Numerical PDE methods for exotic options
|
|
- Timothy Hutchinson
- 7 years ago
- Views:
Transcription
1 Lecture 8 Numerical PDE methods for exotic options Lecture Notes by Andrzej Palczewski Computational Finance p. 1
2 Barrier options For barrier option part of the option contract is triggered if the asset price hits some barrier S = X, at some time prior to expiry. Depending on the way the hitting time is monitored we distinguish following options: continuous monitoring (called American barrier) hitting moment can be any time between issue and expiry of the option; discrete monitoring security price is monitored only in selected moments of time (say daily or weekly); discrete monitoring called European barrier security price is monitored only at expiry. Computational Finance p. 2
3 Barrier options cont. Depending on the conditions under which the option gets or losses value we have following type of barrier options: up-and-in the option expires worthless unless the barrier S = X is reached from below; down-and-in the option expires worthless unless the barrier S = X is reached from above; up-and-out the option expires worthless if the barrier S = X is reached from below; down-and-out the option expires worthless if the barrier S = X is reached from above. Computational Finance p. 3
4 Barrier options valuation The value of a barrier option can be computed using the Black-Scholes equation. For European options with barriers of European or American type there are explicit analytic formulas (although some of them are quite complicated). But American options or option with the barrier monitored discretely have to be valued numerically. To solve the Black-Scholes equation for barrier options we have to supplement the equation with proper boundary and terminal conditions. Computational Finance p. 4
5 Knock-out options Terminal conditions are as for vanila options modified by limitations coming from boundary conditions. Boundary conditions: Down-and-out option V(S,t) = 0, for S = X, V(S, t) = boundary value for vanila option, for S > X. Up-and-out option V(S, t) = boundary value for vanila option, for S < X, V(S,t) = 0, for S = X. Computational Finance p. 5
6 Double knock-out options A double knock-out option is a barrier option which expires worthless when the price of the underlying asset is to the left of the lower barrier X 1 or to the right of the upper barrier X 2. Terminal conditions are as for vanila options modified by limitations coming from boundary conditions. Boundary conditions: V(S,t) = 0, for S = X 1, V(S,t) = 0, for S = X 2. Computational Finance p. 6
7 Knock-in options Down-and-in option. For S [0,X] this is a plain vanila option. We show conditions for S > X. Terminal condition Boundary condition V(S,T) = 0 for S > X. V(S,t) 0 as S, V(X,t) = C(X,t), where C(S,t) is the price of corresponding vanila option. Computational Finance p. 7
8 Up-and-in option. For S [X, ) this is a plain vanila option. We show conditions for S < X. Terminal condition Boundary condition V(S,T) = 0 for S < X. V(S,t) 0 as S 0, V(X,t) = C(X,t), where C(S,t) is the price of corresponding vanila option. Computational Finance p. 8
9 Discretely monitored barrier A boundary constraint which holds at a point in time can be applied directly in an explicit manner. We compute the solution for a given time level and apply the constraint if necessary. Then move to the next time level. Consider a down-and-out option with barrier X monitored in time moments t α. At time level ν we compute V ν and then apply the boundary constraint V ν i = V(s i,t ν ) = where δ (tν,t α ) is the Kronecker delta. { 0, if s i δ (tν,t α )X, V ν i, otherwise Computational Finance p. 9
10 iscretely monitored barrier cont. Boundary conditions on outer boundaries For discretely monitored barriers we need boundary conditions for S 0 and/or S. To impose these boundary conditions correctly we can use the relation knock-in option + knock-out option = vanilla option (1) which holds as well for continuously and discretely monitored barriers when the monitoring moments are the same for in and out options. Consider again a down-and-out option with barrier monitored discretely. For the upper boundary we have V(S, t) = boundary value for vanila option, for S. Computational Finance p. 10
11 iscretely monitored barrier cont. To impose proper conditions on the lower boundary S = 0 let us observe that for the down-and-in option we have V(S, t) = boundary value for vanila option, for S 0. Using relation (1) we obtain for a down-and-out option V(S,t) = 0, for S 0. Boundary conditions for other options can be obtained in a similar manner. Computational Finance p. 11
12 Lookback options For the basic lookback contracts, the payoff comes in two varieties: the fixed strike and the floating strike. These options have payoffs that are the same as vanilla options except that in the floating strike option the vanilla exercise price is replaced by the maximum or minimum. In the fixed strike option it is the asset value in the vanilla option that is replaced by the maximum or minimum. M T S T, floating put, S T m T, floating call H(S T,J T ) = (M T K) +, fixed call, (K m T ) +, fixed put, where J T denotes M T or m T and M T = max 0 t T S t, m T = min 0 t T S t. Computational Finance p. 12
