x, y gr and all ρ 1; 1. w= v= 1 exp v 2 2 1 ρ 2 2ρvw + w 2} dwdv 3. HE PRICE OF A DUAL SWICH OPION For our purposes we introduce the Brownian motion with drift Y = Y t ; t 0 defined by Y t = W t + µ BS t.
66 A. PECHL Furthermore let F the distribution function of the random variable Γ +, X with i.e. Γ +, X = Ft = P 1 [X t > ] dt = = Γ +, 1 Y, 1 [Y t > 1 ] S0 dt 5 [Γ +, 1 ] S0 Y t. By heorem 1.1 and heorem 3.1 in Pechtl 1997 F is explicitly known and can be represented in the following way, P [Γ +, 1 ] S0 Y t = F P [Γ +, 1 ] S0 Y t = 1 F, t, 1 ; µ BS, t, 1 where the function F τ, θ, κ; µ is defined for all κ < 0 and all 0 θ < τ by, ;, >, 6 F τ, θ, κ; µ = 3 + 2κµ + 2µ 2 τ } exp 2κµ} κ N + µ τ, µ τ θ; 1 θ 7 τ τ κ +N µ τ, µ τ θ; 1 θ τ τ 1 + 2κµ + 2µ 2 θ } κ θ exp 2κµ} N + µ θ N µ τ θ κ θ +N µ θ N µ τ θ 8 θ 2µ 2π exp 1 } 2 κ θ 2 µ θ N µ τ θ τ +2µ 2π exp 1 } 2 κ τ 1 2 µ τ N κ θ 1 τ τ θ 2µ 2π exp 1 } κ θ 2 µ2 τ θ exp 2κµ} N + µ θ. For θ < 0 let be F τ, θ, κ; µ 0 and for θ τ let be F τ, θ, κ; µ 1.
SOME APPLICAIONS OF OCCUPAION IMES OF BROWNIAN MOION 67 θ Furthermore we define for κ < 0 the integral J τ, θ, κ; µ = F τ, v, κ; µ dv v= and calculate J τ, θ, κ; µ = 3 + 2κµ + 2µ 2 τ } κ θ exp 2κµ} N + µ τ, µ τ θ; 1 θ τ τ } 1 + 2µ 2 µ2 τ 2 2τ + κ 2 κ exp 2κµ} N + µ τ, µ τ θ; 1 θ τ τ } + θ 1 2µ κ N 2 µ } µ 2 θ 2 + 2κµθ + θ + κ 2 1 κ θ exp 2κµ} N 2µ + µ θ 2 } + θ κ µ 1 κ θ N 2µ µ θ 2 κ τ µ τ, µ τ θ; 1 θ τ N µ τ θ N µ τ θ 9 } } µθ + κ + 1 θ µ 2π exp 1 2 κ θ 2 µ θ N µ τ θ } } + 2µθ µτ 1 τ µ + κ 2π exp 1 2 κ τ 2 µ 1 τ N κ θ 1 τ } } + µ τ θ + 1 τ θ µ 2π exp 2κµ} exp µ2 κ θ 2 τ θ N + µ θ } } θ τ θ + exp µ2 2π 2 τ θ exp 1 2 κ θ 2 µ θ. his formula can be verified immediately by differentiation with respect to θ which is very tedious. An intuitive argument for the correctness of eq. 8 will be provided in Section 4 where we demonstrate that it coincides in the special case of the price of a single switch option with the result derived independently by using a static replication argument. he reader interested in a direct proof of eq. 8 by integration may be referred to the similar proof of heorem 3.1 in Pechtl 1997. In the following we consider the payout profile Φ 2 = Φ 2 λ 1, λ 2,, ; S of a dual switch option with a total lifetime 1 from t = 0 to t =, i.e. 1 = 0 +, at time t = 0. In the time-interval [ 0 ; 0] the price-process S respectively the no-arbitrage process X is described by its historical data. hen Φ 2 λ 1, λ 2,, ; S is determined by } + Φ 2 λ 1, λ 2,, ; S = λ 1 1 [S t > ] dt λ 2 1 [S t ] dt. t= 0 t= 0 From this we immediately obtain Φ 2 λ 1, λ 2,, ; S = λ 1 + λ 2 Γ + 1, S λ 2 1 } +. 10
68 A. PECHL For λ 1 + λ 2 = 0 and if λ 2 > 0 we trivially have Φ 2 λ 1, λ 2,, ; S = 0, i.e. π BS Φ 2 λ 1, λ 2,, ; S = 0, if λ 2 0 we have Φ 2 λ 1, λ 2,, ; S = λ 2 1, i.e. π BS Φ 2 λ 1, λ 2,, ; S = r λ 2 1, respectively. hus for further discussion we only take the case λ 1 + λ 2 0 into consideration. Let now be γ 0 = Γ + 0, S = hen we have for λ 1 + λ 2 > 0 0 t= 0 1 [S t > ] dt and τ 0 = λ 2 λ 1 + λ 2 1 γ 0. Φ 2 λ 1, λ 2,, ; S = λ 1 + λ 2 Γ +, S τ 0 } 1 [Γ +, S > τ 0 ], respectively for λ 1 + λ 2 < 0 Φ 2 λ 1, λ 2,, ; S = λ 1 + λ 2 Γ +, S τ 0 } 1 [Γ +, S < τ 0 ]. We use those representations of Φ 2 to calculate the corresponding prices in the Black & Scholes model. i For λ 1 + λ 2 > 0 we