Windings of Planar Processes and Applications to the Pricing of Asian Options

Motivated by a common Mathematical Finance topic, this paper surveys several results concerning windings of 2-dimensional processes, including planar Brownian motion, complex-valued Ornstein-Uhlenbeck processes and planar stable processes. In particular, we present Spitzer’s asymptotic Theorem for each case. We also relate this study to the pricing of Asian options.


Introduction
Windings of 2-dimensional processes, and especially of planar Brownian motion have several applications, namely in Finance.More precisely, in Financial Mathematics, the exponential functionals of Brownian motion are of special interest.A fundamental example is the pricing of Asian options (see e.g.[8,11,31,33]), where the payout of an Asian call option is given by: where (β u , u ≥ 0) is a real Brownian motion, ν ∈ R and K ∈ R + is the strike price.It is easy to show (for further details, see e.g.[11]) that the computation of this expectation simplifies to the computation of In particular, in [31] one can find a more detailed discussion for the distribution of the exponential functional A (ν) t := t 0 ds exp(β s + νs) taken up to a random time T λ which follows the exponential distribution with parameter λ > 0.More precisely, Yor [30] obtained that where Q 1,a ∼ Beta(1, a), G b ∼ Gamma(b), U ∼ U[0, 1], a = (ν/2) + (1/2) √ 2λ + ν 2 , b = a − ν and the random variables are independent.
In this paper, we will study this exponential functional in terms of planar Brownian motion, i.e. taken up to a random time different from the case mentioned above, that is (we suppose now that ν = 0 but at the end we will also discuss the case ν = 0): where T γ c = inf{u ≥ 0 : γ u = c}, c > 0, and (γ u , u ≥ 0) another real Brownian motion independent from β.For more precise connection with the windings, see Proposition 2 below.
We consider the following processes: • (Z t , t ≥ 0) a planar Brownian motion (BM), • (V t , t ≥ 0) a complex-valued Ornstein-Uhlenbeck (OU) process, i.e.: with λ ≤ 0, or equivalently, with (B t , t ≥ 0) another planar Brownian motion starting from V 0 : and all of them starting from a point different from 0 (without loss of generality we may consider all of them start from 1).
It is well-known [13] that since Z 0 = 0, (Z t , t ≥ 0) does not visit a.s. the point 0 but keeps winding around it infinitely often.Hence, its continuous winding process θ Z t = Im( t 0 dZ s Z s ), t ≥ 0 is well defined.We also recall the skew-product representation of planar BM (see also e.g.[19]): with (β u + iγ u , u ≥ 0) denoting another planar Brownian motion starting from log 1 + i0 = 0 (for the Bessel clock H Z , see also [29]).We easily deduce that the two σ-fields σ{|Z t | , t ≥ 0} and σ{β u , u ≥ 0} are identical, whereas (γ u , u ≥ 0) is independent from (|Z t | , t ≥ 0).Note that the inverse of H Z is: Similarly, for the OU process V we have that its continuous winding process, i.e.
Concerning now the Stable process U (see also [3,7,14]), contrary to planar Brownian motion, we cannot define its winding number directly.However, we can consider a path on a finite time interval [0, t] and "fill in" the gaps with line segments in order to obtain the curve of a continuous function f : [0, 1] → C with f (0) = 1.0 is polar and V has no jumps across 0 a.s., thus we have f (u) = 0 for every u ∈ [0, 1] and the process of the winding number of around 0, θ U = (θ U t , t ≥ 0) is well-defined, it has cadlag paths of absolute length greater than π and, for all t ≥ 0, We also introduce the clock: and its inverse: For each process, we will study the exit times from a cone of single and of double border,that is: where W = Z, V or U.Moreover, we will study the asymptotic behavior of each winding process.The rest of this paper is organized as follows: In Section 2 we discuss windings and the associated version of Spitzer's Asymptotic Theorem (that correspond to the large time asymptotics) (i) for planar Brownian motion, (ii) for complex-valued Ornstein-Uhlenbeck processes, and (iii) for planar Stable processes.In particular, in Subsection 2.1 we characterize the distribution of the exit times from a single and from a double border cone via their Gauss-Laplace transform, that we prove by using Bougerol's identity in law, and we also illustrate the formula for the second random time by giving two examples.We also comment briefly to the small time asymptotics case.Section 3 deals with applications of the previous results to the pricing of Asian options.More precisely, we discuss separately the case of exponential functionals of Brownian motion and of Lévy processes.Finally, in the Appendix A we give a comparative table concerning the clocks associated to each process, i.e.H Z , H V and H U and Spitzer's asymptotic Theorem for planar Brownian motion, for complex-valued Ornstein-Uhlenbeck and for planar Stable processes, followed by some comments and remarks.

