Saddlepoint Approximation Methods for Pricing. Financial Options on Discrete Realized Variance
|
|
- Blake Lamb
- 5 years ago
- Views:
Transcription
1 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 a joint work with Wendong Zheng. 1
2 Overview The saddlepoint method derives approximation formulas for the density function and the cumulative distribution function of a random variable based on knowledge of its moment generating function (mgf). The method is applicable to a large class of Markov processes for which the mgf or characteristic function, but not the transition function, can be found in closed form. The mgf of the discrete realized variance of a stock price process is not available in closed form. We price options on discrete realized variance via small time asymptotic approximation to the moment generating function of the quadratic variation process and discretely sampled realized variance. 2
3 Complex moment generating functions Let F U (u) and f U (u) = F U (u) denote the cumulative distribution function and continuous density function of a random variable U. Define the complex moment generating function of U by M(z) = ezu f(u) du, where z = x + iy. (i) When y = 0, we recover the ordinary real moment generating function M(x). (ii) When x = 0, M(iy) = eiyu f(u) du is the characteristic function. We assume M(z) to be analytic in some open vertical strip G containing the imaginary axis. In this way, the characteristic function is also analytic. 3
4 G = vertical strip where M(z) is analytic I = set of points at which the real moment generating function M(x) exists = intersection of G with the real axis 4
5 Complex cumulant generating function We define κ(z) = log M(z) to be the complex cumulant generating function. We commonly assume the existence of an analytic form of the cumulant generating function (CGF) so that analytic expressions for the derivatives of the CGF of various orders can be obtained. The CGF κ(z) is commonly assumed to be finite in some open vertical strip {z : α < Re(z) < α + } in the complex plane that contains the imaginary axis, where α < 0 and α + > 0; and both α and α + can be infinite. 5
6 In terms of the complex CGF κ(z), for b > 0, we have b+i density function: f U (u) = 1 exp(κ(z) zu) du 2πi b i tail probability: F U (u) = P [U > u] = 1 b+i exp(κ(z) zu) 2πi b i z dz. An inversion formula similar to those for densities and tail probabilities also exists for E[(U u) + ]. For b > 0, we have where E[(U u) + ] = 1 2πi = 1 2πi b+i b i b+i b i exp(κ(z) zu) z 2 dz exp( κ(z) zu) dz κ(z) = κ(z) log z 2. 6
7 Saddlepoint approximation formula for the density function Consider the Bromwich integral: f(y) = 1 2πi b+i b i exp(κ(u) yu) du. Set û be the solution in u of the equation: κ (u) = y. With regard to the convexity in u of the function κ(u) (a property that is observed for most Markov processes), solution of the above equation exists and is unique also. Consider the Taylor expansion about û, where κ(u) yu = κ(û) yû We set u = û + iv, v R, so that f(y) 1 2π = 2 κ(u) u 2 exp(κ(û) yû) exp 1 2 κ(u) 2 u 2 ( 1 exp v2 2π 2κ (û) exp(κ(û) yû) 2π ( u=û (u û) 2 +. ) v 2 ) u=û dv = dv exp(κ(û) yû). 2π κ (û) 7
8 Lugannani-Rice formula Lugannani and Rice (1980) derive a saddlepoint approximation to the tail probability as follows: P {X x} = 1 Φ(s) + ϕ(s) 1 1 t nκ ( t) s + O(n 3/2 ) with s = sgn( t) 2n x t κ( t), and t satisfies κ ( t) = x. Here, Φ and ϕ are the CDF and PDF of the standard normal random variable, respectively. The proof of the above formula can be performed in a similar manner by approximating κ(t) xt in an interval containing both t = 0 and t = t by a quadratic function, where [κ(t) xt] [κ( t) x t] = 1 2 (w ŵ)2, with ŵ 2 /2 = κ( t) x t. Finally, we change the integration variable from t to w. 8
9 Pricing options on discrete realized variance The advantage of the saddlepoint approximation is more appreciated when we consider approximation formulas for pricing highly path dependent derivatives, like options on discrete realized variance. The terminal payoff of a put option on the discrete realized variance is given by max(k V d (0, T ; N), 0), where K is the strike price and V d (0, T ; N) = A N N i=1 ( ln S t i S ti 1 The annualized factor A is taken to be 252 for daily monitoring and {t 0, t 1,..., t N } is the set of monitoring instants. ) 2. The moment generating function of V d (0, T ; N) cannot be found readily under stochastic volatility models, unlike that of the continuous realized variance. 9
