Boundary Conditions For MeanReverting Square Root Process


 Abraham Dixon
 3 years ago
 Views:
Transcription
1 Boundary Conditions For MeanReverting Square Root Process by Jonathan AquanAssee A research paper presented to the University of Waterloo in fulfillment of the requirement for the degree of Master of Mathematics in Computational Mathematics supervisors: Peter Forsyth and George Labahn Waterloo, Ontario, Canada, 2009 c Jonathan AquanAssee 2009
2 I hereby declare that I am the sole author of this paper. This is a true copy of the paper, including any required final revisions, as accepted by my examiners. I understand that my paper may be made electronically available to the public. ii
3 Abstract The CoxIngersollRoss (CIR) interest rate model is used to model interest rates and interest rate derivatives. We study the term structure equation for singlefactor models that predict nonnegative interest rates. It is shown using finite difference techniques that if the boundary is attainable, then this boundary behaviour serves as a boundary condition and guarantees a uniqueness of solutions. However, if the boundary is nonattainable, then the boundary condition is not needed to guarantee uniqueness. The finite difference solution is verified by use of nonnegative numerical approximations. iii
4 Acknowledgements I would like to thank Peter Forsyth and George Labahn for helping make this paper possible. iv
5 Contents List of Tables List of Figures vii viii 1 Introduction 1 2 Problem Formulation Analytical positivity and boundedness Boundary Conditions Numerical Methods Finite Difference Discretization Methods CrankNicolson Discretization Discretization of the Boundary condition Discretization of stochastic numerical methods Numerical positivity Numerical discretization Experiments and Results Weak Convergence Criterion Convergence and Verification of methods Conclusions 35 v
6 APPENDICES 36 A Probability distribution of the CIR process 37 B Analytical solution to the term structure equation 39 C Algorithm for squareroot diffusion 40 References 42 vi
7 List of Tables 4.1 CIR model parameters FD simulation of bond price using CIR Model when boundary is attainable FD simulation of bond price using CIR model when boundary is not attainable MC simulation of bond price using CIR Model when boundary is attainable MC simulation of bond price using CIR model when boundary is not attainable Weak error in numerical approximation schemes vii
8 List of Figures 4.1 Convergence of bond prices when using FD FD simulation of bond Price using CIR model MC bond value versus computation cost Comparison of CIR distribution viii
9 Chapter 1 Introduction The meanreverting squareroot process is a stochastic differential equation (SDE) that has found considerable use as a model for volatility, interest rates, and other financial quantities. The equation has no general, explicit solution, although its transition density can be characterized. Some standard expectations can be analytically calculated which can be useful for calibrating the parameters. These processes were initially introduced to model the short interest rate by Cox, Ingersoll, and Ross [1], and are now widely used in modeling because they present interesting features like nonnegativity and mean reversion. The equation for this process is dr(t) = κ(b R(t))dt + σ R(t)dW (t), (1.1) R(t) will denote a CoxIngersollRoss (CIR) process. The parameters κ, b, and σ are all nonnegative and dw (t) denotes the increments of a Brownian motion. Notice that the drift term in equation (1.1) is positive if R(t) < b and negative if R(t) > b. Thus R(t) is pulled toward the level b, a property generally referred to as mean reversion. We may interpret b as the longrun interest rate level and κ as the speed at which R(t) is pulled toward b. In this paper, we assume that κ, b, and σ do not vary with time. Under the above assumption on the parameters, that we will suppose valid through all the paper, it is well known that this SDE has a nonnegative solution, and this solution is pathwise unique (see [2] and [3]). Also, under the conditions the process is always positive. 2κb > σ 2 and R(0) > 0 (1.2) It is well known that the increments of the CIR process are noncentral chisquared random variables that can be simulated exactly. However, the exact simulation in general 1
10 requires more computational cost than a simulation with approximation schemes. It may also be restrictive if one wishes to study a generalized version of the mean reverting process. For these reasons, studying approximation schemes are relevant. The CIR process is frequently used for pricing of different interest rate derivatives such as bonds and bond options. For valuing interest rate derivatives, stochastic methods seem to be more commonly used than finite difference (FD) methods. When using FD methods it is necessary to use the hyperbolic partial differential equation (PDE) called the term structure equation, for valuing bond prices. The authors of [4] believe that stochastic methods are preferred over FD methods because the pricing equation with appropriate boundary conditions is not fully developed from a mathematical sense. Moreover, in [5], the authors claim that there are multiple solutions to the term structure equation that satisfy the same boundary conditions. In this paper we compare stochastic and FD methods for pricing of bonds using the CIR process. We will show using simulation that the unique solution to the term structure equation is the one given using FD methods subject to the correct boundary conditions. We verify these results by performing a Monte Carlo (MC) simulation using a nonnegative approximation of the CIR equation. We are interested in the solution of the term structure equation for vanishing interest rates (i.e as r 0) and the behavior of the solution at the boundary using different boundary conditions. Our hypothesis is that if the boundary is attainable, then this boundary behavior serves as a boundary condition and guarantees uniqueness of solutions. However, if the boundary is nonattainable, then the boundary condition is not needed to guarantee uniqueness, but it is nevertheless very useful from a numerical perspective. The project is structured as follows: Chapter 2 develops the methods for simulating the stochastic interest rate model given by the CIR process. A discussion of boundary conditions for the model is presented. Chapter 3 details the numerical schemes for the CIR model. Discretization methods for stochastic integration schemes are discussed. A CrankNicolson method is also developed for the CIR model. Chapter 4 presents simulation results of the integration schemes developed previously. The theoretical results are verified using the numerical tests. Finally, Chapter 5 concludes by pointing out the main results of the paper. 2
