Sensitivity Analysis on Long-term Cash flows
|
|
- Herbert Skinner
- 6 years ago
- Views:
Transcription
1 Sensitivity Analysis on Long-term Cash flows Hyungbin Park Worcester Polytechnic Institute 19 March 2016 Eastern Conference on Mathematical Finance Worcester Polytechnic Institute, Worceseter, MA 1 / 49
2 Table of contents Introduction Martingale extraction Sensitivity Analysis Examples of Options Examples of Expected Utilities Examples of LEFTs Conclusion 2 / 49
3 Introduction 3 / 49
4 Introduction Related articles Borovicka, J., Hansen, L.P., Hendricks, M., Scheinkman, J.A.: Risk price dynamics. Journal of Financial Econometrics 9(1), 3-65 (2011) Fournie, E., Lasry J., Lebuchoux, J., Lions P., Touzi, N.: Applications of Mallivin calculus to Monte Carlo methods in finance. Finance Stoch. 3, (1999) Hansen, L.P., Scheinkman, J.A.: Long-term risk: An operator approach. Econometrica 77, (2009) Hansen, L.P., Scheinkman, J.A.: Pricing growth-rate risk. Finance Stoch. 16(1), 1-15 (2012) 4 / 49
5 Introduction Let W t = (W 1 (t), W 2 (t),, W d (t)) be a standard d-dimensional Brownian motion. Assumption An underlying process X t is a conservative d-dimensional time-homogeneous Markov diffusion process with the following form: dx t = b(x t ) dt + σ(x t ) dw t, X 0 = ξ. Here, b is a d-dimensional column vector and σ is a d d matrix with some technical conditions. 5 / 49
6 Introduction In finance, we often encounter the quantity of the form: p T := E Q [e T 0 r(xt) dt f (X T )]. Purpose: to study a sensitivity analysis for the quantity p T with respect to the perturbation of X t for large T. This sensitivity is useful for long-term static investors and for long-dated option prices. 6 / 49
7 Introduction Let Xt ɛ be a perturbed process of X t (with the same initial value ξ = X 0 = X0 ɛ ) of the form: dx ɛ t = b ɛ (X ɛ t ) dt + σ ɛ (X ɛ t ) dw t. (1.1) The perturbed quantity is given by p ɛ T := EQ [e T 0 r(x ɛ s ) ds f (X ɛ T ) ]. For the sensitivity analysis, we compute ɛ pt ɛ ɛ=0 and investigate the behavior of this quantity for large T. 7 / 49
8 Introduction For the sensitivity w.r.t. the initial value X 0 = ξ, we compute p T ξ and investigate the behavior of this quantity for large T. 8 / 49
9 Martingale extraction 9 / 49
10 Martingale extraction We denote the infinitesimal generator corresponding to the operator f p T = E Q [e T 0 r(xt) dt f (X T )] by L : L := 1 2 where a = σσ. d 2 a ij (x) + x i x j i,j=1 d i=1 b i (x) x i r(x) 10 / 49
11 Martingale extraction Let (λ, φ) be an eigenpair of Lφ = λφ with positive function φ. It is easily checked that is a local martingale. M t := e λt t 0 r(xs)ds φ(x t ) φ 1 (ξ) Definition When the local martingale M t is a martingale, we say that (X t, r) admits the martingale extraction with respect to (λ, φ). e t 0 r(xs)ds = M t e λt φ 1 (X t ) φ(ξ) : martingale M t is extracted from the discount factor 11 / 49
12 Martingale extraction When M t is a martingale, we can define a new measure P by P(A) := A M T dq = E Q [I A M T ] for A F T, that is, M T = dp dq The measure P is called the transformed measure from Q with respect to (λ, φ). p T can be expressed by FT p T = E Q [e T 0 r(xs)ds f (X T )] = φ(ξ) e λt E Q [M t (φ 1 f )(X t )] = φ(ξ) e λt E P [(φ 1 f )(X T )]. 12 / 49
13 Martingale extraction We have p T = E Q [e T 0 r(xs)ds f (X T )] = φ(ξ) e λt E P [(φ 1 f )(X T )]. This relationship implies that the quantity p T can be expressed in a relatively more manageable manner. The term E P [(φ 1 f )(X T )] depends on the final value of X T, whereas E Q [e T 0 r(xs)ds f (X T )] depends on the whole path of X t at 0 t T. This advantage makes it easier to analyze the sensitivity of long-term cash flows. 13 / 49
14 Martingale extraction In general, there are infinitely many ways to extract the martingale. We choose a special one. Definition Consider a martingale extraction such that E P [(φ 1 f )(X T )] converges to a nonzero constant as T. We say this is a martingale extraction stabilizing f. In this case, 1 T ln p T = λ. For example, if X t has an invariant distribution µ under P, then E P [(φ 1 f )(X T )] (φ 1 f )(z) dµ(z) for suitably nice f. 14 / 49
15 Sensitivity Analysis 15 / 49