10 Our contributions to the saddlepoint approximation method for pricing options on discrete realized variance We develop a viable approach such that the saddlepoint approximation can be derived even when κ(z) is defined only in the left half complex plane not including the imaginary axis. When the analytic expression of the CGF is not available, we deduce useful analytic approximations using the small time asymptotic approximation of the Laplace transform of the discrete realized variance as a control. 10
11 Let κ(θ) and κ 0 (θ) denote the CGF of the random discretely sampled realized variance I and I K, respectively, where K represents the fixed strike. The two CGFs are related by κ 0 (θ) = κ(θ) Kθ. We write X = I K and F 0 (x) as the distribution function of X. Recall that the Fourier transform of the payoff of a call option on discrete realized variance E[X1 {X>0} ] is eκ 0 (t) t 2. Consider the following tail expectations expressed in terms of Bromwich integrals: Ξ 1 = E[X1 {X>0} ] = 1 2πi Ξ 2 = E[X1 {X<0} ] = 1 2πi τ1 +i e κ 0(t) τ 1 i τ2 +i e κ 0(t) τ 2 i t 2 dt, τ 1 (0, α + ) where α + > 0; t 2 dt, τ 2 (α, 0) where α < 0. The contour is taken to be along a vertical line parallel to the imaginary axis. We write the integrand as e κ 0(t) 2 ln t. 11
12 First order saddlepoint approximation The first order saddlepoint approximation to Ξ j, j = 1, 2, is given by Ξ j ˆΞ j = 2π e κ 0(ˆt j ) /ˆt 2 j [ 2 ˆt 2 j ], j = 1, 2, + κ (2) 0 (ˆt j ) where ˆt 1 > 0 (ˆt 2 < 0) is the positive (negative) root in (α, α + ) of the saddlepoint equation: κ 0 (t) 2/t = 0. Note that Ξ 1 Ξ 2 = µ X, which is consistent with the put-call parity in option pricing theory. Suppose both roots ˆt 1 and ˆt 2 exist, we can use either the saddlepoint approximation ˆΞ 1 ( Ξ 1 ) or µ X + ˆΞ 2 (µ X + Ξ 2 ) to approximate the value of the call option. 12
13 Second order Saddlepoint approximation By performing the Taylor expansion of κ(t) xt up to the fourth order, we manage to derive the second order saddlepoint approximation formulas. The second order saddlepoint approximation to Ξ j is given by Ξ j = ˆΞ j (1 + R j ), j = 1, 2, where the adjustment term R j is given by R j = 1 8 κ (4) 0 (ˆt j ) + 12ˆt 4 j 0 (ˆt j ) + 2ˆt 2 ] 2 [κ (2) j 5 24 [κ (3) 0 (ˆt j ) 4ˆt 3 j ] 2 [κ (2) 0 (ˆt j ) + 2ˆt 2 j ] 3, j = 1, 2. 13
14 Small time asymptotic approximation to MGF We consider the small time asymptotic approximation to the MGFs of the quadratic variation process I(0, T ; ) = 1 T [ln S T, ln S T ] and discretely sampled realized variance I(0, T ; N). The asymptotic limit of V d (0, T ; N) is gamma distributed with shape parameter N/2 and scale parameter 2V 0 /N. For any u 0, we obtain lim M T 0 + I(0,T ; ) (u) = euv 0, lim M T 0 + I(0,T ;N) (u) = ( 1 2V 0u N ) N/2. 14
15 Difference in the corresponding asymptotic approximation terms as a control The difference M I(0,T ;N) (u) M I(0,T ; ) (u) is seen to be almost invariant with respect to T, we use the above difference as a control and propose the following approximate MGF formula: ˆM I(0,T ;N) (u) = M I(0,T ; ) (u) + ( 1 2V 0u N ) N/2 e uv 0, u C. Under an affine stochastic volatility model, M I(0,T ; ) (u) can be derived analytically by solving a Riccati system of equations. 15
16 Analytic formulas for the approximate cumulant generating function and its higher order derivatives ˆκ I(0,T ;N) (u) = ln ˆM I(0,T ;N) (u), ˆκ I(0,T ;N) (u) = M I(0,T ; ) (u) + f 1(u), M I(0,T ;N) (u) ˆκ (2) I(0,T ;N) (2) M I(0,T ; ) (u) = (u) + f 2(u) ˆM I(0,T ;N) (u) [M I(0,T ; ) (u) + f 1(u)] 2 [ ˆM I(0,T ;N) (u)] 2, where the sequence of functions f n (u) is defined by ( N N2 f n (u) = V0 k ) (N 2 + n ) ( ( ) N2 n 1 2V ) N/2 n 0u, n = 1, 2,. N 16
17 Heston affine stochastic volatility models with simultaneous jumps (SVSJ) Under a pricing measure Q, the joint dynamics of stock price S t and its instantaneous variance V t under the affine SVSJ model assumes the form ds t S t = (r λm) dt + V t dw S t + (e JS 1) dn t, dv t = κ(θ V t ) dt + ε V t dw V t + J V dn t, where Wt S and Wt V are a pair of correlated standard Brownian motions with dwt SdW t V = ρ dt, and N t is a Poisson process with constant intensity λ that is independent of the two Brownian motions. J S and J V denote the random jump sizes of the log price and variance, respectively. These random jump sizes are assumed to be independent of W S t, W V t and N t. 17
