Financial Risk Management
|
|
- Virgil Ellis
- 5 years ago
- Views:
Transcription
1 Risk-neutrality in derivatives pricing University of Oulu - Department of Finance Spring 2018
2 Portfolio of two assets Value at time t = 0 Expected return Value at time t = 1 Asset A Asset B % 10% } E [ P 1 = Let us consider a portfolio of two assets with their current values of A 0 = e10.00 and B 0 = e We assume the risk-free rate of 3% and the market risk premium of 5%, both in terms of a simple annual rate. The beta of the asset A is β A = 0.80, and thus the expected return of the asset A is ER A ) = 3% + 5% 0.80 = 7%. Correspondingly, the expected value of the asset A at the end of a one-year period is EA 1 ) = 1 + 7%) = The beta of the asset B is β B = 1.40, and thus the expected return of the asset B is ER B ) = 3% + 5% 1.40 = 10%. Correspondingly, the expected value of the asset B at the end of a one-year period is EB 1 ) = %) = The expected value of the portfolio is E [ P 1 = = e43.70.
3 Portfolio of two assets Value at time t = 0 Expected return Value contribution at time t = 1 Asset A Asset B % 9.25% } E [ P 1 = Let us apply an alternative approach to evaluate the same portfolio. The beta of the portfolio of the two assets is β P = w A β A + w B β B = = Correspondingly, the expected return of the portfolio of the two assets is ER P ) = 3% + 5% 1.25 = 9.25%. We are able to calculate the portfolio-specific value contributions of the two assets as follows: ÊA 1 ) = %) = ÊA 2 ) = %) = The value contributions sum to the expected value of portfolio: E [ P 1 = ÊA1 ) + ÊB 1 ) = = e43.70.
4 Portfolio of two assets Asset A Asset B Value at time t = 0 Expected return Value contribution at time t = 1 } % E [ P 1 = Assume that we know the current price A 0 of the asset A, but do not know the current price B 0 of the asset B. Assume further that we know the expected return of the portfolio. Furthermore, somehow we know the expected end-of-period value E [ P 1 of the portfolio. E [ P 1 = The expected return of the portfolio is 9.25% and thus the contribution of the asset A to the value of the portfolio is ÊA 1 ) = %) = The contribution of the asset B to the value of the portfolio depends on Ê[A 1, and is Ê [ P 1 A 1 = P1 Ê[ A 1 = = The current value of the asset B is obtained by discounting the contribution with the expected return of the portfolio: B 0 = Ê[B % = % = e30.00.
5 Portfolio of a forward contract and the underlying asset Underlying asset Shorted forward contract Value at time t = 0 Expected return Value contribution at time t = T } S 0 r Ê [ S T = S0 e rt P T = K Ê [ e rt Ê [ K S T K S T Let us follow the same approach in a case of a portfolio of an asset and a shorted forward contract. The current price of the underlying asset is S 0. The end-of-period value P T of the portfolio is known to be P T = S T + K S T ) = K. The portfolio is risk-free and thus the contribution of the underlying asset to the value of the portfolio is Ê [ S T = S0 e rt. The contribution of the shorted forward contract to the value of the portfolio depends on Ê[S T, and is Ê[K S T, where Ê[S T = S 0 e rt. The current value of the contract is obtained by discounting the contribution with the expected return of the portfolio: f0 = e rt Ê [ K S T.
6 Portfolio of options and the underlying asset Value at time t = 0 Expected return Value contribution at time t = T Underlying asset Put option Shorted call option S 0 e rt Ê [ max0, X S T ) e rt Ê [ max0, S T X ) r Ê [ S T = S0 e rt Ê [ max0, X S T ) Ê [ max0, S T X ) } P T = X Let us follow the same approach in a case of a portfolio of an asset, a put option and a shorted call option. The end-of-period value P T of the portfolio is known to be P T = S T + max0, X S T ) max0, S T X ) = X. The portfolio is risk-free and thus the contribution of the underlying asset to the value of the portfolio is Ê [ S T = S0 e rt. The contributions of the options to the value of the portfolio depend on Ê[S T, and are Ê [ max0, X S T ) and Ê [ max0, S T X ), where Ê [ S T = S0 e rt. The current values of the options is obtained by discounting the contributions with the expected return of the portfolio: p 0 = e rt Ê [ max0, X S T ) and c 0 = e rt Ê [ max0, S T X ), where Ê [ S T = S0 e rt.