The planar Brownian motion case
First, we recall our main tool, that is Bougerol's celebrated identity in law [6], stating that with (β u , u ≥ 0) and ( βu , u ≥ 0) denoting two independent linear Brownian motions for t ≥ 0 fixed, For the proof and other developments of this identity, see [23] and the references therein.We will study Bougerol's identity in law in terms of planar Brownian motion, which is strongly related to exponential functionals of Brownian motion as one can see below.To that end, we shall also need the following exit times for the BM γ associated to θ Z : A first result is the following: Proposition 2. We have the following relations: , and ), using the fact that A Z is the inverse of H Z (see also ( 4)): The second relation follows similarly.
From now on, all the results may be stated either for ).For the sake of applications in the Mathematical Finance framework, we will mostly use the first notation.

Proof. First,
For c → ∞, taking logarithms on both sides of (15) and dividing by c, the LHS gives: 1 c log T θ c − 2 c log c, and the RHS writes: Invoking the classical Laplace argument: Hence: which can be equivalently stated as (recall that β T γ The LHS of ( 17) equals: where t = exp(cx).Recalling now the fact that |C 1 | we get: hence we obtain precisely Spitzer's Theorem (14).
Remark 1. Spitzer's Asymptotic Theorem can be equivalently stated in terms of the clock H Z , i.e.: Following [21,22], we have: Remark 2. From e.g.formula (22) with some analytic computations, we can obtain the density function of . For further details, see e.g.[21,22].

Examples
We first illustrate formula (22) by 2 examples that come essentially from [25]: We will verify this formula by simple computations.Indeed, we have that: where (X 0 t , t ≥ 0) is another real BM starting from 0. However, with N ∼ N (0, 1), = 1 N 2 , hence, the LHS of (23) gives: thus we get directly (23).Note that on the RHS of ( 23) we have the Laplace transform of an exponential variable of parameter 1, denoted by e 1 .
2. m = 2 ⇒ c = π 4 Similarly, (22) writes: With (N, Ñ ∼ N (0, 1)), where for every x, T x = inf{t, β t = x} and T is an independent copy of T. We remark that = 1 Ñ2 , which yields (C is a constant): Using Fubini's theorem we get (25).Note that now, on the RHS of (25) we have the Laplace transform of the variable N 2 2 + 2e 1 Proof of Proposition 4. We apply Bougerol's identity (12) for T which yields that for fixed c > 0: , where (C c , c ≥ 0) is a standard Cauchy process.We identify the densities of the two variables: LHS: 1 and we remark that a(y) = arg sinh(y) = log(y + 1 + y 2 ).Changing the variables x = y 2 we get (21).Formula ( 22) follows by Bougerol's identity for T |γ| c and we apply the same arguments as previously, recalling that the density of β e my +e −my (see e.g.[4]).
For other results and variants concerning properties of the random times , the interested reader is addressed to [25,26] and to the references therein.
We can also obtain the analogue of Spitzer's Theorem for OU processes:

Theorem 6. (Spitzer's Theorem for OU processes)
We have that: where, C σ stands for a Cauchy variable with parameter σ.
Proof.Equation (5) yields: We remark that: and applying Spitzer's Theorem (14), we obtain (29).Remark 3. In terms of the clock H V , Spitzer's Theorem for OU processes may be also stated as: 1