18 Joint moment generating function Let X t = ln S t. The joint moment generating function of X t and V t is defined to be E[exp(ϕX T + bv T + γ)], where ϕ, b and γ are constant parameters. Let U(X t, V t, t) denote the non-discounted time-t value of a contingent claim with the terminal payoff function: U T (X T, V T ), where T is the maturity date. Let τ = T t, U(X, V, τ) is governed by the following partial integro-differential equation (PIDE): U τ = ( r mλ V 2 ) U X + κ(θ V ) U V + V 2 U 2 X 2 + ε2 V 2 U 2 V 2 + ρεv 2 U X V + λe [ U(X + J S, V + J V, τ) U(X, V, τ) ]. 18
19 Riccati system of ordinary differential equations The parameter functions B(Θ; τ, q), Γ(Θ; τ, q) and Λ(Θ; τ, q) satisfy the following Riccati system of ordinary differential equations: B τ = 1 2 (ϕ ϕ2 ) (κ ρεϕ)b + ε2 2 B2 Γ τ = rϕ + κθb Λ τ = λ ( E[exp(ϕJ S + BJ V ) 1] mϕ ) with the initial conditions: B(0) = b, Γ(0) = γ and Λ(0) = 0. Canonical jump distributions Suppose we assume that J V exp(1/η) and J S follows J S J V Normal(ν + ρ J J V, δ 2 ), which is the Gaussian distribution with mean ν + ρ J J V and variance δ 2, we obtain provided that ηρ J < 1. m = E[e JS 1] = eν+δ2 /2 1 ηρ J 1, 19
20 Under the above assumptions on J S and J V, the parameter functions can be found to be B(Θ; τ, q) = b(ξ e ζτ + ξ + ) (ϕ ϕ 2 )(1 e ζτ ) (ξ + + ε 2 b)e ζτ + ξ ε 2, b Γ(Θ; τ, q) = rϕτ + γ κθ [ ε 2 ξ + τ + 2 ln (ξ + + ε 2 b)e ζτ + ξ ε 2 b 2ζ Λ(Θ; τ, q) = λ(mϕ + 1)τ + λe ϕν+δ2 ϕ 2 /2 k 2 k 4 τ 1 ζ ( k1 k 3 k 2 k 4 ) ln k 3e ζτ + k 4 k 3 + k 4 with q = (ϕ b γ) T and ζ = (κ ρεϕ) 2 + ε 2 (ϕ ϕ 2 ), ξ ± = ζ (κ ρεϕ), k 1 = ξ + + ε 2 b, k 2 = ξ ε 2 b, k 3 = (1 ϕρ J η)k 1 η(ϕ ϕ 2 + ξ b), k 4 = (1 ϕρ J η)k 2 η[ξ + b (ϕ ϕ 2 )]., ], 20
21 Numerical results maturity (days) strike (OTM) SPA SPA MC SE strike (ATM) SPA SPA MC SE strike (ITM) SPA SPA MC SE The prices of put options on the daily sampled realized variance with varying strike prices and maturities under the SVSJ model. The first order approximation SPA1 is seen to outperform the second order approximation SPA2 for short-maturity or out-of-the-money put options. 21
22 Plots of the percentage errors against moneyness for the one-month (20 days) put options on daily sampled realized variance with different model parameters of the SVSJ model. 22
23 Conclusion We use the small time asymptotic approximation of the Laplace transform of discrete realized variance to obtain analytic approximation formulas using the saddlepoint approximation method for pricing options on discrete realized variance. The second order saddlepoint approximation formulas provide sufficiently good approximation (with a small percentage error) to the values of options on discrete realized variance even when the option is deep out-of-the-money and under the choices of extreme parameter values (high values of jump intensity λ and volatility of variance ε). 23
Analytic Pricing of the Third Generation Discrete Variance Derivatives
Analytic Pricing of the Third Generation Discrete Variance Derivatives Yue Kuen KWOK Department of Mathematics Hong Kong University of Science and Technology Hong Kong * This is a joint work with Wendong
More informationSaddlepoint approximation methods for pricing derivatives on discrete realized variance
Saddlepoint approximation methods for pricing derivatives on discrete realized variance Wendong Zheng Department of Mathematics, Hong Kong University of Science and Technology E-mail: wdzheng@ust.hk Yue
More informationSADDLEPOINT APPROXIMATIONS TO OPTION PRICES 1. By L. C. G. Rogers and O. Zane University of Bath and First Chicago NBD
The Annals of Applied Probability 1999, Vol. 9, No. 2, 493 53 SADDLEPOINT APPROXIMATIONS TO OPTION PRICES 1 By L. C. G. Rogers and O. Zane University of Bath and First Chicago NBD The use of saddlepoint
More informationAdvanced Topics in Derivative Pricing Models. Topic 4 - Variance products and volatility derivatives
Advanced Topics in Derivative Pricing Models Topic 4 - Variance products and volatility derivatives 4.1 Volatility trading and replication of variance swaps 4.2 Volatility swaps 4.3 Pricing of discrete
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 informationM5MF6. Advanced Methods in Derivatives Pricing
Course: Setter: M5MF6 Dr Antoine Jacquier MSc EXAMINATIONS IN MATHEMATICS AND FINANCE DEPARTMENT OF MATHEMATICS April 2016 M5MF6 Advanced Methods in Derivatives Pricing Setter s signature...........................................
