DISTRIBUTIONS OF OCCUPATION TIMES OF BROWNIAN MOTION WITH DRIFT

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1 c Journal of Applied Mathematics & Decision Sciences, 3, Reprints Available directly from the Editor. Printed in New Zealand. DISTRIBUTIONS OF OCCUPATION TIMES OF BROWNIAN MOTION WITH DRIFT ANDREAS PECHTL Center of Asset Pricing and Financial Products Development Deutsche Genossenschaftsbank Frankfurt am Main Am Platz der Republik, D 6035 Frankfurt am Main, Germany Abstract. The purpose of this paper is to present a survey of recent developments concerning the distributions of occupation times of Brownian motion and their applications in mathematical finance. The main result is a closed form version for Akahori s generalized arc-sine law which can be exploited for pricing some innovative types of options in the Black & Scholes model. Moreover a straightforward proof for Dassios representation of the α-quantile of Brownian motion with drift shall be provided. Keywords: options. Brownian motion with drift, occupation times, Black & Scholes model, quantile. INTRODUCTION Problems of pricing derivative securities in the traditional Black & Scholes framework are often closely connected to the knowledge of distributions induced by application of measurable functionals to Brownian motion. Familiar types of measurable functionals such as passage times and maximum or minimum functionals were already considered in early investigations of Brownian motion and from a stochastic point of view are responsible for popular exotic option constructions like barrier options and look-back options. In the context of this paper we concentrate on applications of occupation times of Brownian motion to problems in mathematical finance. For instance the distribution of occupation times is decisive for the pricing of socalled α-quantile options, a certain class of average options. Although occupation times of Brownian motion had been a subject of intensive research in stochastic calculus years ago e.g. a nice proof of Lévy s famous arc-sine law is presented in Billingsley 968 the interest of financial economists and applied mathematicians in questions concerning quantile options has caused a renaissance in this topic see Akahori 995, Dassios 995. This paper intends to present a survey and a summary of familiar and recent results with respect to the distribution of occupation times of Brownian motion with drift in Section. The main result of this section is Theorem., where a representation of the occupation times density by the partial derivatives of a convolution integral is provided. In Section we demonstrate how this represen-

2 4 A. PECHTL tation can be used to derive by elementary methods one of the most interesting results of recent research Dassios identity in law for occupation times. Then, in Section 3, a closed form representation of the distribution in question is introduced which is appropriate for the calculation of option pricing formulae in the Black & Scholes framework. Such an application we present in Section 4 where the explicit formula of a quantile option is discussed. Finally, in the Appendix miscellaneous remarks and computational aspects of proofs are provided for those readers who are interested in technical details.. OCCUPATION TIMES AND A GENERALIZED ARC-SINE LAW Let W t, t 0 be a one-dimensional standard Brownian motion. With ν gr we introduce the Brownian motion with drift Z defined by Z t = W t + νt, t 0. In the framework of this paper we consider the restriction of Z to the time-interval 0; T, i.e. Z is a random variable with values in the measurable space C0; T ; C 0;T, the space of all continuous functions on 0; T endowed with the σ-algebra of Borel sets see Billingsley 968, Chapter. Then by application of the measurable functional Γ + T, k : C 0; T gr defined by Γ + T, kz := T t=0 zt > k dt, z C 0; T, to Z we obtain the real-valued random variable occupation time Γ + T, k Z of k; up to time T see Karatzas & Shreve 988, Example In the driftless case ν = 0 with k = 0 the distribution of Γ + T, 0W is a familiar example in the literature on Brownian motion and provides a well-known arc-sine law. As an application of the central limit theorem of random walk Billingsley presents an elementary derivation for the joint density P Γ + T, 0W du; W T dx see Billingsley 968, Chapter, pp The joint density P Γ + T, 0W du; W T dx is thus given by the formula, x T exp x T t dtdudx, x < 0; π t=u P Γ + T, 0W du; W T dx = t T t 3 x T exp x T t dtdudx, x > 0. π t=t u t T t 3 With respect to the following explicit calculations of distributions and their applications in mathematical finance we denote the univariate and the bivariate standard normal distribution functions by N x = π x v= exp v dv