11 Chapter 2 Problem Formulation This chapter develops the methods for simulating the stochastic interest rate model given by the CIR process in equation (1.1). The CIR process models the dynamics of an instantaneous continuously compounded interest rate R(t). As shown in [6], the interest rate for the CIR model is given by R(t) = R(0) + t 0 κ(b R(s))ds + t 0 σ R(s)W (s)ds which is just an integral form of the differential equation in (1.1). An investment in a money market account earning interest at a rate R(t) at time t grows from a value of one at time zero to a value of D(t) = e R t 0 R(u)du at time t. The price at time zero of a derivative security that pays X at time T is the expectation of X/D(T ), or In particular, we define: E [e R ] t 0 R(u)du X. Definition 1 (Zero coupon bond) A financial contract promising to pay a certain face amount, which we take to be one, at a fixed maturity date T is called a zero coupon bond. Prior to the expiration date the bond makes no payments. 3
12 Given the stochastic process to the interest rate R(t) as shown in equation (1.1), the value of the zero coupon bond V (r, t) is given by [ V (r, t) = E Q e R ] T t R(s)ds R(t) = r (2.1) with E Q denoting the expectation operator under the riskneutral probability measure Q and r is the interest rate for the chosen dynamics R(t) conditional on information available at time t. This pricing formula is a martingale under the riskneutral measure. As shown in [6], by applying Itô s lemma to the expectation in equation (2.1), it is possible to derive the term structure PDE satisfied by V (r, t) as given by V t (r, t) + κ(b r)v r (r, t) σ2 rv rr (r, t) = rv (r, t). (2.2) 2.1 Analytical positivity and boundedness The SDE in equation (1.1) is defined on the domain [0, ). An important concept in the study of SDEs is the concept of positivity or nonnegativity. We must therefore define the following concept Definition 2 Assuming the process X t is defined on the domain [L, ) and X 0 > L, then the process X t has a nonattainable boundary L, if P (X t > L for all t > 0) = 1. The process X t has an attainable boundary L, if P (X t L for all t > 0) = 1 and P ( t > 0 : X t = L) = 1. For L = 0, attainability is equivalent to positivity and nonnegativity. Throughout the rest of the paper we will make use of Definition 2. 4
13 2.2 Boundary Conditions The boundary behavior for the SDE shown in equation (1.1) has been studied at great detail by Feller in [7]. For the process in (1.1), Feller showed that the boundary at the origin is instantaneously reflecting, regular, and attainable if σ 2 > 2κb, and unattainable, nonattracting, boundary otherwise. As can be seen in Appendix A, interesting properties related to the boundary behavior for the SDE (1.1) can be uncovered by looking at the probability distribution for the process. The results of Feller can be used to derive appropriate boundary conditions for finite differencing schemes of the term structure equation given in (2.2). For example, it is possible that the case σ 2 < 2κb can be handled by simple Dirichlet boundary conditions at r = 0, while the same cannot be done if σ 2 > 2κb. This can be generalized with the following definition. Definition 3 (Feller Condition) Given a process X defined by a SDE of the form dx(t) = β(x(t), t)dt + c(x(t), t)dw. If the condition ( lim β(x) 1 ) c 2 x 0 2 x (x) 0 (2.3) is true, then the boundary at the origin is nonattainable for the process X. According to [4], the appropriate boundary condition for the term structure equation in (2.2) can be obtained by plugging in r = 0 into the equation. The resulting condition is then V t (0, t) + κbv r (0, t) = 0. (2.4) This boundary condition will result in a unique solution to the term structure equation when σ 2 > 2κb. However, when σ 2 < 2κb, the boundary condition (2.4) is not needed to guarantee a unique solution and is thus redundant from a mathematical perspective [4]. At the upper boundary we must truncate the domain to [0, r max ]. Consequently, it is necessary to impose a boundary condition at r = r max. The theoretical upper boundary condition V = 0 as r is of critical use if we retain r as the coordinate. In [8] the authors propose a boundary technique that does not require additional linearity assumptions. This FD method involves using onesided derivatives for the spatial discretization of the PDE. 5
14 In this paper we take the approach of extending the computational domain in order to minimize any boundary errors at r max. In this way the boundary condition at r max can be handled with a simple Dirichlet boundary condition with V (r max, t) = 0. (2.5) 6
15 Chapter 3 Numerical Methods It is well known that an analytical formula for the term structure equation (2.2) exists (see Appendix B). However, recall from the introduction that it has been claimed by previous studies, that multiple solutions to the term structure equation exist that satisfy the same boundary conditions. All numerical methods seek to find a solution for the pricing relationship given in equation (2.1). In this chapter we will examine the numerical methods that will be used to price zero coupon bonds in the CIR model. Using these numerical methods we will be able to examine the effects of different boundary conditions for the term structure equation. 3.1 Finite Difference Discretization Methods In this section we will derive the discretized equations using an approach to approximate the PDE (2.2) using finite differencing techniques. The pricing equation as shown in [9], is given by the following equation V τ = 1 2 σ2 r 2 V r 2 + κ(b r) V r rv, 0 < r < r max, t > 0, (3.1) where V (r, τ) is the bond price, r is the spot interest rate, τ = T t, T is the expiry time of the bond, t is the time (in the forward direction). This is a firstorder hyperbolic equation and it must be augmented with an initial condition and boundary conditions in order to define a valid initial boundary value problem. In this case we define them as V (r max, t) = 0 and V (r, T ) = 1. (3.2) 7
16 If we take the limit as r 0, we get the following boundary condition at r = 0 V τ κb V r = 0, if σ2 > 2κb. (3.3) We can then implement a number of finite differencing discretization schemes. In particular, we will use a forwardbackward differencing scheme. Define a grid of points in the (r, τ) plane r 0, r 1,..., r m τ n = n τ 0 r i r max 0 τ n N τ = T and let Let V (r i, τ n ) = V n i. r i+1/2 = r i+1 r i and r i 1/2 = r i r i 1. Equation (3.1) can be then approximated by ( ) n ( ) V σ 2 r 2 n = τ i 2 V rr + (κ(b r)v r ) n i (rv )n i, (3.4) i where the time derivative in equation (3.4) is approximated by ( ) n V τ i V n+1 i τ V n i The second order derivative term is approximated by. (3.5) (V rr ) n i ( ) V n i+1 Vi n r i+1/2 ( ) V n i Vi 1 n r i 1/2 r i+1/2 + r i 1/2 2 (3.6) 8