16 Sensitivity Analysis The rho and the vaga Let Xt ɛ be a perturbed process of X t (with the same initial value ξ = X 0 = X0 ɛ ) of the form: dx ɛ t = b ɛ (X ɛ t ) dt + σ ɛ (X ɛ t ) dw t. (3.1) The perturbed quantity is given by p ɛ T := EQ [e T 0 r(x ɛ s ) ds f (X ɛ T ) ]. For the sensitivity analysis, we compute ɛ ln pt ɛ ɛ=0 and investigate the behavior of this quantity for large T. 16 / 49
17 Sensitivity Analysis Assume that (X ɛ t, r) also admits the martingale extraction that stabilizes f, then p ɛ T = φ ɛ(ξ) e λ(ɛ)t E Pɛ [(φ 1 ɛ f )(X ɛ T )]. Differentiate with respect to ɛ and evaluate at ɛ = 0, then ɛ ɛ=0 pt ɛ ɛ = ɛ=0 φ ɛ (ξ) λ ɛ (0) + ɛ=0 E P [(φ 1 ɛ f )(X T )] T p T T φ(ξ) T E P [(φ 1 f )(X T )] + ɛ ɛ=0 E Pɛ [(φ 1 f )(XT ɛ )] T E P [(φ 1 f )(X T )] Here, E P [(φ 1 f )(X T )] (a nonzero constant) as T.. 17 / 49
18 Sensitivity Analysis The long-term behavior of the rho and the vaga Under some conditions, the first, third and the last terms go to zero as T, thus we have 1 T ɛ ln pt ɛ = ɛ ɛ=0 pt ɛ = λ (0) ɛ=0 T p T 18 / 49
19 Sensitivity Analysis The dynamics under the transformed measure Let ϕ := σ φ φ (Girsanov kernel) then t B t := W t ϕ(x s ) ds 0 is a Brownian motion under P. The P-dynamics of X t are dx t = b(x t ) dt + σ(x t ) dw t = (b(x t ) + σ(x t )ϕ(x t )) dt + σ(x t ) db t. 19 / 49
20 Sensitivity Analysis The rho Want to control: 1 T The perturbed process X ɛ t expressed by ɛ ɛ=0 E Pɛ [(φ 1 f )(XT ɛ )] 0 as dx ɛ t = b ɛ (X ɛ t ) dt + σ(x ɛ t ) dw t = (σ 1 b ɛ + ϕ ɛ )(X t ) dt + σ(x ɛ t ) db ɛ t = k ɛ (X ɛ t ) dt + σ(x ɛ t ) db ɛ t. 20 / 49
21 Sensitivity Analysis The rho Assume that there exists a function g : R d R with k ɛ (x) ɛ g(x) on (ɛ, x) I R d for an open interval I containing 0 such that (i) there exists a positive number ɛ 0 such that T )] E [exp (ɛ P 0 g 2 (X t ) dt c e at 0 for some constants a and c on 0 < T <. (ii) for each T > 0, there is a positive number ɛ 1 such that E P T 0 g 2+ɛ 1 (X t ) dt is finite. (iii) 1 T EP [(φ 1 f ) 2 (X T )] 0 as T. 21 / 49
22 Sensitivity Analysis Then, E Pɛ [(φ 1 f )(XT ɛ )] is differentiable at ɛ = 0 and 1 T ɛ E Pɛ [(φ 1 f )(XT ɛ )] 0. ɛ=0 22 / 49
23 Sensitivity Analysis The vega One way: The method of Fournie et. al. with bounded-derivative coefficients Fournie et. al.: Applications of Mallivin calculus to Monte Carlo methods in finance. inance Stoch. 3, (1999) The perturbed process X ɛ t : The P ɛ dynamics of X ɛ t are dx ɛ t = b(x ɛ t ) dt + (σ + ɛσ)(x ɛ t ) dw t dx ɛ t = (b + (σ + ɛσ)ϕ ɛ )(X ɛ t ) dt + (σ + ɛσ)(x ɛ t ) db ɛ t 23 / 49
24 Sensitivity Analysis The vega We take apart two perturbations by the chain rule. dx ρ t = (b + (σ + ρσ)ϕ ρ )(X ρ t ) dt + σ(x ρ t ) db ρ t, dx ν t = (b + σϕ)(x ν t ) dt + (σ + νσ)(x ν t ) db ν t. Then we have ɛ E Pɛ [(φ 1 f )(XT ɛ )] ɛ=0 = ρ E Pρ [(φ 1 f )(X ρ T )] + ρ=0 ν E Pν [(φ 1 f )(XT ν )]. ν=0 24 / 49
25 Sensitivity Analysis Let Z t be the variation process given by dz t = (b + σϕ) (X t )Z t dt + σ(x t )db t + d σ i(x t )Z t db i,t, Z 0 = 0 d i=1 where σ i is the i-th column vector of σ and 0 d is the d-dimensional zero column vector. Theorem Suppose that b + σϕ and φ 1 f are continuously differentiable with bounded derivatives. If 1 T EP [ Z T ] 0 as T, then 1 T ν E Pν [(φ 1 f )(XT ν )] = 0. ν=0 25 / 49
26 Sensitivity Analysis The vega One way : the Lamperti transform for univariate processes The perturbed process is expressed by dxt ɛ = b ɛ (Xt ɛ ) dt + σ ɛ (Xt ɛ ) dw t, X0 ɛ = X 0 = ξ, (3.2) Define a function u ɛ (x) := x ξ σ 1 ɛ (y) dy. (3.3) 26 / 49
27 Sensitivity Analysis The vega Then we have du ɛ (X ɛ t ) = (σ 1 ɛ b ɛ 1 2 σ ɛ)(x ɛ t ) dt + dw t, u ɛ (X ɛ 0) = u ɛ (ξ) = 0. Set U ɛ t := u ɛ (X ɛ t ), then du ɛ t = δ ɛ (U ɛ t ) dt + dw t, U ɛ 0 = 0, (3.4) where δ ɛ ( ) := (( σ 1 ɛ b ɛ 1 2 σ ɛ) vɛ ) ( ). 27 / 49
28 Sensitivity Analysis The delta Let p T := E Q ξ [e T 0 r(xs)ds f (X T )] The quantity of interest is for large T From the martingale extraction ξ p T. p T = E Q [e T 0 r(xs)ds f (X T )] = φ(ξ) e λt E P [(φ 1 f )(X T )], it follows that ξ p T p T = ξ φ φ(ξ) + ξ E P ξ [(φ 1 f )(X T )] E P. ξ [(φ 1 f )(X T )] 28 / 49
29 Sensitivity Analysis The long-term behavior of the delta We have if ξ p T p T = ξ φ φ(ξ) ξ E P ξ [(φ 1 f )(X T )] / 49
30 Sensitivity Analysis Let Y t be the first variation process defined by dy t = (b + σϕ) (X t )Y t dt + d σ i(x t )Y t db i,t, Y 0 = I d where σ i is the i-th column vector of σ and I d is the d d identity matrix. Corollary Assume that the functions b + σϕ and σ are continuously differentiable with bounded derivatives. If E P ξ (φ 1 f ) 2 (X T ) and E P ξ σ 1 (X T )Y T 2 are bounded on 0 < T <, then E P ξ [(φ 1 f )(X T )] is differentiable by ξ and ξ E P ξ (φ 1 f )(X T ) 0 as T. Here, is the matrix 2-norm. i=1 30 / 49