3 DISTRIBUTIONS OF OCCUPATION TIMES OF BROWNIAN MOTION 43 for all x gr and by N x, y; ρ = π ρ x v= y w= exp v ρ ρvw + w dwdv for all x, y gr and all ρ ;. By Girsanov s theorem we obtain the joint density P Γ + T, 0Z du; Z T dx = exp ν T + νx P Γ + T, 0W du; W T dx and by application of Fubini s theorem the first result of Akahori s generalized arc-sine law see Akahori 995, Theorem. i, = P Γ + T, 0Z du 3 π exp ν T ν + exp ν T u N ν u u T u π T u ν exp ν π u u N ν T u ν N ν u N ν T u du. Actually for ν = 0 this result yields Lévy s classical arc-sine law for occupation times P Γ + T, 0W u = u dt = u π t T t π arcsin T t=0 for 0 u T. With eq. 3 we immediately obtain a double integral representation for P Γ + T, kz t. First we introduce the functional passage time T k by and consider T k Z with distribution T k z = inf t 0 z t = k P T k Z t = P min Z s k 0 s t for k < 0 and with distribution P T k Z t = P max Z s k 0 s t kt = N ν t + 4 kt exp kν N + ν t = N k + ν t + 5 t exp kν N k ν t t

4 44 A. PECHTL for k > 0. Obviously the following density formula is valid. P T k Z dt = h t, k; ν dt, where h t, k; ν = k exp k νt see Karatzas & Shreve 988, p. πt 3 t 96f. Furthermore we denote φ T, t; ν = P Γ + T, 0Z t, 0 t T, and since Z has independent increments we have for all k < 0 and all 0 t < T the relation P Γ + T, kz t = t and by the obvious identity in law φ T s, t s; ν h s, k; ν ds, 6 Γ + T, k W + ν D = T Γ + T, k W ν the corresponding relation for all k > 0 and all 0 t T P Γ + T, kz t = T t φ T s, T t s; ν h s, k; ν ds.7 Eq.s 6 and 7 correspond with Akahori s Theorem. ii see Akahori 995. As our first original result in this paper we now present a single integral version for the distribution of Γ + T, kz and thus an explicit formula for its density. For values 0 θ τ,, µ gr we define the functions F τ, θ, ; µ and D τ, θ, ; µ by and by F τ, θ, ; µ = D τ, θ, ; µ = θ θ φ τ s, θ s; µ h s, ; µ ds 8 N µ τ s N s sign + µ s ds. 9 We shall demonstrate in the following theorem that the distribution of occupation times of Brownian motion with drift can be described completely by the integral D and its partial derivatives with respect to and τ. Theorem.. Let F and D be defined as in eqs. 8 and 9. Then the distribution of Γ + T, kz is determined on 0; T by P Γ + T, kz t = F T, t, k; ν, k < 0; P Γ + T, kz t = F T, T t, k; ν, k > 0,

5 DISTRIBUTIONS OF OCCUPATION TIMES OF BROWNIAN MOTION 45 and for < 0 the function F can be represented in the following way. F τ, θ, ; µ = µ exp µ D τ, θ, ; µ µ exp µ D τ, θ, ; µ 4 exp µ τ D τ, θ, ; µ 4 exp µ µ D τ, θ, ; µ. τ For t < 0 we trivially have P Γ + T, k Z t = 0 and for t T we have P Γ + T, k Z t =. Proof. For < 0 we rewrite eq. 8 as F τ, θ, ; µ = θ θ s u=0 φ τ s, u; µ h s, ; µ duds u and obtain the partial derivative of F with respect to θ by where F τ, θ, ; µ = θ θ φ τ s, θ s; µ h s, ; µ ds, u φ τ s, u; µ is determined by eq. 3. Thus we have the following u representation for F τ, θ, ; µ. θ where we define J θ, ; µ = θ 4 F τ, θ, ; µ = θ i= G i τ, θ, ; µ, µ N µs θ s exp ds, πs 3 s G τ, θ, ; µ = µ N µ J θ, ; µ, G τ, θ, ; µ = 4N µ J θ, ; µ θ G 3 τ, θ, ; µ = 4 τ N µ J θ, ; µ, G 4 τ, θ, ; µ = 8 µ τ N µ J θ, ; µ θ πθ exp 3 πθ exp 3 µθ, θ µθ. θ We only have to calculate J θ, ; µ and by partial differentiation we obtain explicit versions for G i τ, θ, ; µ, i 4, and thus for F τ, θ, ; µ. θ Since J θ, ; µ = θ exp µ µ exp θ π θ 3 θ + µ θ we have θ J θ, ; µ = exp µ N + µ θ.