13 Lookback options cont. Depending on the way the maximum or minimum is monitored we distinguish: continuous monitoring security price is monitored continuously between the issue and expiry of the option; discrete monitoring security price is monitored only in selected moments of time (say weekly or monthly). The pair (S t,j t ) is a Markov process, and the price at time t of the lookback option is equal to V(t,S t,j t ) where the functionv is defined fort [0,T] and(s,j) in an appropriate set depending whether J = M or J = m { {(S,M) R 2 + : 0 S M}, for J = M, (S,J) {(S,m) R 2 + : 0 m S}, for J = m. Computational Finance p. 13
14 Lookback options valuation The value of a lookback option can be computed using an extended version of the Black-Scholes equation. We derive this equation assuming J = M. Let ( t( ) ) n 1/n. M n (t) = Sτ We have 0 lim M n(t) = max S τ = M t. n 0 τ t Next we consider the function V(S,M n,t), which can be think of as a price of an instrument depending on the variable M n. Computational Finance p. 14
15 Black-Scholes equation Using the Feynman-Kac theorem and the differential dm n (t) = 1 n S n t ( Mn (t) ) n 1 dt we obtain V(S,M n,t) t +rs V(S,M n,t) S + 1 n S n M n 1 n V(S,M n,t) M n σ2 S 2 2 V(S,M n,t) S 2 rv(s,m n,t) = 0. Computational Finance p. 15
16 Black-Scholes equation cont. Since S t max 0 τ t S τ = M t then for S < M lim n 1 n S n ( Mn ) n 1 = 0 and we obtain the Black-Scholes equation V(S,M,t) t +rs V(S,M,t) S σ2 S 2 2 V(S,M,t) S 2 rv(s,m,t) = 0. Observe that the above equation is the standard onedimensional Black-Scholes equation in which the variable M plays the role of a parameter. This parameter enters however into boundary and terminal conditions. Computational Finance p. 16
17 Black-Scholes equation cont. At expiry we have V(S,M,T) = H(S,M). We require a boundary condition on the line S = M. Consider the stochastic process S t close to current maximum. Since the probability that the current maximum is still the maximum at expiry is zero, the option price must be insensitive to small changes of M when S is close to M. Hence V(S,M,t) M = 0 for S = M. Computational Finance p. 17
18 Black-Scholes equation cont. Additional boundary conditions depend on the type of the option. For a floating put option we have V(0,M,t) = Me r(t t). For a floating call option we have boundary conditions as for a vanila option V(S,m,t) S for S very large. Observe that in all these cases we have to solve a 3- dimensional PDE. Computational Finance p. 18
19 Similarity reduction For floating strike lookback options the 3-dimensional Black- Scholes equation can be reduced to 2 dimensions. We perform all calculations for a floating put option. We introduce the new variable x = M S. When the payoff can be written as (this is the case of the floating put and call) H(S,M) = Mh(x), then we search for a solution of the Black-Scholes equation assuming V(S,M,t) = MW(x,t). The equation for W has the form W(x,t) t +rx W(x,t) x σ2 x 2 2 W(x,t) x 2 rw(x,t) = 0. Computational Finance p. 19
20 Similarity reduction cont. The terminal and boundary conditions are as follows: W(x,T) = max(1 x,0), W(0,t) = e r(t t), W x = W at x = 1. Computational Finance p. 20
21 Asian options For the basic Asian contracts, the payoff comes in two varieties: the average price and the average strike. These options have payoffs that are the same as vanilla options except that in the average strike option the vanilla exercise price is replaced by the average of asset prices. In the average price option it is the asset value in the vanilla option that is replaced by the average of asset prices. (Ŝ K) +, average price call, (K Ŝ) +, average price put, (S T Ŝ) +, average strike call, (Ŝ S T ) +, average strike put, where Ŝ denotes the average of asset prices. Computational Finance p. 21
22 Asian options cont. There are several ways the average of past values of S t can be formated: arithmetic average continuously monitored Ŝ = T 1 T 0 S tdt, arithmetic average discretely monitored Ŝ = n 1 n i=1 S t i, geometric ( average continuously monitored Ŝ = exp 1T ) T 0 log(s t)dt, geometric average discretely monitored Ŝ = ( n i=1 S t i ) 1/n, Computational Finance p. 22
23 Asian options cont. Since for geometric average Asian options we have analytic expressions for the price, we are only interested in calculation of arithmetic average Asian option. In what follows, we shall describe the PDE method for the calculation of an arithmetic average continuously monitored Asian option. This problem leads to a 3-dimensional Black- Scholes equation. For an arithmetic average continuously monitored Asian option this equation can be reduced to two dimensions and solved numerical by finite difference algorithms. For an arithmetic average discretely monitored Asian option we can use straightforwardly the known 2-dimensional Black-Scholes equation appropriately modifying initial conditions at the time of monitoring. Computational Finance p. 23
