CHANGE OF TIME METHOD IN MATHEMATICAL FINANCE

Transcription

1 CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 15, Number 3, Fall 27 CHANGE OF TIME METHOD IN MATHEMATICAL FINANCE ANATOLIY SWISHCHUK ABSTRACT. In his paper, we consider applicaions of he change of ime mehod o sudy he following hree models in finance: he geomerical Brownian moion model for sock prices, he mean-revering model for commodiy asse prices and Heson s sochasic volailiy model for sock prices. We apply he change of ime mehod o derive he well-known Black-Scholes formula for European call opions and o derive an explici opion pricing formula for a European call opion for a meanrevering model for commodiy prices. We also derive explici formulas for variance and volailiy swaps for financial markes wih a sochasic volailiy following a Cox-Ingersoll-Ross process [21]. Two numerical examples are presened. One is he S&P 6 Canada Index (January 1997 February 22) o price variance and volailiy swaps for he Heson model and he oher is he AECO Naural Gas Index, (1 May April 1999), o price a European call opion for he mean-revering asse model. 1 Inroducion One probabilisic mehod which is useful in solving sochasic differenial equaions (SDE) arising in finance is he change of ime mehod. 1.1 Change of ime mehod The general heory of ime changes for maringales and semimaringales (see Ikeda and Waanabe [39]) is well known. Le (Ω, F, P ) be a given probabiliy space wih a righ coninuous filraion (F ). Suppose M is a square inegrable local coninuous maringale such ha lim + M () = + almos sure (a.s.), τ := inf{u : M (u) > } and F = F τ. Then he ime-changed process Research is suppored by NSERC gran RPG Keywords: change of ime mehod, maringales, semimaringales, sochasic differenial equaions, geomeric Brownian moion, mean-revering asse, Cox-Ingersol- Ross process, opion pricing formula, Heson model, variance and volailiy swaps, S&P 6 Canada Index, AECO Naural Gas Index. Copyrigh c Applied Mahemaics Insiue, Universiy of Albera. 299

2 3 ANATOLIY SWISHCHUK B() := M(τ ) is an F -Brownian moion. Also, M() = B(< M > ()). Here, defines predicable quadraic variaion. If φ is a change of ime process (i.e., any coninuous F -adaped process such ha φ =, φ is sricly increasing and lim + φ = + a.s.) and if X is an F -adaped semimaringale, hen he process X := X τ is an F - adaped semimaringale, where τ := inf{u : φ u > }, and F := F τ. X is called he ime change of X by φ. We noe, ha Clark [19] inroduced Bochner s [11] ime-changed Brownian moion B τ ino financial economics. He wroe down a model for he log-price M as M = B τ, where B is Brownian moion, τ is a ime-change independen of B. The change of ime mehod is closely associaed wih he embedding problem: o embed a process (X, F ) in Brownian moion is o find a Wiener process W and an increasing family of sopping imes τ such ha W τ has he same join disribuions as X. The embedding problem was firs reaed by Skorokhod [53], who showed ha he sum of any sequence of independen random variables wih mean zero and finie variance could be embedded in Brownian moion using sopping imes. Dambis [22] and Dubins and Schwarz [23] had shown ha every coninuous maringale could be ime-changed ino Brownian moion. Huff [37] showed ha every process of pahwise bounded variaion could be embedded in Brownian moion. Monroe [44] had proved ha every righ coninuous maringale could be embedded in a Brownian moion process wih a sufficienly large σ-field. In Monroe [45] i was shown ha a process can be embedded in Brownian moion if and only if his process is a local semimaringale. Geman, Madan and Yor [31] consider pure jump Lévy processes of finie variaion wih an infinie arrival rae of jumps as models for he logarihm of asse prices. These processes may be also wrien as ime-changed Brownian moion. This paper exhibis he explici ime change for each of a wide class of Lévy processes and shows ha he ime change is a weighed price move measure of ime. We also menion variance gamma process (Madan and Senea [42]) ha can be inerpreed as a Brownian moion wih drif, ime changed by a gamma process. In his paper we give a brief descripion of he change of ime mehod in he following seings: maringales and sochasic differenial equaions. The change of ime mehod is used o solve he following SDE: dx = α(, X )db() wih B() being a Brownian moion and α(, x) being a good funcion of and x R. Having solved he equaion we can also solve he

3 CHANGE OF TIME METHOD 31 general SDE dx = β(, X )d + γ(, X )db() wih drif β(, X ) using he mehod of ransformaion of drif (he Girsanov ransformaion) (see Ikeda and Waanabe [39, Chaper IV, Secion 4, p. 176]). 1.2 Subordinaors Feller [29] inroduced a subordinaed process X τ for a Markov process X and τ a process wih independen incremens. τ was called a randomized operaional ime. Increasing Lévy processes can also be used as a ime change for oher Lévy processes (see [1, 2, 6, 8, 2, 49]). Lévy processes of his kind are called subordinaors. They are very imporan ingrediens for building Lévy based models in finance (see [2, 49]). If S is a subordinaor, hen is rajecories are almos surely increasing, and S can be inerpreed as a ime deformaion and used o ime change oher Lévy processes. Roughly, if (X ) is a Lévy process and (S ) is a subordinaor independen of X, hen he process (Y ) defined by Y := X S is a Lévy process (see [2]). This ime scale has he financial inerpreaion of business ime (see [32]), ha is, he inegraed rae of informaion arrival. The ime change mehod was used o inroduce sochasic volailiy ino a Lévy model o achieve he leverage effec and a long-erm skew (see [15]). In he Baes model [7] he leverage effec and long-erm skew were achieved using correlaed sources of randomness in he price process and he insananeous volailiy. The sources of randomness are hus required o be Brownian moions. In he Barndorff-Nielsen and Shephard model [4, 5] he leverage effec and long-erm skew are generaed using he same jumps in he price and volailiy wihou a requiremen for he sources of randomness o be Brownian moions. Anoher way o achieve he leverage effec and long-erm skew is o make he volailiy govern he ime scale of he Lévy process driving jumps in he price. Carr e al. [15] suggesed he inroducion of sochasic volailiy ino an exponenial-lévy model via a ime change. The generic model here is S = exp(x ) = exp(y v ), where v := σ2 sds. The volailiy process should be posiive and mean-revering (i.e., an Ornsein-Uhlenbeck or Cox-Ingersoll-Ross processes). Barndorff-Nielsen e al. [6] reviewed and placed in he conex some of heir recen work on sochasic volailiy models including he relaionship beween subordinaion and sochasic volailiy. The main difference beween he change of ime mehod and he subordinaor mehod is ha in he former case he change of ime process φ depends on he process X, bu in he laer case, he subordinaor