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 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 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 informationRough Heston models: Pricing, hedging and microstructural foundations
Rough Heston models: Pricing, hedging and microstructural foundations Omar El Euch 1, Jim Gatheral 2 and Mathieu Rosenbaum 1 1 École Polytechnique, 2 City University of New York 7 November 2017 O. El Euch,
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 informationQuadratic hedging in affine stochastic volatility models
Quadratic hedging in affine stochastic volatility models Jan Kallsen TU München Pittsburgh, February 20, 2006 (based on joint work with F. Hubalek, L. Krawczyk, A. Pauwels) 1 Hedging problem S t = S 0
More informationAn Analytical Approximation for Pricing VWAP Options
.... An Analytical Approximation for Pricing VWAP Options Hideharu Funahashi and Masaaki Kijima Graduate School of Social Sciences, Tokyo Metropolitan University September 4, 215 Kijima (TMU Pricing of
More informationM.I.T Fall Practice Problems
M.I.T. 15.450-Fall 2010 Sloan School of Management Professor Leonid Kogan Practice Problems 1. Consider a 3-period model with t = 0, 1, 2, 3. There are a stock and a risk-free asset. The initial stock
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 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 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 informationUnified Credit-Equity Modeling
Unified Credit-Equity Modeling Rafael Mendoza-Arriaga Based on joint research with: Vadim Linetsky and Peter Carr The University of Texas at Austin McCombs School of Business (IROM) Recent Advancements
More informationUSC Math. Finance April 22, Path-dependent Option Valuation under Jump-diffusion Processes. Alan L. Lewis
USC Math. Finance April 22, 23 Path-dependent Option Valuation under Jump-diffusion Processes Alan L. Lewis These overheads to be posted at www.optioncity.net (Publications) Topics Why jump-diffusion models?
More informationBIRKBECK (University of London) MSc EXAMINATION FOR INTERNAL STUDENTS MSc FINANCIAL ENGINEERING DEPARTMENT OF ECONOMICS, MATHEMATICS AND STATIS- TICS
BIRKBECK (University of London) MSc EXAMINATION FOR INTERNAL STUDENTS MSc FINANCIAL ENGINEERING DEPARTMENT OF ECONOMICS, MATHEMATICS AND STATIS- TICS PRICING EMMS014S7 Tuesday, May 31 2011, 10:00am-13.15pm
More informationVariable Annuities with Lifelong Guaranteed Withdrawal Benefits
Variable Annuities with Lifelong Guaranteed Withdrawal Benefits presented by Yue Kuen Kwok Department of Mathematics Hong Kong University of Science and Technology Hong Kong, China * This is a joint work
More informationLikelihood Estimation of Jump-Diffusions
Likelihood Estimation of Jump-Diffusions Extensions from Diffusions to Jump-Diffusions, Implementation with Automatic Differentiation, and Applications Berent Ånund Strømnes Lunde DEPARTMENT OF MATHEMATICS
More informationHedging Under Jump Diffusions with Transaction Costs. Peter Forsyth, Shannon Kennedy, Ken Vetzal University of Waterloo
Hedging Under Jump Diffusions with Transaction Costs Peter Forsyth, Shannon Kennedy, Ken Vetzal University of Waterloo Computational Finance Workshop, Shanghai, July 4, 2008 Overview Overview Single factor
More informationNear-expiration behavior of implied volatility for exponential Lévy models
Near-expiration behavior of implied volatility for exponential Lévy models José E. Figueroa-López 1 1 Department of Statistics Purdue University Financial Mathematics Seminar The Stevanovich Center for
More informationValuing volatility and variance swaps for a non-gaussian Ornstein-Uhlenbeck stochastic volatility model
Valuing volatility and variance swaps for a non-gaussian Ornstein-Uhlenbeck stochastic volatility model 1(23) Valuing volatility and variance swaps for a non-gaussian Ornstein-Uhlenbeck stochastic volatility
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 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 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 informationPricing and hedging with rough-heston models
Pricing and hedging with rough-heston models Omar El Euch, Mathieu Rosenbaum Ecole Polytechnique 1 January 216 El Euch, Rosenbaum Pricing and hedging with rough-heston models 1 Table of contents Introduction
More informationRough volatility models
Mohrenstrasse 39 10117 Berlin Germany Tel. +49 30 20372 0 www.wias-berlin.de October 18, 2018 Weierstrass Institute for Applied Analysis and Stochastics Rough volatility models Christian Bayer EMEA Quant