The planar Stable process case
We turn now our interest to planar Stable processes.Bertoin and Werner following [12] obtained the following two Lemmas for α ∈ (0, 2) (for the proofs see [3]): Concerning now U, we use the analogue of the skew product representation for planar BM which is the Lamperti correspondence for stable processes.Hence, there exist two real-valued Lévy processes (ξ u , u ≥ 0) and (ρ u , u ≥ 0), the first one non-symmetric whereas the second one symmetric, both starting from 0, such that: For further discussion on the independence, see e.g.[16], where is shown that an isotropic α-self-similar Markov process has a skew-product structure if and only if its radial and its angular part do not jump at the same time.
Bertoin and Werner in [3] obtained the analogue of Spitzer's asymptotic Theorem 3 for isotropic stable Lévy processes of index α ∈ (0, 2): Theorem 9.The family of processes ) endowed with the Skorohod topology, as c → ∞, to r(α)β t , t ≥ 0 , where (β s , s ≥ 0) is a real valued Brownian motion and Proof.We refer to two different proofs: 1. Bertoin and Werner (1996) [3], using an "Ornstein-Uhlenbeck type" process and ergodicity arguments, and 2. Doney and Vakeroudis (2012) [7], using the continuity of the composition function ρ H U (•) (see [27]).[3] showed also that the clock H U (t) satisfies the following a.s.convergence: For discussions concerning the small time asymptotic behavior (t → 0) of the windings of a Stable process, we address the interested reader to [7] for the case where our process is issued from a point different from the origin and to [14] when it is issued from the origin.For the latter, one can either apply classical scaling arguments, or use a novel method appealing to tools from the theory of self-similar Markov processes involving path transformations and time changes.This approach may give access to windings of some conditioned versions of stable processes and to the so-called one-dimensional windings, that is the upcrossings of stable processes over the origin.

Asian options and exponential functionals of Brownian motion
In this Subection, we return to the initial Financial Mathematics problem, that is the characterisation of the distribution of To that end, one may use the previously stated results to access the distribution of A t via Williams' so called 'pinching method' [17,28].Loosely speaking, when Williams studied windings of Brownian motion, instead of working out directly the asymptotics of the winding process θ, he studied the asymptotic behaviour of this process taken at a random time, depending on θ (for similar results but with the use of a random time independent of θ, see e.g.[22]).Next, one simply remarks that the difference between the initial winding process and the subordinated process is finite, and renormalising appropriately, this difference converges to 0. Hence, the asymptotic study of the renormalised subordinated process yields similar results for the renormalised initial one.We propose here to mimic this method for our benefit, by invoking the time changes discussed in the previous Sections.
Proposition 10.The following convergence in law holds where C 1 is a standard Cauchy random variable.The distribution of A t may also be characterized by the following result due to Dufresne [8].

Proposition 4 .
The distributions of A Z T the following Gauss-Laplace transforms (x ≥ 0 and m

Lemma 7 .Lemma 8 .
The time-changed process (θ U A U (u) , u ≥ 0) is a real-valued symmetric Lévy process, say ρ.It has no Gaussian component and its Lévy measure has support in [−π, π].Let us denote now by dz the Lebesgue measure on C.Then, for every complex number z = 0, φ(z) stands for the determination of its argument valued in ( − π, π ].The Lévy measure of θ A U (•) is the image of the Lévy measure ν of U by the mapping z

Remark 4 .
|Z| and Z A U (•) /|Z A U (•) | are NOT independent.Indeed, the processes |Z A U (•) | and Z A U (•) /|Z A U (•) | jump at the same times hence they cannot be independent.Moreover, A U (•) depends only upon |Z|, hence |Z| and Z A U (•) /|Z A U (•) | are not independent.

preprints.org) | NOT PEER-REVIEWED | Posted: 24 October 2016 doi:10.20944/preprints201610.0097.v1
[19]e, studying asymptotically t −1 log A Z T γ t as t → ∞ would yield similar results for t −1 log A Z t .Following[26], by applying the scaling property of Brownian motion and changing variables we have that (recall that T the fact that the p-norm converges to the ∞-norm when p → ∞, the latter converges for t → ∞ towards 2 sup 0≤u≤T γ 1 β u .We recall that invoking the reflexion principle (see e.g.[19]) t→∞ 0. Preprints (www.=|C 1 |.and the result for A Z t follows immediately.