4 32 ANATOLIY SWISHCHUK S and Lévy process X are independen. 1.3 Black-Scholes by change of ime mehod In he early 197s, Black and Scholes [1] made a major breakhrough by deriving a pricing formula for vanilla opion wrien on a sock. Their model and is exensions assume ha he probabiliy disribuion of he underlying cash flow a any given fuure ime is lognormal. There are a leas hree proofs of heir resul, including parial differenial equaion and maringale approach (see [26, 61]). One of he aims of his paper is o give ye one more derivaion of he Black-Scholes resul by he change of ime mehod. 1.4 An opion pricing formula for a mean-revering asse model using a change of ime mehod Some commodiy prices, like oil and gas, exhibi mean reversion. This means ha hey end over ime o reurn o some long-erm mean. This mean-revering model is a one-facor version of he wo-facor model made popular in he conex of energy modeling by Pilipovic [48]. Black s model [9] and Schwarz s model [5] have become sandard ools o price opions on commodiies. These models have he advanage ha hey give rise o closed-form soluions for some ypes of opion (see [6]). Bos, Ware and Pavlov [12] presened a mehod for he evaluaion of he price of a European opion using a semi-specral mehod. This did no have he convenience of a closed-form soluion. However, hey showed ha opion values of cerain ypes may be found exremely efficienly. The working paper of Swishchuk [57] presens an explici expression for a European opion price for a mean-revering asse using he change of ime mehod under boh physical and risk-neural measures. We noe, ha he recen book by Geman [3] discusses hard and sof commodiies, (ha is, energy, agriculure and meals) and also presens an analysis economic and geopoliical issues in commodiies markes. One of he aims of his paper is o obain an explici expression for a European call opion price on a mean-revering model of commodiy asse using he change of ime mehod. As we can see, if he meanrevering level is zero hen he opion price coincides wih Black-Scholes resul. 1.5 Swaps by change of ime mehod: Heson model Volailiy swaps are forward conracs on he fuure realized sock volailiy, variance swaps are similar conrac on variance, he square of he fuure volailiy. Boh hese insrumens provide a way for invesors o obain exposure o he fuure level of volailiy. A sock s volailiy is

5 CHANGE OF TIME METHOD 33 he simples measure of is risk or uncerainy. Formally, he volailiy σ R is he annualized sandard deviaion of he sock s reurns during he period of ineres, where he subscrip R denoes he observed or realized volailiy. The easy way o rade volailiy is o use volailiy swaps, someimes called realized volailiy forward conracs, because hey provide pure exposure o volailiy, (and only o volailiy). Demeerfi e al. [24] explained he properies and he heory of boh variance and volailiy swaps. Javaheri e al. [4] discussed he valuaion and hedging of a GARCH(1,1) sochasic volailiy model. They used a general and flexible PDE approach o deermine he firs wo momens of he realized variance in a coninuous or discree conex. Then hey approximae he expeced realized volailiy via a convexiy adjusmen. Brockhaus and Long [13] provided an analyical approximaion for he valuaion of volailiy swaps and analyzed oher opions wih volailiy exposure. The working paper by Théore e al. [59] presened an analyical soluion for pricing volailiy swaps, proposed by Javaheri e al. [4] They priced volailiy swaps wihin framework of a GARCH(1,1) sochasic volailiy model and applied he analyical soluion o price a swap on volailiy of he S&P 6 Canada Index (5-year hisorical period: ). Alhough opions marke paricipans alk of volailiy, i is variance, or volailiy squared, ha has more fundamenal significance (see [24]). Modeling and pricing of variance, volailiy, covariance and correlaion swaps for Heson model has been considered in [54]. Variance swaps for financial markes wih underlying asse and sochasic volailiies wih delay were modelled and priced in [56]. Variance swaps for sochasic volailiy wih delay is very similar o variance swaps for sochasic volailiy in Heson model (see [54]), bu simplier o model and price. Variance swaps for he muli-facor sochasic volailiy models wih delay have been sudied in [58]. The pricing of variance swaps in Markov-modulaed Brownian markes was considered in [27, 28] One of he aim of his paper is o value variance and volailiy swaps for Heson model [34] using he change of ime mehod. Remark 1.1. An exensive review of he lieraure on sochasic volailiy is given in [51, 52]. A deailed inroducion o variance and volailiy swaps, including a hisory and marke producs, may be found in [14, 24]. The pricing of a range of volailiy derivaives, including volailiy and variance swaps and swapions, is sudied in [36]. This paper also conains many volailiy models, including some wih jumps. A volailiy model wih jumps was firs considered in [46].

6 34 ANATOLIY SWISHCHUK Remark 1.2. The fac ha sochasic volailiy models, such he Heson model and ohers, are able o fi skews and smiles, while simulaneously providing sensible Greeks, have made hese models a popular choice in he pricing of opions and swaps. Some ideas how o calculae he Greeks for volailiy conracs may be found in [36]. 1.6 Organizaion of he paper We give a brief descripion of he change of ime mehod for maringales and sochasic differenial equaions (Secion 2). We apply he change of ime mehod o derive he well-known Black-Scholes formula for European call opion (Secion 3) and o derive an explici opion pricing formula for a European call opion on a mean-revering model for commodiy prices (Secion 4). We derive explici formulas for variance and volailiy swaps for financial markes whose sochasic volailiy follows a Cox-Ingersoll-Ross process (Secion 5). The conclusion, fuure work and references are presened in Secions 6, 7 and 8, respecively. Two numerical examples are presened in Secion 9. They are on he S&P 6 Canada Index (January 1997 February 22) o price variance and volailiy swaps for he Heson model and on AECO Naural Gas Index (1 May April 1999) o price European call opion for mean-revering asse model. 2 Change of ime mehod In his secion we give a brief descripion of he change of ime mehod for he maringales and sochasic differenial equaions. Throughou in his paper we consider (Ω, F, F, P ) o be a probabiliy space wih a righ coninuous filraion (F ). 2.1 Change of ime mehod in maringale seing In his secion, we describe he change of ime mehod for a maringale M() M c,loc 2, he space of local square inegrable coninuous maringales (see [39, Theorem 7.2, Chaper 2]). If M() M c,loc 2, lim + M () = + a.s., τ := inf{u : M (u) > } and F := F τ, hen he following process wih changed ime W () := M(τ ) is an F -Brownian moion (or sandard Wiener process). Consequenly, we can express a local maringale M() using an F - Brownian moion W () and an F -sopping ime, (since { M () u} = {τ u } F τu = F u ) M() = W ( M ()).

