On the quantiles of the Brownian motion and their hitting times.

Transcription

1 On the quantiles of the Brownian motion and their hitting times. Angelos Dassios London School of Economics May 23 Abstract The distribution of the α-quantile of a Brownian motion on an interval [, t] has been obtained motivated by a problem in financial mathematics. In this paper we generalise these results by calculating an explicit expression for the joint density of the α-quantile of a standard Brownian motion, its first and last hitting times and the value of the process at time t. Our results can be easily generalised for a Brownian motion with drift. It is shown that the first and last hitting times follow a transformed arcsine law. 1 Introduction Let X s, s be a real valued stochastic process on a probability space Ω, F, Pr. For < α < 1, define the α quantile of the path of X s, s up to a fixed time t by M X α, t = inf { x : t } 1 X s x ds > αt. 1 The study of the quantiles of various stochastic processes has been recently undertaken as a response to a problem arising in the field of mathematical finance, the pricing of a particular path-dependent financial option; see Miura [7], Akahori [1] and Dassios [3]. This involves calculating quantities such as E h M X α, t, where h x = e x b + or some other appropriate function. This requires obtaining the distribution of X t. In the case where X s, s is a Lévy process having stationary and independent increments the following result was obtained: 1

2 Proposition 1 Let X 1 s and X 2 s be independent copies of X s. Then, MX α, t law sup s αt X = 1 s + inf s 1 αt X 2 s X t X 1 αt + X α t When X s, s is a Brownian motion, we can use this result and obtain an explicit formula for the joint density of M X α, t and X t. This result was first proved for a Brownian motion with drift; see Dassios [3] and Embrechts, Rogers and Yor [5] and for Lévy processes by Dassios [4]. There is also a similar result for discrete time random walks first proved by Wendel [8]. We now let be the first, and L X α, t = inf {s [, t] : X s = M X α, t} K X α, t = sup {s [, t] : X s = M X α, t}, the last time the process hits M X α, t. One can now introduce a barrier element to the financial application by making the option worthless if the quantile is hit too early or too late. For example, this can involve calculating quantities such as E h M X α, t 1 L X α, t > v, K X α, t < u. The first study of these quantities can be found in Chaumont [2]. By using combinatorial arguments he derives of the same type as Proposition 1 that are extensions to Wendel s results in discrete time. In the case where the random walk steps can only take the values +1 or -1, a represenation for the analogues of L X α, t and K X α, t is obtained. Finally he derives a continuous time representation for the triple law of M X α, t, L X α, t and X t, extending Proposition 1 when X t is a Brownian motion. We will adopt a direct approach that seems better suited to obtaining explicit expressions for the densities involved. We will also derive alternative representations and prove a remarkable arc-sine law. For the rest of the paper we assume that X s, s is a standard Brownian motion. We will derive the joint density of M X α, t, L X α, t, K X α, t and X t. If we denote this density by f y, x, u, v, our results can be generalised for a Brownian motion with drift m, using a Cameron- Martin-Girsanov transformation. The corresponding density will be f y, x, u, v exp mx m 2 t/2. Before we obtain the density of M X α, t, L X α, t, K X α, t, X t, we will first show that the law of L X α, t and K X α, t is a transformed arcsine law. 2

3 2 An arcsine law for L X α, t. Let S X t = sup s t {X s} and θ X t = sup {s [, t] : X s = S X t}. We prove the following theorem: Theorem 1 For u >, Pr L X α, t > u = Pr u < θ X t αt + Pr u < θ X t 1 α t 3 and 1 u αt + 1 u 1 α t Pr L X α, t du = π du. 4 u t u Furthermore, K X α, t has the same distribution as t L X α, t. Proof We will first prove that We observe that Pr M X α, t >, L X α, t > u = Pr u < θ X t αt. 5 Pr M X α, t >, L X α, t > u = Pr M X α, t > S X u = t Pr 1 X s S X u ds < αt = t Pr 1 X s X u S X u X u ds < αt. 6 u Let X s = X u + s X u. X s, s is a standard Brownian motion which is independent of X s, s u. We condition on S X u X u = c, and set τ c = inf {s > : X s = c} and X s = X τ c + s c. X s, s is a standard Brownian motion which is independent of both X s, s u and X s, s τ c. We have that t u Pr 1 X s c ds < αt u = αt u Pr τ c dr Pr t u r 1 X s ds < αt u r t u r and since 1 X s ds has the same arcsine law as θ X t u r, this is equal to αt u Pr τ c dr Pr θ X t u r < αt u r = 3