More informationIEOR E4703: Monte-Carlo Simulation
IEOR E4703: Monte-Carlo Simulation Simulating Stochastic Differential Equations Martin Haugh Department of Industrial Engineering and Operations Research Columbia University Email: martin.b.haugh@gmail.com
More informationLecture 2: Rough Heston models: Pricing and hedging
Lecture 2: Rough Heston models: Pricing and hedging Mathieu Rosenbaum École Polytechnique European Summer School in Financial Mathematics, Dresden 217 29 August 217 Mathieu Rosenbaum Rough Heston models
More informationVolatility Smiles and Yield Frowns
Volatility Smiles and Yield Frowns Peter Carr NYU CBOE Conference on Derivatives and Volatility, Chicago, Nov. 10, 2017 Peter Carr (NYU) Volatility Smiles and Yield Frowns 11/10/2017 1 / 33 Interest Rates
More informationEquity correlations implied by index options: estimation and model uncertainty analysis
1/18 : estimation and model analysis, EDHEC Business School (joint work with Rama COT) Modeling and managing financial risks Paris, 10 13 January 2011 2/18 Outline 1 2 of multi-asset models Solution to
More informationThe Black-Scholes Model
The Black-Scholes Model Liuren Wu Options Markets Liuren Wu ( c ) The Black-Merton-Scholes Model colorhmoptions Markets 1 / 18 The Black-Merton-Scholes-Merton (BMS) model Black and Scholes (1973) and Merton
More informationPricing Variance Swaps under Stochastic Volatility Model with Regime Switching - Discrete Observations Case
Pricing Variance Swaps under Stochastic Volatility Model with Regime Switching - Discrete Observations Case Guang-Hua Lian Collaboration with Robert Elliott University of Adelaide Feb. 2, 2011 Robert Elliott,
More informationThe Black-Scholes Model
The Black-Scholes Model Liuren Wu Options Markets (Hull chapter: 12, 13, 14) Liuren Wu ( c ) The Black-Scholes Model colorhmoptions Markets 1 / 17 The Black-Scholes-Merton (BSM) model Black and Scholes
More informationModeling the Implied Volatility Surface. Jim Gatheral Global Derivatives and Risk Management 2003 Barcelona May 22, 2003
Modeling the Implied Volatility Surface Jim Gatheral Global Derivatives and Risk Management 2003 Barcelona May 22, 2003 This presentation represents only the personal opinions of the author and not those
More informationOption Pricing for a Stochastic-Volatility Jump-Diffusion Model with Log-Uniform Jump-Amplitudes
Option Pricing for a Stochastic-Volatility Jump-Diffusion Model with Log-Uniform Jump-Amplitudes Floyd B. Hanson and Guoqing Yan Department of Mathematics, Statistics, and Computer Science University of
More informationVolatility Smiles and Yield Frowns
Volatility Smiles and Yield Frowns Peter Carr NYU IFS, Chengdu, China, July 30, 2018 Peter Carr (NYU) Volatility Smiles and Yield Frowns 7/30/2018 1 / 35 Interest Rates and Volatility Practitioners and
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 informationAn Overview of Volatility Derivatives and Recent Developments
An Overview of Volatility Derivatives and Recent Developments September 17th, 2013 Zhenyu Cui Math Club Colloquium Department of Mathematics Brooklyn College, CUNY Math Club Colloquium Volatility Derivatives
More informationDr. Maddah ENMG 625 Financial Eng g II 10/16/06
Dr. Maddah ENMG 65 Financial Eng g II 10/16/06 Chapter 11 Models of Asset Dynamics () Random Walk A random process, z, is an additive process defined over times t 0, t 1,, t k, t k+1,, such that z( t )
More informationLecture 8: The Black-Scholes theory
Lecture 8: The Black-Scholes theory Dr. Roman V Belavkin MSO4112 Contents 1 Geometric Brownian motion 1 2 The Black-Scholes pricing 2 3 The Black-Scholes equation 3 References 5 1 Geometric Brownian motion
More informationWKB Method for Swaption Smile
WKB Method for Swaption Smile Andrew Lesniewski BNP Paribas New York February 7 2002 Abstract We study a three-parameter stochastic volatility model originally proposed by P. Hagan for the forward swap
More informationBrownian Motion. Richard Lockhart. Simon Fraser University. STAT 870 Summer 2011
Brownian Motion Richard Lockhart Simon Fraser University STAT 870 Summer 2011 Richard Lockhart (Simon Fraser University) Brownian Motion STAT 870 Summer 2011 1 / 33 Purposes of Today s Lecture Describe
More informationStochastic Volatility (Working Draft I)
Stochastic Volatility (Working Draft I) Paul J. Atzberger General comments or corrections should be sent to: paulatz@cims.nyu.edu 1 Introduction When using the Black-Scholes-Merton model to price derivative