6 46 A. PECHTL We set d τ, θ, ; µ := µ θ D τ, θ, ; µ = N θ N + µ θ, and obtain the assertion of Theorem. immediately. Remark.. The joint density in eq. can be expressed explicitly by = P Γ + T, 0W du, W T dx x T u x πt exp + u T u π exp x T x x u N T dudx T T 5 T u for x < 0 and by P Γ + T, 0W du, W T dx x u = πt T u exp x + u π exp x T x N x T u dudx T T u T 5 for x > 0. An alternative approach to the distribution of occupation times was presented by Karatzas & Shreve 984, 988 who provided a density formula for P Γ + T, 0W du, W T dx, L T W db where L denotes the local time of Brownian motion see Karatzas & Shreve 988, Proposition Remark.3. For convenience of some calculations the partial derivatives of d τ, θ, ; µ are provided. d τ, θ, ; µ = exp θ πθ + µ θ N µ, τ d τ, θ, ; µ = µ π exp τ d τ, θ, ; µ = µ 4π θ exp µ θ N + µ θ, µ µ θ τ Remark.4. Though in Akahori 995 a different notation is used it is evident that in Akahori s Theorem. ii are some slight incorrectnesses beside a few minor.

7 DISTRIBUTIONS OF OCCUPATION TIMES OF BROWNIAN MOTION 47 typographic errors. Otherwise F T, t, k; ν = F T, T t, k; ν would hold for all values k > 0, ν gr and all 0 t T. To demonstrate this contradiction choose ν = 0 and prove by using Remark A. that in general θ F T, t, k; 0 F T, T t, k; 0. θ Remark.5. The function P Γ + T, kz t is continuous in the crucial point k = 0, thus it is continuous for all k gr. This can be easily proved by Theorem., Remark.3 and eq. 3. Furthermore P Γ + T, kz t is continuous for all t T if k < 0 since P Γ + T, kz = T = P T k Z T > 0. Analogously P Γ + T, kz t is continuous for all t 0 if k > 0. Evidently it is continuous on gr for k = A STRAIGHTFORWARD PROOF FOR DASSIOS IDENTITY IN LAW A simple but remarkable relationship between the distribution of occupation times and passage times was discovered by Dassios see Dassios 995 who pursued the investigations of Akahori on the so-called α-quantile or α-percentile options, a problem in mathematical finance which we shall illuminate in Section 4. In the language of this paper we present the result of Dassios in the following way. T P W s + νs > k ds t = P max W s + νs + min W s + νs k 0 s T t 0 s t 0 for all k gr and all t 0; T where W and W are independent standard Brownian motions. Dassios proved this property using Feynman & Kac computations, two further proofs based on stochastic properties of Brownian motion were provided by Embrechts, Rogers and Yor 995. In this paper we point out a direct proof of Dassios identity in law using the explicit density of Γ + T, kz and the knowledge of the distribution of passage times. Before we prove eq. 0 we provide a representation of F τ, θ, ; µ as θ a convolution integral. Referring to this convolution integral Dassios result can be derived by relatively elementary methods. Proof of Dassios theorem. By Theorem. the following representation holds for < 0,