obtain using eqs. 6 to 8 = r λ 1 + λ 2 π BS Φ 2 λ 1, λ 2,, ; S = r E λ 1 + λ 2 Γ +, X τ 0 } 1 [Γ +, X > τ 0 ] = r λ 1 + λ 2 t τ 0 dft t τ 0; ] +r λ 1 + λ 2 = r λ 1 + λ 2 = r λ 1 + λ 2 t τ 0; ] 1 F, t, 1 ; µ BS t [0; τ 0 +r λ 1 + λ 2 J F [ t J ii For λ 1 + λ 2 < 0 we obtain anaously, t, 1 t τ 0; ] ], t, 1 ; µ BS, τ 0, 1 1 Ft dt } dt 1 [ ] dt1 [ > ] 1 [ ] t=τ 0 1 [ > ]. π BS Φ 2 λ 1, λ 2,, ; S = r E λ 1 + λ 2 Γ +, X τ 0 } 1 [Γ +, X < τ 0 ] = r λ 1 + λ 2 t τ 0 dft t [0;τ 0 = r λ 1 + λ 2 = r λ 1 + λ 2 F, t, 1 t [0;τ 0 ; µ BS t [0;τ 0 Ftdt dt1 [ ]
SOME APPLICAIONS OF OCCUPAION IMES OF BROWNIAN MOION 69 r λ 1 + λ 2 1 F, t, 1 t τ 0; ] = r λ 1 + λ 2 J, τ 0, 1 ; µ BS } dt1 [ > ] 1 [ ] [ r λ 1 + λ 2 t J, t, 1 S ] 0 1 [ > ]. t= τ 0 Concluding we obtain the main result of this paper, the Black & Scholes formula for dual switch options. heorem 3.1. Let be Φ 2 λ 1, λ 2,, ; S be the payout profile of a dual switch option as defined by eq. 9. Furthermore let J τ, θ, κ; µ be defined by eq. 8. hen for λ 1 + λ 2 0 the Black & Scholes price of such an option is calculated by where π BS Φ 2 λ 1, λ 2,, ; S = r λ 1 + λ 2 J, θ τ 0, ;, κ ; ; µ ; θ τ 0, ; e λ 1, λ 2 ;, J,, κ ; ; µ ; e λ 1, λ 2, ; }, λ 2 τ 0 = 1 Γ + 0, S, λ 1 + λ 2 θ τ 0, ; = τ 0 1 [ ] + τ 0 1 [ > ], κ ; = 1 1 [ ] 1 [ > ]}, µ ; = µ BS 1 [ ] 1 [ > ]}, e λ 1, λ 2, ; = 1 [λ 1 λ 2 ] 1 [ ] + 1 [λ 1 < λ 2 ] 1 [ > ]. Remark 3.2. he formula for π BS Φ 2 λ 1, λ 2,, ; S remains valid even for λ 0 1 + λ 0 2 = 0 if the option s price is interpreted as limit of the pricing formula given in heorem 3.1 for λ 1 + λ 2 0 with λ 1 + λ 2 > 0. 4. APPLICAION OF HE RESUL O SINGLE SWICH OPIONS An immediate consequence of heorem 3.1 is the explicit formula for the Black & Scholes price of a single switch option, i.e. an option with payout profile Φ 1 λ,, ; S = λ 1 [S t > ] dt t= 0 11 with total lifetime from t = 0 to t = and λ > 0. Obviously this payout profile can be interpreted by Φ 2 λ, 0,, ; S and we can easily provide the Black & Scholes price. With the notations of section 3 we have τ 0 = γ 0 and we obtain
70 A. PECHL i for π BS Φ 1 λ,, ; S = r λ ii for > π BS Φ 1 λ,, ; S = r λj + γ 0 J,, 1 ; µ } BS, = r λγ 0 + r λj, + γ 0, 1 We conclude this result in the following corollary. 2µ 2 BS,, 1 Corollary 4.1. he Black & Scholes price π BS Φ 1 λ,, ; S of a single switch option with a total lifetime from t = 0 to t = and [ 0 ; 0] already passed is calculated by π BS Φ 1 λ,, ; S = r λγ 0 + r λ N + µ BS S0 +r λ sign + 2 S0 N sign µ BS µ BS r λ sign +r λ µ BS 2 2µ 2 BS 2π exp 1 2 µ 2 BS S0 2 N sign 2 + µ BS,. S0 + µ BS where λγ 0 is the amount of payout determined by the historical path of S in [ 0 ; 0]. Remark 4.2. he Black & Scholes price formula for a single switch option can be also derived directly using the nice idea that the payout profile Φ 1 λ,, ; S can be interpreted as a portfolio of simple digital options with increasing lifetimes t and common strike, where each of them pays out an amount of λdt or nothing at time. hen by application of Fubini s theorem we obtain π BS Φ 1 λ,, ; S = r λγ 0 + r λe = r λγ 0 + r λ N 1 [X t > ] dt t + µ BS t dt. his elementary idea was presented without proof of the result in the framework of a general approach to classify derivative securities by static replication of options portfolios in Pechtl 1995.