7 Risk-neutrality in derivatives pricing When pricing any derivative security we 1. Determine the risk-neutral expected payoff Ê[. of the contract, 2. Calculate the current value of the security by discounting the risk-neutral expected payoff with the risk-free rate. Risk-neutral does not mean risk-free, but it means that we are able to replace the expected return µ of the underlying asset with the risk-free rate r. The risk, or volatility σ, of the underlying asset still remains and has to be considered in the determination of the expected payoff), but no reward is paid on it or any risk), because it can be eliminated by an appropriate diversification. In the case of a forward contract the risk-neutral expected payoff is Ê [ S T K = Ê[ S T K = S0 e rt K. Correspondingly, the current value of the risk-neutral expected payoff is f 0 = e rt Ê [ S T K = e rt [ S 0 e rt K = S 0 Ke rt. In this case, the risk-neutrality appears explicitly in the risk-neutral expected price S 0 e rt of the underlying asset, where the asset-specific expected return µ is replaced with the risk-free rate r. The volatility σ of the asset does not play any role in the pricing of a forward contract, but is not ignored by any means.
8 Risk-neutrality in derivatives pricing When pricing any derivative security we 1. Determine the risk-neutral expected payoff Ê[. of the contract, 2. Calculate the current value of the security by discounting the risk-neutral expected payoff with the risk-free rate. Risk-neutral does not mean risk-free, but it means that we are able to replace the expected return µ of the underlying asset with the risk-free rate r. The risk, or volatility σ, of the underlying asset still remains and has to be considered in the determination of the expected payoff), but no reward is paid on it or any risk), because it can be eliminated by an appropriate diversification. In the case of an option the risk-neutral expected payoffs are c T = Ê[ max0, S T X ) max 0, Ê[ ) p T = Ê[ max0, X S T ) S T X, max 0, X Ê [ ) S T. In this case the solution is not as straightforward as in the case of a forward contract, and we are to make considerably much more effort to determine an explicit solution for the risk-neutral expected payoff. The payoff will be evaluated under an assumption that the underlying asset earns the risk-free rate r instead of the expected return µ of its own, and in this case the volatility σ of the underlying asset plays an important role. Correspondingly, the current values of the risk-neutral expected payoffs are c 0 = e rt Ê [ max0, S T X ), p 0 = e rt Ê [ max0, X S T ).
9 Risk-neutrality in derivatives pricing When pricing derivatives portfolio of a derivative and the underlying asset is considered, underlying asset is evaluated and priced as a part of the created portfolio, derivative is evaluated and priced as a part of the created portfolio. The underlying asset price expectation is based on the portfolio s expected return, portfolio is risk-free, and the expected return of it equals to the risk-free rate, analysis results to a risk-neutral price expectation of the asset price. The derivative derivative is priced on the basis of the underlying asset s risk-neutral price expectation, derivative s risk-neutral price expectation is discounted with the risk-free rate. the resulted derivatives price is a fair price in terms of reward-to-risk.
10 Itô s Lemma Risk-neutrality in derivatives pricing If a random variable x follows an Itô-process dx = ax, t)dt + bx, t)dz, then, according to Itô s lemma, any random variable y, the value of which is determined by the random variable x and time t, follows the Itô-process dy = y x a + y t ) y x 2 b2 dt + y x bdz. If asset price S follow the process ds = µsdt + σsdz, then, any derivative price y, the value of which is determined by the underlying asset price S and time t, follows y y dy = µs + S t ) y 2 S 2 σ2 S 2 dt + S σsdz.