8 48 A. PECHTL where θ F τ, θ, ; µ = g τ, θ, ; µ + g τ, θ, ; µ, g τ, θ, ; µ = exp µ µ d τ, θ, ; µ + µ d τ, θ, ; µ + d τ, θ, ; µ, τ g τ, θ, ; µ = 4 exp µ d τ, θ, ; µ. µ τ Since d τ, θ, ; µ is a product of two functions converging to 0 for it can be interpreted as a convolution integral. d τ, θ, ; µ = N µ z = z N θ + µ θ N z= + z= z N z µ θ N + µ θ z µ dz zθ N + µ θ dz. Partial differentiation with respect to yields the representation of d τ, θ, ; µ as convolution. The following relation is the crucial idea of this proof. It is obtained by integration by parts and Remark.3 to d τ, θ, ; µ, namely τ µ d τ, θ, ; µ = µ We arrive at µ z= z= z z N z N d τ, θ, ; µ. τ g τ, θ, ; µ z = µ exp π 3 z= θ + µ θ N z µ z µ dz z µ zθ N + µ θ dz zθ exp zµ N + µ z θ dz + µ exp zθ z= πθ 3 µ θ

9 DISTRIBUTIONS OF OCCUPATION TIMES OF BROWNIAN MOTION 49 exp z µ N z µ dz. θ + µ θ can be repre- Similarly d τ, θ, ; µ = τ τ N µ N sented as convolution. We obtain = g τ, θ, ; µ z exp π 3 z= exp zθ πθ µ θ z µ dz z= exp z µ π z exp zθ πθ 3 µ θ dz. Since by eqs. 4 and 5 P P min Ws + µs dz 0 s θ max W s + µs d z 0 s τ θ θ P min Ws + µs z 0 s θ θ P max 0 s τ θ Ws + µs z zθ = µ exp zµ N + µ θ dz + exp zθ πθ µ θ dz, = µ exp z µ N z µ dz exp z µ dz, π = z exp zθ πθ 3 µ θ, z = exp z µ, π 3 we obtain a final convolution integral representation by θ F τ, θ, ; µ = τ θ P W s + µs > ds θ = z= θ P max W s + µs z P min W s + µs dz 0 s τ θ 0 s θ z= θ P min W s + µs z P max W s + µs d z 0 s θ 0 s τ θ = θ P max W s + µs + min W s + µs. 0 s τ θ 0 s θ

10 50 A. PECHTL From this and by the fact that eq. 0 holds trivially for k < 0 and t = 0 eq. 0 is generally valid for k < 0. For k 0 we use Theorem. and obtain T P W s + νs > k ds t T = P W s νs > k ds T t = P max W s νs + min W s νs k 0 s t 0 s T t = P max W s + νs + min W s + νs k. 0 s T t 0 s t This completes the proof. 4. A CLOSED FORM DISTRIBUTION FUNCTION In Section, Theorem., we provided a single integral version for the distribution of Γ + T, k Z and thus an explicit representation for its density. With respect to various applications in mathematical finance an explicit formula for this distribution would be a comfortable result. By eq. 8 and Theorem. it is sufficient to obtain an explicit version of the function F. Theorem 3.. The function F defined by eq. 8 can be represented explicitly for all values < 0 in the following way. F τ, θ, ; µ = 3 + µ + µ τ τ exp µ N + µ τ, µ ; θ τ +N µ τ, µ ; θ τ τ + µ + µ θ θ exp µ N + µ θ N µ θ +N µ θ N µ θ µ π exp θ µ θ N µ τ +µ π exp τ µ τ N θ τ