7 CHANGE OF TIME METHOD Change of ime mehod in sochasic differenial equaion (SDE) seing We consider he following generalizaion of he previous resuls o a SDE of he following form (wihou a drif) (2.1) dx() = α(, X()) dw (), where W () is a Brownian moion and α(, X) is a coninuous and measurable by and X funcion on [, + ) R. The reason we consider his equaion is if we solve he equaion, hen we can solve and more general equaion wih a drif β(, X) using he Girsanov ransformaion (see [39, Chaper 4, Secion 4]). Theorem 2.1. ([39, Chaper IV, Theorem 4.3]) Le W () be an onedimensional F -Wiener process wih W () =, given on a probabiliy space (Ω, F, (F ), P ) and le X() be an F -adoped random variable. Define a coninuous process V = V () by (2.2) V () = X() + W (). Le φ be he change of ime process (see Secion 2.3): (2.3) φ = If α 2 (φ s, X() + W (s))ds. (2.4) X() := V (φ 1 ) = X() + W (φ 1 ) and F := F φ 1, hen here exiss F -adoped Wiener process W = W () such ha (X(), W ()) is a soluion of (1) on probabiliy space (Ω, F, F, P ). The proof of his heorem may be found in [39, Chaper IV, Theorem 4.3]. We noe ha in his case, (2.5) M() := W (φ 1 ) is a maringale wih quadraic variaion φ 1 (2.6) M () = φ 1 = α 2 (φ s, X)dφ s = α(s, X) 2 ds,

8 36 ANATOLIY SWISHCHUK and φ 1 saisfies he equaion (2.7) φ 1 = We also remark, ha (2.8) W () = α 2 (s, X() + W (φ 1 s )) ds. α 1 (s, X(s))d W (φ 1 s ) = α 1 (s, X(s)) dm(s) and X() = X() + α(s, X) dw (s). Corollary 2.1. The soluion of he following SDE (2.9) dx() = a(x()) dw () may be presened in he following form X() = X() + W (φ 1 ), where a(x) is a coninuous measurable funcion, W () is an one-dimensional F -Wiener process wih W () =, given on a probabiliy space (Ω, F, (F ), P ) and X() is an F -adoped random variable. In his case, (2.1) φ = and (2.11) φ 1 = a 2 (X() + W (s)) ds, a 2 (X() + W (φ 1 s )) ds. (See [39, Chaper IV, Example 4.2].) We noe ha M() := W (φ 1 ) is a maringale wih quadraic variaion φ 1 M () = φ 1 = a 2 (X) dφ s = a(x) 2 ds.

9 CHANGE OF TIME METHOD 37 We also remark, ha W () = a 1 (X(s)) d W (φ 1 s ) = a 1 (X() + W (φ 1 s ))) d W (φ 1 s ) and X() = X() + a(x(s)) dw (s). Corollary 2.2. (Soluion for Ornshein-Uhlenbeck (OU) Process Using Change of Time). Le S saisfy he following SDE: ds = αs d + σ dw. Then S may be presened in he following form using he change of ime mehod: S = e α [S + W (φ 1 )], where φ 1 saisfies φ 1 = σ 2 (e αs (S + W (φ 1 s ))) 2 ds. Corollary 2.3 (Soluion for Vasicek Process Using Change of Time). Le S saisfy he following SDE: ds = α(b S ) d + σ dw. Then S may be presened in he following form using he change of ime mehod S = e α [S b + W (φ 1 )], where φ 1 saisfies φ 1 = σ 2 (e αs (S b + W (φ 1 )) + b) 2 ds. Remark 2.1. We are going o apply Theorem 2.1 above o solve Cox- Ingersoll-Ross [21] equaion and mean-reversion equaion for commodiy price (Pilipovich model [48]), and Corollary 2.1 o solve SDE for sock price in Black-Scholes [1] seing. s

10 38 ANATOLIY SWISHCHUK 3 Black-Scholes formula by change of ime mehod Le (Ω, F, F, P ) be a probabiliy space wih a sample space Ω, σ-algebra of Borel ses F and probabiliy P. The filraion F, [, T ], is he naural filraion of a sandard Brownian moion W, [, T ], and F T = F. 3.1 Black-Scholes formula The well-known Black-Scholes formula [1] saes ha if we have (B, S)-securiy marke consising of riskless asse B() wih iners rae r as a consan (3.1) db() = rb() d, B() >, r >, and risky asse (sock) S() (3.2) ds() = µs() d + σs() dw (), S() >, where µ R is an appreciaion rae, σ > is a volailiy, hen opion price formula for European call opion wih pay-off funcion f(t ) = max(s(t ) K, ) (K > is a srike price) has he following form (3.3) C(T ) = S()Φ(y + ) e rt KΦ(y ), where (3.4) y ± := and ln( S() K (3.5) Φ(y) := 1 2π y σ2 ) + (r ± 2 )T σ T e x2 2 dx. 3.2 Soluion of SDE for geomeric Brownian moion using change of ime mehod Lemma 3.1. The soluion of he equaion (3.2) has he following look: (3.6) S() = e µ (S() + W (φ 1 s )), where W () is an one-dimensional Wiener process, and φ 1 = σ 2 [S() + W (φ 1 )] 2 ds φ = σ 2 [S() + W (s)] 2 ds. s

