= { < exp (-(r - y)(t - t)
|
|
- Lorraine Alexander
- 6 years ago
- Views:
Transcription
1 Stochastic Modelling in Finance - Solutions Consider the Black & Scholes Model with volatility a > 0, drift /i, interest rate r and initial stock price Sq. Take a put-option with strike /C > 0 and maturity T > 0. The value of this option is V(t, s) = EQ[(K- Sr)+e-r(r-f) St = s]. Without reference to the put-call parity, show that the value of the put option at time t>0 satisfies V(t, St) = tfe-r<t-"(l - *(<L(T - t, St))) - 5((1 - *(d+(r - t, St))) where <[> denotes the CDF of the Normal distribution and d±(ttx) are defined as This is pretty much identical to the calculation done in lecture: In the Black & Scholes model ST = exp((r - \a2)t + awt) where WT ~ JV(O, T) and St = St exp I f r -a2 ) (T t) + awr-t ) Hence, a) = Q[(tf - Sr)4"e"r(T- \St = s] C _ o p-r{t-t) J?\Q tt I C _ el where X ~ A^(0,1). For the final equality we calculate that {K > ST] = ik> 5texp ((r - j)(t - t) = { < exp (-(r - y)(t - t) = {in ( ) + (r - y)(t - t) < = {d.(t-t,st)<x} for -Wr-t/x/T3! = X ~ N{0,1) under Q. Hence, Eq{Ik>St \St = s] = EQ%t_(T-t,.)<x] = -^= / e"1'/2 dx = 1 - ^(d_(t - *, s)) V27T Jd-(T-t,s)
2 and» S, = s] = 1 f = s / e V27T Jd-{T-t,s) 1 r = s == / e V27T Jd-(T-t,s) = s f= / e~y ^2 c?2/ Hence V(«, St) = Ke'^-^il - $(d_(t - i, S«))) - St(l - $(^+(T - t, 50))- 7.2 Consider the Black &; Scholes Model with volatility a > 0, drift fi, interest rate r and initial stock price So- Take a 'power call-option' with power n > 1, strike K > 0 and maturity T > 0. The value of this option is V(t, s) = EQ [(5? - K)+e-riT-l) \ Show that the value of this option at time t = 0 satisfies 1/(0, So) = SJc(n-l)rTe2l!^ffaT$(d^(r,So) H-r where $ denotes the CDF of the standard Normal distribution and d^l{t^x) is defined as Let V(0, So) = e-rteq{s^>k] - Ke-rTEQ[Isf>K]. (0.1) Next we look more closely at the event {SJ > K} and write > exp {-awt - (r - ^2) t) } Note that -Wr ~ N(0,T) and hence X := -WT/\/T ~ W(0,1) so {in (J^j + (r- \a^ T > -owr} = {<P.(T,So) > X},
3 Using this we can rewrite the second term in (0.1) as Ke-rTEQ[lSn>K} = Ke-rrEQ[ = ^=- \ V^TT J- The first term is: = SSEQ [exp (nawt - ^cj2t) lsn>k\ ' n.(n-l)rt Eg [exp [-navfx - -a2t) r<r_{t,s0) exp (-nvvty - ^2t) e^2/2dy Next we need to 'complete the square' in this exponential, i.e. note that and thus 1= V27T / d-(t,st) Changing variables using z = y -\- no\/t we get qnp(n-l)rtpn(n-l)a2t/2 ^/2 (r, 5b) + nay/f). So we have derived that the price of the power call option is given by V(0, Sb) = SSe{n-l)rTe2il*2*aaT${<rL(T, So) + navt) It was shown in an example in Lecture that in the Black &, Scholes Model that the value of a call option with strike K > 0 and maturity T > 0 is = St*(d+(T - i, St)) - Ke-^T-l^(d_{T - t, 5t)). Suppose that u(t, s) is a solution to the PDE d d Id2 u(t, s) + rs u(t, s) + -crv u(, s) - ru(t, s) = 0 at as 2 us* subject to the terminal condition u(t, s) = (s K)+. 3
4 (i) Show that limt_>t V(Tt s) = (s - K) + Recall that so lim d±(t-i,s) = lim v 1 (, ^ T^7 = lim - t +00 s > K oo s < K. 0 s = K Hence lim$(d±(r-i,a))= i s> 0 s < and lim V(t, s) = li - t, ft)) - - t St))) s-k s>k 0 s<k o s = /r so we can conclude that \imt->t V(t, s) = (s /C)+ as required. (ii) Show that V{t,x) = u(tfx) by substituting the explicit expression for V(t,s) into the PDE determining u(tts). First we need to calculate the derivatives ^V(t, s), ^V(t,s) and ^?V(t, s). You can take these from chapter 3 of the background reading or directly calculate: ^V(t, s) = - - t,a)) - - t, a)),