More informationOption Pricing for a Stochastic-Volatility Jump-Diffusion Model
Option Pricing for a Stochastic-Volatility Jump-Diffusion Model Guoqing Yan and Floyd B. Hanson Department of Mathematics, Statistics, and Computer Science University of Illinois at Chicago Conference
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 informationTime-changed Brownian motion and option pricing
Time-changed Brownian motion and option pricing Peter Hieber Chair of Mathematical Finance, TU Munich 6th AMaMeF Warsaw, June 13th 2013 Partially joint with Marcos Escobar (RU Toronto), Matthias Scherer
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 informationSensitivity Analysis on Long-term Cash flows
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
More informationSTOCHASTIC INTEGRALS
Stat 391/FinMath 346 Lecture 8 STOCHASTIC INTEGRALS X t = CONTINUOUS PROCESS θ t = PORTFOLIO: #X t HELD AT t { St : STOCK PRICE M t : MG W t : BROWNIAN MOTION DISCRETE TIME: = t < t 1
More informationShort-time asymptotics for ATM option prices under tempered stable processes
Short-time asymptotics for ATM option prices under tempered stable processes José E. Figueroa-López 1 1 Department of Statistics Purdue University Probability Seminar Purdue University Oct. 30, 2012 Joint
More informationStochastic Volatility and Jump Modeling in Finance
Stochastic Volatility and Jump Modeling in Finance HPCFinance 1st kick-off meeting Elisa Nicolato Aarhus University Department of Economics and Business January 21, 2013 Elisa Nicolato (Aarhus University
More informationPricing Parisian options using numerical inversion of Laplace transforms
using numerical inversion of Laplace transforms Jérôme Lelong (joint work with C. Labart) http://cermics.enpc.fr/~lelong Tuesday 23 October 2007 J. Lelong (MathFi INRIA) Tuesday 23 October 2007 1 / 33
More information7 pages 1. Premia 14
7 pages 1 Premia 14 Calibration of Stochastic Volatility model with Jumps A. Ben Haj Yedder March 1, 1 The evolution process of the Heston model, for the stochastic volatility, and Merton model, for the
More informationThe rth moment of a real-valued random variable X with density f(x) is. x r f(x) dx
1 Cumulants 1.1 Definition The rth moment of a real-valued random variable X with density f(x) is µ r = E(X r ) = x r f(x) dx for integer r = 0, 1,.... The value is assumed to be finite. Provided that
More informationTwo hours. To be supplied by the Examinations Office: Mathematical Formula Tables and Statistical Tables THE UNIVERSITY OF MANCHESTER
Two hours MATH20802 To be supplied by the Examinations Office: Mathematical Formula Tables and Statistical Tables THE UNIVERSITY OF MANCHESTER STATISTICAL METHODS Answer any FOUR of the SIX questions.
More informationA Closed-form Solution for Outperfomance Options with Stochastic Correlation and Stochastic Volatility
A Closed-form Solution for Outperfomance Options with Stochastic Correlation and Stochastic Volatility Jacinto Marabel Romo Email: jacinto.marabel@grupobbva.com November 2011 Abstract This article introduces
More informationManaging Systematic Mortality Risk in Life Annuities: An Application of Longevity Derivatives
Managing Systematic Mortality Risk in Life Annuities: An Application of Longevity Derivatives Simon Man Chung Fung, Katja Ignatieva and Michael Sherris School of Risk & Actuarial Studies University of
More informationNEWCASTLE UNIVERSITY SCHOOL OF MATHEMATICS, STATISTICS & PHYSICS SEMESTER 1 SPECIMEN 2 MAS3904. Stochastic Financial Modelling. Time allowed: 2 hours
NEWCASTLE UNIVERSITY SCHOOL OF MATHEMATICS, STATISTICS & PHYSICS SEMESTER 1 SPECIMEN 2 Stochastic Financial Modelling Time allowed: 2 hours Candidates should attempt all questions. Marks for each question
More informationCEV Implied Volatility by VIX
CEV Implied Volatility by VIX Implied Volatility Chien-Hung Chang Dept. of Financial and Computation Mathematics, Providence University, Tiachng, Taiwan May, 21, 2015 Chang (Institute) Implied volatility
More informationFrom Discrete Time to Continuous Time Modeling
From Discrete Time to Continuous Time Modeling Prof. S. Jaimungal, Department of Statistics, University of Toronto 2004 Arrow-Debreu Securities 2004 Prof. S. Jaimungal 2 Consider a simple one-period economy
More informationINDIAN INSTITUTE OF SCIENCE STOCHASTIC HYDROLOGY. Lecture -5 Course Instructor : Prof. P. P. MUJUMDAR Department of Civil Engg., IISc.