17 with the discounting term in equation (3.4) as (V ) n i V n i (3.7) while the derivative term can be approximated by either central, forward, or backward differencing. A central difference is ( (V r ) n i V n i+1 V n i 1 r i+1/2 + r i 1/2 ), (3.8) a forward difference is and a backward difference is ( ) V (V r ) n n i i+1 Vi n, (3.9) (V r ) n i ( V n i r i+1/2 V n i 1 r i 1/2 ). (3.10) It is well known that central differencing is preferred because it is second order, while forward and backwards differencing are first order. Unfortunately central differencing can result in problems if used in all cases. Using any of (3.8), (3.9), or (3.10) gives V n+1 i Then α i and β i are defined as = V n i (1 (α i + β i + r i ) τ) + V n i 1 τα i + V n i+1 τβ i. (3.11) [ αi central σ 2 ri 2 = (r i r i 1 )(r i+1 r i 1 ) κ(b r i) r i+1 r i 1 ) [ β central i = σ 2 r 2 i (r i+1 r i )(r i+1 r i 1 ) + κ(b r i) r i+1 r i 1 ) ] ] in the case of central differencing, or 9
18 [ α forward σ 2 ri 2 i = (r i r i 1 )(r i+1 r i 1 ) [ β forward i = σ 2 ri 2 (r i+1 r i )(r i+1 r i 1 ) + κ(b r i) r i+1 r i ) ] ] in the case of forward differencing, or [ αi backward σ 2 ri 2 = (r i r i 1 )(r i+1 r i 1 ) κ(b r i) r i r i 1 ) [ ] β backward i = σ 2 r 2 i (r i+1 r i )(r i+1 r i 1 ) ] in the case of backwards differencing. For stability reasons it is vital that all α i and β i values be always positive. We can then decide between the central or upstream (i.e. forward or backward) discretization by: For i = 0,..., n 1 If αi central 0 and βi central 0 α i = αi central β i = βi central ElseIf β forward i 0 α i = α forward i β i = β forward i Else α i = α backward i β i = β backward i EndIf EndFor The above algorithm guarantees that α i and β i are nonnegative. As shown by [10], requiring that all α i and β i be nonnegative has important ramifications for the stability of scheme (3.11). 10
19 Remark 1 (Unconditional stability) As shown by [10], the discretization method (3.11) is unconditionally stable provided that α i and β i are nonnegative. Remark 2 (Convergence to the viscosity solution) As shown by [10], requiring that all α i and β i are nonnegative has important ramifications for the convergence of scheme (3.11). It is possible to show that the discretization (3.11) is monotone and consistent. Since it is also unconditionally stable, then the discretized solution converges to the viscosity solution CrankNicolson Discretization In this section we consider a CrankNicolson timestepping strategy for the FD discretization in equation (3.11). Ordinary explicit FD schemes are known to become unstable if certain timestep conditions are not satisfied. For these reasons we will study the CrankNicolson method which is known to be unconditionally stable. In [4] the authors proposed a backward differencing formula of order two (BDF2), however we found that this method produced oscillations in the solution when the boundary condition (2.4) was not explicitly defined. The FD equation (3.11) can be written into the CrankNicolson timestepping form as follows [ V n+1 i 1 + (α i + β i + r i ) τ ] τ 2 2 β iv n+1 i+1 τ 2 α iv n+1 i 1 [ = Vi n 1 (α i + β i + r i ) τ ] 2 + τ 2 β iv n i+1 + τ 2 α iv n i 1 (3.12) which is an implicit equation rather than explicit. Let entries ˆM be a tridiagonal matrix with [ ˆMV n ] i = τα i V n 2 i 1 + τ(α i + β i + r i ) 2 V n i τβ i 2 V n i+1 (3.13) that is, ˆM is defined by γ 1 γ 2 0 ˆM = τ α 2 r 2 + α 2 + β 2 β α m 1 r m 1 + α m 1 + β m 1 β m (3.14) 11
20 where m is the number of grid points, γ 1 and γ 2 are the boundary points at r = 0. Note that the values γ 1 and γ 2 depend on which boundary conditions we impose at r = 0. In Section 3.1.2, we will show the discretization for γ 1 and γ 2 using two different boundary conditions. The last row of zeros in matrix ˆM is due to the Dirichlet boundary condition being imposed at r = r max, which is further discussed in section Then we can write equation (3.12) as where V n is the value of the solution at the nth time step. [I + ˆM]V n+1 = [I ˆM]V n (3.15) Theorem 1 (Necessary conditions for stability) Let the CrankNicolson timestepping be defined as in equation (3.15), where ˆM is of size (m + 1) (m + 1), where there are m + 1 nodes in the discretization, and ˆM has the properties 1. The offdiagonal entries of ˆM are all strictly negative, 2. The diagonal entries of ˆM are nonnegative, 3. ˆMis row diagonally dominant. Then CrankNicolson timestepping satisfies the necessary conditions for unconditional stability. Proof 1 The proof follows similarly as in [11]. Assume that ˆM has m + 1 linearly independent eigenvectors. If X k is the k th eigenvector of ˆM, then ˆMX k = λ k X k where λ k is the eigenvalue associated with X k. Then, rewrite equation (3.15 as V n+1 = BV n, B = [I + ˆM] 1 [I ˆM]. (3.16) If V 0 is the initial condition, then after n timesteps we have Assuming ˆM has a complete set of eigenvectors, then V n = B n V 0. (3.17) 12
21 V 0 = k C k X k (3.18) for some coefficients C k. Substituting equation (3.18) into equation (3.17) then gives V n = k [ ] n 1 λk C k X k X 1 + λ k. 0 (3.19) k The requirement of equation (3.19) to remain bound as n, is [ ] 1 λk 1 k, (3.20) 1 + λ k which will be true as long as Re(λ k ) 0 k. (3.21) From the Gershgorin circle theorem, we know that every eigenvalue λ of least one of the inequalities ˆM satisfies at ˆM ii λ j i ˆM ii ˆM ij since ˆM is diagonally dominant. Therefore, in the complex plane λ = x + 1y, every λ k lies in some disk with center x = ˆM ii and radius strictly less then ˆM ii, and therefore Re(λ k ) 0 k. Theorem 1 states that the CN method satisfies the necessary conditions for stability, however this does not prove that it satisfies the sufficient conditions for stability. The precise definition of stability is 13
22 V n = bounded. (3.22) However, it takes a bit of work to prove (3.22) for the CN method, so the idea of algebraic stability has been introduced [10]. Definition 4 (Stability of CrankNicolson) If n is the number of timesteps, m is the number of grid nodes, and B is defined as in equation (3.16), then given an arbitrary norm, we say that B is algebraic stable if B n Cn α m β, n, m. (3.23) where C, α, β are independent of n, m. This is often referred to as bounding the power of a matrix. We remark that algebraic stability is a weaker condition than strong stability or strict stability. It can be shown that for the CN method, algebraic stability is guaranteed and is summarized in the following theorem. Theorem 2 Algebraic Stability of CrankNicolson The CrankNicolson discretization is algebraically stable in the sense that, the CN method satisfies equation (3.23) with β = 0, α = 1/2. Proof 2 The proof is the same as in [10]. All the Gershgorin disks of ˆM are in the right half of the complex plane Discretization of the Boundary condition The spatial discretization given by matrix (3.14) depends on γ 1 and γ 2, the discretization of the boundary condition imposed at r = 0. In a financial context, we would like to insure convergence to the viscosity solution. Viscosity solutions have been discussed in [12]. It is therefore important that the discretization of the boundary conditions are monotone in order for the solution to converge to the viscosity solution. Boundary condition at r = 0: 14