11 Portfolio of an underlying asset and a derivative security Portfolio of an underlying asset and a derivative security n s shares of an underlying asset, n f contracts of a derivative security on the asset. The underlying asset: n s = S ds = µsdt + σsdz The derivative security: n f = 1 df = µs + S t ) f 2 S 2 σ2 S 2 dt + S σsdz. The portfolio: P = n s S + n f f [ dp = n s [µsdt + σsdz + n f µs + S t ) f S 2 σ2 S 2 dt + S σsdz
12 Portfolio of an underlying asset and a derivative security The number n s of shares needed varies over time. The number n f of derivatives remains constant over time. The stochastic terms dz behave identically in the both components. dp = [ [ µsdt + σsdz 1 µs + S S t ) f 2 S 2 σ2 S 2 dt + S σsdz = µsdt + σsdz µsdt S S S t dt 1 2 = t ) f 2 S 2 σ2 S 2 dt 2 f S 2 σ2 S 2 dt S σsdz The resulting price process does not contain dz, and is thus purely deterministic in nature. The result is based on a process of the form ds = µsdt + σsdz, but is independent of the level µ of the expected return of the asset. A practical application of the above is the dynamic delta-hedging of a sold derivative position.
13 Black-Scholes differential equation The risk-free portfolio of the underlying asset and a derivative is allowed to earn the risk-free rate r only! dp = t ) f 2 S 2 σ2 S 2 dt = rpdt The risk-free rate r is earned on the portfolio value P over a period of dt. t ) f 2 S 2 σ2 S 2 ) dt = r n s S + n f f dt t ) f 2 S 2 σ2 S 2 dt = n s rsdt + n f rfdt }{{} t ) f 2 S 2 σ2 S 2 dt = rsdt rfdt S The original price process of the underlying asset must be replaced with the risk-neutral price process ds = rsdt + σsdz. t f 2 S 2 σ2 S 2 + rs rf = 0 S
14 Derivatives pricing under risk-neutrality Forward contract: Price process of the underlying asset: Price process of the underlying asset under risk-neutrality: ds = µsdt + σsdz ds = rsdt + σsdz Expected payoff under risk-neutrality: ÊS T K) = ÊS T ) K = S t e rt t) K Current value of the contract: f t = e rt t) rt t) ÊS T K) = S t Ke Call option: Price process of the underlying asset: Price process of the underlying asset under risk-neutrality: ds = µsdt + σsdz ds = rsdt + σsdz Expected payoff under risk-neutrality: Ê[max0, S T X ) =? Current value of the contract: c t = e rt t) Ê[max0, S T X ) =? Put option: Price process of the underlying asset: Price process of the underlying asset under risk-neutrality: ds = µsdt + σsdz ds = rsdt + σsdz Expected payoff under risk-neutrality: Ê[max0, X S T ) =? Current value of the contract: p t = e rt t) Ê[max0, X S T ) =?
15 Forward contract and Black-Scholes differential equation Forward contract: Pricing model: rt t) f = S Ke Partial derivatives: S = 1 2 f S 2 = 0 t = rke rt t) Black-Scholes differential equation: t f 2 S 2 σ2 S 2 + rs rf S = rke rt t) σ2 S 2 rt t)) + rs r S Ke = rke rt t) + rs rs + rke rt t) = 0
16 An exotic forward contract An exotic forward contract pays off the difference of the squared asset price ST 2 and the contracted delivery price K at the maturity T of the contract. Price processes and expected values: µt t) ds = µsdt + σsdz ES T ) = S t e ds 2 ) = 2µ + σ 2 )S 2 dt + 2σS 2 dz ES 2 T ) = S2 t e2µ+σ2 )T t) Price processes and expected values under risk-neutrality: rt t) ds = rsdt + σsdz ÊS T ) = S t e ds 2 ) = 2r + σ 2 )S 2 dt + 2σS 2 dz ÊS 2 T ) = S2 t e2r+σ2 )T t) Expected payoff under risk-neutrality: ÊS 2 T K) = ÊS2 T ) K = S2 t e2r+σ2 )T t) K Current value of the contract: f t = e rt t) ÊST 2 K) = e rt t)[ St 2 e2r+σ2 )T t) K = St 2 er+σ2 )T t) Ke rt t)
17 An exotic forward contract and BS differential equation An exotic forward contract: Pricing model: f = S 2 e r+σ2 )T t) Ke rt t) Partial derivatives: S = 2 )T t) 2 f 2Ser+σ S 2 = 2 )T t) 2er+σ t = r + σ2 )S 2 e r+σ 2 )T t) rt t) rke Black-Scholes differential equation: t f 2 S 2 σ2 S 2 + rs rf = 0 S = r + σ 2 )S 2 e r+σ2 )T t) rt t) 1 rke )T t)σ 2er+σ 2 2 r+σ 2 )T t) S + 2Se rs r S 2 e r+σ2 )T t) rt t) ) Ke = 0
Black-Scholes-Merton Model
Black-Scholes-Merton Model Weerachart Kilenthong University of the Thai Chamber of Commerce c Kilenthong 2017 Weerachart Kilenthong University of the Thai Chamber Black-Scholes-Merton of Commerce Model
More informationFinancial Risk Management
Extensions of the Black-Scholes model University of Oulu - Department of Finance Spring 018 Dividend payment Extensions of the Black-Scholes model ds = rsdt + σsdz ÊS = S 0 e r S 0 he risk-neutral price