11 DISTRIBUTIONS OF OCCUPATION TIMES OF BROWNIAN MOTION 5 µ π exp θ µ exp µ N + µ θ. Since the proof of Theorem 3. is very technical with respect to the explicit calculation of F τ, θ, ; µ we omit it here and refer the reader interested in the details of the proof to the appendix of the paper. 5. THE PRICE OF A QUANTILE OPTION In the last few years a lot of problems concerning the application of measurable functionals to Brownian motion with drift have been stimulated by the rapid development of mathematical finance based on the fundamental ideas of Black & Scholes. In this paper we do not want to discuss economic models and the reader interested in those questions should be referred to a number of excellent surveys on option pricing theory e.g. Harrison & Pliska 98, Duffie 988, et al.. For our purposes it is sufficient to know that an option is determined uniquely by a measurable functional Φ on a certain underlying security S with initial price S 0, the so-called payout profile, and that the price of this option can be calculated in the Black & Scholes model by the discounted expectation r T EΦX using the no-arbitrage price-process X which is described by X t = S 0 exp σw t + µ BS t, 0 t T, where 0; T is the considered time-horizon, σ the volatility of the underlying security, r> is one plus the risk-free interest rate and µ BS = log r σ the risk-neutral drift determined by the assumptions of Black & Scholes. The option types we want to discuss here are so-called average options, especially the class of α-quantile options or in Akahori s terminology α-percentile options. As mentioned before Dassios found his identity in law when investigating α-quantile options. To illustrate the properties of such an option let z be a continuous function on the time-interval T 0 ; T with T 0, T 0. Then for 0 < α < the α-quantile M α, T 0 + T of z is defined by T M α, T 0 + T z = inf k gr z τ k dτ > α T 0 + T. τ= T 0 Now we represent the price-process S of the underlying security by S τ = S 0 exp σsτ, T 0 τ T,

12 5 A. PECHTL where the values S τ respectively sτ in T 0 ; 0 are given by historical data. Given a certain strike K the payout profile Φ 0 of an α-quantile call option is described by Φ 0 S = M α, T 0 + T S K + using the identity = S 0 exp σm α, T 0 + T s K + M α, T 0 + T S = S 0 exp σm α, T 0 + T s. For pricing an option dependent on the future time-interval 0; T in the Black & Scholes model we use the process X and define X τ = S τ for all τ T 0 ; 0. Then the price of this α-quantile option is computed by π BS Φ 0 = r T E S 0 exp σm α, T 0 + T Z K + 3 with Z defined by s on T 0 ; 0 and by Z τ = W τ + µ BS τ for all τ 0; T. By the σ evident relation T M α, T 0 + T Z k = Z τ > k dτ α T 0 + T 4 τ= T 0 we obtain the connection between α-quantiles and occupation times. Furthermore 0 the function γ + T 0, k := Z τ > k dτ is a deterministic function in k. τ= T 0 Since Z is considered to be a continuous function γ + T 0, k is strictly decreasing on the closed interval rangez = Z τ T 0 τ 0. For all values k < min Z τ T 0 τ 0 we have γ + T 0, k = T 0 and for all values k max Z τ we have γ + T 0, k = 0. T 0 τ 0 Finally γ + T 0, k is right-continuous in k with left limits. Defining tt 0, k by tt 0, k = α T 0 + T γ + T 0, k 5 we have the property M α, T 0 + T Z k = Γ + T, kz tt 0, k. 6 Then we obtain the distribution of M α, T 0 + T Z, F α k : = P M α, T 0 + T Z k = P Γ + T, kz tt 0, k, 7 and by integration by parts we arrive at the integral version of the option s price, π BS Φ 0 = r T S 0 σ F α k exp σk dk. 8 k> σ log K S 0

13 DISTRIBUTIONS OF OCCUPATION TIMES OF BROWNIAN MOTION 53 Since F α depends on the arbitrary character of the historical function tt 0, k the further computation of the α-quantile option s price will be in general a question of numerical calculus. Nevertheless some properties of the option s price shall be discussed. To do so we introduce the following boundaries for F α, namely kt 0 = sup k gr tt 0, k < 0 and kt 0 = inf k gr tt 0, k > T, such that F α is concentrated on kt 0 ; kt 0. Then we have for σ log K S 0 kt 0 kt 0 kt0 π BS Φ 0 = r T S 0 σ F α k exp σk dk 9 k= σ log K S 0 = r T S 0 exp σkt 0 K + r T S 0 σ F αk exp σk dk, 0 k kt 0;kT 0 respectively for kt 0 < σ log K S 0 < kt 0 π BS Φ 0 = r T S 0 σ k F α k exp σk dk. σ log K S0 ;kt0 It is evident that for kt 0 σ log K S 0 the option ceases to exist. Remark 4.. It can be further proved that kt 0 > is equivalent to the condition T < α α T 0 and kt 0 < is equivalent to T < α α T 0 respectively. To describe the behaviour of the option s price let us denote for a moment the starting point of the option by t 0 and its total lifetime by T = T 0 + T. Then tt 0, k can be rewritten by t0+t 0 tt 0, k = α T Z τ > k dτ. τ=t 0 Then tt 0, k is decreasing for T 0 T, i.e. T 0. Thus kt 0 is increasing in T 0 and kt 0 decreases since tt 0, k > T is equivalent to t0+t 0 T 0 Z τ > k dτ > αt. τ=t 0 Obviously for T 0 = T we have kt = kt and F α is concentrated at this point, the α-quantile of Z in the time-period t 0 ; t 0 + T. Remark 4.. For T 0 = 0 we have k0 = and k0 =. Thus the option s price is