23 At the lower boundary point we solve the initial value problem using boundary condition from equation (2.4). In addition, we will alternatively use an arbitrary Neumann boundary condition V τ = 0. (3.24) We will now refer to equation (2.4) and equation (3.24) as BC1 and BC2 respectively. BC1 is discretized using first order forward differencing as Therefore γ 1 and γ 2 of matrix (3.14) are ( ) V (V r ) n n 0 1 V0 n. (3.25) r 0+1/2 1 γ 1 = κb, r 0+1/2 1 γ 2 = κb, r 0+1/2 and the discretization for BC2 is γ 1 = 0, γ 2 = 0. Boundary condition at r = r max : We can implement the Dirichlet boundary condition V (r max, t) = 0, by computing the solution on the grid ˆM (3.14) with the last row and last column removed. Therefore, the matrix ˆM becomes (m 1) (m 1), where m is the number of grid points. 3.2 Discretization of stochastic numerical methods In this section we discuss numerical schemes for the solution of the meanreverting square root process given by equation (1.1). We are examining stochastic numerical methods 15
24 because these methods are able to naturally implement boundary conditions that are imposed in stochastic differential equations. We therefore seek to compare the solution of our FD approach to that of a MonteCarlo simulation with stochastic numerical methods. In the numerical solution of SDEs, the convergence and numerical stability properties of the schemes play a fundamental role as well as in a deterministic framework. As discussed in [13], the region of absolute stability defines possible restrictions on the maximum allowed step size. However, based on the fact that the coefficients of (1.1) are nonlinear and nonlipschitzian it is not possible to appeal to standard convergence theory for numerical simulations to deduce the numerically computed paths are accurate for small stepsizes. These issues were discussed by [14], where the authors showed that a natural EulerMaruyama discretization provides qualitatively correct approximations to the first and second moments of the SDE (1.1) Numerical positivity It is well known that the domain for the SDE in equation (1.1) is [0, ). In order to simulate this process using numerical approximation we must be assured of not getting negative values. The concept of positivity or nonnegativity for this process has been discussed in [15] and [16]. Definition 5 Let R(t) be a stochastic process defined on a domain of [L, ) with P (R(t) > L for all t > 0) = 1. Then the stochastic integration scheme possesses an eternal life time if Otherwise it has a finite life time. P (R n+1 > L R n > L) = 1. Based on the above definition we are therefore interested in numerical methods that have an eternal life time Numerical discretization We will consider six numerical discretization schemes: EulerMaruyama, 16
25 Milstein, Secondorder Milstein, Implicit Milstein, Balanced implicit method, Exact transition distribution. A natural way to simulate the process in (1.1) is with the explicit EulerMaruyama scheme Definition 6 (EulerMaruyama method) R n+1 = R n + κ(b R n ) t + σ R n t(z n+1 ), (3.26) where the timesteps are discretized as t 0 < t 1 <... < t n < T, R n is the interest rate value at the n th timestep, t = t n+1 t n, and Z 1, Z 2,... are independent, mdimensional standard normal random vectors. The EulerMaruyama scheme can lead to negative values since the Gaussian increment is not bounded from below. Therefore, the standard Euler Maruyanam scheme has a finite life time. A natural fix adopted in [14], is to replace the method with the computationally safer method R n+1 = R n + κ(b R n ) t + σ R n t(z n+1 ). (3.27) Theorem 3 For SDE (1.1) the first moment is and second moment is given by lim E[R(t)] = b (3.28) t lim t E[R(t)2 ] = b 2 + σ2 b 2κ. (3.29) Proof 3 As shown in [14], the first moment results by taking expectations of equation (1.1). The second moment result can be obtained by applying the Ito formula to R(t) 2 and taking expectations, using the result for E[R(t)]. 17
26 Corollary 1 We can use properties (3.28) and (3.29) to estimate the stepsize needed to obtain qualitatively correct solutions. We can rewrite (3.27) as R n+1 = R n (1 κ t) + κb t + σ R n t(z n+1 ). (3.30) Using induction, the exception of (3.30) can be shown to be and hence E(R n ) = (1 κ t) n (E(R 0 ) b) + b for t < 2/κ, E(R n ) b as n, for t = 2/κ, E(R n ) = ( 1) n E(R 0 ) + (( 1) n+1 + 1)b, for t > 2/κ, E(R n ) as n. Proof 4 As shown in [14], the proof follows after taking expected values in (3.30). Theorem 3 and Corollary 1 show that as t n the correct mean is produced if and only if the stepsize satisfies t < 2/κ. This result will be used in stepsize selection for the following methods. In addition to the Euler method, we will consider five other numerical schemes for the solution of SDEs. First, we consider the Milstein method which offers an improvement over the Euler scheme. The Euler scheme approximation expands the drift to O( t) but the diffusion term only to O( t). In order to ease notation we define the following a = κ(b R n ) and c = σ R n, where a, c, and their derivatives are all evaluated at time t. Definition 7 (Milstein method) R n+1 = R n + a t + c tz n c c t(z 2 n+1 1). (3.31) The approximation method in (3.31) adds a term to the Euler scheme. It expands both the drift and diffusion terms to O( t). The new term 1 2 c c t(z 2 n+1 1), has mean zero and is uncorrelated with the Euler terms because Z 2 n+1 1 and Z n+1 are uncorrelated. Next, we consider a higher order version of the Milstein method. We consider a simplified version of the scheme as shown in [13]. 18
27 Definition 8 (Secondorder Milstein method) R n+1 = R n + a t + c tz n (a c + ac + 12 ) 2 c2 c t tz n cc [Z 2 n+1 t] + (aa c2 a ) 1 2 t2. (3.32) This method was introduced by Milstein and has a weak order convergence of 2 under the conditions that a and c be six times continuously differentiable with uniformly bounded derivatives. This scheme should produce more accurate results than the Euler method, but it is computationally more expensive. Next, we consider an implicit Milstein method. As noted in [13], implicit Euler schemes can reveal better stability properties. This implicit scheme is obtained by making implicit only the purely deterministic term of the equation, while at each time step, the coefficients of the random part of the equation are retained from the previous step. Note for the implicit method we use a(r n+1 ) = κ(b R n+1 ). Definition 9 (Implicit Milstein method) The implicit Milstein method is defined by R n+1 = R n + a(r n+1 ) t + c tz n c c t(z 2 n+1 1). (3.33) Removing the implicitness and expanding a(r n+1 ) and c yields R n+1 = R n + κb + σ R n tz n σ2 t(zn+1 2 1). (3.34) 1 + κ t The Milstein methods discussed previously fall under the category of dominating methods. Another idea to guarantee nonnegativity for an SDE is to use balancing methods. The balancing method that we will examine was discussed in [16]. Definition 10 (Balanced implicit method (BIM)) The integration scheme for the BIM is given as follows Removing the implicitness yields R n+1 = R n + a t + c tz i+1 + (R n R n+1 )Q n (R n ), (3.35) Q n (R n ) = q 0 (R n ) + q 1 (R n ) tz i+1. (3.36) 19