5 Inserting these into the PDE gives we should have sa d 0 = - tt a)) - - t, s)) 1 say/t tdx - r{s$(d+{t - t, s)) - Canceling terms on the right-hand side validates that the PDE indeed holds. 7.4 The price of a put-option with strike price K > 0 in the Black & Scholes model is M) = EQ [(K - St = s] - tt St))) - St(l - - t, St))) Find explicit expressions for the following sensitivities: (i) The delta of this put option is denned as Explain how the delta of a option in the Black k, Scholes model can be used for hedging. This one looks complicated but once we have proven the statement (0.2) will be much more straightforward. the others say/t t We can calculate using the chain-rule that - t, s)) cf>(d+(t - tt s)) = - tt so\/t t ojt -t
6 where (f>(x) = -^${x). Furthermore, = -{d+{t - i, s))2 - a2{t - t) + 2d+{T - t, s)a = -(d+(t-t,s))2-a2(t-t) -^) +2r{T-t)+ a2(t - t) K tt s)) K 2r(T - t). so d -t s)- d+{t-tts)-oyjt-t dy -oo -1 1 exp{-(d+{t -t,s) - osjt - if ser(t-t) ser{t-t) 4(d+(T-tya)). Thus we have found that d d (0.2) and thus ds We saw in lecture that the price process associated with the put option can be decomposed as i = tt(p)o + / &udsu where (At)t>o is the replicating strategy. Applying the Ito formula to V(t, s) and using that (e~7't7r(c)i)t>o is a martingale we can see that Thus the delta of the option calculated in this question is the number of units of stock held in a replicating portfolio. (ii) The gamma of this put option, which is denned as
7 Differentiating the answer given in (i) gives S(J Vf where Six) := -j-^(x). (iii) The theta of this put option, which is denned as First we note that d_(t tts) - d+(t t, s) o\/t i, showing that Applying the chain rule we see that dd (T-ta)-±d lt-ta)+ " Ft ( *'5)-at+u t's' + IV(t, s) = (T - t, a))) - s(l - $(d+(t - t, s))) - t, s))) + scj>(d+(t - t, St)))^d+(T - i, a) - Ke-T{T-t]<t>{d-{T - t, a))^-d_(r - t, s) C(7 - t, a))) /// Q U'+\1 L> */ t Due to (0.2) the second term is zero so we have derived that ft C1* (iv) The vega of this put option, which is defined as First we note that cl(t t,s) = d+(t t,s) oy/t t, showing that -d.(t -tis) = d+(t - t, s) - 00 ocr
8 Applying the chain rule we see that = s4>(d+(t - i, St)))-^ ^d.{t - t} s) t, St))) - - tt s)))^-d+(t - t, s). Due to (0.2) the second term is zero so we have derived that d V{t,s) = (v) The rho of this put option, which is defined as p(t,s):=-v(t,s). First we differentiate d±(t t, s), showing that Applying the chain rule we see that -V(t, s) = -K(T - (scp(d+(t-t,st))- - t, Due to (0.2) the final term is zero so V{t,s) = -K(T- - t,
last 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 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 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 9 Lecture 9 9.1 The Greeks November 15, 2017 Let
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 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 Pricing Black-Scholes Formula Lecture 19. Dr. Vasily Strela (Morgan Stanley and MIT)
Risk Neutral Pricing Black-Scholes Formula Lecture 19 Dr. Vasily Strela (Morgan Stanley and MIT) Risk Neutral Valuation: Two-Horse Race Example One horse has 20% chance to win another has 80% chance $10000
More informationRho and Delta. Paul Hollingsworth January 29, Introduction 1. 2 Zero coupon bond 1. 3 FX forward 2. 5 Rho (ρ) 4. 7 Time bucketing 6
Rho and Delta Paul Hollingsworth January 29, 2012 Contents 1 Introduction 1 2 Zero coupon bond 1 3 FX forward 2 4 European Call under Black Scholes 3 5 Rho (ρ) 4 6 Relationship between Rho and Delta 5
More informationCourse MFE/3F Practice Exam 2 Solutions
Course MFE/3F Practice Exam Solutions The chapter references below refer to the chapters of the ActuarialBrew.com Study Manual. Solution 1 A Chapter 16, Black-Scholes Equation The expressions for the value