14 54 A. PECHTL with π BS = r T S 0 σ F α k exp σk dk k> σ log K S 0 F α k = P Γ + T, kz α T 3 F T, αt, k; µ BS, k 0; = σ F T, α T, k; µ BS, k < 0. σ Now for values < 0 we introduce the function I by I τ, θ, ; ν, σ = s= F τ, θ, s; µ exp σs ds. Using Lemma A.3 I can be calculated explicitly. We obtain the following representation, I τ, θ, ; µ, σ = µ + µ τ + 3 µ exp µ + σ N + µ τ, µ ; θ µ + σ µ + σ τ τ 4 µ + σ σ µ + σ exp µ + σ σ τ N µ + σ τ, µ + σ ; θ τ τ 4µ µ + σ σ µ + σ exp µ + σ σ θ θ N µ + σ θ N µ µ + µ θ σ θ + exp µ + σ N µ + σ µ + σ + µ θ N µ µ θ µ + σ π exp µ + σ σ θ exp θ µ + σ θ N µ + µ τ µ + σ π exp µ + σ σ τ exp τ µ + σ τ N θ τ + σ exp σ N µ τ, µ ; θ τ τ + θ σ exp σ N µ θ N µ µ µ + σ π exp µ θ exp µ + σ N + µ θ.

15 DISTRIBUTIONS OF OCCUPATION TIMES OF BROWNIAN MOTION 55 Then for S 0 K the Black & Scholes price of the option is determined by π BS Φ 0 = r T S 0 σi T, αt, σ log S 0 K ; µ BS σ, σ, 4 and for S 0 > K by π BS Φ 0 = r T S 0 K + r T S 0 σi T, α T, σ log K ; µ BS S 0 σ +r T S 0 σ I T, αt, 0; µ BS σ, σ I, σ T, α T, 0; µ BS σ, σ. 5 Remark 4.3. Though Akahori 995 and Dassios 995 consider the Black & Scholes price of an α-quantile option they do not provide an explicit solution which can be exploited successfully for option pricing in practice. Especially when dealing with path-dependent options well-known approximative methods, e.g. numerical integration, Monte Carlo simulation or lattice approximation, often consume enormous resources with respect to accuracy and computation time whereas explicit formulae if available provide reliable prices nearly real time. 6. CONCLUSION The main results of this paper are explicit representations for the density and the distribution of occupation times of Brownian motion with drift. By application of these distribution formulae to types of derivative securities where the payout depends on a certain amount of time-units exceeding a given boundary e.g. quantile options are in this class of derivative securities closed form solutions for the corresponding Black & Scholes prices can be obtained which are appropriate for direct implementation in daily option trading. Remark 5.. The author of this paper was motivated to his investigations in 994 by the problem of an explicit pricing formula for a median option, i.e. a quantile option with quantile α =, independently from the research of Akahori and Dassios. He presented his results, i.e. Theorem. and Theorem 3. of this paper, at the conference Derivatives 96, produced by RISK Publications, Geneva, 5 and 6 January, 996. Appendix A.. Auxiliary Results In this appendix we provide some integration details for certain relations in the previous sections and auxiliary results used in this paper.