28 R n+1 = R n + a t + c tz i+1 + R n (q 0 (R n ) t + q 1 (R n ) tz i q 0 (R n ) t + q 1 (R n ). (3.37) tz i+1 In this method the functions q 0 and q 1 are called control functions. To guarantee convergence of the method, the control functions must be bounded and have to satisfy 1 + q 0 (R n ) t + q 1 (R n ) tz i+1 > 0. Finally, we consider a scheme of sampling the exact transition distribution (ETD) of the CIR process as shown in [17]. The SDE (1.1) is not explicitly solvable, however the transition density for the process is known. Based on work by Feller in [7], the distribution of R(t) given R(u) for some t > u is, up to a scale factor, a noncentral chisquared distribution. This property can then be used to simulate the CIR process. The approach to simulate the process is suggested in [17]. Further information on the CIR distribution can be seen in Appendix A. Definition 11 (Exact Transition Distribution (ETD)) Given R(u) and t > u, R(t) is distributed as σ 2 ( 1 e κ(t u)) /(4κ) times a noncentral chisquare random variable χ 2 d (λ) with d degrees of freedom and noncentrality parameter Therefore, λ = 4κe κ(t u) σ 2 (1 e κ(t u) ) R(u). where ( R(t) = σ2 1 e κ(t u)) χ 2 4κ d ( ) 4κe κ(t u) σ 2 (1 e κ(t u) ) R(u), t > u, d = 4κb σ 2. The algorithm that performs the ETD method is shown in Appendix C. 20
The Behavior of Bonds and Interest Rates. An Impossible Bond Pricing Model. 780 w Interest Rate Models
780 w Interest Rate Models The Behavior of Bonds and Interest Rates Before discussing how a bond marketmaker would deltahedge, we first need to specify how bonds behave. Suppose we try to model a zerocoupon
More informationNumerical methods for American options
Lecture 9 Numerical methods for American options Lecture Notes by Andrzej Palczewski Computational Finance p. 1 American options The holder of an American option has the right to exercise it at any moment
More informationPricing Barrier Options under Local Volatility
Abstract Pricing Barrier Options under Local Volatility Artur Sepp Mail: artursepp@hotmail.com, Web: www.hot.ee/seppar 16 November 2002 We study pricing under the local volatility. Our research is mainly
More informationMonte Carlo Methods in Finance
Author: Yiyang Yang Advisor: Pr. Xiaolin Li, Pr. Zari Rachev Department of Applied Mathematics and Statistics State University of New York at Stony Brook October 2, 2012 Outline Introduction 1 Introduction
More informationNumerical Methods for Option Pricing
Chapter 9 Numerical Methods for Option Pricing Equation (8.26) provides a way to evaluate option prices. For some simple options, such as the European call and put options, one can integrate (8.26) directly
More informationARBITRAGEFREE OPTION PRICING MODELS. Denis Bell. University of North Florida
ARBITRAGEFREE OPTION PRICING MODELS Denis Bell University of North Florida Modelling Stock Prices Example American Express In mathematical finance, it is customary to model a stock price by an (Ito) stochatic
More informationNumerical Analysis Lecture Notes
Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number
More informationA SemiLagrangian Approach for Natural Gas Storage Valuation and Optimal Operation
A SemiLagrangian Approach for Natural Gas Storage Valuation and Optimal Operation Zhuliang Chen Peter A. Forsyth October 2, 2006 Abstract The valuation of a gas storage facility is characterized as a
More informationPricing and calibration in local volatility models via fast quantization
Pricing and calibration in local volatility models via fast quantization Parma, 29 th January 2015. Joint work with Giorgia Callegaro and Martino Grasselli Quantization: a brief history Birth: back to
More informationMathematical Finance
Mathematical Finance Option Pricing under the RiskNeutral Measure Cory Barnes Department of Mathematics University of Washington June 11, 2013 Outline 1 Probability Background 2 Black Scholes for European
More informationQUANTIZED INTEREST RATE AT THE MONEY FOR AMERICAN OPTIONS
QUANTIZED INTEREST RATE AT THE MONEY FOR AMERICAN OPTIONS L. M. Dieng ( Department of Physics, CUNY/BCC, New York, New York) Abstract: In this work, we expand the idea of Samuelson[3] and Shepp[,5,6] for
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationPricing European and American bond option under the Hull White extended Vasicek model
1 Academic Journal of Computational and Applied Mathematics /August 2013/ UISA Pricing European and American bond option under the Hull White extended Vasicek model Eva Maria Rapoo 1, Mukendi Mpanda 2
More informationThe BlackScholesMerton Approach to Pricing Options
he BlackScholesMerton Approach to Pricing Options Paul J Atzberger Comments should be sent to: atzberg@mathucsbedu Introduction In this article we shall discuss the BlackScholesMerton approach to determining
More informationMonte Carlo Methods and Models in Finance and Insurance
Chapman & Hall/CRC FINANCIAL MATHEMATICS SERIES Monte Carlo Methods and Models in Finance and Insurance Ralf Korn Elke Korn Gerald Kroisandt f r oc) CRC Press \ V^ J Taylor & Francis Croup ^^"^ Boca Raton
More informationOn the mathematical theory of splitting and Russian roulette
On the mathematical theory of splitting and Russian roulette techniques St.Petersburg State University, Russia 1. Introduction Splitting is an universal and potentially very powerful technique for increasing
More informationConvergence Remedies For NonSmooth Payoffs in Option Pricing
Convergence Remedies For NonSmooth Payoffs in Option Pricing D. M. Pooley, K. R. Vetzal, and P. A. Forsyth University of Waterloo Waterloo, Ontario Canada NL 3G1 June 17, 00 Abstract Discontinuities in
More informationFast solver for the threefactor HestonHull/White problem. F.H.C. Naber Floris.Naber@INGbank.com tw1108735
Fast solver for the threefactor HestonHull/White problem F.H.C. Naber Floris.Naber@INGbank.com tw8735 Amsterdam march 27 Contents Introduction 2. Stochastic Models..........................................
More informationBINOMIAL OPTIONS PRICING MODEL. Mark Ioffe. Abstract
BINOMIAL OPTIONS PRICING MODEL Mark Ioffe Abstract Binomial option pricing model is a widespread numerical method of calculating price of American options. In terms of applied mathematics this is simple
More informationSimulating Stochastic Differential Equations
Monte Carlo Simulation: IEOR E473 Fall 24 c 24 by Martin Haugh Simulating Stochastic Differential Equations 1 Brief Review of Stochastic Calculus and Itô s Lemma Let S t be the time t price of a particular
More informationMoreover, under the risk neutral measure, it must be the case that (5) r t = µ t.
