AN EULER-TYPE METHOD FOR THE STRONG APPROXIMATION OF THE COX-INGERSOLL-ROSS PROCESS

Transcription

1 AN ULR-TYP MTHOD FOR TH STRONG APPROXIMATION OF TH COX-INGRSOLL-ROSS PROCSS STFFN DRICH, ANDRAS NUNKIRCH AND LUKASZ SZPRUCH Abstract. We analyze the strong approximation of the Cox-Ingersoll-Ross CIR process in the regime where the process does not hit zero by a positivity preserving drift-implicit uler-type method. As an error criterion we use the p-th mean of the maximum distance between the CIR process and its approximation on a finite time interval. We show that under mild assumptions on the parameters of the CIR process the proposed method attains, up to a logarithmic term, the convergence of order /2. This agrees with the standard rate of the strong convergence for global approximations of stochastic differential equations SDs with Lipschitz coefficients despite the fact that the CIR process has a non-lipschitz diffusion coefficient. Keywords. Cox-Ingersoll-Ross process; strong global approximation; square-root coefficient; Lamperti transformation; drift-implicit method; one-sided Lipschitz condition Mathematics Subject Classification 2. 65C3; 6H35. Introduction and Main Result In computational finance a lot of effort has been given to the so called Cox-Ingersoll- Ross process CIR recently. The CIR process under consideration has the following form dx t = κλ X t dt + θ X t dw t, X = x, t. Here W = W t t, is a one-dimensional Brownian motion, κ, λ, θ > and x >. It is well known that equation admits a unique strong solution which is non-negative, see e.g. Chapter 5 in [24]. The CIR process was originally proposed by Cox et al. [9] in 985 as a model for short-term interest rates. Nowadays, this model is widely used in financial modeling, e.g. as volatility process in the Heston model [8]. One of the main objectives in mathematical finance is the pricing of path-dependent derivatives. If the asset prices or the interest rates dynamics are modeled by a d- dimensional SD with solution S t t [,T ], then this corresponds to the quadrature problem p = F S where F : C[, T ], R d R is the discounted payoff of the derivative. Typically, explicit formulae for such quantities are unknown and have to be approximated by Monte Carlo methods that are based on approximate solutions of the stochastic differential equation on [, T ]. Here, the knowledge of the global strong convergence rate of the approximation is important, in particular if the efficient Multi-level Monte-Carlo method [2, 3] is to be used. Date: October 27, 2.

2 2 STFFN DRICH, ANDRAS NUNKIRCH AND LUKASZ SZPRUCH The strong global approximation of equation has been studied in several articles. Strong convergence without a rate or with a logarithmic rate of several discretization schemes has been shown in [2, 7,, 7, 9]. In [2], a general framework for the analysis of strong approximation of the CIR process is presented along with extensive simulation studies. Moreover, the strong approximation of more general Ait-Sahalia-type interest rate models is analyzed in [2]. However, only in [4] non-logarithmic convergence rates are obtained. In [4] it is shown that a symmetrized uler method has strong convergence order /2 under restrictive assumptions on the parameters of the equation, see Section 2. The difficulties to obtain strong convergence rates for the approximations of the CIR process are due to its square-root coefficient. Thus the standard theory which relies on the global Lipschitz assumption does not apply [25, 26]. In this article we focus on the regime where the CIR process does not hit zero, i.e. where PX t > for t = an assumption which is often fulfilled in interest rate models. By the Feller test, it is true if and only if 2κλ, see e.g. Chapter 5 in [24]. We will not directly do a numerical analysis for the CIR process, but for a coordinate transformation thereof. We consider the process Y t = X t, which by Itô s formula satisfies dy t = α Y t dt + βy t dt + γ dw t, t, Y = x, 2 with 4κλ θ2 α =, β = κ 8 2, γ =. This transformation, known as Lamperti transformation, allows us to shift the nonlinearity from the diffusion coefficient into the drift coefficient. Note that the drift fx = α + βx, x >, x satisfies for α >, β R the one-sided Lipschitz condition x yfx fy βx y 2, x, y >. This property is crucial to control the error propagation of drift-implicit uler schemes, see [2, 2]. The drift-implicit uler method with stepsize > for equation 2 leads to the numerical scheme α y k+ = y k + + βy k+ + γ k W, k =,,... 3 y k+ with y = x and k W = W k+ W k, k =,,.... Recalling that α, γ > and β <, equation 3 has the unique positive solution y k+ = y k + γ k W 2 β + y k + γ k W 2 + α 4 β 2 β, which we call drift-implicit square-root uler method. Transforming back, i.e. x k = y 2 k, k =,,...,