11 CHANGE OF TIME METHOD 39 Proof. Se (3.7) V () = e µ S(), where S() is defined in (3.2). Applying Iô formula o V () we obain (3.8) dv () = σv () dw (). Equaion (3.8) looks like equaion (2.9) wih a(x) = σx. In his way, he soluion of he equaion (3.8) using change of ime mehod (see Corollary 2.1, Secion 2.2, (2.1) and (2.11)) is (3.9) V () = S() + W (φ 1 ), where W () is an one-dimensional Wiener process, and φ 1 = σ 2 [S() + W (φ 1 )] 2 ds φ = σ 2 [S() + W (s)] 2 ds. From (3.7) and (3.9) i follows ha he soluion of he equaion (3.2) has he represenaion (3.6). 3.3 Properies of he process W (φ 1 ) Lemma 3.2. Process W (φ 1 ) is a mean-zero maringale wih quadraic variaion < W (φ 1 ) >= φ 1 and has he following represenaion (3.1) W (φ 1 ) = S()(e s = σ 2 [S() + W (φ 1 )]2 ds σ 2 σw () 2 1). Proof. The resul follows from Corollary 2.1, Secion 2.4. s

12 31 ANATOLIY SWISHCHUK We noe ha E W (φ 1 s ) = and E[ W (φ 1 )] 2 = S 2 ()(e σ2 1), where E := E P is an expecaion under physical measure P. Since σ2 σw () (3.11) E[e 2 ] n = e σ2 2 n(n 1), we can obain all he momens for he process W (φ 1 s ) : (3.12) E[ W (φ 1 )] n = S n () n k= C k n e σ2 2 k(k 1) ( 1) n k, where C k n := (n!)/(k!(n k)!), n! := n. Corollary 3.1. From Lemma 3.2 (see (3.6), (3.1) and (3.11)) i follows ha we can also obain all he momen for he asse price S() in (3.9), since (3.13) E[S()] n = e nµ E[S() + W (φ 1 s )]n = e nµ S n ()E[e σw () σ2 2 ] n = e nµ S n ()e σ2 2 n(n 1). For example, variance of S() is going o be V ars() = ES 2 () (ES()) 2 = S 2 ()e 2µ (e σ2 1), where ES() = S()e µ (see (3.13)). 3.4 Black-Scholes formula by change of ime mehod In riskneural world he dynamic of sock price S() has he following look: (3.14) ds() = rs() d + σs() dw (), where (3.15) W () := W () + µ r σ. Following Secion 3.2, from (3.6) we have he soluion of equaion (3.14) (3.16) S() = e r [S() + W (φ 1 )],

13 CHANGE OF TIME METHOD 311 where (3.17) W (φ 1 ) = S()(e σw () σ 2 2 1) and W () is defined in (3.15). Le E P be an expecaion under risk-neural measure (or maringale measure) P (i.e., process e rt S() is a maringale under he measure P ). Then he opion pricing formula for European call opion wih pay-off funcion f(t ) = max[s(t ) K, ] has he following form (3.18) C(T ) = e rt E P [f(t )] = e rt E P [max(s(t ) K, )]. Proposiion 3.1. (3.19) C(T ) = S()Φ(y + ) Ke rt Φ(y ), where y ± and Φ(y) are defined in (3.4) and (3.5). Proof. Using change of ime mehod we have he following represenaion for he process S() (see (3.16)) S() = e r [S() + W (φ 1 )], where W (φ 1 ) is defined in (3.17). From (3.14) (3.18), afer subsiuing W (φ 1 ) ino (3.16), and S(T ) ino (3.18), i follows ha (3.2) C(T ) = e rt E P [max(s(t ) K, )] = e rt E P [max(e r [S() + W (φ 1 )] K, )] = e rt E P [max(e r S()e σw (T ) σ2 T 2 K, )] = e rt E P [max(s()e σw (T )+(r σ2 2 )T K, )] = e rt 1 2π + max[s()e σu T +(r σ2 2 )T K, ]e u2 2 du.

14 312 ANATOLIY SWISHCHUK Le y be a soluion of he following equaion namely, S()e σy T +(r σ 2 /2)T = K, y = ln( K S() ) (r σ2 /2)T σ. T Then (3.2) may be presened in he following form (3.21) C(T ) = e rt 1 + (S()e σu T +(r σ 2 2 )T K)e u2 2 du. 2π y Finally, sraighforward calculaion of he inegral in he righ-hand side of (3.21) gives us he Black-Scholes resul C(T ) = 1 + 2π y = S() 2π + y σ T S()e σu T σ 2 T 2 e u 2 /2 du Ke rt [1 Φ(y )] e u2 /2 du Ke rt [1 Φ(y )] = S()[1 Φ(y σ T )] Ke rt [1 Φ(y )] = S()Φ(y + ) Ke rt Φ(y ), where y ± and Φ(y) are defined in (3.4) and (3.5). 4 Explici opion pricing formula for mean-revering asse model (MRAM) by change of ime mehod In his secion, we consider a risky asse S following he mean-revering sochasic process given by he following sochasic differenial equaion (4.1) ds = a(l S ) d + σs dw, where W is an F -measurable one-dimensional sandard Wiener process, σ > is he volailiy, consan L is called he long-erm mean of he process, o which i revers over ime, and a > measures he srengh of mean reversion. We find explici soluion of equaion (4.1) using change of ime mehod, give some properies of he mean-revering asse S and presen explici opion pricing formula for European call opion for his mean-revering asse model of commodiy price.

15 CHANGE OF TIME METHOD Explici soluion of SDE for MRAM Lemma 4.1. Equaion in (4.1) has he following soluion ds = a(l S )d + σs dw, (4.2) S = e a [S L + W (φ 1 )] + L, where W (φ 1 ) is an one-dimensional Wiener process and (4.3) φ 1 = σ 2 (S L + W (φ 1 ) + e as L) 2 ds Proof. The resul follows from he subsiuion (4.4) V := e a (S L). and Theorem 2.1, Secion Explici expression for he process W (φ 1 ) I urns ou ha we can find he explici expression for he process W (φ 1 ). Lemma 4.2. Process W (φ 1 ) has he following form (4.5) W (φ 1 ) = S()(e σ 2 σw () + ale σw () σ ) + L(1 e a ) e as e σw (s)+ σ2 s 2 ds. Proof. The resul follows from he following presenaion (4.6) W () = σ 1 [S() L + Le a + W (φ 1 s )] 1 d W (φ 1 s ). Remark 4.1. If L = in (4.5), hen he expression for W (φ 1 ) in (4.5) is coincided wih he expression W (φ 1 ) in (3.1).