31 Examples of Options 31 / 49
32 Examples of Options The CIR short-interest rate model Under a risk-neutral measure Q, the interest rate r t follows with 2θ > σ 2. dr t = (θ ar t ) dt + σ r t dw t The short-interest rate option price p T := E Q [e T 0 rt dt f (r T )] Want to know the behavior for large T of p T θ, p T a, p T σ 32 / 49
33 Examples Assume f (r) is a nonnegative continuous function on r [0, ), which is not identically zero, and that the growth rate at infinity is equal to or less than e mr with m < a σ 2. The associated second-order equation is Lφ(r) = 1 2 σ2 rφ (r) + (θ ar)φ (r) rφ(r) = λφ(r). a 2 +2σ 2 a Set κ :=. It can be shown that the martingale σ 2 extraction with respect to (λ, φ(r)) := (θκ, e κr ) stabilizes f. 33 / 49
34 Examples For large T, we have that 1 T ln p T = θκ, 1 T ln p T = θ 1 T ln p T a 1 T ln p T σ r 0 ln p T = a 2 + 2σ 2 a σ 2, = θ( a 2 + 2σ 2 a) σ 2 a 2 + 2σ 2, = θ( a 2 + 2σ 2 a) 2 σ 3, a 2 + 2σ 2 a 2 + 2σ 2 a σ / 49
35 Examples The quadratic term structure model Let X t be a d-dimensional OU process under the risk-neutral measure Q dx t = (b + BX t ) dt + σ dw t where b is a d-dimensional vector, B is a d d matrix, and σ is a non-singular d d matrix. The short interest rate : r(x) = β + α, x + Γx, x where the constant β, vector α and symmetric positive definite Γ are taken to be such that r(x) is non-negative for all x R d. 35 / 49
36 Examples Interested in: p T = E Q [e T 0 r(xt) dt f (X T )] with suitably nice payoff function f. Let V be the stabilizing solution (the eigenvalues of B 2aV have negative real parts) of 2VaV B V VB Γ = 0, and let u = (2Va B ) 1 (2Vb + α). The martingale extraction stabilizes f is (λ, φ(x)) = (β 1 2 u au + tr(av ) + u b, e u,x Vx,x ). 36 / 49
37 Examples We have that for 1 i, j d, 1 T ln p T = β u au tr(av ) u b, 1 T ln p T = λ b i b i, 1 T ln p T B ij = λ, B ij 1 T ln p T σ i = λ, σ i ξ p T p T = u 2V ξ. 37 / 49
38 Examples of Expected Utilities 38 / 49
39 Examples The geometric Brownian motion Assume that S t satisfies ds t = µs t dt + σs t dw t with µ 1 2 σ2 > 0. Let Q be the objective measure. Interested in p T = E Q [e rt S α T ]. The corresponding infinitesimal generator: (Lφ)(s) = 1 2 σ2 s 2 φ (s) + µsφ (s) rφ(s). The stabilizing martingale extraction: (λ, φ(s)) := (r µα 1 2 σ2 α(α 1), s α ) 39 / 49
40 Examples The geometric Brownian motion With this (λ, φ), we conclude that 1 T ln p T = r + µα σ2 α(α 1), 1 T µ ln p T = λ µ = α, 1 T σ ln p T = λ = σα(α 1), σ ln p T = φ (S 0 ) S 0 φ(s 0 ) = α. S 0 40 / 49
41 Examples The Heston model An asset X t follows dx t = µx t dt + v t X t dz t, dv t = (γ βv t ) dt + δ v t dw t, where Z t and W t are two standard Brownian motions with Z, W t = ρt for the correlation 1 ρ 1. Interested in: p T := E Q [u(x T )] = E Q [X α T ] 41 / 49
42 Examples The Heston model 1 T µ ln p T = α 1 T γ ln p T = 1 (β ραδ) 2 α(1 α) 2 + δ 2 α(1 α) β + ραδ δ 2 1 (β ραδ) T β ln p T = 2 + δ 2 α(1 α) β + ραδ δ 2 (β ραδ) 2 + δ 2 α(1 α) 42 / 49
43 Examples The Heston model 1 T 1 T (β ραδ) δ ln p T = ρα 2 + δ 2 α(1 α) β + ραδ δ 2 (β ραδ) 2 + δ 2 α(1 α) ρ ln p T = α ln p T = α X 0 X 0 ln p T v 0 + ( (β ραδ) 2 + δ 2 α(1 α) β + ραδ) 2 δ 3 (β ραδ) 2 + δ 2 α(1 α) (β ραδ) 2 + δ 2 α(1 α) αβ + ρα 2 δ δ (β ραδ) 2 + δ 2 α(1 α) = 1 2 α(1 α) (β ραδ) 2 + δ 2 α(1 α) β + ραδ δ / 49
44 Examples of LEFTs 44 / 49
45 Examples of LEFTs The sensitivity analysis of the expected utility and the return of an exchange-traded fund (ETF) is explored. The leveraged ETF (LETF) L t can be written by ( ) L β t Xt β(β 1) r(β 1)t t = e 2 0 σ2 (X u)/xu 2 du. L 0 X 0 We consider a power utility function of the form u(c) = c α, 0 < α 1. Interested in the sensitivity analysis of p T := E Q [u(l T )] 45 / 49
46 Examples of LEFTs The 3/2 model The dynamics of X t follows the 3/2 model dx t = (θ ax t )X t dt + σx 3/2 t dw t with θ, a, σ > 0 under the objective measure Q. The expected utility and the return of LETF is p T := E Q [u(l T )] = E Q [e αβ(β 1)σ2 2 Interested in the behavior for large T of T 0 Xu du X αβ T ] e rα(β 1)T. p T θ, p T a, p T σ 46 / 49