3 AN ULR-TYP MTHOD FOR TH CIR PROCSS 3 gives a strictly positive approximation of the original CIR process. This scheme has already been suggested in [2], but convergence results have not been established. Using piecewise linear interpolation, i.e. x t = k + t x k + t k x k+, t [k, k + ], we obtain a global approximation x t t [,T ] of the CIR process on [, T ]. xploiting the structure of SD 2 we establish the following theorem. Theorem.. Let 2κλ >, x > and T >. Then, for all p < 2κλ there exists a constant K p > such that /p max X t x t p K p log, t [,T ] for all, /2]. Hence we obtain the optimal strong convergence rate for the approximation of SDs with Lipschitz coefficients, see [27]. The only price to be paid is the restriction on p that arises from the need to control the inverse p-th moments of the CIR process, which are infinite for p 2κλ. The remainder of this article is structured as follows: In the next section we give a short overview on discretization schemes for the CIR process, while the proof of Theorem. is given in Section Numerical Methods for the CIR Process Discretization schemes for the CIR process were proposed in numerous articles, among these are [2, 3, 4, 7,, 5, 7, 9, 23, 28]. In this short summary we mainly focus on uler-type methods. In [4] the authors study a symmetrized uler method for the approximation of equation, i.e. x k+ = x k + κλ x k + θ x k k W 4 for k =,,.... Under the assumption 2κλ > + { } κ 8 max 6p, 6p 2 θ they found that max X k x k 2p C p p, where the constant C p > depends only p, κ, λ, θ, x and T. As a consequence of Lemma 3.5 in Section 3 the piecewise linear interpolation of this scheme satisfies the same error estimate with respect to the p-th mean maximum distance as our drift-implicit squareroot uler scheme. However for the symmetrized uler scheme the assumptions on the parameters of the CIR process are clearly more restrictive. Moreover, it is shown in [7] that the symmetrized uler scheme converges weakly with rate if 2κλ 2.

4 4 STFFN DRICH, ANDRAS NUNKIRCH AND LUKASZ SZPRUCH The truncated uler scheme x k+ = x k + κλ x k + θ x + k kw, k =,,... 5 was analyzed in [], while the scheme x k+ = x k + κλ x k + θ x k k W, k =,,... 6 was studied in [9]. Both schemes do not preserve positivity, but satisfy max X k x k 2 for without further restrictions on the parameters of the equation. In particular an application of Theorem 3. in [7] yields a logarithmic convergence rate for the uler schemes 5 and 6. Moreover, if 2κλ it follows from [6] that the uler schemes 4 6 have a pathwise convergence rate of /2 ε for all ε >, i.e. we have max X /2 ε k x k P a.s. for. The asymptotic error distribution of these schemes can be deduced from [29]: For 2κλ it holds that max X L k x k max t t [,T ] Φ t db s 8 Φ s for, where B t t is a Brownian motion independent of W t t and Φ t t is given by t Φ t = exp κt θ2 ds + θ t dw s, t. 8 X s 2 Xs While no explicit solution of equation is known, the finite dimensional distributions can be characterized in terms of a non-central chi-square distribution, see e.g. [9]. Thus, equation can be simulated exactly at a finite number of time points using the Markov property, see e.g. [8, 4]. However, the algorithms for the exact simulation of the CIR process are strongly problem dependent: The number of degrees of freedom of the non-central chi-square random variable, which has to be simulated in each step, is 4κλ/. Thus the computational cost of the algorithms depends strongly on κ, λ and θ. The same problem, i.e. strong dependence of the computational cost of the algorithm on the parameters of the equation, arises also for the exact sampling algorithm introduced in [5], which can be also applied to equation, see [6]. While for the simulation of the CIR process at a single point the exact simulation methods are useful, discretization schemes remain erior if a full sample path of the CIR process has to be simulated or if the CIR process is part of a system of stochastic differential equations, see e.g. [9]. Moreover, results on strong convergence rates for approximations of equation have an interest of its own, since it is one of the most prominent examples for a stochastic differential equation, whose coefficients do not satisfy the standard global Lipschitz assumption.

5 AN ULR-TYP MTHOD FOR TH CIR PROCSS 5 3. Proof of Theorem. In the following we will denote by c constants regardless of their value. 3.. Preliminaries. For our error analysis we need to control the inverse moments of the CIR process. Since X t follows a non-central chi-square distribution, we have 2κ X p t = x exp κt p x expκt 2κλ Γ + p θ 2 F Γ 2κλ p p, 2κλ, 2κ x expκt for p > 2κλ, where F denotes the confluent hypergeometric function, and else, see e.g. Theorem 3. in [22]. Since F a, b, z = X p t = Γb Γb a z a + O z, z, see formula 3..5 on page 54 in [], it follows for p > 2κλ that t X p t is bounded on [, T ], i.e. X p t < for p > 2κλ t [,T ] θ. 7 2 Moreover, from Theorem 3. [22] or Lemma A.2 in [7] we also have T exp ε Xs ds < 2κλ if and only if ε θ2 2 8 θ. In general, one can estimate polynomial moments 2 against exponential moments, since for q, ε > there exists c > such that x q ce εx for x. Hence we arrive at Lemma 3.. Let 2κλ >, T > and q. It holds T q Xs ds <. Further we need a smoothness result for equation 2: Lemma 3.2. Let 2κλ > and T >. Then, for all q we have and s<t T, t s Yt Y s q c t s q/2, for s, t [, T ], Y t Y s q c log q/2, for, /2], Y t q <. t [,T ]