16 314 ANATOLIY SWISHCHUK 4.3 Properies of he process W (φ 1 ) We noe ha process W (φ 1 is F := F φ 1-measurable (see Secion 2.2) wih zero mean: (4.7) E W (φ 1 ) =. Also, process W (φ 1 ) ) is a maringale wih quadraic variaion φ 1. In his way we can find is second momen. Le us calculae he second momen of W (φ 1 ): (4.8) E W 2 (φ 1 ) = E < W (φ 1 ) >= Eφ 1 = σ 2 E(S L + W (φ 1 s ) + eas L) 2 ds [ = σ 2 (S L) 2 + 2L(S L)(e a 1) a + L2 (e 2a 1) 2a + E W 2 (φ 1 s ) ds ]. From (4.8), solving his linear ordinary nonhomogeneous differenial equaion wih respec o E W 2 (φ 1 ), de W 2 (φ 1 ) d = σ 2 [(S L) 2 + 2L(S L)e a + L 2 e 2a + E W 2 (φ 1 )], we obain E W 2 (φ 1 ) = σ 2 [(S L) 2 eσ 2 1 (4.9) σ 2 + 2L(S L)(e a e σ2 ) a σ 2 + L2 (e 2a e σ 2 ) 2a σ 2 ]. Using maringale properies of he process W (φ 1 ) and explici expression (4.5) we can do more and calculae all he momen for his process W (φ 1 ). I means ha we shall know he disribuion of he process W (φ 1 ). Lemma 4.3. Process W (φ 1 (4.1) E[ W (φ 1 )] n = ) has he following momens n Cn k [L(1 ea ) S()] n k k= k j= C j k Eηj 1 ()ηk j 2 (),

17 CHANGE OF TIME METHOD 315 where (4.11) (4.12) E[η l 1 ()ηm 2 ()] = mal η 1 () := S()e σw () σ2 2 e as E[η l 1 (s)ηm 1 2 (s)] ds, l, m 1, and (4.13) η 2 () := ale σw () σ2 2 e as e σw (s)+ σ2 s 2 ds. Proof. The resul follows from he following presenaion (4.14) W (φ 1 ) = S()e σ 2 σw () 2 + ale σw () σ2 2 e as e σw (s)+ σ2 s 2 ds + [L(1 e a ) S()]. Remark 4.2. I is easy o see ha he process W (φ 1 ) in (4.5) consiss of wo zero-mean maringales and S()(e σw () σ2 2 1) L(1 e a ) + ale σw () σ2 2 From here we can see, for example, ha where η 2 () is defined in (4.13). Eη n 2 () = L n (e a 1) n, e as e σw (s)+ σ2 s 2 ds. 4.4 Some properies of he mean-revering asse S From (4.2) we obain he mean value of he firs momen for mean-revering asse S : ES() = e a [S() L] + L. I means ha ES() L when + (mean-revering propery).

18 316 ANATOLIY SWISHCHUK Using formulae (4.2) and (4.12) we can calculae he second momen of S() : ES 2 () = (e a (S() L) + L) 2 + σ 2 e 2a [ (S() L) 2 eσ2 1 σ 2 + 2L(S() L)(ea e σ2 ) a σ 2 + L2 (e 2a e σ 2a σ 2 Combining he firs and he second momens we have he variance of S : V ar(s()) = ES() 2 (ES()) 2 = σ 2 e 2a [(S() L) 2 eσ2 1 σ 2 + 2L(S() L)(ea e σ2 ) a σ 2 2 ) ]. + L2 (e 2a e σ 2 ) 2a σ 2 ]. However, knowing all he momens of he process W (φ 1 ) (see Lemma 4.3) and explici expression for S() hrough W (φ 1 ) (see Lemma 4.1) we can also know all he momens for our mean-revering asse S(). Lemma 4.4. Process S() has he following momens: (4.15) E[S()] n = e na E[η 1 () + η 2 ()] n = e na n k= C k n E[ηk 1 ηn k 2 ], where η 1 () and η 2 () are defined in (4.12) and (4.13), respecively, and E[η l 1 ()ηm 2 ()] = mal e as E[η l 1 (s)ηm 1 2 (s)] ds, l, m 1. Proof. The resul follows from he following presenaion (4.16) S() = e a [S L + W (φ 1 )] + L = S()e a e σw () σ2 2 + ale a e σw () σ2 2 = e a [η 1 () + η 2 ()], e as e σw (s)+ σ2 s 2 ds where η 1 () and η 2 () are defined in (4.12) and (4.13), respecively.

19 CHANGE OF TIME METHOD 317 The characerisic funcion of he mean-revering asse is equal o φ (λ) = + n= (iλ) n E[S()] n, n! where E[S()] n is defined in (4.15) and i := 1. For example, characerisic funcion of he process η 2 () is equal o φ (λ) = + n= (iλ) n E[η 2 ()] n = n! + n= (iλ) n L n (e a 1) n = e iλl(ea 1), n! since Eη n 2 () = Ln (e a 1) n. Knowing characerisic funcion of he process and applying inverse Fourier ransform o he funcion we can obain he disribuion of he mean-revering asse S(). 4.5 Explici opion pricing formula for European call opion under risk-neural measure In his secion, we are going o obain explici opion pricing formula for European call opion under riskneural measure P using change of ime mehod Mean-revering risk-neural asse model Consider our model (4.1) (4.17) ds = a(l S ) d + σs dw. We wan o find a probabiliy P equivalen o P, under which he process e r S is a maringale, where r > is a consan ineres rae. Lemma 4.5. In risk-neural world he model in (4.17) akes he following form: (4.18) ds = a (L S ) d + σs dw, where (4.19) (4.2) a := a + λσ, L := al a + λσ, W = W + λ and λ R is a marke price of risk. S(u) du Proof. The resul follows from Girsanov heorem (see [26]). Now, we are going o apply our mehod of changing of ime o he model (4.18) o obain he explici opion pricing formula.