INDIAN INSTITUTE OF SCIENCE STOCHASTIC HYDROLOGY Lecture -5 Course Instructor : Prof. P. P. MUJUMDAR Department of Civil Engg., IISc. Summary of the previous lecture Moments of a distribubon Measures of
More informationPricing Variance Swaps on Time-Changed Lévy Processes
Pricing Variance Swaps on Time-Changed Lévy Processes ICBI Global Derivatives Volatility and Correlation Summit April 27, 2009 Peter Carr Bloomberg/ NYU Courant pcarr4@bloomberg.com Joint with Roger Lee
More informationConditional Density Method in the Computation of the Delta with Application to Power Market
Conditional Density Method in the Computation of the Delta with Application to Power Market Asma Khedher Centre of Mathematics for Applications Department of Mathematics University of Oslo A joint work
More informationHomework Assignments
Homework Assignments Week 1 (p. 57) #4.1, 4., 4.3 Week (pp 58 6) #4.5, 4.6, 4.8(a), 4.13, 4.0, 4.6(b), 4.8, 4.31, 4.34 Week 3 (pp 15 19) #1.9, 1.1, 1.13, 1.15, 1.18 (pp 9 31) #.,.6,.9 Week 4 (pp 36 37)
More informationarxiv: v2 [q-fin.pr] 23 Nov 2017
VALUATION OF EQUITY WARRANTS FOR UNCERTAIN FINANCIAL MARKET FOAD SHOKROLLAHI arxiv:17118356v2 [q-finpr] 23 Nov 217 Department of Mathematics and Statistics, University of Vaasa, PO Box 7, FIN-6511 Vaasa,
More informationOptimally Thresholded Realized Power Variations for Lévy Jump Diffusion Models
Optimally Thresholded Realized Power Variations for Lévy Jump Diffusion Models José E. Figueroa-López 1 1 Department of Statistics Purdue University University of Missouri-Kansas City Department of Mathematics
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 informationThe Lognormal Interest Rate Model and Eurodollar Futures
GLOBAL RESEARCH ANALYTICS The Lognormal Interest Rate Model and Eurodollar Futures CITICORP SECURITIES,INC. 399 Park Avenue New York, NY 143 Keith Weintraub Director, Analytics 1-559-97 Michael Hogan Ex
More informationChapter 15: Jump Processes and Incomplete Markets. 1 Jumps as One Explanation of Incomplete Markets
Chapter 5: Jump Processes and Incomplete Markets Jumps as One Explanation of Incomplete Markets It is easy to argue that Brownian motion paths cannot model actual stock price movements properly in reality,
More informationMASM006 UNIVERSITY OF EXETER SCHOOL OF ENGINEERING, COMPUTER SCIENCE AND MATHEMATICS MATHEMATICAL SCIENCES FINANCIAL MATHEMATICS.
MASM006 UNIVERSITY OF EXETER SCHOOL OF ENGINEERING, COMPUTER SCIENCE AND MATHEMATICS MATHEMATICAL SCIENCES FINANCIAL MATHEMATICS May/June 2006 Time allowed: 2 HOURS. Examiner: Dr N.P. Byott This is a CLOSED
More informationExtrapolation analytics for Dupire s local volatility
Extrapolation analytics for Dupire s local volatility Stefan Gerhold (joint work with P. Friz and S. De Marco) Vienna University of Technology, Austria 6ECM, July 2012 Implied vol and local vol Implied
More informationHedging Credit Derivatives in Intensity Based Models
Hedging Credit Derivatives in Intensity Based Models PETER CARR Head of Quantitative Financial Research, Bloomberg LP, New York Director of the Masters Program in Math Finance, Courant Institute, NYU Stanford
More informationHints on Some of the Exercises
Hints on Some of the Exercises of the book R. Seydel: Tools for Computational Finance. Springer, 00/004/006/009/01. Preparatory Remarks: Some of the hints suggest ideas that may simplify solving the exercises
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 informationNo-arbitrage theorem for multi-factor uncertain stock model with floating interest rate
Fuzzy Optim Decis Making 217 16:221 234 DOI 117/s17-16-9246-8 No-arbitrage theorem for multi-factor uncertain stock model with floating interest rate Xiaoyu Ji 1 Hua Ke 2 Published online: 17 May 216 Springer
More informationLocal vs Non-local Forward Equations for Option Pricing
Local vs Non-local Forward Equations for Option Pricing Rama Cont Yu Gu Abstract When the underlying asset is a continuous martingale, call option prices solve the Dupire equation, a forward parabolic
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 informationParametric Inference and Dynamic State Recovery from Option Panels. Torben G. Andersen
Parametric Inference and Dynamic State Recovery from Option Panels Torben G. Andersen Joint work with Nicola Fusari and Viktor Todorov The Third International Conference High-Frequency Data Analysis in
More informationSolving the Black-Scholes Equation
Solving the Black-Scholes Equation An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2014 Initial Value Problem for the European Call The main objective of this lesson is solving