24 Asian options valuation The value of an arithmetic average continuously monitored Asian option can be computed using an extended version of the Black-Scholes equation. Let us consider a generalized arithmetic average A t = t 0 f(s τ,τ)dτ, where f(x,t) depends on the type of averaging (for a simple arithmetic average f(x,t) = x up to a constant factor). The pair (S t,a t ) is a Markov process, and the price at time t of an arithmetic average Asian option is equal to V(t,S t,a t ) where the function V is defined for t [0,T], S t > 0 and A t > 0. Computational Finance p. 24
25 Black-Scholes equation Using the Feynman-Kac theorem and the differential we obtain da t = f(s t,t)dt V(S,A,t) t +rs V(S,A,t) S +f(s,t) V(S,A,t) A σ2 S 2 2 V(S,A,t) S 2 rv(s,a,t) = 0. + Computational Finance p. 25
26 Similarity reduction We consider an arithmetic average strike call with payoff ( S T 1 ) ( + T A T = ST 1 1 TS T T 0 S τ dτ) +. This form of the payoff suggest introduction of an auxiliary variable R t = 1 t S τ dτ. S t Then the payoff is 0 S T (1 1 T R T) +, and we can look for the factorization V(S,A,t) = SH(R,t). Computational Finance p. 26
27 Similarity reduction cont. The equation for H has the form H(R, t) t +(1 rr) H(R,t) R σ2 R 2 2 H(R,t) R 2 = 0. The terminal condition follows from the payoff H(R T,T) = ( 1 1 T R T) +. Since S t is integrable, then, from the definition of R t, for R we have S 0. Remembering that for S 0 call option is not exercised, we obtain H(R,t) = 0 for R. Computational Finance p. 27
28 Similarity reduction cont. To obtain the boundary condition on the boundary R = 0 we return to the PDE which H fulfills. For R 0 this equation gives H(R, t) t + H(R,t) R = 0 for R 0. Finally we obtain the following boundary value problem H t +(1 rr) H R σ2 R 2 2 H R 2 = 0, H = 0 for R, H t + H = 0 for R 0, R ( H(R T,T) = T) T R Computational Finance p. 28
29 Numerical scheme We solve the 2-dimensional Black-Scholes PDE of the previous slides (barrier, lookback and options) without transformation of variables. This approach is very usefull because the boundary conditions can be easily implemented. The equation reads u(x, t) t +α(x) u(x,t) x σ2 x 2 2 u(x,t) x 2 ρu(x,t) = 0, with appropriate terminal and boundary conditions. Computational Finance p. 29
30 Finite difference approximation Letv i,ν be an approximation ofu(x i,t ν ) andη i approximation 2α(x of i ) x i+1 x i 1. Then v i,ν 1 v i,ν δt = θa i v i 1,ν 1 +θb i v i,ν 1 +θc i v i+1,ν 1 + +(1 θ)a i v i 1,ν +(1 θ)b i v i,ν +(1 θ)c i v i+1,ν, (2) where A i = 1 2 (σ2 i 2 η i ), B i = (σ 2 i 2 +ρ), C i = 1 2 (σ2 i 2 +η i ). The choice of θ gives the explicit (θ = 0), the implicit (θ = 1) and the Crank-Nicolson (θ = 1 2 ) scheme. Computational Finance p. 30
31 Spurious oscillations Numerical scheme (2), even if the grid is chosen properly to avoid instabilities, can be a subject of spurious oscillations. Computational Finance p. 31
32 Spurious oscillations cont. To understand the phenomenon of spurious oscillations let us consider a model equation u t +a u x = b 2 u x 2. The source of oscillations is the first order term a u x. To see this let us perform the von Neumann stability analysis. To this end we approximate the above equation by finite differences w j,ν+1 w jν t +a w j+1,ν w j 1,ν 2 x = b w j+1,ν 2w jν +w j 1,ν ( x) 2. Computational Finance p. 32
33 Spurious oscillations cont. Let us express the approximations w jν of the ν-th time level by a sum of Fourier modes w jν = k c (ν) k eikj x, The linearity of the numerical scheme allows to find a relation c (ν+1) k = G k c (ν) k, where G k is the growth factor. For G k 1 it is guaranteed that the modes e ik x are not amplified. Computational Finance p. 33
34 Spurious oscillations cont. For the model equation we arrive at and where G k = 1 2λ+2λcosk x iβλsink x G k 2 = (1 4λs 2 ) 2 +4β 2 λ 2 s 2 (1 s 2 ), λ = b t a x ( x) 2, β =, s = sin k x b 2. A straightforward analysis of the polynomial G k 2 reveals that G k 1 for 0 λ 1 2, and, in addition, for β > 2 we need λβ2 2. Computational Finance p. 34
35 Spurious oscillations cont. The inequality 0 λ 1/2 brings back the stability criterion of the heat equation. β is called a mesh Péclet number and corresponds to the ratio of the convection term a u x to the diffusion term u b 2 x in 2 the model equation. It is clear that controlling oscillations means that β must be small. Computational Finance p. 35
36 purious oscillations for BS equation Theorem In order to prevent the formation of spurious oscillations in the numerical scheme (2), the following conditions must be satisfied (x i x i 1 ) 2 x 2 i < σ2 η i and 1 (1 θ)δt > x 2 σ2 i (x i+1 x i )(x i x i 1 ) +ρ. Computational Finance p. 36