16 56 A. PECHTL A.. Miscellaneous comments to Section In the following some additional information concerning the results of Akahori 995 is presented. Remark A.. A straightforward derivation of Akahori s Theorem.i, i.e. eq. 3, is pointed out in detail. Using eq. and Girsanov s theorem we obtain by Fubini s theorem P Γ + T, 0Z du = exp ν T exp νx P Γ + T, 0W du; W T dx x gr T T = exp ν dt dt T + du π 3 3 t=u t T t t=t u t T t T T + ν exp ν dt t N ν T t + exp ν t π 3 t=u t t=t u + ν exp ν T du ν exp ν u du + ν N ν T πt πu N ν T t du ν N ν u du. dt 3 t du By T T t=r dt t 3 T t T π exp ν T t=u = T r dt r t 3 T t + we obtain T t=t u By integration by parts we obtain small ν T π t=r ν3 π T t=r exp ν t N N exp ν t Then we use the relations T t=t u ν3 π T dt t T t = t=t u ν3 π u +ν N u t=0 ν dt T t ν T t dt t T t, dt t 3 T t t 3 dt t = exp ν t N ν dt T t = t exp ν t N ν dt T t t N ν u ν N ν T t=0 ν T u du = π exp ν T du. u T u ν exp ν πt ν π exp ν T t N T t=r ν T T t t=r dt t T t.

17 DISTRIBUTIONS OF OCCUPATION TIMES OF BROWNIAN MOTION 57 and the property ν 3 T π t=0 to conclude the final expression P Γ + T, 0Z du = exp ν t N ν dt T t t exp ν T π π ν exp ν u u u T u N ν = ν N + π T u ν T ν exp ν T ν exp ν T u N ν u T u ν N ν u N ν T u du. Remark A.. It can be shown that the representation of F τ, θ, ; µ for < 0 in Theorem. holds for > 0, respectively. A.. Miscellaneous comments to Sections 3 and 4 In Section 3 we omitted the technical proof of Theorem 3.. In the following we provide the ideas behind the explicit computations of all partial derivatives of the function D introduced by eq. 9. As mentioned before the distribution of occupation times can be represented completely by a linear combination of those partial derivatives. Proof of Theorem 3.. For < 0 we know the following representation for F by Theorem., F τ, θ, ; µ = µ exp µ D τ, θ, ; µ µ exp µ D τ, θ, ; µ 4 exp µ τ D τ, θ, ; µ 4 exp µ µ D τ, θ, ; µ. τ Furthermore the corresponding partial derivatives of d τ, θ, ; µ = D τ, θ, ; µ θ are known by Remark.3. We start with the evaluation of D τ, θ, ; µ and τ calculate the integral H := π θ exp ds = τ s s s π τ w= θ w + exp w + dw. τ By differentiation of this integral representation to we obtain H = πτ exp N τ θ τ

18 58 A. PECHTL and by Lemma A.3, eq. A9, H = 4N, 0; θ. Thus we have τ τ D τ, θ, ; µ = µ exp µ τ τ exp µ N, 0; θ τ τ In the next step we calculate D τ, θ, ; µ by integration with respect to applying Lemma A.3, eq. A4. We obtain explicitly τ D τ, θ, ; µ = exp µ τ τ exp µ N, 0; θ τ τ N + µ τ, µ ; θ. τ τ. Similarly to the crucial idea in our proof of Dassios identity in law we use integration by parts for the computation of D τ, θ, ; µ to avoid integration with respect to τ which appears to be somewhat complicated. We obtain the following relation which shows the interesting relation between D τ, θ, ; µ and D τ, θ, ; µ, τ D τ, θ, ; µ = µ By Lemma A.3, eq. θ N + µ θ + exp µ N + µ θ N µ θ + D τ, θ, ; µ + exp µ D τ, θ, ; µ µ τ τ exp µ N µ τ. A4, D τ, θ, ; µ, i.e. τ τ D τ, θ, ; µ = N N D τ, θ, ; µ can be calculated analogously to τ µ N µ τ µ θ, µ ; θ θ τ + exp µ τ exp µ N, 0; θ τ τ.