20 318 ANATOLIY SWISHCHUK Explici soluion for mean-revering risk-neural asse model Lemma 4.6. Explici soluion for our risk-neural model (4.18) has he following look (4.21) S = e a [S L + W ((φ ) 1 )] + L, where W () is an F -measurable sandard one-dimensional Wiener process under measure P, (φ ) 1 is an inverse funcion o φ : (4.22) φ = σ 2 (S L + W (s) + e a φ s L ) 2 ds. Proof. Applying Lemma 4.1 (see (4.2) (4.6)) o our model (4.18) we obain he explici soluion (4.21) for our risk-neural model (4.18). We noe ha (4.23) (φ ) 1 = σ 2 (S L + W ((φ ) 1 ) + e a s L ) 2 ds, where a and L are defined in (4.19) Explici expression for he process W (φ 1 ) I urns ou ha we can find he explici expression for he process W (φ 1 ) using he following equaion for V (see Lemma 4.2) namely, or V = S L + W (φ 1 ), W () = σ 1 [S() L + Le a + W (φ 1 )] 1 d W (φ 1 W (φ 1 ) = σ [S() L + Le a + W (φ 1 s I is linear SDE and we can solve i explicily. following form: s )] dw (). s ) The soluion has he (4.24) W (φ 1 ) = S() ( e σw () σ ) + L(1 e a ) + ale σw () σ2 2 e as e σw (s)+ σ2 s 2 ds.

21 CHANGE OF TIME METHOD Some properies of he process W ((φ ) 1 ) Using he same argumen as in Secion 4.3, we obain he following properies of he process W ((φ ) 1 ) in (4.24): (4.25) E W ((φ ) 1 ) =, E [ W ((φ ) 1 )] 2 = σ 2 [(S L ) 2 eσ2 1 σ 2 + 2L (S L )(e a e σ2 ) a σ 2 + (L ) 2 (e 2a e σ2 ) 2a σ 2 ], where E is he expecaion wih respec o he probabiliy P, and a, L and (φ ) 1 are defined in (4.19) and (4.23), respecively. Also, process W (φ 1 ) has he following momens (see Lemma 4.3) (4.26) where E[ W (φ 1 )] n = n Cn[L k (1 e a ) S()] n k k= E[η l 1 ()ηm 2 ()] = ma L k j= C j k Eηj 1 ()ηk j 2 (), e a s E[η l 1 (s)ηm 1 2 (s)] ds, l, m Some properies of he mean-revering risk-neural asse Using he same argumen as in Secion 4.4, we obain he following properies of he mean-revering risk-neural asse S in (4.21): E S = e a [S L ] + L V ar (S ) := E S 2 (E S ) 2 (4.27) = σ 2 e 2a [(S L ) 2 eσ2 1 σ 2 + 2L (S L )(e a e σ2 ) a σ 2 + (L ) 2 (e 2a e σ2 ) 2a σ 2 ],

22 32 ANATOLIY SWISHCHUK where E is he expecaion wih respec o he probabiliy P, and a, L and (φ ) 1 are defined (4.2). From he expression for W (φ 1 ) (see (4.24)) and for S() in (4.21) we can find he explici expression for S() hrough W () : (4.28) S() = e a [S L + W (φ 1 )] + L = S()e a e σw () σ2 2 + a L e a e σw () σ2 2 e a s e σw (s)+ σ2 s 2 ds Also, from (4.28) and Lemma 4.4 i follows ha process S() in (4.28) has he following momens E[S()] n = e na E[η 1 () + η 2 ()]n = e na where η 1() and η 2() are defined as follows and η 1 () := S()eσW () σ2 2, η2 () := a L e σw () σ2 2 E[η l 1 ()ηm 2 ()] = ma L n Cn k E[(η 1 )k (η2 )n k ], k= e a s e σw (s)+ σ2 s 2 ds, e a s E[η l 1 (s)ηm 1 2 (s)] ds, l, m Explici opion pricing formula for European call opion under risk-neural measure The payoff funcion f T for European call opion equals o f T = (S T K) + := max(s T K, ), where S T is an asse price, T is an expiraion ime (mauriy) and K is a srike price. In his way (see (4.28)), (4.29) f T = [e at (S L + W (φ 1 T )) + L K]+ = [S()e a T e σw (T ) σ2 T 2 + a L e a T e σw (T ) σ2 T 2 T e a s e σw (s)+ σ2 s 2 ds K] +.

23 CHANGE OF TIME METHOD 321 Theorem 4.1. Explici opion pricing formula for European call opion under risk-neural measure for mean-revering asse S() in (4.29) has he following form (4.3) C T = e (r+a )T S()Φ(y + ) e rt KΦ(y ) + L e (r+a )T [(e a T 1) where y is he soluion of he following equaion y zf T (dz)], (4.31) (4.32) y = ln( K S() ) + ( σ2 2 + a )T σ T ln(1 + a L S() T ea s e σy s+ σ 2 s 2 ds) σ T y + := σ T y and y := y, a := a + λσ, L := al a + λσ, λ is he marke price of risk and FT (dz) is he disribuion wih characerisic funcion φ λ(t ) = e iλ(ea T 1), i := 1, λ C. Proof. The resul follows from he expression (4.33) C T := e rt E P f T = e rt E P [e a T (S L + W (φ 1 T )) + L K] + = 1 + e rt 2π + ale a T e σy T σ2 T 2 T and he above-menioned resuls. max[s()e a T e σy T σ2 T 2 e a s e σy s+ σ2 s 2 ds K, ]e y2 2 dy.

24 322 ANATOLIY SWISHCHUK 4.6 Connecion wih Black-Scholes resul: L = and a = r and Black-Scholes formula follows! Corollary 4.1. If L = and a = r hen we obain from Theorem 4.1 (4.34) C T = S()Φ(y + ) e rt KΦ(y ), where (4.35) y + := σ T y and y := y, and y is he soluion of he following equaion (see (4.47)) or S()e rt e σy T σ 2 T 2 = K (4.36) y = ln( K S() ) + ( σ2 2 r)t σ T, and Φ(x) = 1 2π x e y2 2 dy. Bu (4.34) (4.36) is exacly he well-known Black-Scholes resul! (see Secion 3.1, (3.3) (3.5), Proposiion 3.1, (3.2)). Remark 4.3. In his way, we can see ha he opion pricing formula in (4.3) for mean-revering asse S() consiss of a Balck-Scholes par and addiional par due o mean reverion. 5 Variance and volailiy swaps for Heson model by change of ime mehod 5.1 Variance and volailiy swaps A sock volailiy swap is a forward conrac on he annualized volailiy. Is payoff a expiraion is equal o N(σ R (S) K vol ),