37 Upwind scheme The source of oscillations is the first order term a u x. Let us consider the model equation u t +a u x = 0. The solution of this equation isf(x at), wheref(y) = u 0 (y) is the initial condition for our equation u(x,0) = u 0 (x). For a > 0, we approximate the above equation by finite differences w j,ν+1 w jν t +a w j,ν w j 1,ν x This scheme is called upwind scheme. = 0 Computational Finance p. 37
38 Upwind scheme cont. To explain the name we return to the solution F(x at). As t increases the profile F(y) drifts in positive x-directions. We say that the wind blows to the right. In agreement with that blowing direction we approximate the derivative u x w j,ν w j 1,ν, x i.e. the information flows from downstream to upstream nodes. The stability analysis for this scheme gives G k = 1 γ +γe ik x, where γ = a t x. Then G k 1 for γ 1. Computational Finance p. 38
39 Upwind scheme cont. For a < 0, the upwind scheme is w j,ν+1 w jν t +a w j+1,ν w j,ν x = 0. The stability analysis for this scheme gives G k 1 for γ 1. Computational Finance p. 39
Numerical 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 informationLecture 6 Black-Scholes PDE
Lecture 6 Black-Scholes PDE Lecture Notes by Andrzej Palczewski Computational Finance p. 1 Pricing function Let the dynamics of underlining S t be given in the risk-neutral measure Q by If the contingent
More informationMore Exotic Options. 1 Barrier Options. 2 Compound Options. 3 Gap Options
More Exotic Options 1 Barrier Options 2 Compound Options 3 Gap Options More Exotic Options 1 Barrier Options 2 Compound Options 3 Gap Options Definition; Some types The payoff of a Barrier option is path
More informationPath-dependent options
Chapter 5 Path-dependent options The contracts we have seen so far are the most basic and important derivative products. In this chapter, we shall discuss some complex contracts, including barrier options,
More informationOption Pricing. Chapter 9 - Barrier Options - Stefan Ankirchner. University of Bonn. last update: 9th December 2013
Option Pricing Chapter 9 - Barrier Options - Stefan Ankirchner University of Bonn last update: 9th December 2013 Stefan Ankirchner Option Pricing 1 Standard barrier option Agenda What is a barrier option?
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 informationThe Black-Scholes pricing formulas
The Black-Scholes pricing formulas Moty Katzman September 19, 2014 The Black-Scholes differential equation Aim: Find a formula for the price of European options on stock. Lemma 6.1: Assume that a stock
More informationThe Evaluation of Barrier Option Prices Under Stochastic Volatility. BFS 2010 Hilton, Toronto June 24, 2010
The Evaluation of Barrier Option Prices Under Stochastic Volatility Carl Chiarella, Boda Kang and Gunter H. Meyer School of Finance and Economics University of Technology, Sydney School of Mathematics
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 Black-Scholes
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 informationPricing Barrier Option Using Finite Difference Method and MonteCarlo Simulation
Pricing Barrier Option Using Finite Difference Method and MonteCarlo Simulation Yoon W. Kwon CIMS 1, Math. Finance Suzanne A. Lewis CIMS, Math. Finance May 9, 000 1 Courant Institue of Mathematical Science,
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 informationDouble Barrier Cash or Nothing Options: a short note
Double Barrier Cash or Nothing Options: a short note Antonie Kotzé and Angelo Joseph May 2009 Financial Chaos Theory, Johannesburg, South Africa Mail: consultant@quantonline.co.za Abstract In this note
More informationTABLE OF CONTENTS. A. Put-Call Parity 1 B. Comparing Options with Respect to Style, Maturity, and Strike 13
TABLE OF CONTENTS 1. McDonald 9: "Parity and Other Option Relationships" A. Put-Call Parity 1 B. Comparing Options with Respect to Style, Maturity, and Strike 13 2. McDonald 10: "Binomial Option Pricing:
More informationHedging. An Undergraduate Introduction to Financial Mathematics. J. Robert Buchanan. J. Robert Buchanan Hedging
Hedging An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2010 Introduction Definition Hedging is the practice of making a portfolio of investments less sensitive to changes in
More informationPricing European Barrier Options with Partial Differential Equations
Pricing European Barrier Options with Partial Differential Equations Akinyemi David Supervised by: Dr. Alili Larbi Erasmus Mundus Masters in Complexity Science, Complex Systems Science, University of Warwick
More informationPricing Formulae for Foreign Exchange Options 1
Pricing Formulae for Foreign Exchange Options Andreas Weber and Uwe Wystup MathFinance AG Waldems, Germany www.mathfinance.com 22 December 2009 We would like to thank Peter Pong who pointed out an error
More informationBarrier Options. Peter Carr
Barrier Options Peter Carr Head of Quantitative Financial Research, Bloomberg LP, New York Director of the Masters Program in Math Finance, Courant Institute, NYU March 14th, 2008 What are Barrier Options?