47 Examples of LEFTs The 3/2 model The corresponding infinitesimal operator is 1 2 σ2 x 3 φ (x) + (θ ax)xφ (x) 1 2 αβ(β 1)σ2 xφ(x) = λφ(x). Set l := (1 2 + a σ 2 ) 2 ( 1 + αβ(β 1) 2 + a ) σ 2 It can be shown that the martingale extraction with respect to (λ, φ(x)) := (θl, x l) stabilizes f (x) := x αβ when β / 49
48 Examples of LEFTs The 3/2 model a If + 1 αβ > 0, then σ 2 (1 1 T ln p T = θ 2 + a σ 2 (1 1 T ln p T = θ 2 + a σ 2 ) 2 ( 1 + αβ(β 1) 2 + a ) σ 2, ) 2 ( 1 + αβ(β 1) 2 + a ) σ 2, 1 T ln p T a 1 T ln p T σ = = θ ( ( a σ 2 ) 2 + αβ(β 1) ( σ 2 + a)) σ 2 ( a σ 2 ) 2 + αβ(β 1), 2aθ ( ( a σ 2 ) 2 + αβ(β 1) ( a σ 2 ) ) σ 3 ( a σ 2 ) 2 + αβ(β 1). 48 / 49
49 Conclusion Thank you! 49 / 49
AMH4 - ADVANCED OPTION PRICING. Contents
AMH4 - ADVANCED OPTION PRICING ANDREW TULLOCH Contents 1. Theory of Option Pricing 2 2. Black-Scholes PDE Method 4 3. Martingale method 4 4. Monte Carlo methods 5 4.1. Method of antithetic variances 5
More informationLecture 4. Finite difference and finite element methods
Finite difference and finite element methods Lecture 4 Outline Black-Scholes equation From expectation to PDE Goal: compute the value of European option with payoff g which is the conditional expectation
More informationGirsanov s Theorem. Bernardo D Auria web: July 5, 2017 ICMAT / UC3M
Girsanov s Theorem Bernardo D Auria email: bernardo.dauria@uc3m.es web: www.est.uc3m.es/bdauria July 5, 2017 ICMAT / UC3M Girsanov s Theorem Decomposition of P-Martingales as Q-semi-martingales Theorem
More informationMLEMVD: A R Package for Maximum Likelihood Estimation of Multivariate Diffusion Models
MLEMVD: A R Package for Maximum Likelihood Estimation of Multivariate Diffusion Models Matthew Dixon and Tao Wu 1 Illinois Institute of Technology May 19th 2017 1 https://papers.ssrn.com/sol3/papers.cfm?abstract
More informationRisk, Return, and Ross Recovery
Risk, Return, and Ross Recovery Peter Carr and Jiming Yu Courant Institute, New York University September 13, 2012 Carr/Yu (NYU Courant) Risk, Return, and Ross Recovery September 13, 2012 1 / 30 P, Q,
More informationThe Use of Importance Sampling to Speed Up Stochastic Volatility Simulations
The Use of Importance Sampling to Speed Up Stochastic Volatility Simulations Stan Stilger June 6, 1 Fouque and Tullie use importance sampling for variance reduction in stochastic volatility simulations.
More informationStochastic Dynamical Systems and SDE s. An Informal Introduction
Stochastic Dynamical Systems and SDE s An Informal Introduction Olav Kallenberg Graduate Student Seminar, April 18, 2012 1 / 33 2 / 33 Simple recursion: Deterministic system, discrete time x n+1 = f (x
More informationRoss Recovery theorem and its extension
Ross Recovery theorem and its extension Ho Man Tsui Kellogg College University of Oxford A thesis submitted in partial fulfillment of the MSc in Mathematical Finance April 22, 2013 Acknowledgements I am
More informationSTOCHASTIC CALCULUS AND BLACK-SCHOLES MODEL
STOCHASTIC CALCULUS AND BLACK-SCHOLES MODEL YOUNGGEUN YOO Abstract. Ito s lemma is often used in Ito calculus to find the differentials of a stochastic process that depends on time. This paper will introduce
More informationThe stochastic calculus
Gdansk A schedule of the lecture Stochastic differential equations Ito calculus, Ito process Ornstein - Uhlenbeck (OU) process Heston model Stopping time for OU process Stochastic differential equations
More informationRoss Recovery with Recurrent and Transient Processes
Ross Recovery with Recurrent and Transient Processes Hyungbin Park Courant Institute of Mathematical Sciences, New York University, New York, NY, USA arxiv:4.2282v5 [q-fin.mf] 7 Oct 25 23 September 25
More informationLecture Note 8 of Bus 41202, Spring 2017: Stochastic Diffusion Equation & Option Pricing
Lecture Note 8 of Bus 41202, Spring 2017: Stochastic Diffusion Equation & Option Pricing We shall go over this note quickly due to time constraints. Key concept: Ito s lemma Stock Options: A contract giving
More informationRobust Pricing and Hedging of Options on Variance
Robust Pricing and Hedging of Options on Variance Alexander Cox Jiajie Wang University of Bath Bachelier 21, Toronto Financial Setting Option priced on an underlying asset S t Dynamics of S t unspecified,
More informationValuation of derivative assets Lecture 8
Valuation of derivative assets Lecture 8 Magnus Wiktorsson September 27, 2018 Magnus Wiktorsson L8 September 27, 2018 1 / 14 The risk neutral valuation formula Let X be contingent claim with maturity T.