More informationValuation of Volatility Derivatives. Jim Gatheral Global Derivatives & Risk Management 2005 Paris May 24, 2005
Valuation of Volatility Derivatives Jim Gatheral Global Derivatives & Risk Management 005 Paris May 4, 005 he opinions expressed in this presentation are those of the author alone, and do not necessarily
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 informationA Continuity Correction under Jump-Diffusion Models with Applications in Finance
A Continuity Correction under Jump-Diffusion Models with Applications in Finance Cheng-Der Fuh 1, Sheng-Feng Luo 2 and Ju-Fang Yen 3 1 Institute of Statistical Science, Academia Sinica, and Graduate Institute
More informationComputer Exercise 2 Simulation
Lund University with Lund Institute of Technology Valuation of Derivative Assets Centre for Mathematical Sciences, Mathematical Statistics Fall 2017 Computer Exercise 2 Simulation This lab deals with pricing
More informationProbability in Options Pricing
Probability in Options Pricing Mark Cohen and Luke Skon Kenyon College cohenmj@kenyon.edu December 14, 2012 Mark Cohen and Luke Skon (Kenyon college) Probability Presentation December 14, 2012 1 / 16 What
More informationESTIMATION OF UTILITY FUNCTIONS: MARKET VS. REPRESENTATIVE AGENT THEORY
ESTIMATION OF UTILITY FUNCTIONS: MARKET VS. REPRESENTATIVE AGENT THEORY Kai Detlefsen Wolfgang K. Härdle Rouslan A. Moro, Deutsches Institut für Wirtschaftsforschung (DIW) Center for Applied Statistics
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 informationSYSM 6304 Risk and Decision Analysis Lecture 2: Fitting Distributions to Data
SYSM 6304 Risk and Decision Analysis Lecture 2: Fitting Distributions to Data M. Vidyasagar Cecil & Ida Green Chair The University of Texas at Dallas Email: M.Vidyasagar@utdallas.edu September 5, 2015
More informationValue at Risk Ch.12. PAK Study Manual
Value at Risk Ch.12 Related Learning Objectives 3a) Apply and construct risk metrics to quantify major types of risk exposure such as market risk, credit risk, liquidity risk, regulatory risk etc., and
More informationThe Black-Scholes Model
IEOR E4706: Foundations of Financial Engineering c 2016 by Martin Haugh The Black-Scholes Model In these notes we will use Itô s Lemma and a replicating argument to derive the famous Black-Scholes formula
More informationLimit Theorems for the Empirical Distribution Function of Scaled Increments of Itô Semimartingales at high frequencies
Limit Theorems for the Empirical Distribution Function of Scaled Increments of Itô Semimartingales at high frequencies George Tauchen Duke University Viktor Todorov Northwestern University 2013 Motivation
More informationPricing and Risk Management with Stochastic Volatility. Using Importance Sampling
Pricing and Risk Management with Stochastic Volatility Using Importance Sampling Przemyslaw Stan Stilger, Simon Acomb and Ser-Huang Poon March 2, 214 Abstract In this paper, we apply importance sampling
More informationHeston vs Heston. Antoine Jacquier. Department of Mathematics, Imperial College London. ICASQF, Cartagena, Colombia, June 2016
Department of Mathematics, Imperial College London ICASQF, Cartagena, Colombia, June 2016 - Joint work with Fangwei Shi June 18, 2016 Implied volatility About models Calibration Implied volatility Asset
More informationHedging of swaptions in a Lévy driven Heath-Jarrow-Morton framework
Hedging of swaptions in a Lévy driven Heath-Jarrow-Morton framework Kathrin Glau, Nele Vandaele, Michèle Vanmaele Bachelier Finance Society World Congress 2010 June 22-26, 2010 Nele Vandaele Hedging of
More informationSingular Stochastic Control Models for Optimal Dynamic Withdrawal Policies in Variable Annuities
1/ 46 Singular Stochastic Control Models for Optimal Dynamic Withdrawal Policies in Variable Annuities Yue Kuen KWOK Department of Mathematics Hong Kong University of Science and Technology * Joint work
More informationA Consistent Pricing Model for Index Options and Volatility Derivatives
A Consistent Pricing Model for Index Options and Volatility Derivatives 6th World Congress of the Bachelier Society Thomas Kokholm Finance Research Group Department of Business Studies Aarhus School of
More informationEssays in Financial Engineering. Andrew Jooyong Ahn
Essays in Financial Engineering Andrew Jooyong Ahn Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Graduate School of Arts and Sciences COLUMBIA UNIVERSITY
More information