SOME APPLICATIONS OF OCCUPATION TIMES OF BROWNIAN MOTION WITH DRIFT IN MATHEMATICAL FINANCE

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.


INTRODUCTION AND DESCRIPTION OF THE 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 ), Karatzas & 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.To 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).The α-quantile M (α, T ) (z) of a measurable function z on [0; T ] is defined Questions concerning the distribution of M (α, T ) (X) are closely related to questions of distribution concerning occupation times of Brownian motion with drift The distribution of M (α, T ) (Z) was studied successfully by Akahori (1995) and Dassios (1995).Though 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 + (T, k) (Z) [see Pechtl (1997), Theorem 1.1 and Theorem 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.The payout profile of a traditional single corridor option is given by with T, T 0 ≥ 0, λ > 0 and −∞ ≤ K 1 < K 2 ≤ ∞, whereas the more complex structure of a dual corridor option is given by The aim of this paper is to provide a general pricing formula for dual switch options, i.e. dual corridor options with K 2 = ∞, and related derivative securities in the well-known Black & Scholes model.
The paper is structured as follows.In Section 2 the Black & Scholes assumptions and the notations used in the paper are introduced briefly.The main results are 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.

PRELIMINARY REMARKS AND NOTATIONS
Throughout 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 with a one-dimensional standard Brownian motion W = (W t , t ≥ 0), the initial price S 0 of the underlying security, its volatility σ, r(> 1) defined by one plus the risk-free interest rate and the drift 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; T ] ; C [0;T ] , the linear space of all continuous functions on the time-interval [0; T ] endowed with the σ-algebra of Borel sets.Then the Black & Scholes price π BS (Φ) is calculated by For the following explicit calculations we introduce the univariate and the bivariate standard normal distribution functions by for all x ∈ gR and by for all x, y ∈ gR and all ρ ∈ (−1; 1).
Furthermore we define for κ < 0 the integral J (τ, θ, κ; µ) = θ v=−∞ F (τ, v, κ; µ) dv and calculate This 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.The reader interested in a direct proof of eq. ( 8) by integration may be referred to the similar proof of Theorem 3.1 in Pechtl (1997).
Thus for further discussion we only take the case λ 1 + λ 2 = 0 into consideration.
Let now be Then we have for 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) (ii) For λ 1 + λ 2 < 0 we obtain analogously Concluding we obtain the main result of this paper, the Black & Scholes formula for dual switch options.

APPLICATION OF THE RESULT TO SINGLE SWITCH OPTIONS
An immediate consequence of Theorem 3.1 is the explicit formula for the Black & Scholes price of a single switch option, i.e. an option with payout profile with total lifetime from t = −T 0 to t = T and λ > 0. Obviously this payout profile can be interpreted by Φ 2 (λ, 0, T, K; 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 We conclude this result in the following corollary.
Corollary 4.1.The Black & Scholes price π BS (Φ 1 (λ, T, K; S)) of a single switch option with a total lifetime from t = −T 0 to t = T and [−T 0 ; 0] already passed is calculated by Remark 4.2.The 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, T, K; S) can be interpreted as a portfolio of simple digital options with increasing lifetimes t and common strike K, where each of them pays out an amount of λdt or nothing at time T .Then by application of Fubini's theorem we obtain This 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).

,
where λγ 0 is the amount of payout determined by the historical path of S in [−T 0 ; 0].