More informationExam Quantitative Finance (35V5A1)
Exam Quantitative Finance (35V5A1) Part I: Discrete-time finance Exercise 1 (20 points) a. Provide the definition of the pricing kernel k q. Relate this pricing kernel to the set of discount factors D
More informationIntroduction to Affine Processes. Applications to Mathematical Finance
and Its Applications to Mathematical Finance Department of Mathematical Science, KAIST Workshop for Young Mathematicians in Korea, 2010 Outline Motivation 1 Motivation 2 Preliminary : Stochastic Calculus
More informationVariance Reduction for Monte Carlo Simulation in a Stochastic Volatility Environment
Variance Reduction for Monte Carlo Simulation in a Stochastic Volatility Environment Jean-Pierre Fouque Tracey Andrew Tullie December 11, 21 Abstract We propose a variance reduction method for Monte Carlo
More informationApplication of Stochastic Calculus to Price a Quanto Spread
Application of Stochastic Calculus to Price a Quanto Spread Christopher Ting http://www.mysmu.edu/faculty/christophert/ Algorithmic Quantitative Finance July 15, 2017 Christopher Ting July 15, 2017 1/33
More informationOptimal robust bounds for variance options and asymptotically extreme models
Optimal robust bounds for variance options and asymptotically extreme models Alexander Cox 1 Jiajie Wang 2 1 University of Bath 2 Università di Roma La Sapienza Advances in Financial Mathematics, 9th January,
More informationSPDE and portfolio choice (joint work with M. Musiela) Princeton University. Thaleia Zariphopoulou The University of Texas at Austin
SPDE and portfolio choice (joint work with M. Musiela) Princeton University November 2007 Thaleia Zariphopoulou The University of Texas at Austin 1 Performance measurement of investment strategies 2 Market
More informationRohini Kumar. Statistics and Applied Probability, UCSB (Joint work with J. Feng and J.-P. Fouque)
Small time asymptotics for fast mean-reverting stochastic volatility models Statistics and Applied Probability, UCSB (Joint work with J. Feng and J.-P. Fouque) March 11, 2011 Frontier Probability Days,
More informationMartingales & Strict Local Martingales PDE & Probability Methods INRIA, Sophia-Antipolis
Martingales & Strict Local Martingales PDE & Probability Methods INRIA, Sophia-Antipolis Philip Protter, Columbia University Based on work with Aditi Dandapani, 2016 Columbia PhD, now at ETH, Zurich March
More informationHelp Session 2. David Sovich. Washington University in St. Louis
Help Session 2 David Sovich Washington University in St. Louis TODAY S AGENDA Today we will cover the Change of Numeraire toolkit We will go over the Fundamental Theorem of Asset Pricing as well EXISTENCE
More informationMonte Carlo Simulations
Monte Carlo Simulations Lecture 1 December 7, 2014 Outline Monte Carlo Methods Monte Carlo methods simulate the random behavior underlying the financial models Remember: When pricing you must simulate
More informationRMSC 4005 Stochastic Calculus for Finance and Risk. 1 Exercises. (c) Let X = {X n } n=0 be a {F n }-supermartingale. Show that.
1. EXERCISES RMSC 45 Stochastic Calculus for Finance and Risk Exercises 1 Exercises 1. (a) Let X = {X n } n= be a {F n }-martingale. Show that E(X n ) = E(X ) n N (b) Let X = {X n } n= be a {F n }-submartingale.
More informationReal Options and Free-Boundary Problem: A Variational View
Real Options and Free-Boundary Problem: A Variational View Vadim Arkin, Alexander Slastnikov Central Economics and Mathematics Institute, Russian Academy of Sciences, Moscow V.Arkin, A.Slastnikov Real
More informationValuing power options under a regime-switching model
6 13 11 ( ) Journal of East China Normal University (Natural Science) No. 6 Nov. 13 Article ID: 1-5641(13)6-3-8 Valuing power options under a regime-switching model SU Xiao-nan 1, WANG Wei, WANG Wen-sheng
More informationMath 416/516: Stochastic Simulation
Math 416/516: Stochastic Simulation Haijun Li lih@math.wsu.edu Department of Mathematics Washington State University Week 13 Haijun Li Math 416/516: Stochastic Simulation Week 13 1 / 28 Outline 1 Simulation
More informationEnlargement of filtration
Enlargement of filtration Bernardo D Auria email: bernardo.dauria@uc3m.es web: www.est.uc3m.es/bdauria July 6, 2017 ICMAT / UC3M Enlargement of Filtration Enlargement of Filtration ([1] 5.9) If G is a
More informationChapter 3: Black-Scholes Equation and Its Numerical Evaluation
Chapter 3: Black-Scholes Equation and Its Numerical Evaluation 3.1 Itô Integral 3.1.1 Convergence in the Mean and Stieltjes Integral Definition 3.1 (Convergence in the Mean) A sequence {X n } n ln of random
More informationStochastic Processes and Brownian Motion
A stochastic process Stochastic Processes X = { X(t) } Stochastic Processes and Brownian Motion is a time series of random variables. X(t) (or X t ) is a random variable for each time t and is usually
More informationEconomathematics. Problem Sheet 1. Zbigniew Palmowski. Ws 2 dw s = 1 t
Economathematics Problem Sheet 1 Zbigniew Palmowski 1. Calculate Ee X where X is a gaussian random variable with mean µ and volatility σ >.. Verify that where W is a Wiener process. Ws dw s = 1 3 W t 3
More informationStock Loan Valuation Under Brownian-Motion Based and Markov Chain Stock Models
Stock Loan Valuation Under Brownian-Motion Based and Markov Chain Stock Models David Prager 1 1 Associate Professor of Mathematics Anderson University (SC) Based on joint work with Professor Qing Zhang,
More informationAnalytical formulas for local volatility model with stochastic. Mohammed Miri
Analytical formulas for local volatility model with stochastic rates Mohammed Miri Joint work with Eric Benhamou (Pricing Partners) and Emmanuel Gobet (Ecole Polytechnique Modeling and Managing Financial
More informationCredit Risk Models with Filtered Market Information
Credit Risk Models with Filtered Market Information Rüdiger Frey Universität Leipzig Bressanone, July 2007 ruediger.frey@math.uni-leipzig.de www.math.uni-leipzig.de/~frey joint with Abdel Gabih and Thorsten
More informationAsian Options under Multiscale Stochastic Volatility
Contemporary Mathematics Asian Options under Multiscale Stochastic Volatility Jean-Pierre Fouque and Chuan-Hsiang Han Abstract. We study the problem of pricing arithmetic Asian options when the underlying
More information25857 Interest Rate Modelling
25857 UTS Business School University of Technology Sydney Chapter 20. Change of Numeraire May 15, 2014 1/36 Chapter 20. Change of Numeraire 1 The Radon-Nikodym Derivative 2 Option Pricing under Stochastic
More informationExtended Libor Models and Their Calibration
Extended Libor Models and Their Calibration Denis Belomestny Weierstraß Institute Berlin Vienna, 16 November 2007 Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Vienna, 16 November
More informationA No-Arbitrage Theorem for Uncertain Stock Model
Fuzzy Optim Decis Making manuscript No (will be inserted by the editor) A No-Arbitrage Theorem for Uncertain Stock Model Kai Yao Received: date / Accepted: date Abstract Stock model is used to describe
More informationStochastic modelling of electricity markets Pricing Forwards and Swaps
Stochastic modelling of electricity markets Pricing Forwards and Swaps Jhonny Gonzalez School of Mathematics The University of Manchester Magical books project August 23, 2012 Clip for this slide Pricing
More informationStochastic Volatility
Stochastic Volatility A Gentle Introduction Fredrik Armerin Department of Mathematics Royal Institute of Technology, Stockholm, Sweden Contents 1 Introduction 2 1.1 Volatility................................