More informationWITH SKETCH ANSWERS. Postgraduate Certificate in Finance Postgraduate Certificate in Economics and Finance
WITH SKETCH ANSWERS BIRKBECK COLLEGE (University of London) BIRKBECK COLLEGE (University of London) Postgraduate Certificate in Finance Postgraduate Certificate in Economics and Finance SCHOOL OF ECONOMICS,
More informationLecture 3. Sergei Fedotov Introduction to Financial Mathematics. Sergei Fedotov (University of Manchester) / 6
Lecture 3 Sergei Fedotov 091 - Introduction to Financial Mathematics Sergei Fedotov (University of Manchester) 091 010 1 / 6 Lecture 3 1 Distribution for lns(t) Solution to Stochastic Differential Equation
More information4. Black-Scholes Models and PDEs. Math6911 S08, HM Zhu
4. Black-Scholes Models and PDEs Math6911 S08, HM Zhu References 1. Chapter 13, J. Hull. Section.6, P. Brandimarte Outline Derivation of Black-Scholes equation Black-Scholes models for options Implied
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 informationAspects of Financial Mathematics:
Aspects of Financial Mathematics: Options, Derivatives, Arbitrage, and the Black-Scholes Pricing Formula J. Robert Buchanan Millersville University of Pennsylvania email: Bob.Buchanan@millersville.edu
More informationFE610 Stochastic Calculus for Financial Engineers. Stevens Institute of Technology
FE610 Stochastic Calculus for Financial Engineers Lecture 13. The Black-Scholes PDE Steve Yang Stevens Institute of Technology 04/25/2013 Outline 1 The Black-Scholes PDE 2 PDEs in Asset Pricing 3 Exotic
More informationPricing theory of financial derivatives
Pricing theory of financial derivatives One-period securities model S denotes the price process {S(t) : t = 0, 1}, where S(t) = (S 1 (t) S 2 (t) S M (t)). Here, M is the number of securities. At t = 1,
More informationThe Black-Scholes Equation
The Black-Scholes Equation MATH 472 Financial Mathematics J. Robert Buchanan 2018 Objectives In this lesson we will: derive the Black-Scholes partial differential equation using Itô s Lemma and no-arbitrage
More informationOption Pricing Models for European Options
Chapter 2 Option Pricing Models for European Options 2.1 Continuous-time Model: Black-Scholes Model 2.1.1 Black-Scholes Assumptions We list the assumptions that we make for most of this notes. 1. The underlying
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 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 informationLecture 11: Ito Calculus. Tuesday, October 23, 12
Lecture 11: Ito Calculus Continuous time models We start with the model from Chapter 3 log S j log S j 1 = µ t + p tz j Sum it over j: log S N log S 0 = NX µ t + NX p tzj j=1 j=1 Can we take the limit
More informationContinuous time; continuous variable stochastic process. We assume that stock prices follow Markov processes. That is, the future movements in a
Continuous time; continuous variable stochastic process. We assume that stock prices follow Markov processes. That is, the future movements in a variable depend only on the present, and not the history
More informationIntroduction to Financial Derivatives
55.444 Introduction to Financial Derivatives Weeks of November 18 & 5 th, 13 he Black-Scholes-Merton Model for Options plus Applications 11.1 Where we are Last Week: Modeling the Stochastic Process for
More informationDerivation and Comparative Statics of the Black-Scholes Call and Put Option Pricing Formulas
Derivation and Comparative Statics of the Black-Scholes Call and Put Option Pricing Formulas James R. Garven Latest Revision: February 27, 2012 Abstract This paper provides an alternative derivation of
More information2.3 Mathematical Finance: Option pricing
CHAPTR 2. CONTINUUM MODL 8 2.3 Mathematical Finance: Option pricing Options are some of the commonest examples of derivative securities (also termed financial derivatives or simply derivatives). A uropean
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 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 informationIntroduction to Financial Derivatives
55.444 Introduction to Financial Derivatives Weeks of November 19 & 6 th, 1 he Black-Scholes-Merton Model for Options plus Applications Where we are Previously: Modeling the Stochastic Process for Derivative
More informationReading: You should read Hull chapter 12 and perhaps the very first part of chapter 13.
FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 Asset Price Dynamics Introduction These notes give assumptions of asset price returns that are derived from the efficient markets hypothesis. Although a hypothesis,
More informationOptions. An Undergraduate Introduction to Financial Mathematics. J. Robert Buchanan. J. Robert Buchanan Options
Options An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2014 Definitions and Terminology Definition An option is the right, but not the obligation, to buy or sell a security such
More information1 Implied Volatility from Local Volatility
Abstract We try to understand the Berestycki, Busca, and Florent () (BBF) result in the context of the work presented in Lectures and. Implied Volatility from Local Volatility. Current Plan as of March
More informationA Moment Matching Approach To The Valuation Of A Volume Weighted Average Price Option
A Moment Matching Approach To The Valuation Of A Volume Weighted Average Price Option Antony Stace Department of Mathematics and MASCOS University of Queensland 15th October 2004 AUSTRALIAN RESEARCH COUNCIL
More informationFinancial Derivatives Section 5
Financial Derivatives Section 5 The Black and Scholes Model Michail Anthropelos anthropel@unipi.gr http://web.xrh.unipi.gr/faculty/anthropelos/ University of Piraeus Spring 2018 M. Anthropelos (Un. of
More informationAMH4 - 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 informationBluff Your Way Through Black-Scholes
Bluff our Way Through Black-Scholes Saurav Sen December 000 Contents What is Black-Scholes?.............................. 1 The Classical Black-Scholes Model....................... 1 Some Useful Background
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 informationDerivation and Comparative Statics of the Black-Scholes Call and Put Option Pricing Formulas
Derivation and Comparative Statics of the Black-Scholes Call and Put Option Pricing Formulas James R. Garven Current Version: November 15, 2017 Abstract This paper provides an alternative derivation of
More informationMAS452/MAS6052. MAS452/MAS Turn Over SCHOOL OF MATHEMATICS AND STATISTICS. Stochastic Processes and Financial Mathematics
t r t r2 r t SCHOOL OF MATHEMATICS AND STATISTICS Stochastic Processes and Financial Mathematics Spring Semester 2017 2018 3 hours t s s tt t q st s 1 r s r t r s rts t q st s r t r r t Please leave this
More information7.1 Volatility Simile and Defects in the Black-Scholes Model
Chapter 7 Beyond Black-Scholes Model 7.1 Volatility Simile and Defects in the Black-Scholes Model Before pointing out some of the flaws in the assumptions of the Black-Scholes world, we must emphasize
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 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 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 informationBasics of Asset Pricing Theory {Derivatives pricing - Martingales and pricing kernels
Basics of Asset Pricing Theory {Derivatives pricing - Martingales and pricing kernels Yashar University of Illinois July 1, 2012 Motivation In pricing contingent claims, it is common not to have a simple
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 informationPricing Financial Derivatives Using Stochastic Calculus. A Thesis Presented to The Honors Tutorial College, Ohio University
Pricing Financial Derivatives Using Stochastic Calculus A Thesis Presented to The Honors Tutorial College, Ohio University In Partial Fulfillment of the Requirements for Graduation from the Honors Tutorial
More informationStochastic Differential Equations in Finance and Monte Carlo Simulations
Stochastic Differential Equations in Finance and Department of Statistics and Modelling Science University of Strathclyde Glasgow, G1 1XH China 2009 Outline Stochastic Modelling in Asset Prices 1 Stochastic
More informationSOA Exam MFE Solutions: May 2007
Exam MFE May 007 SOA Exam MFE Solutions: May 007 Solution 1 B Chapter 1, Put-Call Parity Let each dividend amount be D. The first dividend occurs at the end of months, and the second dividend occurs at
More informationSimulation Analysis of Option Buying
Mat-.108 Sovelletun Matematiikan erikoistyöt Simulation Analysis of Option Buying Max Mether 45748T 04.0.04 Table Of Contents 1 INTRODUCTION... 3 STOCK AND OPTION PRICING THEORY... 4.1 RANDOM WALKS AND