LECTURE 7: BLACK SCHOLES THEORY 1. Introduction: The Black Scholes Model In 1973 Fisher Black and Myron Scholes ushered in the modern era of derivative securities with a seminal paper 1 on the pricing
More informationAN INTRODUCTION TO NUMERICAL METHODS AND ANALYSIS
AN INTRODUCTION TO NUMERICAL METHODS AND ANALYSIS Revised Edition James Epperson Mathematical Reviews BICENTENNIAL 0, 1 8 0 7 z ewiley wu 2007 r71 BICENTENNIAL WILEYINTERSCIENCE A John Wiley & Sons, Inc.,
More informationWhen to Refinance Mortgage Loans in a Stochastic Interest Rate Environment
When to Refinance Mortgage Loans in a Stochastic Interest Rate Environment Siwei Gan, Jin Zheng, Xiaoxia Feng, and Dejun Xie Abstract Refinancing refers to the replacement of an existing debt obligation
More informationBias in the Estimation of Mean Reversion in ContinuousTime Lévy Processes
Bias in the Estimation of Mean Reversion in ContinuousTime Lévy Processes Yong Bao a, Aman Ullah b, Yun Wang c, and Jun Yu d a Purdue University, IN, USA b University of California, Riverside, CA, USA
More informationDoes BlackScholes framework for Option Pricing use Constant Volatilities and Interest Rates? New Solution for a New Problem
Does BlackScholes framework for Option Pricing use Constant Volatilities and Interest Rates? New Solution for a New Problem Gagan Deep Singh Assistant Vice President Genpact Smart Decision Services Financial
More informationMaster of Mathematical Finance: Course Descriptions
Master of Mathematical Finance: Course Descriptions CS 522 Data Mining Computer Science This course provides continued exploration of data mining algorithms. More sophisticated algorithms such as support
More informationx if x 0, x if x < 0.
Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about the
More information5. Convergence of sequences of random variables
5. Convergence of sequences of random variables Throughout this chapter we assume that {X, X 2,...} is a sequence of r.v. and X is a r.v., and all of them are defined on the same probability space (Ω,
More informationCS 522 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options
CS 5 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options 1. Definitions Equity. The common stock of a corporation. Traded on organized exchanges (NYSE, AMEX, NASDAQ). A common
More informationStochastic Modelling and Forecasting
Stochastic Modelling and Forecasting Department of Statistics and Modelling Science University of Strathclyde Glasgow, G1 1XH RSE/NNSFC Workshop on Management Science and Engineering and Public Policy
More informationSome Polynomial Theorems. John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.
Some Polynomial Theorems by John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.com This paper contains a collection of 31 theorems, lemmas,
More informationThe Basics of Interest Theory
Contents Preface 3 The Basics of Interest Theory 9 1 The Meaning of Interest................................... 10 2 Accumulation and Amount Functions............................ 14 3 Effective Interest
More informationNonparametric adaptive age replacement with a onecycle criterion
Nonparametric adaptive age replacement with a onecycle criterion P. CoolenSchrijner, F.P.A. Coolen Department of Mathematical Sciences University of Durham, Durham, DH1 3LE, UK email: Pauline.Schrijner@durham.ac.uk
More information5 =5. Since 5 > 0 Since 4 7 < 0 Since 0 0
a p p e n d i x e ABSOLUTE VALUE ABSOLUTE VALUE E.1 definition. The absolute value or magnitude of a real number a is denoted by a and is defined by { a if a 0 a = a if a
More informationt := maxγ ν subject to ν {0,1,2,...} and f(x c +γ ν d) f(x c )+cγ ν f (x c ;d).
1. Line Search Methods Let f : R n R be given and suppose that x c is our current best estimate of a solution to P min x R nf(x). A standard method for improving the estimate x c is to choose a direction
More informationImplied Vol Constraints
Implied Vol Constraints by Peter Carr Bloomberg Initial version: Sept. 22, 2000 Current version: November 2, 2004 File reference: impvolconstrs3.tex I am solely responsible for any errors. I Introduction
More informationExplicit Option Pricing Formula for a MeanReverting Asset in Energy Markets
Explicit Option Pricing Formula for a MeanReverting Asset in Energy Markets Anatoliy Swishchuk Mathematical & Computational Finance Lab Dept of Math & Stat, University of Calgary, Calgary, AB, Canada
More informationModels Used in Variance Swap Pricing
Models Used in Variance Swap Pricing Final Analysis Report Jason Vinar, Xu Li, Bowen Sun, Jingnan Zhang Qi Zhang, Tianyi Luo, Wensheng Sun, Yiming Wang Financial Modelling Workshop 2011 Presentation Jan
More informationMonte Carlo Simulation of Stochastic Processes
Monte Carlo Simulation of Stochastic Processes Last update: January 10th, 2004. In this section are presented the steps to perform the simulation of the main stochastic processes used in real options applications,
More informationFrom CFD to computational finance (and back again?)
computational finance p. 1/21 From CFD to computational finance (and back again?) Mike Giles mike.giles@maths.ox.ac.uk Oxford University Mathematical Institute OxfordMan Institute of Quantitative Finance
More informationProperties of the SABR model
U.U.D.M. Project Report 2011:11 Properties of the SABR model Nan Zhang Examensarbete i matematik, 30 hp Handledare och examinator: Johan Tysk Juni 2011 Department of Mathematics Uppsala University ABSTRACT
More informationAsian Option Pricing Formula for Uncertain Financial Market
Sun and Chen Journal of Uncertainty Analysis and Applications (215) 3:11 DOI 1.1186/s446715357 RESEARCH Open Access Asian Option Pricing Formula for Uncertain Financial Market Jiajun Sun 1 and Xiaowei
More informationLectures. Sergei Fedotov. 20912  Introduction to Financial Mathematics. No tutorials in the first week
Lectures Sergei Fedotov 20912  Introduction to Financial Mathematics No tutorials in the first week Sergei Fedotov (University of Manchester) 20912 2010 1 / 1 Lecture 1 1 Introduction Elementary economics
More informationStochastic Inventory Control
Chapter 3 Stochastic Inventory Control 1 In this chapter, we consider in much greater details certain dynamic inventory control problems of the type already encountered in section 1.3. In addition to the
More information1 Error in Euler s Method
1 Error in Euler s Method Experience with Euler s 1 method raises some interesting questions about numerical approximations for the solutions of differential equations. 1. What determines the amount of
More informationSOLVING LINEAR SYSTEMS
SOLVING LINEAR SYSTEMS Linear systems Ax = b occur widely in applied mathematics They occur as direct formulations of real world problems; but more often, they occur as a part of the numerical analysis
More informationStephane Crepey. Financial Modeling. A Backward Stochastic Differential Equations Perspective. 4y Springer
Stephane Crepey Financial Modeling A Backward Stochastic Differential Equations Perspective 4y Springer Part I An Introductory Course in Stochastic Processes 1 Some Classes of DiscreteTime Stochastic