More informationSparse Wavelet Methods for Option Pricing under Lévy Stochastic Volatility models
Sparse Wavelet Methods for Option Pricing under Lévy Stochastic Volatility models Norbert Hilber Seminar of Applied Mathematics ETH Zürich Workshop on Financial Modeling with Jump Processes p. 1/18 Outline
More informationLogarithmic derivatives of densities for jump processes
Logarithmic derivatives of densities for jump processes Atsushi AKEUCHI Osaka City University (JAPAN) June 3, 29 City University of Hong Kong Workshop on Stochastic Analysis and Finance (June 29 - July
More informationRisk Neutral Measures
CHPTER 4 Risk Neutral Measures Our aim in this section is to show how risk neutral measures can be used to price derivative securities. The key advantage is that under a risk neutral measure the discounted
More informationSlides 4. Matthieu Gomez Fall 2017
Slides 4 Matthieu Gomez Fall 2017 How to Compute Stationary Distribution of a Diffusion? Kolmogorov Forward Take a diffusion process dx t = µ(x t )dt + σ(x t )dz t How does the density of x t evolves?
More informationTHE MARTINGALE METHOD DEMYSTIFIED
THE MARTINGALE METHOD DEMYSTIFIED SIMON ELLERSGAARD NIELSEN Abstract. We consider the nitty gritty of the martingale approach to option pricing. These notes are largely based upon Björk s Arbitrage Theory
More informationSupplementary Appendix to The Risk Premia Embedded in Index Options
Supplementary Appendix to The Risk Premia Embedded in Index Options Torben G. Andersen Nicola Fusari Viktor Todorov December 214 Contents A The Non-Linear Factor Structure of Option Surfaces 2 B Additional
More informationAD in Monte Carlo for finance
AD in Monte Carlo for finance Mike Giles giles@comlab.ox.ac.uk Oxford University Computing Laboratory AD & Monte Carlo p. 1/30 Overview overview of computational finance stochastic o.d.e. s Monte Carlo
More information1.1 Basic Financial Derivatives: Forward Contracts and Options
Chapter 1 Preliminaries 1.1 Basic Financial Derivatives: Forward Contracts and Options A derivative is a financial instrument whose value depends on the values of other, more basic underlying variables
More informationQueens College, CUNY, Department of Computer Science Computational Finance CSCI 365 / 765 Fall 2017 Instructor: Dr. Sateesh Mane.
Queens College, CUNY, Department of Computer Science Computational Finance CSCI 365 / 765 Fall 2017 Instructor: Dr. Sateesh Mane c Sateesh R. Mane 2017 14 Lecture 14 November 15, 2017 Derivation of the
More information13.3 A Stochastic Production Planning Model
13.3. A Stochastic Production Planning Model 347 From (13.9), we can formally write (dx t ) = f (dt) + G (dz t ) + fgdz t dt, (13.3) dx t dt = f(dt) + Gdz t dt. (13.33) The exact meaning of these expressions
More informationRough volatility models: When population processes become a new tool for trading and risk management
Rough volatility models: When population processes become a new tool for trading and risk management Omar El Euch and Mathieu Rosenbaum École Polytechnique 4 October 2017 Omar El Euch and Mathieu Rosenbaum
More informationExact Sampling of Jump-Diffusion Processes
1 Exact Sampling of Jump-Diffusion Processes and Dmitry Smelov Management Science & Engineering Stanford University Exact Sampling of Jump-Diffusion Processes 2 Jump-Diffusion Processes Ubiquitous in finance
More informationGeneralized Multi-Factor Commodity Spot Price Modeling through Dynamic Cournot Resource Extraction Models
Generalized Multi-Factor Commodity Spot Price Modeling through Dynamic Cournot Resource Extraction Models Bilkan Erkmen (joint work with Michael Coulon) Workshop on Stochastic Games, Equilibrium, and Applications
More informationSimulating Stochastic Differential Equations
IEOR E4603: Monte-Carlo Simulation c 2017 by Martin Haugh Columbia University Simulating Stochastic Differential Equations In these lecture notes we discuss the simulation of stochastic differential equations
More informationComputational Finance
Path Dependent Options Computational Finance School of Mathematics 2018 The Random Walk One of the main assumption of the Black-Scholes framework is that the underlying stock price follows a random walk
More informationExtended Libor Models and Their Calibration
Extended Libor Models and Their Calibration Denis Belomestny Weierstraß Institute Berlin Haindorf, 7 Februar 2008 Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Haindorf, 7 Februar
More informationStochastic Volatility Effects on Defaultable Bonds
Stochastic Volatility Effects on Defaultable Bonds Jean-Pierre Fouque Ronnie Sircar Knut Sølna December 24; revised October 24, 25 Abstract We study the effect of introducing stochastic volatility in the
More informationAsymptotic Method for Singularity in Path-Dependent Option Pricing
Asymptotic Method for Singularity in Path-Dependent Option Pricing Sang-Hyeon Park, Jeong-Hoon Kim Dept. Math. Yonsei University June 2010 Singularity in Path-Dependent June 2010 Option Pricing 1 / 21
More informationGood Deal Bounds. Tomas Björk, Department of Finance, Stockholm School of Economics, Lausanne, 2010
Good Deal Bounds Tomas Björk, Department of Finance, Stockholm School of Economics, Lausanne, 2010 Outline Overview of the general theory of GDB (Irina Slinko & Tomas Björk) Applications to vulnerable
More informationTangent Lévy Models. Sergey Nadtochiy (joint work with René Carmona) Oxford-Man Institute of Quantitative Finance University of Oxford.