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 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 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 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 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 informationTEACHING NOTE 98-04: EXCHANGE OPTION PRICING
TEACHING NOTE 98-04: EXCHANGE OPTION PRICING Version date: June 3, 017 C:\CLASSES\TEACHING NOTES\TN98-04.WPD The exchange option, first developed by Margrabe (1978), has proven to be an extremely powerful
More informationAsset-or-nothing digitals
School of Education, Culture and Communication Division of Applied Mathematics MMA707 Analytical Finance I Asset-or-nothing digitals 202-0-9 Mahamadi Ouoba Amina El Gaabiiy David Johansson Examinator:
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 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 informationChapter 3: Black-Scholes Equation and Its Numerical Evaluation
Chapter 3: Black-Scholes Equation and Its Numerical Evaluation 3.1 Itô Integral 3.1.1 Convergence in the Mean and Stieltjes Integral Definition 3.1 (Convergence in the Mean) A sequence {X n } n ln of random
More informationFinancial Risk Management
Risk-neutrality in derivatives pricing University of Oulu - Department of Finance Spring 2018 Portfolio of two assets Value at time t = 0 Expected return Value at time t = 1 Asset A Asset B 10.00 30.00
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 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 informationEuropean call option with inflation-linked strike
Mathematical Statistics Stockholm University European call option with inflation-linked strike Ola Hammarlid Research Report 2010:2 ISSN 1650-0377 Postal address: Mathematical Statistics Dept. of Mathematics
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 informationRMSC 4005 Stochastic Calculus for Finance and Risk. 1 Exercises. (c) Let X = {X n } n=0 be a {F n }-supermartingale. Show that.
1. EXERCISES RMSC 45 Stochastic Calculus for Finance and Risk Exercises 1 Exercises 1. (a) Let X = {X n } n= be a {F n }-martingale. Show that E(X n ) = E(X ) n N (b) Let X = {X n } n= be a {F n }-submartingale.
More informationMartingale Approach to Pricing and Hedging
Introduction and echniques Lecture 9 in Financial Mathematics UiO-SK451 Autumn 15 eacher:s. Ortiz-Latorre Martingale Approach to Pricing and Hedging 1 Risk Neutral Pricing Assume that we are in the basic
More informationDynamic Hedging and PDE Valuation
Dynamic Hedging and PDE Valuation Dynamic Hedging and PDE Valuation 1/ 36 Introduction Asset prices are modeled as following di usion processes, permitting the possibility of continuous trading. This environment
More informationHedging under Arbitrage
Hedging under Arbitrage Johannes Ruf Columbia University, Department of Statistics Modeling and Managing Financial Risks January 12, 2011 Motivation Given: a frictionless market of stocks with continuous
More 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 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 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 informationSYSM 6304: Risk and Decision Analysis Lecture 6: Pricing and Hedging Financial Derivatives
SYSM 6304: Risk and Decision Analysis Lecture 6: Pricing and Hedging Financial Derivatives M. Vidyasagar Cecil & Ida Green Chair The University of Texas at Dallas Email: M.Vidyasagar@utdallas.edu October
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 information1 Interest Based Instruments
1 Interest Based Instruments e.g., Bonds, forward rate agreements (FRA), and swaps. Note that the higher the credit risk, the higher the interest rate. Zero Rates: n year zero rate (or simply n-year zero)
More informationDerivative Securities Section 9 Fall 2004 Notes by Robert V. Kohn, Courant Institute of Mathematical Sciences.