More informationPricing Options with Discrete Dividends by High Order Finite Differences and Grid Stretching
Pricing Options with Discrete Dividends by High Order Finite Differences and Grid Stretching Kees Oosterlee Numerical analysis group, Delft University of Technology Joint work with Coen Leentvaar, Ariel
More informationAn Introduction to Exotic Options
An Introduction to Exotic Options Jeff Casey Jeff Casey is entering his final semester of undergraduate studies at Ball State University. He is majoring in Financial Mathematics and has been a math tutor
More informationNumerical Methods for Pricing Exotic Options
Imperial College London Department of Computing Numerical Methods for Pricing Exotic Options by Hardik Dave - 00517958 Supervised by Dr. Daniel Kuhn Second Marker: Professor Berç Rustem Submitted in partial
More information10.2 Series and Convergence
10.2 Series and Convergence Write sums using sigma notation Find the partial sums of series and determine convergence or divergence of infinite series Find the N th partial sums of geometric series and
More informationENGINEERING AND HEDGING OF CORRIDOR PRODUCTS - with focus on FX linked instruments -
AARHUS SCHOOL OF BUSINESS AARHUS UNIVERSITY MASTER THESIS ENGINEERING AND HEDGING OF CORRIDOR PRODUCTS - with focus on FX linked instruments - AUTHORS: DANIELA ZABRE GEORGE RARES RADU SIMIAN SUPERVISOR:
More informationOption Pricing. Chapter 12 - Local volatility models - Stefan Ankirchner. University of Bonn. last update: 13th January 2014
Option Pricing Chapter 12 - Local volatility models - Stefan Ankirchner University of Bonn last update: 13th January 2014 Stefan Ankirchner Option Pricing 1 Agenda The volatility surface Local volatility
More informationLecture 1: Stochastic Volatility and Local Volatility
Lecture 1: Stochastic Volatility and Local Volatility Jim Gatheral, Merrill Lynch Case Studies in Financial Modelling Course Notes, Courant Institute of Mathematical Sciences, Fall Term, 2002 Abstract
More informationThe market for exotic options
The market for exotic options Development of exotic products increased flexibility for risk transfer and hedging highly structured expression of expectation of asset price movements facilitation of trading
More informationPackage fexoticoptions
Version 2152.78 Revision 5392 Date 2012-11-07 Title Exotic Option Valuation Package fexoticoptions February 19, 2015 Author Diethelm Wuertz and many others, see the SOURCE file Depends R (>= 2.4.0), methods,
More informationValuation, Pricing of Options / Use of MATLAB
CS-5 Computational Tools and Methods in Finance Tom Coleman Valuation, Pricing of Options / Use of MATLAB 1.0 Put-Call Parity (review) Given a European option with no dividends, let t current time T exercise
More informationHedging Barriers. Liuren Wu. Zicklin School of Business, Baruch College (http://faculty.baruch.cuny.edu/lwu/)
Hedging Barriers Liuren Wu Zicklin School of Business, Baruch College (http://faculty.baruch.cuny.edu/lwu/) Based on joint work with Peter Carr (Bloomberg) Modeling and Hedging Using FX Options, March
More informationEuropean Call Option Pricing using the Adomian Decomposition Method
Advances in Dynamical Systems and Applications ISSN 0973-5321, Volume 9, Number 1, pp. 75 85 (2014) http://campus.mst.edu/adsa European Call Option Pricing using the Adomian Decomposition Method Martin
More informationBlack-Scholes Option Pricing Model
Black-Scholes Option Pricing Model Nathan Coelen June 6, 22 1 Introduction Finance is one of the most rapidly changing and fastest growing areas in the corporate business world. Because of this rapid change,
More informationThe Heston Model. Hui Gong, UCL http://www.homepages.ucl.ac.uk/ ucahgon/ May 6, 2014
Hui Gong, UCL http://www.homepages.ucl.ac.uk/ ucahgon/ May 6, 2014 Generalized SV models Vanilla Call Option via Heston Itô s lemma for variance process Euler-Maruyama scheme Implement in Excel&VBA 1.