More informationThe Black-Scholes PDE from Scratch
The Black-Scholes PDE from Scratch chris bemis November 27, 2006 0-0 Goal: Derive the Black-Scholes PDE To do this, we will need to: Come up with some dynamics for the stock returns Discuss Brownian motion
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 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 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 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 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 informationPractical Hedging: From Theory to Practice. OSU Financial Mathematics Seminar May 5, 2008
Practical Hedging: From Theory to Practice OSU Financial Mathematics Seminar May 5, 008 Background Dynamic replication is a risk management technique used to mitigate market risk We hope to spend a certain
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 informationBlack-Scholes Option Pricing
Black-Scholes Option Pricing The pricing kernel furnishes an alternate derivation of the Black-Scholes formula for the price of a call option. Arbitrage is again the foundation for the theory. 1 Risk-Free
More informationStochastic Differential equations as applied to pricing of options
Stochastic Differential equations as applied to pricing of options By Yasin LUT Supevisor:Prof. Tuomo Kauranne December 2010 Introduction Pricing an European call option Conclusion INTRODUCTION A stochastic
More informationOption Pricing in Continuous-Time: The Black Scholes Merton Theory and Its Extensions
Chapter 2 Option Pricing in Continuous-Time: The Black Scholes Merton Theory and Its Extensions This chapter is organized as follows: 1. Section 2 provides an overview of the option pricing theory in the
More informationlast problem outlines how the Black Scholes PDE (and its derivation) may be modified to account for the payment of stock dividends.
224 10 Arbitrage and SDEs last problem outlines how the Black Scholes PDE (and its derivation) may be modified to account for the payment of stock dividends. 10.1 (Calculation of Delta First and Finest
More informationEconomics has never been a science - and it is even less now than a few years ago. Paul Samuelson. Funeral by funeral, theory advances Paul Samuelson
Economics has never been a science - and it is even less now than a few years ago. Paul Samuelson Funeral by funeral, theory advances Paul Samuelson Economics is extremely useful as a form of employment
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 information3.1 Itô s Lemma for Continuous Stochastic Variables
Lecture 3 Log Normal Distribution 3.1 Itô s Lemma for Continuous Stochastic Variables Mathematical Finance is about pricing (or valuing) financial contracts, and in particular those contracts which depend
More informationDerivative Securities
Derivative Securities he Black-Scholes formula and its applications. his Section deduces the Black- Scholes formula for a European call or put, as a consequence of risk-neutral valuation in the continuous
More informationDerivative Securities Fall 2012 Final Exam Guidance Extended version includes full semester
Derivative Securities Fall 2012 Final Exam Guidance Extended version includes full semester Our exam is Wednesday, December 19, at the normal class place and time. You may bring two sheets of notes (8.5
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 informationFinancial Economics & Insurance
Financial Economics & Insurance Albert Cohen Actuarial Sciences Program Department of Mathematics Department of Statistics and Probability A336 Wells Hall Michigan State University East Lansing MI 48823
More informationExtensions to the Black Scholes Model
Lecture 16 Extensions to the Black Scholes Model 16.1 Dividends Dividend is a sum of money paid regularly (typically annually) by a company to its shareholders out of its profits (or reserves). In this
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 informationGeometric Brownian Motion
Geometric Brownian Motion Note that as a model for the rate of return, ds(t)/s(t) geometric Brownian motion is similar to other common statistical models: ds(t) S(t) = µdt + σdw(t) or response = systematic