Tangent Lévy Models Sergey Nadtochiy (joint work with René Carmona) Oxford-Man Institute of Quantitative Finance University of Oxford June 24, 2010 6th World Congress of the Bachelier Finance Society Sergey
More informationLarge Deviations and Stochastic Volatility with Jumps: Asymptotic Implied Volatility for Affine Models
Large Deviations and Stochastic Volatility with Jumps: TU Berlin with A. Jaquier and A. Mijatović (Imperial College London) SIAM conference on Financial Mathematics, Minneapolis, MN July 10, 2012 Implied
More informationLIBOR models, multi-curve extensions, and the pricing of callable structured derivatives
Weierstrass Institute for Applied Analysis and Stochastics LIBOR models, multi-curve extensions, and the pricing of callable structured derivatives John Schoenmakers 9th Summer School in Mathematical Finance
More informationMultiname and Multiscale Default Modeling
Multiname and Multiscale Default Modeling Jean-Pierre Fouque University of California Santa Barbara Joint work with R. Sircar (Princeton) and K. Sølna (UC Irvine) Special Semester on Stochastics with Emphasis
More informationSaddlepoint Approximation Methods for Pricing. Financial Options on Discrete Realized Variance
Saddlepoint Approximation Methods for Pricing Financial Options on Discrete Realized Variance Yue Kuen KWOK Department of Mathematics Hong Kong University of Science and Technology Hong Kong * This is
More informationAdvanced topics in continuous time finance
Based on readings of Prof. Kerry E. Back on the IAS in Vienna, October 21. Advanced topics in continuous time finance Mag. Martin Vonwald (martin@voni.at) November 21 Contents 1 Introduction 4 1.1 Martingale.....................................
More informationGeneralized Affine Transform Formulae and Exact Simulation of the WMSV Model
On of Affine Processes on S + d Generalized Affine and Exact Simulation of the WMSV Model Department of Mathematical Science, KAIST, Republic of Korea 2012 SIAM Financial Math and Engineering joint work
More informationRisk Neutral Valuation
copyright 2012 Christian Fries 1 / 51 Risk Neutral Valuation Christian Fries Version 2.2 http://www.christian-fries.de/finmath April 19-20, 2012 copyright 2012 Christian Fries 2 / 51 Outline Notation Differential
More informationIEOR E4703: Monte-Carlo Simulation
IEOR E4703: Monte-Carlo Simulation Generating Random Variables and Stochastic Processes Martin Haugh Department of Industrial Engineering and Operations Research Columbia University Email: martin.b.haugh@gmail.com
More informationHedging under Arbitrage
Hedging under Arbitrage Johannes Ruf Columbia University, Department of Statistics Modeling and Managing Financial Risks January 12, 2011 Motivation Given: a frictionless market of stocks with continuous
More informationPAPER 27 STOCHASTIC CALCULUS AND APPLICATIONS
MATHEMATICAL TRIPOS Part III Thursday, 5 June, 214 1:3 pm to 4:3 pm PAPER 27 STOCHASTIC CALCULUS AND APPLICATIONS Attempt no more than FOUR questions. There are SIX questions in total. The questions carry
More informationIn chapter 5, we approximated the Black-Scholes model
Chapter 7 The Black-Scholes Equation In chapter 5, we approximated the Black-Scholes model ds t /S t = µ dt + σ dx t 7.1) with a suitable Binomial model and were able to derive a pricing formula for option
More informationNumerical schemes for SDEs
Lecture 5 Numerical schemes for SDEs Lecture Notes by Jan Palczewski Computational Finance p. 1 A Stochastic Differential Equation (SDE) is an object of the following type dx t = a(t,x t )dt + b(t,x t
More informationMartingale representation theorem
Martingale representation theorem Ω = C[, T ], F T = smallest σ-field with respect to which B s are all measurable, s T, P the Wiener measure, B t = Brownian motion M t square integrable martingale with
More informationAll Investors are Risk-averse Expected Utility Maximizers
All Investors are Risk-averse Expected Utility Maximizers Carole Bernard (UW), Jit Seng Chen (GGY) and Steven Vanduffel (Vrije Universiteit Brussel) AFFI, Lyon, May 2013. Carole Bernard All Investors are
More informationIntroduction Taylor s Theorem Einstein s Theory Bachelier s Probability Law Brownian Motion Itô s Calculus. Itô s Calculus.
