CURRENCY OPTION PRICING II
|
|
- Thomas Shepherd
- 8 years ago
- Views:
Transcription
1 Jones Grauate School Rice University Masa Watanabe INTERNATIONAL FINANCE MGMT 657 Calibrating the Binomial Tree to Volatility Black-Scholes Moel for Currency Options Properties of the BS Moel Option Sensitivity Analysis Delta Gamma Vega Theta Rho
2 Calibrating the Binomial Tree Instea of u an, you will usually obtain the volatility, σ, either as Historical volatility Compute the stanar eviation of aily log return (ln(s t / S t- )) Multiply the aily volatility by 5 to get annual volatility Implie volatility In reality, both types of volatility are available from, e.g., Reuters. In fact, OTC options are quote by implie volatility. Euro options quotes, Reuters, /7/003 To construct a binomial tree that is consistent with a given volatility, set u σ t = e, σ t = e, where t = /n is the length of a time step (a subtree) an n is the number of time steps (subtrees) until maturity.
3 Example. Let us construct a one-perio tree that is consistent with a 0% volatility. Using this tree, we will price call an put options. Spot S =.5$/ Strike K =.5$/ Domestic interest rate i $ =.% (continuously compoune) Foreign interest rate i =.% (continuously compoune) Volatility σ = 0% (annualize) Time to maturity = 6 months ( = 0.5) u = exp(..5) =.0773 = exp(-..5) = q = [exp((.0.0).5) ]/(u ) = Risk neutral pricing gives C = [.0846 q + 0 ( q)] exp(-.0.5) = P = [0 q ( q)] exp(-.0.5) =.0435 Spot rate ($/ ) q= q= Call.0846 Put
4 As the number of time steps increases (n an t 0), the binomial moel price converges to the Black-Scholes price. Time step n BS Call Put Graphically: Convergence of the Binomial to BS Moel Call Option # Time Steps Tree BS 4
5 Black-Scholes Moel for Currency Options To price currency options, you can use the Black-Scholes formula on a ivien-paying stock with the ivien yiel replace by the foreign interest rate. Notation Toay is ate t, the maturity of the option is on a future ate T. S /f t or S: Spot rate on ate t, value of currency f in currency F : Forwar rate K : Strike price i : Interest rate on currency (continuously compoune) i f : Interest rate on currency f (continuously compoune) = T t : Time to maturity σ : Volatility of the spot rate (annualize) C : Call P : Put f i i i i C = Se N( ) Ke N( ), P = Ke N( ) Se N( ) () where f f ln( S / K) + i i + σ ln( S / K) + i i σ, () σ σ = σ f This is known as the Garman-Kohlhagen moel 5
6 Note that, in the FX context, you can write the formula in terms of the forwar rate so that the foreign interest rate (or even the spot rate!) oes not appear. Since F ( i i ) = Se, f i i C = e [ FN( ) KN ( )], P = e [ KN ( ) FN( )] (3) where ln( F / K) + σ ln( F / K) σ, (4) σ σ = σ Discounting by the risk-free rate in Equation (3) inicates that the terms in the square brackets are certainty equivalent of the option payoff at maturity. Note: you o have to use the forwar rate that correspons to the maturity of the option. Example. Spot S =.5$/ Strike K =.5$/ Domestic interest rate i $ =.% (continuously compoune) Foreign interest rate i =.% (continuously compoune) (Or you might observe the forwar rate, F =.443$/. Then use (3)-(4)) Volatility σ = 0% Time to maturity = 6 months = [ln(.5/.5) + ( /).5]/(..5) = =..5 = N( ) =.48590, N(- ) = N( ) =.540 N( ) =.44776, N(- ) = N( ) =.544 C =.5 e e =.0939 P =.5 e e =
7 Properties of the Black-Scholes Moel for Currency Options These are also the properties of the BS moel on a ivien-paying stock.. The B-S moel assumes the future spot rate is istribute lognormally. Without this strong assumption we can still put a lower boun on a European call option: C Max(S exp(-i f ) K exp(-i ), 0) To see this, let us first erive the following important result: Result The present value of receiving S T at maturity is S t exp(-i f ). Note: this is NOT S t. The value of foreign interest is subtracte. This can be seen by consiering the following investment strategy (take currency = $, currency f = ): Now: invest $S t exp(-i f ) = exp(-i f ) in a euro eposit. Reinvest the interest continuously in the eposit itself. At maturity: you will receive (=exp(-i f ) exp(i f )) or equivalently $S T. The payoff of a call option at maturity is the larger of S T K or 0. The PV of receiving S T K at maturity is, from the above result, S t exp(-i f ) Kexp(-i ). Since the value of a call option is never negative, we have the above inequality. Graphically: 7
8 Call Spot Rate Call Lower Boun. The prices of forwar ATM call an put options are the same. Graphically, recall that we can create a synthetic forwar by a call an a put. Since the forwar costs nothing, the price of the call an the put must balance. Long Forwar Long Call Short Put Payoff Payoff Payoff F S T /f = + K=F S T /f K=F S T /f Mathematically, since we always have N( ) N( ) = N( ) [ N( )] = N( ) N( ), if K = F in (3), 8
9 9. )] ( ) ( [ )] ( ) ( [ )], ( ) ( [ C N N F e N N F e P N N F e C i i i = = = = Note: Equation (4) also becomes very simple: / = σ, / = = σ. Thus, N( ) = N( ) = N( ) an therefore we can further write ] ) ( [ = = N F e P C i.