More informationMathematical Finance
Mathematical Finance Option Pricing under the Risk-Neutral Measure Cory Barnes Department of Mathematics University of Washington June 11, 2013 Outline 1 Probability Background 2 Black Scholes for European
More informationPricing Parisian Options
Pricing Parisian Options Richard J. Haber Mathematical Institute, 24-29 St Giles, Oxford OX1 3LB Philipp J. Schönbucher University of Bonn, Statistical Department, Adenaueralle 24-42, 53113 Bonn, Germany
More informationSensitivity Analysis of Options. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 264
Sensitivity Analysis of Options c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 264 Cleopatra s nose, had it been shorter, the whole face of the world would have been changed. Blaise Pascal
More informationOption Pricing Methods for Estimating Capacity Shortages
Option Pricing Methods for Estimating Capacity Shortages Dohyun Pak and Sarah M. Ryan Department of Industrial and Manufacturing Systems Engineering Iowa State University Ames, IA 500-64, USA Abstract
More informationOPTION PRICING FOR WEIGHTED AVERAGE OF ASSET PRICES
OPTION PRICING FOR WEIGHTED AVERAGE OF ASSET PRICES Hiroshi Inoue 1, Masatoshi Miyake 2, Satoru Takahashi 1 1 School of Management, T okyo University of Science, Kuki-shi Saitama 346-8512, Japan 2 Department
More informationStock Price Dynamics, Dividends and Option Prices with Volatility Feedback
Stock Price Dynamics, Dividends and Option Prices with Volatility Feedback Juho Kanniainen Tampere University of Technology New Thinking in Finance 12 Feb. 2014, London Based on J. Kanniainen and R. Piche,
More informationS 1 S 2. Options and Other Derivatives
Options and Other Derivatives The One-Period Model The previous chapter introduced the following two methods: Replicate the option payoffs with known securities, and calculate the price of the replicating
More informationPricing of an Exotic Forward Contract
Pricing of an Exotic Forward Contract Jirô Akahori, Yuji Hishida and Maho Nishida Dept. of Mathematical Sciences, Ritsumeikan University 1-1-1 Nojihigashi, Kusatsu, Shiga 525-8577, Japan E-mail: {akahori,
More informationConvergence Remedies For Non-Smooth Payoffs in Option Pricing
Convergence Remedies For Non-Smooth 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 informationStatic Hedging and Model Risk for Barrier Options
Static Hedging and Model Risk for Barrier Options Morten Nalholm Rolf Poulsen Abstract We investigate how sensitive different dynamic and static hedge strategies for barrier options are to model risk.
More information1 The Black-Scholes model: extensions and hedging
1 The Black-Scholes model: extensions and hedging 1.1 Dividends Since we are now in a continuous time framework the dividend paid out at time t (or t ) is given by dd t = D t D t, where as before D denotes
More informationAdditional questions for chapter 4
Additional questions for chapter 4 1. A stock price is currently $ 1. Over the next two six-month periods it is expected to go up by 1% or go down by 1%. The risk-free interest rate is 8% per annum with
More informationDoes Black-Scholes framework for Option Pricing use Constant Volatilities and Interest Rates? New Solution for a New Problem
Does Black-Scholes 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 informationThe Black-Scholes-Merton Approach to Pricing Options
he Black-Scholes-Merton Approach to Pricing Options Paul J Atzberger Comments should be sent to: atzberg@mathucsbedu Introduction In this article we shall discuss the Black-Scholes-Merton approach to determining
More informationPricing Formula for 3-Period Discrete Barrier Options
Pricing Formula for 3-Period Discrete Barrier Options Chun-Yuan Chiu Down-and-Out Call Options We first give the pricing formula as an integral and then simplify the integral to obtain a formula similar
More informationTHE BLACK-SCHOLES MODEL AND EXTENSIONS
THE BLAC-SCHOLES MODEL AND EXTENSIONS EVAN TURNER Abstract. This paper will derive the Black-Scholes pricing model of a European option by calculating the expected value of the option. We will assume that
More informationBarrier Options. 0.1 Introduction
Barrier Options This note is several years old and very preliminary. It has no references to the literature. Do not trust its accuracy! Note that there is a lot of more recent literature, especially on
More informationPRICING DISCRETELY SAMPLED PATH-DEPENDENT EXOTIC OPTIONS USING REPLICATION METHODS
PRICING DISCRETELY SAMPLED PATH-DEPENDENT EXOTIC OPTIONS USING REPLICATION METHODS M. S. JOSHI Abstract. A semi-static replication method is introduced for pricing discretely sampled path-dependent options.
More informationα α λ α = = λ λ α ψ = = α α α λ λ ψ α = + β = > θ θ β > β β θ θ θ β θ β γ θ β = γ θ > β > γ θ β γ = θ β = θ β = θ β = β θ = β β θ = = = β β θ = + α α α α α = = λ λ λ λ λ λ λ = λ λ α α α α λ ψ + α =
More informationLecture 10. Finite difference and finite element methods. Option pricing Sensitivity analysis Numerical examples
Finite difference and finite element methods Lecture 10 Sensitivities and Greeks Key task in financial engineering: fast and accurate calculation of sensitivities of market models with respect to model
More informationHedging Options In The Incomplete Market With Stochastic Volatility. Rituparna Sen Sunday, Nov 15
Hedging Options In The Incomplete Market With Stochastic Volatility Rituparna Sen Sunday, Nov 15 1. Motivation This is a pure jump model and hence avoids the theoretical drawbacks of continuous path models.
More informationLecture 12: The Black-Scholes Model Steven Skiena. http://www.cs.sunysb.edu/ skiena
Lecture 12: The Black-Scholes Model Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena The Black-Scholes-Merton Model
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 informationMATH 425, PRACTICE FINAL EXAM SOLUTIONS.