Derivative Securities Section 9 Fall 2004 Notes by Robert V. Kohn, Courant Institute of Mathematical Sciences. Futures, and options on futures. Martingales and their role in option pricing. A brief introduction
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 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 discounted portfolio value of a selffinancing strategy in discrete time was given by. δ tj 1 (s tj s tj 1 ) (9.1) j=1
Chapter 9 The isk Neutral Pricing Measure for the Black-Scholes Model The discounted portfolio value of a selffinancing strategy in discrete time was given by v tk = v 0 + k δ tj (s tj s tj ) (9.) where
More informationBlack-Scholes model: Derivation and solution
III. Black-Scholes model: Derivation and solution Beáta Stehlíková Financial derivatives Faculty of Mathematics, Physics and Informatics Comenius University, Bratislava III. Black-Scholes model: Derivation
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 informationMATH3075/3975 FINANCIAL MATHEMATICS TUTORIAL PROBLEMS
MATH307/37 FINANCIAL MATHEMATICS TUTORIAL PROBLEMS School of Mathematics and Statistics Semester, 04 Tutorial problems should be used to test your mathematical skills and understanding of the lecture material.
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 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 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 informationLecture Quantitative Finance Spring Term 2015
and Lecture Quantitative Finance Spring Term 2015 Prof. Dr. Erich Walter Farkas Lecture 06: March 26, 2015 1 / 47 Remember and Previous chapters: introduction to the theory of options put-call parity fundamentals
More informationCompleteness and Hedging. Tomas Björk
IV Completeness and Hedging Tomas Björk 1 Problems around Standard Black-Scholes We assumed that the derivative was traded. How do we price OTC products? Why is the option price independent of the expected
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 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 informationand K = 10 The volatility a in our model describes the amount of random noise in the stock price. Y{x,t) = -J-{t,x) = xy/t- t<pn{d+{t-t,x))
-5b- 3.3. THE GREEKS Theta #(t, x) of a call option with T = 0.75 and K = 10 Rho g{t,x) of a call option with T = 0.75 and K = 10 The volatility a in our model describes the amount of random noise in the
More informationLecture 17. The model is parametrized by the time period, δt, and three fixed constant parameters, v, σ and the riskless rate r.
Lecture 7 Overture to continuous models Before rigorously deriving the acclaimed Black-Scholes pricing formula for the value of a European option, we developed a substantial body of material, in continuous
More information1 Geometric Brownian motion
Copyright c 05 by Karl Sigman Geometric Brownian motion Note that since BM can take on negative values, using it directly for modeling stock prices is questionable. There are other reasons too why BM is
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 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 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 informationLecture 4. Finite difference and finite element methods
Finite difference and finite element methods Lecture 4 Outline Black-Scholes equation From expectation to PDE Goal: compute the value of European option with payoff g which is the conditional expectation
More information(1) Consider a European call option and a European put option on a nondividend-paying stock. You are given:
(1) Consider a European call option and a European put option on a nondividend-paying stock. You are given: (i) The current price of the stock is $60. (ii) The call option currently sells for $0.15 more