4 αt u Pr τ c dr Pr Pr sup X s > s αt u sup X s > s αt u r sup αt u s t u X s, sup αt u r s t u r X s = sup s αt u X s > c and so 6 is equal to supu s αt X s X u > sup Pr αt s t X s X u, = sup u s αt X s X u > sup s u X s X u Pr u < θ X t αt. Since X s, s is also a standard Brownian motion and M X α, t = M X 1 α, t almost surely, we use X s instead of X s and we get Pr M X α, t <, L X α, t > u = Pr u < θ X t 1 α t. 7 Adding 5 and 7 we get 3, and since θ X t has an arcsine law, 4 follows. To see that K X α, t has the same distribution as L X α, t, set X s = X t s X t. Clearly X s, s t is a standard Brownian motion and we can easily see that M X α, t = M X α, t X t and K X α, t = t L X α, t. We can also extend our result and obtain the joint distribution of M X α, t, L X α, t also of M X α, t X t, t K X α, t. Theorem 2 For b >, and for b <, Pr M X α, t db, L X α, t du = 8 Pr S X t db, θ X t du 1 < u < αt, 9 Pr M X α, t db, L X α, t du = 1 Pr S X t d b, θ X t du 1 < u < 1 α t. 11 Furthermore M X α, t, L X α, t and M X α, t X t, t K X α, t have the same distribution. 4

5 Proof Let b > and u < αt. We then have that Pr M X α, t > b, L X α, t > u = Pr S X u < M X α, t, M X α, t > b = Pr b < S X u < M X α, t + Pr S X u < b < M X α, t. 12 Let τ b = inf {s > : X s = b} and X s = X τ b + s c. X s, s is a standard Brownian motion which is independent of X s, s τ c. Using theorem 1, we have u u Pr b < S X u < M X α, t = t r Pr τ b dr Pr 1 X s S X u r < αt r = Pr τ b dr Pr Furthermore, u M X αt r t r, t r >, L X αt r t r, t r > u r = Pr τ b dr Pr u r < θ X t r < αt r = 13 Pr S X u < b < M X α, t = Pr αt u = Pr u < θ X t < αt, S X u > b. 14 t r Pr τ b dr Pr αt u S X u < b, t 1 X s b ds < αt = 1 X s < αt r Pr τ b dr Pr θ X t r < αt r = Pr u < θ X t < αt, S X u < b, sup u s αt X s > b. 15 Adding 14 and 15 together, we see that 12 is equal to Pr u < θ X t < αt, sup X s > b = Pr u < θ X t < αt, S X t > b u s αt which leads to 9. Since X s, s is also a standard Brownian motion and M X α, t = M X 1 α, t almost surely, we use X s instead of X s and we get that for b <, Pr M X α, t < b, L X α, t > u = Pr u < θ X t 1 α t, S X t > b, 5

6 which leads to 11. To see that t K X α, t, M X α, t X t has the same distribution as L X α, t, M X α, t, set again X s = X t s X t. Clearly X s, s t is a standard Brownian motion and we can easily see that M X α, t = M X α, t X t, and so M X α, t X t = M X α, t and K X α, t = t L X α, t. Remarks 1. The distribution of θ X t, S X t is well known see for example Karatzas and Shreve [6], page 12. From this and theorem 2, we can deduce the density of L X α, t, M X α, t. This is given by b Pr M X α, t db, L X α, t du = π u 3 t u exp b2 2u [1 < u < αt, b > + 1 < u < 1 α t, b < ] dbdu Theorem 2 also leads to an alternative expression for the distribution of M X α, t ; that is for b > and Pr M X α, t db = Pr S X t db, < θ X t < αt, Pr M X α, t db = Pr S X t d b, < θ X t < 1 α t, for b <. 3. Using the argument at the end of the proof, we can generalise the last assertion of the theorem and observe that has the same law as and so we see that and K X α, t, M X α, t X t, X t t L X α, t, M X α, t, X t K X α, t, M X α, t, X t t L X α, t, M X α, t X t, X t, have the same distribution, a fact we will use in the following section. 6