SOME APPLICAIONS OF OCCUPAION IMES OF BROWNIAN MOION 71 Now, the remaining integral can be calculated as follows. Integration by parts yields N + µ BS t dt = N t + µ BS + 1 1 2π 2 µ BS 2 t exp t 1 2 + µ BS t t dt. For further evaluation of the integral we suggest the following helpful Lemma 4.3. Lemma 4.3. For b, ν 0 the following assertions hold. I 1 b, ν, : = 1 2π = 1 N b exp 2νb} N I 2 b, ν, : = 1 2π = signb ν 1 exp 1 ν t b } 2 t dt 12 t 3 2 sign ν sign N exp 2νb} N b + ν b }. 1 exp 1 ν t b } 2 t dt 13 t 2 sign ν sign b ν b }. I 3 b, ν, : = 1 t exp 1 ν t b } 2 t dt 14 2π 2 = b signb ν 2 N sign ν b + exp 2νb} N sign ν b } + signb ν 3 N sign ν b exp 2νb} N sign ν b } 2 ν 2 2π exp 1 ν b } 2. 2
72 A. PECHL Proof. We consider the integrals 1 b, ν, and 2 b, ν, with 1 b, ν, : = 1 2π = 1 ν 2 [ N b t ν t 2 b, ν, : = 1 2π = t + ] 1 ν 2 t, b exp 1 ν t b } 2 t dt t 3 2 b t 3 [ exp 2νb} N b t + ν t Now we obtain eqs. 11 and 12 by Eq. 13 can be easily verified by exp 1 ν t b } 2 t dt 2 ] I 1 = 1 b 1 + 2, I 2 = 1 ν 2 1. I 3 = 1 ν. bi 2 ν I 2. Remark 4.4. Since the payout profile Ψ 1 λ,, ; S of a single corridor option is defined by Ψ 1 λ,, ; S = λ 1 [ 1 < S t 2 ] dt t= 0 = Φ 1 λ,, 1 ; S Φ 1 λ,, ; S the Black & Scholes price of such an option can be immediately obtained by Corollary 4.1. 5. CONCLUSION In this paper an application of the distribution of occupation times of Brownian motion with drift to a certain type of range options is considered, in particular an explicit pricing formula for dual switch options is provided in the Black & Scholes framework. hough the joint density of Brownian motion with drift at time and its occupation times up to is a familiar result of stochastic calculus [see Billingsley 1968] it seems that an explicit version of the distribution of occupation times has not been provided as far as the author knows. However, the explicit knowledge of this distribution is helpful for the pricing of a large class of path-dependent options such as range or quantile options, especially with respect to a practical point of view.
SOME APPLICAIONS OF OCCUPAION IMES OF BROWNIAN MOION 73 References 1. Akahori, J. 1995. Some Formulae for a New ype of Path- Dependent Option. Ann. Appl. Probab. 5 383 388. 2. Billingsley, P. 1968. Convergence of Probability Measures. Wiley, New York. 3. Black, F. and Scholes, M. 1973. he Pricing of Options and Corporate Liabilities. J. Polit. Econom. 81 637 659. 4. Dassios, A. 1995. he Distribution of the Quantiles of a Brownian Motion with Drift and the Pricing of Path-Dependent Options. Ann. Appl. Probab. 5 389 398. 5. Duffie, D. 1988. Security Markets Stochastic Models. Academic Press, New York. 6. Harrison, J. M. and Pliska, S. R. 1981. Martingales and Stochastic Integrals in the heory of Continuous rading. Stochastic Process. Appl. 15 214 260. 7. Hart, I. and Ross, M. 1994. Striking Continuity. RIS 7, N o 6, 51 56. 8. aratzas, I. and Shreve, S. E. 1988. Brownian Motion and Stochastic Calculus. Springer, Berlin. 9. Miura, R. 1992. A Note on Look-Back Options Based on Order Statistics. Hitotsubashi Commerce Management. 27 15 28. 10. Pechtl, A. 1995. Classified Information. RIS 8, N o 6, 59 61. 11. Pechtl, A. 1997. Distributions of Occupation imes of Brownian Motion with Drift. Unpublished manuscript. Deutsche Genossenschaftsbank Frankfurt am Main. 12. Yor, M. 1992. On Some Exponential Functionals of Brownian Motion. Adv. in Appl. Probab. 24 509 531.
Advances in Operations Research Advances in Decision Sciences Applied Mathematics Algebra Probability and Statistics he Scientific World Journal International Differential Equations Submit your manuscripts at International Advances in Combinatorics Mathematical Physics Complex Analysis International Mathematics and Mathematical Sciences Mathematical Problems in Engineering Mathematics Discrete Mathematics Discrete Dynamics in Nature and Society Function Spaces Abstract and Applied Analysis International Stochastic Analysis Optimization