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 informationθ(t ) = T f(0, T ) + σ2 T
1 Derivatives Pricing and Financial Modelling Andrew Cairns: room M3.08 E-mail: A.Cairns@ma.hw.ac.uk Tutorial 10 1. (Ho-Lee) Let X(T ) = T 0 W t dt. (a) What is the distribution of X(T )? (b) Find E[exp(
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 informationValuation of Equity Derivatives
Valuation of Equity Derivatives Dr. Mark W. Beinker XXV Heidelberg Physics Graduate Days, October 4, 010 1 What s a derivative? More complex financial products are derived from simpler products What s
More information2 f. f t S 2. Delta measures the sensitivityof the portfolio value to changes in the price of the underlying
Sensitivity analysis Simulating the Greeks Meet the Greeks he value of a derivative on a single underlying asset depends upon the current asset price S and its volatility Σ, the risk-free interest rate
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 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 informationINSTITUTE OF ACTUARIES OF INDIA
INSTITUTE OF ACTUARIES OF INDIA EXAMINATIONS 23 rd March 2017 Subject CT8 Financial Economics Time allowed: Three Hours (10.30 13.30 Hours) Total Marks: 100 INSTRUCTIONS TO THE CANDIDATES 1. Please read
More informationPath Dependent British Options
Path Dependent British Options Kristoffer J Glover (Joint work with G. Peskir and F. Samee) School of Finance and Economics University of Technology, Sydney 18th August 2009 (PDE & Mathematical Finance
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 217 Instructor: Dr. Sateesh Mane c Sateesh R. Mane 217 13 Lecture 13 November 15, 217 Derivation of the Black-Scholes-Merton
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 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 informationGreek Maxima 1 by Michael B. Miller
Greek Maxima by Michael B. Miller When managing the risk of options it is often useful to know how sensitivities will change over time and with the price of the underlying. For example, many people know
More informationAMERICAN OPTIONS REVIEW OF STOPPING TIMES. Important example: the first passage time for continuous process X:
AMERICAN OPTIONS REVIEW OF STOPPING TIMES τ is stopping time if {τ t} F t for all t Important example: the first passage time for continuous process X: τ m = min{t 0 : X(t) = m} (τ m = if X(t) never takes
More informationBinomial model: numerical algorithm
Binomial model: numerical algorithm S / 0 C \ 0 S0 u / C \ 1,1 S0 d / S u 0 /, S u 3 0 / 3,3 C \ S0 u d /,1 S u 5 0 4 0 / C 5 5,5 max X S0 u,0 S u C \ 4 4,4 C \ 3 S u d / 0 3, C \ S u d 0 S u d 0 / C 4
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 informationAdvanced Stochastic Processes.
Advanced Stochastic Processes. David Gamarnik LECTURE 16 Applications of Ito calculus to finance Lecture outline Trading strategies Black Scholes option pricing formula 16.1. Security price processes,
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 informationd St+ t u. With numbers e q = The price of the option in three months is
Exam in SF270 Financial Mathematics. Tuesday June 3 204 8.00-3.00. Answers and brief solutions.. (a) This exercise can be solved in two ways. i. Risk-neutral valuation. The martingale measure should satisfy
More informationModule 10:Application of stochastic processes in areas like finance Lecture 36:Black-Scholes Model. Stochastic Differential Equation.
Stochastic Differential Equation Consider. Moreover partition the interval into and define, where. Now by Rieman Integral we know that, where. Moreover. Using the fundamentals mentioned above we can easily
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 informationBarrier options. In options only come into being if S t reaches B for some 0 t T, at which point they become an ordinary option.