25 CHANGE OF TIME METHOD 323 where σ R (S) is he realized sock volailiy (quoed in annual erms) over he life of conrac, σ R (S) := 1 T σs T 2ds, σ is a sochasic sock volailiy, K vol is he annualized volailiy delivery price, and N is he noional amoun of he swap in dollar per annualized volailiy poin. Alhough opions marke paricipans alk of volailiy, i is variance, or volailiy squared, ha has more fundamenal significance (see [24]). A variance swap is a forward conrac on annualized variance, he square of he realized volailiy. Is payoff a expiraion is equal o N(σ 2 R(S) K var ), where σr 2 (S) is he realized sock variance(quoed in annual erms) over he life of he conrac, i.e., σ 2 R (S) := 1 T T σ 2 s ds, K var is he delivery price for variance, and N is he noional amoun of he swap in dollars per annualized volailiy poin squared. The holder of variance swap a expiraion receives N dollars for every poin by which he sock s realized variance σr 2 (S) has exceeded he variance delivery price K var. Therefore, pricing he variance swap reduces o calculaing he square of he realized volailiy. Valuing a variance forward conrac or swap is no differen from valuing any oher derivaive securiy. The value of a forward conrac P on fuure realized variance wih srike price K var is he expeced presen value of he fuure payoff in he risk-neural world P var = E{e rt (σ 2 R(S) K var )}, where r is he risk-free discoun rae corresponding o he expiraion dae T, and E denoes he expecaion. Thus, for calculaing variance swaps we need o know only E{σR 2 (S)}, namely, mean value of he underlying variance.

26 324 ANATOLIY SWISHCHUK To calculae volailiy swaps we need more. From Brockhaus-Long approximaion [13] (which is used he second order Taylor expansion for funcion x) we have (see also [4, p. 16]) } E { σ 2R (S) E{V } V ar{v } 8E{V }, 3/2 where V := σ 2 R (S) and (V ar{v })/(8E{V }3/2 ) is he convexiy adjusmen. Thus, o calculae he value of volailiy swaps P vol = {e rt (E{σ R (S)} K vol )} we need boh E{V } and V ar{v }. In his secion, we explicily solve he Cox-Ingersoll-Ross equaion for sochasic volailiy Heson model using he change of ime mehod and presen he formulas o price variance and volailiy swaps for his model. 5.2 Sochasic volailiy: Heson model Le (Ω, F, F, P ) be probabiliy space wih filraion F, [, T ]. Assume ha underlying asse S in he risk-neural world and variance follow he following model [34] (5.1) ds = r d + σ dw 1 S dσ 2 = k(θ2 σ 2) d + γσ dw 2, where r is he deerminisic ineres rae, σ and θ are shor and long volailiy, k > is he reversion speed, γ > is he volailiy (of volailiy) parameer, and w 1 and w 2 are independen sandard Wiener processes. The Heson asse process has a variance σ 2 ha follows a Cox-Ingersoll- Ros process [21], described by he second equaion in (5.1). If he volailiy σ follows Ornsein-Uhlenbeck process (see, for example, [47]), hen Iô s lemma shows ha he variance σ 2 follows he process described exacly by he second equaion in (5.1). We noe, if 2kθ 2 > γ 2, hen σ 2 > wih P = 1 (see [34]). 5.3 Explici expression for variance process σ 2 In his secion we solve he equaion for variance σ 2 in (5.1) explicily, using change of ime mehod (see Secion 2.4).

27 CHANGE OF TIME METHOD 325 Lemma 5.1. The soluion of he following equaion (5.2) dσ 2 = k(θ 2 σ 2 )d + γσ dw 2 akes he following form: (5.3) σ 2 = e k (σ 2 θ 2 + w 2 (φ 1 )) + θ 2, where w 2 () is an F -measurable one-dimensional Wiener process, and φ 1 is an inverse funcion o φ : (5.4) φ = γ 2 {e kφs (σ 2 θ2 + w 2 (s)) + θ 2 e 2kφs } 1 ds. Proof. The resul follows from Theorem 2.1, Secion 2.2, and he following subsiuion (5.5) v := e k (σ 2 θ2 ). Remark 5.1. We noe ha if 2kθ 2 > γ 2, hen σ 2 for example, [34]). From (5.5) i follows ha > wih P = 1 (see, v e k + θ 2 is sricly posiive oo. If we ake he inegrand in he las inegral we shall obain [e kφs (σ 2 θ 2 + w 2 ()) + θ 2 e 2kφs ] 1 = [e 2kφs (e k (σ 2 θ2 + w 2 ())) + θ 2 )] 1 = [e kφs e k (σ 2 θ2 + w 2 ())) + θ 2 ] 2 = [e kφs e k v + θ 2 ] 2, since v = σ 2 θ2 + w 2 ()). The expression under is posiive (see above) and is well-defined, hence he las expression and herefore, he inegrand in he las inegral are sricly posiive.

28 326 ANATOLIY SWISHCHUK 5.4 Properies of he processes w 2 (φ 1 ) and σ 2 The properies of w 2 (φ 1 ) =: b() are he following: (5.6) (5.7) (5.8) Eb() = ; E(b()) 2 = γ 2 { e k 1 k Eb()b(s) = γ 2 { e k( s) 1 k (σ 2 θ 2 ) + e2k 1 θ }; 2 2k (σ 2 θ 2 ) + e2k( s) 1 θ }, 2 2k where s := min(, s). Using represenaion (5.3) and properies (5.6) (5.8) of b() we obain he properies of σ 2. Sraighforward calculaions give us he following resuls: (5.9) Eσ 2 = e k (σ 2 θ2 ) + θ 2 ; Eσ 2 σ 2 s = γ 2 e k(+s) { e k( s) 1 k } (σ 2 θ 2 ) + e2k( s) 1 θ 2 2k + e k(+s) (σ 2 θ 2 ) 2 + e k (σ 2 θ 2 )θ 2 + e ks (σ 2 θ2 )θ 2 + θ Valuing variance and volailiy swaps Theorem 5.1. The value (or price) P var of variance swap is [ ] 1 e (5.1) P var = e rt kt (σ 2 θ 2 ) + θ 2 K var kt and he value (or price) P vol of volailiy swap is approximaely (5.11) {( ) 1 e P vol e rt kt 1/2 (σ 2 θ 2 ) + θ 2 kt ( γ 2 e 2kT 2k 3 T 2 [(2e2kT 4e kt kt 2)(σ 2 θ 2 ) ) + (2e 2kT kt 3e 2kT + 4e kt 1)θ 2 ] [ ( ) 1 e kt 3/2 ] 8 (σ 2 kt θ2 ) + θ 2 K vol }.