More informationMSc Financial Engineering CHRISTMAS ASSIGNMENT: MERTON S JUMP-DIFFUSION MODEL. To be handed in by monday January 28, 2013
MSc Financial Engineering 2012-13 CHRISTMAS ASSIGNMENT: MERTON S JUMP-DIFFUSION MODEL To be handed in by monday January 28, 2013 Department EMS, Birkbeck Introduction The assignment consists of Reading
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 informationMSC FINANCIAL ENGINEERING PRICING I, AUTUMN LECTURE 6: EXTENSIONS OF BLACK AND SCHOLES RAYMOND BRUMMELHUIS DEPARTMENT EMS BIRKBECK
MSC FINANCIAL ENGINEERING PRICING I, AUTUMN 2010-2011 LECTURE 6: EXTENSIONS OF BLACK AND SCHOLES RAYMOND BRUMMELHUIS DEPARTMENT EMS BIRKBECK In this section we look at some easy extensions of the Black
More informationChange of Measure (Cameron-Martin-Girsanov Theorem)
Change of Measure Cameron-Martin-Girsanov Theorem Radon-Nikodym derivative: Taking again our intuition from the discrete world, we know that, in the context of option pricing, we need to price the claim
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 19 11/20/2013. Applications of Ito calculus to finance
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.7J Fall 213 Lecture 19 11/2/213 Applications of Ito calculus to finance Content. 1. Trading strategies 2. Black-Scholes option pricing formula 1 Security
More information1 The continuous time limit
Derivative Securities, Courant Institute, Fall 2008 http://www.math.nyu.edu/faculty/goodman/teaching/derivsec08/index.html Jonathan Goodman and Keith Lewis Supplementary notes and comments, Section 3 1
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 informationLecture 15: Exotic Options: Barriers
Lecture 15: Exotic Options: Barriers Dr. Hanqing Jin Mathematical Institute University of Oxford Lecture 15: Exotic Options: Barriers p. 1/10 Barrier features For any options with payoff ξ at exercise
More informationA new Loan Stock Financial Instrument
A new Loan Stock Financial Instrument Alexander Morozovsky 1,2 Bridge, 57/58 Floors, 2 World Trade Center, New York, NY 10048 E-mail: alex@nyc.bridge.com Phone: (212) 390-6126 Fax: (212) 390-6498 Rajan
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 informationMerton s Jump Diffusion Model
Merton s Jump Diffusion Model Peter Carr (based on lecture notes by Robert Kohn) Bloomberg LP and Courant Institute, NYU Continuous Time Finance Lecture 5 Wednesday, February 16th, 2005 Introduction Merton
More informationLecture 3: Review of mathematical finance and derivative pricing models
Lecture 3: Review of mathematical finance and derivative pricing models Xiaoguang Wang STAT 598W January 21th, 2014 (STAT 598W) Lecture 3 1 / 51 Outline 1 Some model independent definitions and principals
More informationModels of Option Pricing: The Black-Scholes, Binomial and Monte Carlo Methods
Registration number 65 Models of Option Pricing: The Black-Scholes, Binomial and Monte Carlo Methods Supervised by Dr Christopher Greenman University of East Anglia Faculty of Science School of Computing
More informationSolving the Black-Scholes Equation
Solving the Black-Scholes Equation An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2010 Initial Value Problem for the European Call rf = F t + rsf S + 1 2 σ2 S 2 F SS for (S,
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 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 information- 1 - **** d(lns) = (µ (1/2)σ 2 )dt + σdw t
- 1 - **** These answers indicate the solutions to the 2014 exam questions. Obviously you should plot graphs where I have simply described the key features. It is important when plotting graphs to label
More informationLecture 18. More on option pricing. Lecture 18 1 / 21
Lecture 18 More on option pricing Lecture 18 1 / 21 Introduction In this lecture we will see more applications of option pricing theory. Lecture 18 2 / 21 Greeks (1) The price f of a derivative depends
More informationMATH4143: Scientific Computations for Finance Applications Final exam Time: 9:00 am - 12:00 noon, April 18, Student Name (print):
MATH4143 Page 1 of 17 Winter 2007 MATH4143: Scientific Computations for Finance Applications Final exam Time: 9:00 am - 12:00 noon, April 18, 2007 Student Name (print): Student Signature: Student ID: Question