MATH 45, PRACTICE FINAL EXAM SOLUTIONS. Exercise. a Is the operator L defined on smooth functions of x, y by L u := u xx + cosu linear? b Does the answer change if we replace the operator L by the operator
More informationOptions/1. Prof. Ian Giddy
Options/1 New York University Stern School of Business Options Prof. Ian Giddy New York University Options Puts and Calls Put-Call Parity Combinations and Trading Strategies Valuation Hedging Options2
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 informationNumerical Analysis Lecture Notes
Numerical Analysis Lecture Notes Peter J. Olver. Finite Difference Methods for Partial Differential Equations As you are well aware, most differential equations are much too complicated to be solved by
More informationA Semi-Lagrangian Approach for Natural Gas Storage Valuation and Optimal Operation
A Semi-Lagrangian 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 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 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 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 informationA new Feynman-Kac-formula for option pricing in Lévy models
A new Feynman-Kac-formula for option pricing in Lévy models Kathrin Glau Department of Mathematical Stochastics, Universtity of Freiburg (Joint work with E. Eberlein) 6th World Congress of the Bachelier
More informationA Uniform Asymptotic Estimate for Discounted Aggregate Claims with Subexponential Tails
12th International Congress on Insurance: Mathematics and Economics July 16-18, 2008 A Uniform Asymptotic Estimate for Discounted Aggregate Claims with Subexponential Tails XUEMIAO HAO (Based on a joint
More informationParabolic Equations. Chapter 5. Contents. 5.1.2 Well-Posed Initial-Boundary Value Problem. 5.1.3 Time Irreversibility of the Heat Equation
7 5.1 Definitions Properties Chapter 5 Parabolic Equations Note that we require the solution u(, t bounded in R n for all t. In particular we assume that the boundedness of the smooth function u at infinity
More information7. Barrier options, lookback options and Asian options
7. Barrier options, lookback options and Asian options Path dependent options: payouts are related to the underlying asset price path history during the whole or part of the life of the option. The barrier
More informationValuing double barrier options with time-dependent parameters by Fourier series expansion
IAENG International Journal of Applied Mathematics, 36:1, IJAM_36_1_1 Valuing double barrier options with time-dependent parameters by Fourier series ansion C.F. Lo Institute of Theoretical Physics and
More informationOPTION PRICING WITH PADÉ APPROXIMATIONS
C om m unfacsciu niva nkseries A 1 Volum e 61, N um b er, Pages 45 50 (01) ISSN 1303 5991 OPTION PRICING WITH PADÉ APPROXIMATIONS CANAN KÖROĞLU A In this paper, Padé approximations are applied Black-Scholes
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 information4. Option pricing models under the Black- Scholes framework
4. Option pricing models under the Black- Scholes framework Riskless hedging principle Writer of a call option hedges his exposure by holding certain units of the underlying asset in order to create a
More informationOpenGamma Quantitative Research Numerical Solutions to PDEs with Financial Applications
OpenGamma Quantitative Research Numerical Solutions to PDEs with Financial Applications Richard White Richard@opengamma.com OpenGamma Quantitative Research n. 10 First version: September 24, 2012; this
More informationSchonbucher Chapter 9: Firm Value and Share Priced-Based Models Updated 07-30-2007
Schonbucher Chapter 9: Firm alue and Share Priced-Based Models Updated 07-30-2007 (References sited are listed in the book s bibliography, except Miller 1988) For Intensity and spread-based models of default
More informationLecture 7: Finding Lyapunov Functions 1
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 7: Finding Lyapunov Functions 1
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 informationHow To Price A Call Option
Now by Itô s formula But Mu f and u g in Ū. Hence τ θ u(x) =E( Mu(X) ds + u(x(τ θ))) 0 τ θ u(x) E( f(x) ds + g(x(τ θ))) = J x (θ). 0 But since u(x) =J x (θ ), we consequently have u(x) =J x (θ ) = min
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 informationOption Pricing. Chapter 4 Including dividends in the BS model. Stefan Ankirchner. University of Bonn. last update: 6th November 2013
Option Pricing Chapter 4 Including dividends in the BS model Stefan Ankirchner University of Bonn last update: 6th November 2013 Stefan Ankirchner Option Pricing 1 Dividend payments So far: we assumed
More informationSTCE. Fast Delta-Estimates for American Options by Adjoint Algorithmic Differentiation
Fast Delta-Estimates for American Options by Adjoint Algorithmic Differentiation Jens Deussen Software and Tools for Computational Engineering RWTH Aachen University 18 th European Workshop on Automatic
More informationPrivate Equity Fund Valuation and Systematic Risk