29 CHANGE OF TIME METHOD 327 Proof. The resul follows from Lemma 5.1 and Secion 5.4. The same expressions for E[V ] and for V ar[v ] may be found in [13]. Remark 5.2. Using he same approach as in Secion 5 we can obain he prices of covariance and correalion swaps for wo asses ha follow Heson model (see [55] for deails). 6 Conclusion In his paper we presened change of ime mehod o solve several problems arising in mahemaical finance: o find pricing formula for European call opion in (B, S)-securiy marke, o find a pricing formula for a European call opion for a mean-revering asse of commodiy price (if mean revering level equals o zero, hen his formula coincides wih he Black-Scholes formula), o value variance and volailiy swaps for Heson model [34]. Two numerical examples on S&P 6 Canada Index (January 1997 February 22) o price variance and volailiy swaps for Heson model and on AECO Naural Gas Index (1 May April 1999) o price European call opion for meanrevering asse model have been presened. 7 Fuure work As we could see in Secion 4 (see also Figure 4) he main drawbacks of one-facor mean-revering models are: 1) he long-erm mean L remains fixed over ime needs o be recalibraed on a coninuous basis in order o ensure ha he resuling curves are marked o marke; 2) he bigges drawback is in opion pricing: resuls in a model-implied volailiy erm srucure ha has he volailiies going o zero as expiraion ime increases (spo volailiies have o be increased o non-inuiive levels so ha he long erm opions do no lose all he volailiy value-as in he markeplace hey cerainly do no). To eliminae hese drawbacks we are going o consider wo-facor mean-revering model ds = α(l S ) d + σs dw 1 dl = ξl d + ηl dw 2 or ds = α(l S ) d + σs dw 1 dl = ξ(n L ) d + ηl dw 2

30 328 ANATOLIY SWISHCHUK (processes W 1 and W 2 may be correlaed) and he change of ime mehod o obain explici opion pricing formula for hese models. We noe, ha one of a possible way o eliminae hese drawbacks is o consider he mean-revering Markov swiching model, see [16]. Anoher way is o consider above-menioned SDEs wih Lévy processes as driven processes in place of Wiener processes, see [1, 2, 49]. 8 Numerical examples: S&P 6 Canada Index and AECO Naural Gas Index In his secion, we presen wo numerical examples on S&P 6 Canada Index (January 1997 February 22) o price variance and volailiy swaps for Heson model (see Secion 5) and on AECO Naural Gas Index (1 May April 1999) o price European call opion for mean-revering asse model (see Secion 4). 8.1 Numerical example 1: S&P 6 Canada Index (January 1997 February 22) In his secion, we apply he analyical soluions from Secion 5 o price a swap on he volailiy of he S&P6 Canada index for five years (January 1997 February 22). These daa were kindly presened o auhor by Raymond Théore (Universié du Québec à Monréal, Monréal, Québec, Canada) and Pierre Rosan (Analys a he R&D Deparmen of Bourse de Monréal and Universié du Québec à Monréal, Monréal, Québec, Canada). They calibraed he GARCH parameers from five years of daily hisoric S&P 6 Canada Index (from January 1997 o February 22) (see [59]). In he end of February 22, we waned o price he fixed leg of a volailiy swap based on he volailiy of he S&P6 Canada index. The saisics on log reurns S&P6 Canada Index for 5 year (January 1997 February 22) is presened in Table 1: From he hisogram of he S&P6 Canada index log reurns on a 5- year hisorical period (1,3 observaions from January 1997 o February 22) i may be seen lepokurosis in he hisogram. If we ake a look a he graph of he S&P6 Canada index log reurns on a 5-year hisorical period we may see volailiy clusering in he reurns series. These facs indicae abou he condiional heeroscedasiciy. Figure 1 illusraes he non-adjused and adjused volailiy for he same series of mauriies. Figure 2 depics S&P 6 Canada Index Volailiy Swap.

31 CHANGE OF TIME METHOD 329 Saisics on Log Reurns S&P 6 Canada Index Series: Log Reurns S&P 6 Canada Index Sample: 1 13 Observaions: 13 Mean.235 Median.593 Maximum Minimum.1118 Sd. Dev Skewness Kurosis TABLE 1 FIGURE 1: Convexiy Adjusmen for Volailiy

32 33 ANATOLIY SWISHCHUK FIGURE 2: S&P 6 Canada Index Volailiy Swap 8.2 Numerical example 2: AECO Naural Gas Index (1 May April 1999) We shall calculae value of a European call opion on he price of a daily naural gas conrac. To apply our formula from Secion 4 for calculaing his value we need o calibrae he parameers a, L, σ and λ. These parameers may be obained from fuures prices for he AECO Naural Gas Index for he period 1 May 1998 o 3 April 1999 (see [12, p. 34]). The parameers peraining o he opion are he following: Price and Opion Process Parameers T a σ L λ r K 6 monhs From his able we can calculae he values for a and L : a = a + λσ = ,

33 CHANGE OF TIME METHOD 331 and L = al a + λσ = For he value of S we can ake S [1, 6]. Figure 3 depics he dependence of mean value ES on iniial value of sock S and mauriy T for AECO Naural Gas Index (1 May 1998 o 3 April 1999). FIGURE 3: Dependence of ES on S and T (AECO Naural Gas Index (1 May April 1999)) Figure 4 depics he dependence of volailiy of S on iniial value of sock S and mauriy T for AECO Naural Gas Index (1 May 1998 o 3 April 1999). Figure 5 depics he dependence of European Call Opion Price for mean-revering risk-neural asse model (MRRNAM) on mauriy (monhs) for AECO Naural Gas Index (1 May 1998 o 3 April 1999) wih S() = 1 and K = 3.

34 332 ANATOLIY SWISHCHUK FIGURE 4: Dependence of volailiy of S on S and T (AECO Naural Gas Index (1 May April 1999)) FIGURE 5: Dependence of European Call Opion Price on Mauriy (monhs) (S() = 1 and K = 3) (AECO Naural Gas Index (1 May April 1999))

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