19 DISTRIBUTIONS OF OCCUPATION TIMES OF BROWNIAN MOTION 59 Now we summarize all explicit terms of representation, D τ, θ, ; µ = θ N µ + µ θ θ exp µ N µ θ N µ D τ, θ, ; µ for the following explicit + µ exp µ τ exp µ N, 0; θ τ τ µ N + µ τ, µ ; θ τ τ µ exp µ N µ τ, µ ; θ, τ τ and by integration to using Lemma A.3, eqs. A, A3, A4, we have 3 D τ, θ, ; µ = µ µ τ N + µ τ, µ ; θ τ τ + exp µ N µ µ exp µ τ + µ µ + θ τ µ τ, µ ; θ τ exp µ N, 0; θ τ τ θ N + µ θ N µ + exp µ N µ θ µ θ N µ + θ µ π exp θ + µ θ N µ τ µ π exp τ + µ τ N θ τ + µ π exp µ θ N + µ θ. Finally we summarize those results and calculate F τ, θ, ; µ explicitly. For the explicit calculation of integrals related to the normal distribution in Section 3 and Section 4 we referred essentially to the following lemma. Lemma A.3. The following integral relations hold.

20 60 A. PECHTL A A A3 A4 A5 A6 N ax + b dx = x + b N ax + b + a a π exp ax + b. exp cx N ax + b dx = c exp cx N ax + b c c exp a bc a N ax + b, y; ρ dx = x + b N ax + b, y; ρ a + a π exp ax + y ρ ax + b b N ρ + ρ a π exp y ax + b ρy N. ρ exp cx N ax + b, y; ρ dx = c c c exp a bc a exp cx N ax + b, y; ρ N xn ax + b dx = x b + a x exp cx N ax + b dx = ax + b c a, y cρ a ; ρ. N ax + b + ax b a π exp N ax + b c. a ax + b. cx c exp cx N ax + b + a + abc c c a c exp a bc N ax + b c a a + c ac π exp a bc exp ax + b c. a a

21 DISTRIBUTIONS OF OCCUPATION TIMES OF BROWNIAN MOTION 6 A7 A8 xn ax + b, y; ρ dx = x b + a N ax + b, y; ρ + ax b a π exp ax + y ρ ax + b b N ρ ρ b ρy a π exp y ax + b ρy N ρ ρ ρ 4a exp y ax + b ρy exp π ρ. x exp cx N ax + b, y; ρ dx = x c c exp cx N ax + b, y; ρ + a + abc c c a c exp a bc N ax + b c a a, y c a ρ; ρ + ac exp cx exp π ax + y ρ ax + b b N ρ + ρ ac π exp c abc ay cρ a ax + b ρy N c ρ. ρ a With respect to bivariate normal distribution functions the following relation may be sometimes helpful. For all x, y, p gr we have y A9 N x, ; p = x exp t N y + pt dt. + p + p π t= Proof. Lemma A.3 can be easily proved by differentiation with respect to x. References. Akahori, J Some Formulae for a New Type of Path-Dependent Option. Ann. Appl. Probab Billingsley, P Convergence of Probability Measures. Wiley, New York. 3. Dassios, A The Distribution of the Quantiles of a Brownian Motion with Drift and the Pricing of Path-Dependent Options. Ann. Appl. Probab Duffie, D Security Markets Stochastic Models. Academic Press, New York. 5. Embrechts, P., Rogers, L. C. G. and Yor, M A Proof of Dassios Representation of the α-quantile of Brownian Motion with Drift. Ann. Appl. Probab

22 6 A. PECHTL 6. Harrison, J. M. and Pliska, S. R. 98. Martingales and Stochastic Integrals in the Theory of Continuous Trading. Stochastic Process. Appl Karatzas, I. and Shreve, S. E Trivariate Density of Brownian Motion, its Local and Occupation Times, with Application to Stochastic Control. Ann. Probab Karatzas, I. and Shreve, S. E Brownian Motion and Stochastic Calculus. Springer, Berlin.