10 3. Digital (Binary) Options. Asset-or-nothing (AON) an cash-or-nothing (CON) options are not really exotic. They are the basic builing blocks of the Black-Scholes Moel. By efinition, C = AON(S T > K) K CON(S T > K) P = K CON(S T < K) AON(S T < K) Compare with (). We obtain: AON( S T > K) = Se f i N( ), CON( S T > K) = e i N( ) AON( S T < K) = Se f i N( ), CON( S T < K) = e i N( ) From the expressions for CON, we know that the probability of the call ening up in the money (S T > K) is N( ), the put ening up in the money (S T < K) is N( ). Risk-neutral probabilities of the call an the put ening up ITM Prob. Put ITM N( ) Call ITM N( ) K S T 0
11 Option Sensitivity Analysis Delta The elta of an option (or a portfolio),, is the rate of change in the price of the option (or portfolio) with respect to the spot rate. C f i Delta of a European call: Call = = e N( ) S Recall that 0 < Call <. In the B-S worl, a tighter boun obtains: 0 < Call < exp(-i f ) <. P f i Delta of a European put: Put = = e N( ) S - < -exp(-i f ) < Put < 0. Delta of the European call option in the Example Delta Spot Rate As the figure shows, Call exp(-i f ) (closer to ) as S t (ITM). Call 0 as S t 0 (OTM). Q. What is the limit of the elta of a put as S t or S t 0?
12 Delta of ITM, ATM, an OTM calls plotte against time to maturity Delta Time to Maturity Q. In the above example, recall that S =.5 an F =.443. Why oes the elta of the.5 call converge to 0? Why oes the elta of the.05 call become closer to? The elta of a spot contract is (confirm this). The elta of a forwar contract is exp( i f ). Q3. Show this by first writing own the value of a long forwar contract (this is ifferent from the formula for the forwar rate) an then ifferentiating it with respect to the spot rate. The elta of a futures contract is exp((i i f )). This is slightly ifferent from the forwar contract because of the mark-to-market. The mark-to-market enables you to realize the gain or loss cause by the change in the spot rate immeiately (aily). The elta of a portfolio is the sum of the eltas of its component assets.