More informationLECTURE 15: AMERICAN OPTIONS
LECTURE 15: AMERICAN OPTIONS 1. Introduction All of the options that we have considered thus far have been of the European variety: exercise is permitted only at the termination of the contract. These
More informationAnalysis of Load Frequency Control Performance Assessment Criteria
520 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 16, NO. 3, AUGUST 2001 Analysis of Load Frequency Control Performance Assessment Criteria George Gross, Fellow, IEEE and Jeong Woo Lee Abstract This paper presents
More informationmax cx s.t. Ax c where the matrix A, cost vector c and right hand side b are given and x is a vector of variables. For this example we have x
Linear Programming Linear programming refers to problems stated as maximization or minimization of a linear function subject to constraints that are linear equalities and inequalities. Although the study
More informationEXIT TIME PROBLEMS AND ESCAPE FROM A POTENTIAL WELL
EXIT TIME PROBLEMS AND ESCAPE FROM A POTENTIAL WELL Exit Time problems and Escape from a Potential Well Escape From a Potential Well There are many systems in physics, chemistry and biology that exist
More informationOptimal proportional reinsurance and dividend payout for insurance companies with switching reserves
Optimal proportional reinsurance and dividend payout for insurance companies with switching reserves Abstract: This paper presents a model for an insurance company that controls its risk and dividend
More informationModern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh
Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh Peter Richtárik Week 3 Randomized Coordinate Descent With Arbitrary Sampling January 27, 2016 1 / 30 The Problem
More information3. Reaction Diffusion Equations Consider the following ODE model for population growth
3. Reaction Diffusion Equations Consider the following ODE model for population growth u t a u t u t, u 0 u 0 where u t denotes the population size at time t, and a u plays the role of the population dependent
More informationProbability and Statistics
Probability and Statistics Syllabus for the TEMPUS SEE PhD Course (Podgorica, April 4 29, 2011) Franz Kappel 1 Institute for Mathematics and Scientific Computing University of Graz Žaneta Popeska 2 Faculty
More informationTHE DYING FIBONACCI TREE. 1. Introduction. Consider a tree with two types of nodes, say A and B, and the following properties:
THE DYING FIBONACCI TREE BERNHARD GITTENBERGER 1. Introduction Consider a tree with two types of nodes, say A and B, and the following properties: 1. Let the root be of type A.. Each node of type A produces
More informationRoots of Polynomials
Roots of Polynomials (Com S 477/577 Notes) YanBin Jia Sep 24, 2015 A direct corollary of the fundamental theorem of algebra is that p(x) can be factorized over the complex domain into a product a n (x
More information5.2 Accuracy and Stability for u t = c u x
c 006 Gilbert Strang 5. Accuracy and Stability for u t = c u x This section begins a major topic in scientific computing: Initialvalue problems for partial differential equations. Naturally we start with
More informationTHREE DIMENSIONAL GEOMETRY
Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,
More informationNoarbitrage conditions for cashsettled swaptions
Noarbitrage conditions for cashsettled swaptions Fabio Mercurio Financial Engineering Banca IMI, Milan Abstract In this note, we derive noarbitrage conditions that must be satisfied by the pricing function
More information10.3 POWER METHOD FOR APPROXIMATING EIGENVALUES
55 CHAPTER NUMERICAL METHODS. POWER METHOD FOR APPROXIMATING EIGENVALUES In Chapter 7 we saw that the eigenvalues of an n n matrix A are obtained by solving its characteristic equation n c n n c n n...
More informationSome stability results of parameter identification in a jump diffusion model
Some stability results of parameter identification in a jump diffusion model D. Düvelmeyer Technische Universität Chemnitz, Fakultät für Mathematik, 09107 Chemnitz, Germany Abstract In this paper we discuss
More information1 Finite difference example: 1D implicit heat equation
1 Finite difference example: 1D implicit heat equation 1.1 Boundary conditions Neumann and Dirichlet We solve the transient heat equation ρc p t = ( k ) (1) on the domain L/2 x L/2 subject to the following
More information10.4 APPLICATIONS OF NUMERICAL METHODS Applications of Gaussian Elimination with Pivoting
558 HAPTER NUMERIAL METHODS 5. (a) ompute the eigenvalues of A 5. (b) Apply four iterations of the power method with scaling to each matrix in part (a), starting with x,. 5. (c) ompute the ratios for A
More informationON DETERMINANTS AND SENSITIVITIES OF OPTION PRICES IN DELAYED BLACKSCHOLES MODEL
ON DETERMINANTS AND SENSITIVITIES OF OPTION PRICES IN DELAYED BLACKSCHOLES MODEL A. B. M. Shahadat Hossain, Sharif Mozumder ABSTRACT This paper investigates determinantwise effect of option prices when
More informationFinite Differences Schemes for Pricing of European and American Options
Finite Differences Schemes for Pricing of European and American Options Margarida Mirador Fernandes IST Technical University of Lisbon Lisbon, Portugal November 009 Abstract Starting with the BlackScholes
More informationVilnius University. Faculty of Mathematics and Informatics. Gintautas Bareikis
Vilnius University Faculty of Mathematics and Informatics Gintautas Bareikis CONTENT Chapter 1. SIMPLE AND COMPOUND INTEREST 1.1 Simple interest......................................................................
More informationsince by using a computer we are limited to the use of elementary arithmetic operations
> 4. Interpolation and Approximation Most functions cannot be evaluated exactly: x, e x, ln x, trigonometric functions since by using a computer we are limited to the use of elementary arithmetic operations
More informationAdaptive Online Gradient Descent
Adaptive Online Gradient Descent Peter L Bartlett Division of Computer Science Department of Statistics UC Berkeley Berkeley, CA 94709 bartlett@csberkeleyedu Elad Hazan IBM Almaden Research Center 650
More information4: SINGLEPERIOD MARKET MODELS
4: SINGLEPERIOD MARKET MODELS Ben Goldys and Marek Rutkowski School of Mathematics and Statistics University of Sydney Semester 2, 2015 B. Goldys and M. Rutkowski (USydney) Slides 4: SinglePeriod Market
More informationHPCFinance: New Thinking in Finance. Calculating Variable Annuity Liability Greeks Using Monte Carlo Simulation
HPCFinance: New Thinking in Finance Calculating Variable Annuity Liability Greeks Using Monte Carlo Simulation Dr. Mark Cathcart, Standard Life February 14, 2014 0 / 58 Outline Outline of Presentation
More informationOn a comparison result for Markov processes
On a comparison result for Markov processes Ludger Rüschendorf University of Freiburg Abstract A comparison theorem is stated for Markov processes in polish state spaces. We consider a general class of
More information1. R In this and the next section we are going to study the properties of sequences of real numbers.