6 6 STFFN DRICH, ANDRAS NUNKIRCH AND LUKASZ SZPRUCH Proof. For s t, we have t α Y t Y s = du + s Y u By the Cauchy-Schwarz inequality one has Y t Y s γ W t W s + t s /2 T t s α 2 Yu 2 βy u du + γ W t W s. /2 T /2 du + t s /2 β 2 Yu du 2. Since Y u = X u the first assertion follows now from 7, Lemma 3., Minkowski s inequality and the smoothness of Brownian motion in the q-th mean. For the second assertion in addition we use the well known fact that the modulus of continuity of Brownian motion satisfies s<t T, t s W t W s q c log q/2 for, /2], see e.g. []. The third assertion follows from T Y t y + γ W t + t /2 α 2 /2 T du + t /2 and Y 2 u W t q <. t [,T ] 3.2. rror Bound for the implicit uler Scheme for Y. /2 β 2 Yu 2 du Proposition 3.3. Let 2κλ >. For T > and p < 2κλ, there exists c > such that /p Y k y k p c, for, /2]. Proof. Without loss of generality we assume that < T. For the error at time point k we have the recursion with e = e k = Y k y k e k+ = e k + fy k+ fy k+ + r k r k = k+ k Multiplying both sides with e k+ we obtain fy k+ fy t dt. e 2 k+ 2 e2 k + 2 e2 k+ + e k+ fyk+ fy k+ + e k+ r k. Since e k+ fyk+ fy k+ βe 2 k+

7 AN ULR-TYP MTHOD FOR TH CIR PROCSS 7 we have and it follows that n e 2 n 2 e k+ r k, n =, 2,... k= T/ e k 2 r k. 8 k= Now, it remains to analyze the local error k+ r k = fy k+ fy t dt. k Note that for a, b >. Thus fa fb = βa b + α b a ab fa fb c + a b ab and we obtain k+ r k c + k Y t Y k+ Y k+ Y t dt. An application of Hölder s and Minkowski s inequality yields rk p k+ /p pq Yk+ c + Y Y t Y k+ pq t pq for q, q > with q + q for all q >. Since k =. Now, Lemma 3.2 gives r k p /p c k+ k + Y t Y k+ pq Y t Y k+ pq Y t + 2pq Y k+, 2pq pq pq dt dt, 9 applying 7 with q > such that pq < 2κλ yields r k p /p c 3/2, for p < 2κλ and using 8 completes the proof of the Proposition.

8 8 STFFN DRICH, ANDRAS NUNKIRCH AND LUKASZ SZPRUCH 3.3. Moment Bounds for the implicit uler Scheme for Y. Now we show that all moments of the approximation scheme are uniformly bounded. Multiplying 3 with y k+ yields It follows with and we obtain by induction that y 2 k+ = α + βy 2 k+ + y k+ y k + γ k W. for all k =,,..., T/. This allows us to show: y 2 k+ 2α + γ 2 + y 2 k + M k M k = 2γy k k W + γ 2 k W 2, y 2 k+ y 2 + 2α + γ 2 k + 2 Lemma 3.4. Let 2κλ and >, T >. Then for all p we have Proof. From and we obtain that with y k p <. y k 2 c + c k m t dw t 3 m t = 2γy l + 2γ 2 W t W l, t [l, l +. Since t [, T/ ] m t p c + c the Burkholder-Davis-Gundy inequality, i.e. s p m τ dw τ c s [,t] Jensen s inequality and 3 give that l=,..., T/ t y l p m 2 τ dτ p/2 y k 2p c + c y l p. l=,..., T/, So 2 now yields y k 4 < and the assertion follows from an induction procedure in p.

9 AN ULR-TYP MTHOD FOR TH CIR PROCSS rror Bound for the drift-implicit square-root uler Scheme. Now denote by X the piecewise linear interpolation of the CIR process with stepsize >, i.e. X t = k + t t X k + k X k+, t [k, k + ]. Lemma 3.5. Let 2κλ >, T > and, /2]. Then, for all q we have Proof. Combining the equality max t [,T ] X t X t q c log q/2. X t X s = Y t + Y s Y t Y s with the Cauchy-Schwarz inequality and Lemma 3.2 we get s<t T, t s Now the assertion follows from X t X s q c log q/2. X t X t X t X s. t [,T ] s<t T, t s Due to X t x t X t X t + X k yk 2 t [,T ] t [,T ] and the above Lemma it only remains to control /p. X k yk 2 p Since X k yk 2 p Y k + y k p Y k y k, p an application of Hölder s inequality with ε > such that + εp < 2κλ and Proposition 3.3 give X k yk 2 p Y k + y k p +ε ε +ε ε p/2. It remains to apply Lemma 3.2 and Lemma 3.4 to finish the proof of Theorem.. Acknowledgements. The authors would like to thank Martin Altmayer for valuable comments on an earlier version of the manuscript.

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