13 Example, continue. The elta of the above call option is exp( i f )N( ) = exp(.% 0.5) = The elta of the put option is exp( i f )N( ) = exp(.% 0.5) = Q4. A bank has sol the above put option on million. How can the bank make its position elta neutral using the CME futures contract? The size of the CME euro futures contract is 5,000. What cares shoul be taken about this elta hege? Gamma The gamma of an option/portfolio is the rate of change of the option/portfolio s elta with respect to the spot rate. It is the secon partial erivative of the option/portfolio price with respect to the spot rate. It measures the curvature of the relation between the option/portfolio price an the spot rate. The gamma of a European call option an a put option with the same strike price turns out to be the same: C Γ = S P e = = S i f n( ) Sσ where n( ) is the stanar normal ensity function. Q5. Show this equivalence (not the formula) by the put-call parity. The gammas of a spot, forwar, an futures contract are all zero. You shoul be able to erive this. 3
14 Gamma of the European call in the example 6 Gamma Spot Rate Graphically, this is the slope of the graph of the elta. Confirm this. Delta neutrality provies protection against relatively small spot rate movements. Delta-Gamma neutrality provies protection against larger spot price movements. Traers typically make their position elta-neutral at least once a ay. Gamma an vega (see below) are more ifficult to zero away, because they cannot be altere by traing the spot, forwar, or futures contract. Fortunately, as time elapses, options ten to become eep OTM or ITM. Such options have negligible gamma an vegas. 4
15 Vega The vega of an option/portfolio, υ, is the rate of change of the option/portfolio value with respect to the volatility of the spot rate. The vega of a regular European or American option is always positive. Intuitively, the insurance (time) value of an option increases with volatility. The vegas of a European call an a put are the same. Q6. Show this using put-call parity. Vega of the European call in the example Vega Spot Rate 5
16 Theta The theta of an option/portfolio, Θ, is the rate of change of the option/portfolio value with respect to the passage of time. Theta is usually negative for an option, because the passage of time ecreases the time value. For an ITM call option on a currency with a relatively high interest rate, theta can be positive, because the passage of time shortens the time until the receipt of the foreign interest if exercise in the money (see the picture below). Theta is not the same type of hege measure as others Greeks, because there is no uncertainty about the passage of time. Theta of the European call in the example Theta Spot Rate 6
17 Rho The rho of an option/portfolio is the rate of change of the option/portoflio value with respect to the interest rate. For currency options, there are two rhos, one corresponing to the change in the omestic interest rate an the other corresponing to the foreign interest rate. The rho of a European call option with respect to the omestic interest rate is positive. It is negative for a put. The rho of a European call option with respect to the foreign interest rate is negative. It is positive for a put. Q7. Explain the above two points. Rho of the European call w.r.t. the omestic interest rate in the example Rho Spot Rate 7
18 Suggeste solutions to questions Q. Put 0 as S t (OTM). Put -exp(-i f ) (closer to -) as S t 0 (ITM). Q. The elta of the.5 call converges to 0 because, as 0, it becomes progressively sure that the option will en up OTM. On the other han, the elta of the.05 call becomes closer to because it becomes very likely that the option will en up ITM. Q3. The value of a forwar contract is f = S exp( i f ) K exp( i ) Thus, the elta of a forwar contract is f/ S = exp( i f ). Q4. The elta of the futures contract is exp((.%.%) 0.5) = Let x be the amount of futures contracts that the bank shoul hol long. We set ( million) x = 0. x = 0.5 million x / 5,000 = Thus, the bank shoul sell four CME euro futures contracts. As in the Dozier case, this is not a perfect hege. The bank must rebalance ynamically to remain elta-hege. There is a size mismatch. Q5. By the put-call parity, we have C P = S exp( i f ) K exp( i ). (*) Differentiating with respect to S gives C P = e S S That is, the ifference between the eltas of the call an the put equals the foreign iscount factor. Further ifferentiation yiels f i C P = 0. S S This shows the equivalence of the two eltas.. 8
19 Q6. The put-call parity relation (*) above oes not involve σ. Thus, ifferentiating with C P respect to σ gives = 0. σ σ Q7. Equations (3) an (4) can be consiere a B-S moel on a forwar contract. A higher omestic interest rate or a lower foreign interest rate will increase the forwar rate an therefore makes the call option more ITM, an the put more OTM. 9
Risk Management for Derivatives
Risk Management or Derivatives he Greeks are coming the Greeks are coming! Managing risk is important to a large number o iniviuals an institutions he most unamental aspect o business is a process where
More informationHull, Chapter 11 + Sections 17.1 and 17.2 Additional reference: John Cox and Mark Rubinstein, Options Markets, Chapter 5
Binomial Moel Hull, Chapter 11 + ections 17.1 an 17.2 Aitional reference: John Cox an Mark Rubinstein, Options Markets, Chapter 5 1. One-Perio Binomial Moel Creating synthetic options (replicating options)
More informationWeek 13 Introduction to the Greeks and Portfolio Management:
Week 13 Introduction to the Greeks and Portfolio Management: Hull, Ch. 17; Poitras, Ch.9: I, IIA, IIB, III. 1 Introduction to the Greeks and Portfolio Management Objective: To explain how derivative portfolios
More informationLecture 11: The Greeks and Risk Management
Lecture 11: The Greeks and Risk Management This lecture studies market risk management from the perspective of an options trader. First, we show how to describe the risk characteristics of derivatives.