Itô s Calculus Christopher Ting Christopher Ting http://www.mysmu.edu/faculty/christophert/ : christopherting@smu.edu.sg : 6828 0364 : LKCSB 5036 October 21, 2016 Christopher Ting QF 101 Week 10 October
More informationCredit Risk in Lévy Libor Modeling: Rating Based Approach
Credit Risk in Lévy Libor Modeling: Rating Based Approach Zorana Grbac Department of Math. Stochastics, University of Freiburg Joint work with Ernst Eberlein Croatian Quants Day University of Zagreb, 9th
More informationOption Pricing. 1 Introduction. Mrinal K. Ghosh
Option Pricing Mrinal K. Ghosh 1 Introduction We first introduce the basic terminology in option pricing. Option: An option is the right, but not the obligation to buy (or sell) an asset under specified
More informationKing s College London
King s College London University Of London This paper is part of an examination of the College counting towards the award of a degree. Examinations are governed by the College Regulations under the authority
More informationSample Path Large Deviations and Optimal Importance Sampling for Stochastic Volatility Models
Sample Path Large Deviations and Optimal Importance Sampling for Stochastic Volatility Models Scott Robertson Carnegie Mellon University scottrob@andrew.cmu.edu http://www.math.cmu.edu/users/scottrob June
More information( ) since this is the benefit of buying the asset at the strike price rather
Review of some financial models for MAT 483 Parity and Other Option Relationships The basic parity relationship for European options with the same strike price and the same time to expiration is: C( KT
More informationCONTINUOUS TIME PRICING AND TRADING: A REVIEW, WITH SOME EXTRA PIECES
CONTINUOUS TIME PRICING AND TRADING: A REVIEW, WITH SOME EXTRA PIECES THE SOURCE OF A PRICE IS ALWAYS A TRADING STRATEGY SPECIAL CASES WHERE TRADING STRATEGY IS INDEPENDENT OF PROBABILITY MEASURE COMPLETENESS,
More informationRECURSIVE VALUATION AND SENTIMENTS
1 / 32 RECURSIVE VALUATION AND SENTIMENTS Lars Peter Hansen Bendheim Lectures, Princeton University 2 / 32 RECURSIVE VALUATION AND SENTIMENTS ABSTRACT Expectations and uncertainty about growth rates that
More informationAffine term structures for interest rate models
Stefan Tappe Albert Ludwig University of Freiburg, Germany UNSW-Macquarie WORKSHOP Risk: modelling, optimization and inference Sydney, December 7th, 2017 Introduction Affine processes in finance: R = a
More informationDefinition Pricing Risk management Second generation barrier options. Barrier Options. Arfima Financial Solutions
Arfima Financial Solutions Contents Definition 1 Definition 2 3 4 Contenido Definition 1 Definition 2 3 4 Definition Definition: A barrier option is an option on the underlying asset that is activated
More informationWeierstrass Institute for Applied Analysis and Stochastics Maximum likelihood estimation for jump diffusions
Weierstrass Institute for Applied Analysis and Stochastics Maximum likelihood estimation for jump diffusions Hilmar Mai Mohrenstrasse 39 1117 Berlin Germany Tel. +49 3 2372 www.wias-berlin.de Haindorf
More informationReal Options and Game Theory in Incomplete Markets
Real Options and Game Theory in Incomplete Markets M. Grasselli Mathematics and Statistics McMaster University IMPA - June 28, 2006 Strategic Decision Making Suppose we want to assign monetary values to
More information"Pricing Exotic Options using Strong Convergence Properties
Fourth Oxford / Princeton Workshop on Financial Mathematics "Pricing Exotic Options using Strong Convergence Properties Klaus E. Schmitz Abe schmitz@maths.ox.ac.uk www.maths.ox.ac.uk/~schmitz Prof. Mike
More informationAn application of an entropy principle to short term interest rate modelling
An application of an entropy principle to short term interest rate modelling by BRIDGETTE MAKHOSAZANA YANI Submitted in partial fulfilment of the requirements for the degree of Magister Scientiae in the
More informationContinuous Time Finance. Tomas Björk
Continuous Time Finance Tomas Björk 1 II Stochastic Calculus Tomas Björk 2 Typical Setup Take as given the market price process, S(t), of some underlying asset. S(t) = price, at t, per unit of underlying
More informationStructural Models of Credit Risk and Some Applications
Structural Models of Credit Risk and Some Applications Albert Cohen Actuarial Science Program Department of Mathematics Department of Statistics and Probability albert@math.msu.edu August 29, 2018 Outline
More informationOptimal asset allocation under forward performance criteria Oberwolfach, February 2007
Optimal asset allocation under forward performance criteria Oberwolfach, February 2007 Thaleia Zariphopoulou The University of Texas at Austin 1 References Indifference valuation in binomial models (with
More informationA More Detailed and Complete Appendix for Macroeconomic Volatilities and Long-run Risks of Asset Prices
A More Detailed and Complete Appendix for Macroeconomic Volatilities and Long-run Risks of Asset Prices This is an on-line appendix with more details and analysis for the readers of the paper. B.1 Derivation
More informationAll Investors are Risk-averse Expected Utility Maximizers. Carole Bernard (UW), Jit Seng Chen (GGY) and Steven Vanduffel (Vrije Universiteit Brussel)
All Investors are Risk-averse Expected Utility Maximizers Carole Bernard (UW), Jit Seng Chen (GGY) and Steven Vanduffel (Vrije Universiteit Brussel) First Name: Waterloo, April 2013. Last Name: UW ID #:
More informationKØBENHAVNS UNIVERSITET (Blok 2, 2011/2012) Naturvidenskabelig kandidateksamen Continuous time finance (FinKont) TIME ALLOWED : 3 hours
This question paper consists of 3 printed pages FinKont KØBENHAVNS UNIVERSITET (Blok 2, 211/212) Naturvidenskabelig kandidateksamen Continuous time finance (FinKont) TIME ALLOWED : 3 hours This exam paper
More informationSTOCHASTIC VOLATILITY AND OPTION PRICING
STOCHASTIC VOLATILITY AND OPTION PRICING Daniel Dufresne Centre for Actuarial Studies University of Melbourne November 29 (To appear in Risks and Rewards, the Society of Actuaries Investment Section Newsletter)
More information2 Control variates. λe λti λe e λt i where R(t) = t Y 1 Y N(t) is the time from the last event to t. L t = e λr(t) e e λt(t) Exercises
96 ChapterVI. Variance Reduction Methods stochastic volatility ISExSoren5.9 Example.5 (compound poisson processes) Let X(t) = Y + + Y N(t) where {N(t)},Y, Y,... are independent, {N(t)} is Poisson(λ) with
More informationLocal Volatility Dynamic Models
René Carmona Bendheim Center for Finance Department of Operations Research & Financial Engineering Princeton University Columbia November 9, 27 Contents Joint work with Sergey Nadtochyi Motivation 1 Understanding
More informationInsiders Hedging in a Stochastic Volatility Model with Informed Traders of Multiple Levels
Insiders Hedging in a Stochastic Volatility Model with Informed Traders of Multiple Levels Kiseop Lee Department of Statistics, Purdue University Mathematical Finance Seminar University of Southern California
More informationPDE Methods for the Maximum Drawdown
PDE Methods for the Maximum Drawdown Libor Pospisil, Jan Vecer Columbia University, Department of Statistics, New York, NY 127, USA April 1, 28 Abstract Maximum drawdown is a risk measure that plays an
More information