+a 1. R In this and the next section we are going to study the properties of sequences of real numbers. Definition 1.1. (Sequence) A sequence is a function with domain N. Example 1.2. A sequence of real
More informationValuation of American Options
Valuation of American Options Among the seminal contributions to the mathematics of finance is the paper F. Black and M. Scholes, The pricing of options and corporate liabilities, Journal of Political
More informationIterative Techniques in Matrix Algebra. Jacobi & GaussSeidel Iterative Techniques II
Iterative Techniques in Matrix Algebra Jacobi & GaussSeidel Iterative Techniques II Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin
More informationAnalysis of a Production/Inventory System with Multiple Retailers
Analysis of a Production/Inventory System with Multiple Retailers Ann M. Noblesse 1, Robert N. Boute 1,2, Marc R. Lambrecht 1, Benny Van Houdt 3 1 Research Center for Operations Management, University
More informationIntroduction to Flocking {Stochastic Matrices}
Supelec EECI Graduate School in Control Introduction to Flocking {Stochastic Matrices} A. S. Morse Yale University Gif sur  Yvette May 21, 2012 CRAIG REYNOLDS  1987 BOIDS The Lion King CRAIG REYNOLDS
More informationThe Characteristic Polynomial
Physics 116A Winter 2011 The Characteristic Polynomial 1 Coefficients of the characteristic polynomial Consider the eigenvalue problem for an n n matrix A, A v = λ v, v 0 (1) The solution to this problem
More informationA SURVEY ON CONTINUOUS ELLIPTICAL VECTOR DISTRIBUTIONS
A SURVEY ON CONTINUOUS ELLIPTICAL VECTOR DISTRIBUTIONS Eusebio GÓMEZ, Miguel A. GÓMEZVILLEGAS and J. Miguel MARÍN Abstract In this paper it is taken up a revision and characterization of the class of
More informationPractical Numerical Training UKNum
Practical Numerical Training UKNum 7: Systems of linear equations C. Mordasini Max Planck Institute for Astronomy, Heidelberg Program: 1) Introduction 2) Gauss Elimination 3) Gauss with Pivoting 4) Determinants
More information5 Numerical Differentiation
D. Levy 5 Numerical Differentiation 5. Basic Concepts This chapter deals with numerical approximations of derivatives. The first questions that comes up to mind is: why do we need to approximate derivatives
More informationEigenvalues and eigenvectors of a matrix
Eigenvalues and eigenvectors of a matrix Definition: If A is an n n matrix and there exists a real number λ and a nonzero column vector V such that AV = λv then λ is called an eigenvalue of A and V is
More informationSensitivity analysis of European options in jumpdiffusion models via the Malliavin calculus on the Wiener space
Sensitivity analysis of European options in jumpdiffusion models via the Malliavin calculus on the Wiener space Virginie Debelley and Nicolas Privault Département de Mathématiques Université de La Rochelle
More informationLecture. S t = S t δ[s t ].
Lecture In real life the vast majority of all traded options are written on stocks having at least one dividend left before the date of expiration of the option. Thus the study of dividends is important
More informationValuation of the Surrender Option in Life Insurance Policies
Valuation of the Surrender Option in Life Insurance Policies Hansjörg Furrer Marketconsistent Actuarial Valuation ETH Zürich, Frühjahrssemester 2010 Valuing Surrender Options Contents A. Motivation and
More informationLeast Squares Estimation
Least Squares Estimation SARA A VAN DE GEER Volume 2, pp 1041 1045 in Encyclopedia of Statistics in Behavioral Science ISBN13: 9780470860809 ISBN10: 0470860804 Editors Brian S Everitt & David
More informationPricing InterestRate Derivative Securities
Pricing InterestRate Derivative Securities John Hull Alan White University of Toronto This article shows that the onestatevariable interestrate models of Vasicek (1977) and Cox, Ingersoll, and Ross
More informationThe BlackScholes Formula
FIN40008 FINANCIAL INSTRUMENTS SPRING 2008 The BlackScholes Formula These notes examine the BlackScholes formula for European options. The BlackScholes formula are complex as they are based on the
More informationScientific Computing: An Introductory Survey
Scientific Computing: An Introductory Survey Chapter 10 Boundary Value Problems for Ordinary Differential Equations Prof. Michael T. Heath Department of Computer Science University of Illinois at UrbanaChampaign
More informationCOMPLETE MARKETS DO NOT ALLOW FREE CASH FLOW STREAMS
COMPLETE MARKETS DO NOT ALLOW FREE CASH FLOW STREAMS NICOLE BÄUERLE AND STEFANIE GRETHER Abstract. In this short note we prove a conjecture posed in Cui et al. 2012): Dynamic meanvariance problems in
More informationInvariant Option Pricing & Minimax Duality of American and Bermudan Options
Invariant Option Pricing & Minimax Duality of American and Bermudan Options Farshid Jamshidian NIB Capital Bank N.V. FELAB, Applied Math Dept., Univ. of Twente April 2005, version 1.0 Invariant Option
More informationChap 4 The Simplex Method
The Essence of the Simplex Method Recall the Wyndor problem Max Z = 3x 1 + 5x 2 S.T. x 1 4 2x 2 12 3x 1 + 2x 2 18 x 1, x 2 0 Chap 4 The Simplex Method 8 corner point solutions. 5 out of them are CPF solutions.
More informationLOGNORMAL MODEL FOR STOCK PRICES
LOGNORMAL MODEL FOR STOCK PRICES MICHAEL J. SHARPE MATHEMATICS DEPARTMENT, UCSD 1. INTRODUCTION What follows is a simple but important model that will be the basis for a later study of stock prices as
More informationSummary of week 8 (Lectures 22, 23 and 24)
WEEK 8 Summary of week 8 (Lectures 22, 23 and 24) This week we completed our discussion of Chapter 5 of [VST] Recall that if V and W are inner product spaces then a linear map T : V W is called an isometry
More informationThe Heat Equation. Lectures INF2320 p. 1/88
The Heat Equation Lectures INF232 p. 1/88 Lectures INF232 p. 2/88 The Heat Equation We study the heat equation: u t = u xx for x (,1), t >, (1) u(,t) = u(1,t) = for t >, (2) u(x,) = f(x) for x (,1), (3)
More informationDynamics. Figure 1: Dynamics used to generate an exemplar of the letter A. To generate
Dynamics Any physical system, such as neurons or muscles, will not respond instantaneously in time but will have a timevarying response termed the dynamics. The dynamics of neurons are an inevitable constraint
More informationPricing Discrete Barrier Options
Pricing Discrete Barrier Options Barrier options whose barrier is monitored only at discrete times are called discrete barrier options. They are more common than the continuously monitored versions. The
More informationVALUING REAL OPTIONS USING IMPLIED BINOMIAL TREES AND COMMODITY FUTURES OPTIONS
VALUING REAL OPTIONS USING IMPLIED BINOMIAL TREES AND COMMODITY FUTURES OPTIONS TOM ARNOLD TIMOTHY FALCON CRACK* ADAM SCHWARTZ A real option on a commodity is valued using an implied binomial tree (IBT)
More informationAn Introduction to Modeling Stock Price Returns With a View Towards Option Pricing
An Introduction to Modeling Stock Price Returns With a View Towards Option Pricing Kyle Chauvin August 21, 2006 This work is the product of a summer research project at the University of Kansas, conducted
More informationQuasistatic evolution and congested transport
Quasistatic evolution and congested transport Inwon Kim Joint with Damon Alexander, Katy Craig and Yao Yao UCLA, UW Madison Hard congestion in crowd motion The following crowd motion model is proposed
More information