An Equilibrium Approach and Empirical Evidence Axel Buchner 1, Christoph Kaserer 2, Niklas Wagner 3 Santa Clara University, March 3th 29 1 Munich University of Technology 2 Munich University of Technology
More informationComputational Finance Options
1 Options 1 1 Options Computational Finance Options An option gives the holder of the option the right, but not the obligation to do something. Conversely, if you sell an option, you may be obliged to
More informationMetric Spaces. Chapter 7. 7.1. Metrics
Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some
More informationStatistical Machine Learning
Statistical Machine Learning UoC Stats 37700, Winter quarter Lecture 4: classical linear and quadratic discriminants. 1 / 25 Linear separation For two classes in R d : simple idea: separate the classes
More informationGeneration Asset Valuation with Operational Constraints A Trinomial Tree Approach
Generation Asset Valuation with Operational Constraints A Trinomial Tree Approach Andrew L. Liu ICF International September 17, 2008 1 Outline Power Plants Optionality -- Intrinsic vs. Extrinsic Values
More informationAlternative Price Processes for Black-Scholes: Empirical Evidence and Theory
Alternative Price Processes for Black-Scholes: Empirical Evidence and Theory Samuel W. Malone April 19, 2002 This work is supported by NSF VIGRE grant number DMS-9983320. Page 1 of 44 1 Introduction This
More informationAsian Option Pricing Formula for Uncertain Financial Market
Sun and Chen Journal of Uncertainty Analysis and Applications (215) 3:11 DOI 1.1186/s4467-15-35-7 RESEARCH Open Access Asian Option Pricing Formula for Uncertain Financial Market Jiajun Sun 1 and Xiaowei
More informationMartingale Pricing Applied to Options, Forwards and Futures
IEOR E4706: Financial Engineering: Discrete-Time Asset Pricing Fall 2005 c 2005 by Martin Haugh Martingale Pricing Applied to Options, Forwards and Futures We now apply martingale pricing theory to the
More informationNew Pricing Formula for Arithmetic Asian Options. Using PDE Approach
Applied Mathematical Sciences, Vol. 5, 0, no. 77, 380-3809 New Pricing Formula for Arithmetic Asian Options Using PDE Approach Zieneb Ali Elshegmani, Rokiah Rozita Ahmad and 3 Roza Hazli Zakaria, School
More informationMarkovian projection for volatility calibration
cutting edge. calibration Markovian projection for volatility calibration Vladimir Piterbarg looks at the Markovian projection method, a way of obtaining closed-form approximations of European-style option
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 Discrete-Time Stochastic
More informationARBITRAGE-FREE OPTION PRICING MODELS. Denis Bell. University of North Florida
ARBITRAGE-FREE 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 informationOption pricing. Vinod Kothari
Option pricing Vinod Kothari Notation we use this Chapter will be as follows: S o : Price of the share at time 0 S T : Price of the share at time T T : time to maturity of the option r : risk free rate
More informationOn Black-Scholes Equation, Black- Scholes Formula and Binary Option Price
On Black-Scholes Equation, Black- Scholes Formula and Binary Option Price Abstract: Chi Gao 12/15/2013 I. Black-Scholes Equation is derived using two methods: (1) risk-neutral measure; (2) - hedge. II.
More informationCHAPTER 5 American Options
CHAPTER 5 American Options The distinctive feature of an American option is its early exercise privilege, that is, the holder can exercise the option prior to the date of expiration. Since the additional
More informationFast solver for the three-factor Heston-Hull/White problem. F.H.C. Naber Floris.Naber@INGbank.com tw1108735
Fast solver for the three-factor Heston-Hull/White problem F.H.C. Naber Floris.Naber@INGbank.com tw8735 Amsterdam march 27 Contents Introduction 2. Stochastic Models..........................................
More informationRisk/Arbitrage Strategies: An Application to Stock Option Portfolio Management
Risk/Arbitrage Strategies: An Application to Stock Option Portfolio Management Vincenzo Bochicchio, Niklaus Bühlmann, Stephane Junod and Hans-Fredo List Swiss Reinsurance Company Mythenquai 50/60, CH-8022
More informationSOLVING PARTIAL DIFFERENTIAL EQUATIONS RELATED TO OPTION PRICING WITH NUMERICAL METHOD. KENNEDY HAYFORD, (B.Sc. Mathematics)
SOLVING PARTIAL DIFFERENTIAL EQUATIONS RELATED TO OPTION PRICING WITH NUMERICAL METHOD BY KENNEDY HAYFORD, (B.Sc. Mathematics) A Thesis submitted to the Department of Mathematics, Kwame Nkrumah University
More informationAn Analysis of Onion Options and Double-no-Touch Digitals *
wilm6.qxd 7/6/ 3:46 PM Page 68 An Analysis of Onion Options and Double-no-Touch Digitals * tefan Ebenfeld, Matthias R. Mayr and Jürgen Topper d-fine GmbH, Mergenthalerallee 55, 6576 Eschborn/Frankfurt,
More informationSTOCK LOANS. XUN YU ZHOU Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong 1.
Mathematical Finance, Vol. 17, No. 2 April 2007), 307 317 STOCK LOANS JIANMING XIA Center for Financial Engineering and Risk Management, Academy of Mathematics and Systems Science, Chinese Academy of Sciences
More information