7 3 The joint law of L X α, t, K X α, t, M X α, t, X t. From now on we will denote the density of τ b by k, ; that is for v >, Pr τ b dv = k v, b dv = 2 b exp b2 dv. 17 2πv 3 2v We will also denote the joint density of M v X t, t, X t by g,,, ; that is for < v < t, v Pr M X t, t db, X t da = g b, a, v, t dbda. We can calculate g,,, by using the proposition in the introduction. MX v t, t, X t has the same distribution as S X1 v S X2 t v, X 1 v X 2 t v, where X 1 s, s v and X 2 s, s t v are independent standard Brownian motions. The density of S X t, X t is given by Pr S X t db, X t da = b a 2b a2 exp 1 b, b a dadb 19 2πt 3 2t see Karatzas and Shreve [6], p.95. We observe that since 19 is bounded, g,,, is a bounded density. For our results, we need to calculate g,, v, t. This is the same as the value of the density of M X v t, t, M X v t, t X t at,. From 19 we see that Pr S X t dy, S X t X t dx = 2 2 y + x y + x2 exp 1 y, x dydx 21 2πt 3 2t and it is a simple exercise to verify that 2 y + x exp 2πv 3 g,, v, t = y + x2 2v 2 y + x 2π t v 3 exp y + x2 dxdy 2 t v v t v = t We will now obtain a preliminary result. 7

8 Lemma 1 For any u and v, such that < u < v < t, we have that Pr L X α, t > u, M X α, t db, X t da, K X α, t > v = Pr τ b > u, M X α, t db, X t da, K X α, t > v. 23 Proof Since M X α, t = M X 1 α, t, it suffices to prove 23 for b >. We have to prove that 1 lim δ,ε δε {Pr L X α, t > u, M X α, t b, b + δ], X t a, a + ε], K X α, t > v Pr τ b > u, M X α, t b, b + δ], X t a, a + ε], K X α, t > v} =. 24 Let X s = X s + u X u. We then have that Pr L X α, t > u, M X α, t b, b + δ], X t a, a + ε], K X α, t > v Pr τ b > u, M X α, t b, b + δ], X t a, a + ε], K X α, t > v = Pr b < S X u < M X α, t b + δ, X t a, a + ε], K X α, t > v Pr b < S X u < M X α, t b + δ, X t a, a + ε] = b < S X u < b + δ, Pr S X u < M X αt u, t u + X u b + δ,. 25 X t u + X u a, a + ε] Since X s, s t u is independent of X s, s u, and g,,, is bounded, we condition on S X u = y and X u = x and see that there is a constant K, such that y < MX αt u, t u + x b + δ, Pr X Kε b + δ y. t u + x a, a + ε] We therefore conclude that 25 is bounded by KεE b + δ S X u 1 b < S X u < b + δ Kεδ Pr b < S X u < b + δ and by the continuity of the distribution of S X u, we see that the limit in 24 is zero. As a corollary we will obtain the distribution of L X α, t, M X α, t, X t. 8