More informationThe Derivation and Discussion of Standard Black-Scholes Formula
The Derivation and Discussion of Standard Black-Scholes Formula Yiqian Lu October 25, 2013 In this article, we will introduce the concept of Arbitrage Pricing Theory and consequently deduce the standard
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 informationThe Black-Scholes Equation using Heat Equation
The Black-Scholes Equation using Heat Equation Peter Cassar May 0, 05 Assumptions of the Black-Scholes Model We have a risk free asset given by the price process, dbt = rbt The asset price follows a geometric
More informationUtility Indifference Pricing and Dynamic Programming Algorithm
Chapter 8 Utility Indifference ricing and Dynamic rogramming Algorithm In the Black-Scholes framework, we can perfectly replicate an option s payoff. However, it may not be true beyond the Black-Scholes
More informationMath489/889 Stochastic Processes and Advanced Mathematical Finance Solutions to Practice Problems
Math489/889 Stochastic Processes and Advanced Mathematical Finance Solutions to Practice Problems Steve Dunbar No Due Date: Practice Only. Find the mode (the value of the independent variable with the
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 informationHedging Errors for Static Hedging Strategies
Hedging Errors for Static Hedging Strategies Tatiana Sushko Department of Economics, NTNU May 2011 Preface This thesis completes the two-year Master of Science in Financial Economics program at NTNU. Writing
More informationThe Black-Scholes Model
The Black-Scholes Model Liuren Wu Zicklin School of Business, Baruch College Options Markets Liuren Wu (Baruch) The Black-Scholes Model Options Markets 1 / 55 Outline 1 Brownian motion 2 Ito s lemma 3
More informationOULU BUSINESS SCHOOL. Ilkka Rahikainen DIRECT METHODOLOGY FOR ESTIMATING THE RISK NEUTRAL PROBABILITY DENSITY FUNCTION
OULU BUSINESS SCHOOL Ilkka Rahikainen DIRECT METHODOLOGY FOR ESTIMATING THE RISK NEUTRAL PROBABILITY DENSITY FUNCTION Master s Thesis Finance March 2014 UNIVERSITY OF OULU Oulu Business School ABSTRACT
More informationForwards and Futures. Chapter Basics of forwards and futures Forwards
Chapter 7 Forwards and Futures Copyright c 2008 2011 Hyeong In Choi, All rights reserved. 7.1 Basics of forwards and futures The financial assets typically stocks we have been dealing with so far are the
More informationFinance II. May 27, F (t, x)+αx f t x σ2 x 2 2 F F (T,x) = ln(x).
Finance II May 27, 25 1.-15. All notation should be clearly defined. Arguments should be complete and careful. 1. (a) Solve the boundary value problem F (t, x)+αx f t x + 1 2 σ2 x 2 2 F (t, x) x2 =, F
More informationThe Black-Scholes Model
The Black-Scholes Model Liuren Wu Zicklin School of Business, Baruch College Fall, 2007 (Continuous time finance primer) Liuren Wu The Black-Scholes Model Option Pricing, Fall, 2007 1 / 57 Outline 1 Brownian
More informationCHAPTER 12. Hedging. hedging strategy = replicating strategy. Question : How to find a hedging strategy? In other words, for an attainable contingent
CHAPTER 12 Hedging hedging dddddddddddddd ddd hedging strategy = replicating strategy hedgingdd) ddd Question : How to find a hedging strategy? In other words, for an attainable contingent claim, find
More informationReplication and Absence of Arbitrage in Non-Semimartingale Models
Replication and Absence of Arbitrage in Non-Semimartingale Models Matematiikan päivät, Tampere, 4-5. January 2006 Tommi Sottinen University of Helsinki 4.1.2006 Outline 1. The classical pricing model:
More informationYoungrok Lee and Jaesung Lee
orean J. Math. 3 015, No. 1, pp. 81 91 http://dx.doi.org/10.11568/kjm.015.3.1.81 LOCAL VOLATILITY FOR QUANTO OPTION PRICES WITH STOCHASTIC INTEREST RATES Youngrok Lee and Jaesung Lee Abstract. This paper
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 informationFundamentals of Finance
Finance in a nutshell University of Oulu - Department of Finance Fall 2018 Valuation of a future cash-flow Finance is about valuation of future cash-flows is terms of today s euros. Future cash-flow can
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 information