Barrier options A typical barrier option contract changes if the asset hits a specified level, the barrier. Barrier options are therefore path-dependent. Out options expire worthless if S t reaches the
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 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 informationPricing in markets modeled by general processes with independent increments
Pricing in markets modeled by general processes with independent increments Tom Hurd Financial Mathematics at McMaster www.phimac.org Thanks to Tahir Choulli and Shui Feng Financial Mathematics Seminar
More information3 Ito formula and processes
3 Ito formula and processes 3.1 Ito formula Let f be a differentiable function. If g is another differentiable function, we have by the chain rule d dt f(g(t)) = f (g(t))g (t), which in the differential
More informationPAijpam.eu ANALYTIC SOLUTION OF A NONLINEAR BLACK-SCHOLES EQUATION
International Journal of Pure and Applied Mathematics Volume 8 No. 4 013, 547-555 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: http://dx.doi.org/10.173/ijpam.v8i4.4
More informationLecture 9: Practicalities in Using Black-Scholes. Sunday, September 23, 12
Lecture 9: Practicalities in Using Black-Scholes Major Complaints Most stocks and FX products don t have log-normal distribution Typically fat-tailed distributions are observed Constant volatility assumed,
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 informationwhere W( ) := W{i) + JQ M(s)ds defines a Brownian motion (W[t) : t [0, T}) under
CHAPTERS. RISK-NEUTRAL PRICING (b) the discounted stock price S is a [local] martingale with respect to Q. The condition (a) is required because under the risk-neutral measure Q the same events A should
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 informationPDE Methods for the Maximum Drawdown
PDE Methods for the Maximum Drawdown Libor Pospisil, Jan Vecer Columbia University, Department of Statistics, New York, NY 127, USA April 1, 28 Abstract Maximum drawdown is a risk measure that plays an
More 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 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 informationValuation of derivative assets Lecture 6
Valuation of derivative assets Lecture 6 Magnus Wiktorsson September 14, 2017 Magnus Wiktorsson L6 September 14, 2017 1 / 13 Feynman-Kac representation This is the link between a class of Partial Differential
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 informationCourse MFE/3F Practice Exam 1 Solutions
Course MFE/3F Practice Exam 1 Solutions he chapter references below refer to the chapters of the ActuraialBrew.com Study Manual. Solution 1 C Chapter 16, Sharpe Ratio If we (incorrectly) assume that the
More informationStochastic Modelling in Finance
in Finance Department of Mathematics and Statistics University of Strathclyde Glasgow, G1 1XH April 2010 Outline and Probability 1 and Probability 2 Linear modelling Nonlinear modelling 3 The Black Scholes
More informationEstimating the Greeks
IEOR E4703: Monte-Carlo Simulation Columbia University Estimating the Greeks c 207 by Martin Haugh In these lecture notes we discuss the use of Monte-Carlo simulation for the estimation of sensitivities
More informationFIN FINANCIAL INSTRUMENTS SPRING 2008
FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 The Greeks Introduction We have studied how to price an option using the Black-Scholes formula. Now we wish to consider how the option price changes, either
More informationStatistical Methods in Financial Risk Management
Statistical Methods in Financial Risk Management Lecture 1: Mapping Risks to Risk Factors Alexander J. McNeil Maxwell Institute of Mathematical Sciences Heriot-Watt University Edinburgh 2nd Workshop on
More informationErrata, Mahler Study Aids for Exam 3/M, Spring 2010 HCM, 1/26/13 Page 1
Errata, Mahler Study Aids for Exam 3/M, Spring 2010 HCM, 1/26/13 Page 1 1B, p. 72: (60%)(0.39) + (40%)(0.75) = 0.534. 1D, page 131, solution to the first Exercise: 2.5 2.5 λ(t) dt = 3t 2 dt 2 2 = t 3 ]
More informationON AN IMPLEMENTATION OF BLACK SCHOLES MODEL FOR ESTIMATION OF CALL- AND PUT-OPTION VIA PROGRAMMING ENVIRONMENT MATHEMATICA
Доклади на Българската академия на науките Comptes rendus de l Académie bulgare des Sciences Tome 66, No 5, 2013 MATHEMATIQUES Mathématiques appliquées ON AN IMPLEMENTATION OF BLACK SCHOLES MODEL FOR ESTIMATION
More informationKØBENHAVNS UNIVERSITET (Blok 2, 2011/2012) Naturvidenskabelig kandidateksamen Continuous time finance (FinKont) TIME ALLOWED : 3 hours
This question paper consists of 3 printed pages FinKont KØBENHAVNS UNIVERSITET (Blok 2, 211/212) Naturvidenskabelig kandidateksamen Continuous time finance (FinKont) TIME ALLOWED : 3 hours This exam paper
More informationIEOR E4602: Quantitative Risk Management
IEOR E4602: Quantitative Risk Management Basic Concepts and Techniques of Risk Management Martin Haugh Department of Industrial Engineering and Operations Research Columbia University Email: martin.b.haugh@gmail.com
More information