9 Corollary 1 The law of L X α, t, M X α, t, X t is given by Pr L X α, t du, M X α, t db, X t da = { k b, u g, a b, αt u, t u 1 < u < αt dudbda b > k b, u g, a b, αt, t u 1 < u < αt dudbda b < 26 Proof For b >, since X s + τ b X τ b, s t τ b is independent of X s, s τ b, we have that αt v Pr τ b > v, M X α, t b, b + δ], X t a, a + ε = Pr τ b du Pr M X αt u, t u, δ], X t a b, a b + ε. For b <, we use that M X α, t = M X 1 α, t and so g, b a, 1 α t u, t u = g, a b, αt, t u. We can now obtain the law of L X α, t, K X α, t, M X α, t, X t. Theorem 3 Pr L X α, t du, K X α, t dv, M X α, t db, X t da = 2 b b a dudvdbda exp b2 b a2 π 2 v u 2 u 3 t v 3 2u 2 t v v u 1 α t 1 α t1 u >, u + 1 α t < v < t b >, b > a αt u v αt1 < u < αt < v < t b >, b < a. v u αt αt1 u >, u + αt < v < t b <, b > a 1 α t u v 1 α t1 < u < 1 α t < v < t b <, b < a 27 Proof We start with the case b >, b > a. Using 23, and choosing ε such that a + ε < b, we need to look at Pr τ b r, K X α, t v, M X α, t b, b + δ], X t a, a + ε] = τ Pr b r, M X α, t b, b + δ], = X t a, a + ε], M X α, t sup v s t X s 9

10 r Pr τ b du Pr r Pr τ b du Pr M X αt u, t u, δ], X t u a b, a b + ε], M X αt u, t u sup v u s t u X s M X αt u, t u, δ], X t u a b, a b + ε], K X αt u, t u v u Using the last remark of the previous section, we then see that M X αt u, t u, δ], Pr X t u a b, a b + ε], = K X αt u, t u v u Pr M X αt u, t u X t u, δ], X t u a b, a b + ε], L X αt u, t u t v From the previous theorem we see that the density of = L X αt u, t u, M X αt u, t u X t u, X t u at t v,, a b is k b a, t v g,, v u 1 α t, v u 1 < t v < αt u. Combining this with 28 we get that L X α, t, K X α, t, M X α, t, X t has a continuous density at u, v, b, a that is given by k b, u k b a, t v g,, v u 1 α t, v u 3 1 u >, u + 1 α t < v < t. 31 We now look at the case b >, b < a. Using 23, and choosing δ such that b + δ < a, we need to look at Pr τ b r, K X α, t > v, M X α, t b, b + δ], X t a, a + ε] = τ Pr b r, M X α, t b, b + δ], = X t a, a + ε], M X α, t < inf v s t X s r Pr τ b du Pr M X αt u, t u, δ], X t u a b, a b + ε], M X αt u, t u < inf v u s t u X s = 1

11 r Pr τ b du Pr M X αt u, t u, δ], X t u a b, a b + ε], K X αt u, t u < v u Using 29 and the previous theorem we see that the density of. 32 L X αt u, t u, M X αt u, t u X t u, X t u at t v,, a b is k b a, t v g,, αt u, v u 1 αt < v. Combining this with 28 we get that L X α, t, K X α, t, M X α, t, X t has a continuous density at u, v, b, a that is given by k b, u k b a, t v g,, αt u, v u 1 < u < αt < v < t. 33 Substituting 17 and 22 into 31 and 33, we get the first two legs of 27. Considering X s, s t and observing that M X α, t = M X 1 α, t, L X α, t = L X 1 α, t and K X α, t = K X 1 α, t yields the rest of 27. References [1] Akahori, J., 1995, Some formulae for a new type of path-dependent option, Ann. Appl. Prob., 5, [2] Chaumont, L., 1999, A path transformation and its applications to fluctuation theory, J. London Math. Soc. 2, 59, no 2, [3] Dassios, A., 1995, The distribution of the quantiles of a Brownian motion with drift and the pricing of related path-dependent options, Ann. Appl. Prob., 5, [4] Dassios, A., 1996, Sample quantiles of stochastic processes with stationary and independent increments, Ann. Appl. Prob., 6, [5] Embrechts P.,Rogers, L.C.G., and Yor, M., 1995, A proof of Dassios representation of the α-quantile of Brownian motion with drift, Ann. Appl. Prob., 5,

12 [6] Karatzas I. and S.E. Shreve, 1988, Brownian motion and Stochastic Calculus, Springer-Verlag. [7] Miura, R., 1992, A note on a look-back option based on order statistics, Hitosubashi Journal of Commerce and Management, 27, [8] Wendel, J.G., 196, Order statistics of partial sums, Ann. of Math. Stat., 31,