More informationCaput Derivatives: October 30, 2003
Caput Derivatives: October 30, 2003 Exam + Answers Total time: 2 hours and 30 minutes. Note 1: You are allowed to use books, course notes, and a calculator. Question 1. [20 points] Consider an investor
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 informationInstitutional Finance 08: Dynamic Arbitrage to Replicate Non-linear Payoffs. Binomial Option Pricing: Basics (Chapter 10 of McDonald)
Copyright 2003 Pearson Education, Inc. Slide 08-1 Institutional Finance 08: Dynamic Arbitrage to Replicate Non-linear Payoffs Binomial Option Pricing: Basics (Chapter 10 of McDonald) Originally prepared
More informationHedging Illiquid FX Options: An Empirical Analysis of Alternative Hedging Strategies
Hedging Illiquid FX Options: An Empirical Analysis of Alternative Hedging Strategies Drazen Pesjak Supervised by A.A. Tsvetkov 1, D. Posthuma 2 and S.A. Borovkova 3 MSc. Thesis Finance HONOURS TRACK Quantitative
More informationPricing Options: Pricing Options: The Binomial Way FINC 456. The important slide. Pricing options really boils down to three key concepts
Pricing Options: The Binomial Way FINC 456 Pricing Options: The important slide Pricing options really boils down to three key concepts Two portfolios that have the same payoff cost the same. Why? A perfectly
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 information14 Greeks Letters and Hedging
ECG590I Asset Pricing. Lecture 14: Greeks Letters and Hedging 1 14 Greeks Letters and Hedging 14.1 Illustration We consider the following example through out this section. A financial institution sold
More informationBlack-Scholes model: Greeks - sensitivity analysis
VII. Black-Scholes model: Greeks- sensitivity analysis p. 1/15 VII. Black-Scholes model: Greeks - sensitivity analysis Beáta Stehlíková Financial derivatives, winter term 2014/2015 Faculty of Mathematics,
More informationCh 10. Arithmetic Average Options and Asian Opitons
Ch 10. Arithmetic Average Options an Asian Opitons I. Asian Option an the Analytic Pricing Formula II. Binomial Tree Moel to Price Average Options III. Combination of Arithmetic Average an Reset Options
More informationACTS 4302 SOLUTION TO MIDTERM EXAM Derivatives Markets, Chapters 9, 10, 11, 12, 18. October 21, 2010 (Thurs)
Problem ACTS 4302 SOLUTION TO MIDTERM EXAM Derivatives Markets, Chapters 9, 0,, 2, 8. October 2, 200 (Thurs) (i) The current exchange rate is 0.0$/. (ii) A four-year dollar-denominated European put option
More informationFINANCIAL ENGINEERING CLUB TRADING 201
FINANCIAL ENGINEERING CLUB TRADING 201 STOCK PRICING It s all about volatility Volatility is the measure of how much a stock moves The implied volatility (IV) of a stock represents a 1 standard deviation
More informationHow To Understand The Greeks
ETF Trend Trading Option Basics Part Two The Greeks Option Basics Separate Sections 1. Option Basics 2. The Greeks 3. Pricing 4. Types of Option Trades The Greeks A simple perspective on the 5 Greeks 1.
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 informationThe Greeks Vega. Outline: Explanation of the greeks. Using greeks for short term prediction. How to find vega. Factors influencing vega.
The Greeks Vega 1 1 The Greeks Vega Outline: Explanation of the greeks. Using greeks for short term prediction. How to find vega. Factors influencing vega. 2 Outline continued: Using greeks to shield your
More informationInvesco Great Wall Fund Management Co. Shenzhen: June 14, 2008
: A Stern School of Business New York University Invesco Great Wall Fund Management Co. Shenzhen: June 14, 2008 Outline 1 2 3 4 5 6 se notes review the principles underlying option pricing and some of
More informationPart V: Option Pricing Basics
erivatives & Risk Management First Week: Part A: Option Fundamentals payoffs market microstructure Next 2 Weeks: Part B: Option Pricing fundamentals: intrinsic vs. time value, put-call parity introduction
More informationOptions: Valuation and (No) Arbitrage
Prof. Alex Shapiro Lecture Notes 15 Options: Valuation and (No) Arbitrage I. Readings and Suggested Practice Problems II. Introduction: Objectives and Notation III. No Arbitrage Pricing Bound IV. The Binomial
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 informationFX Derivatives Terminology. Education Module: 5. Dated July 2002. FX Derivatives Terminology
Education Module: 5 Dated July 2002 Foreign Exchange Options Option Markets and Terminology A American Options American Options are options that are exercisable for early value at any time during the term
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 informationLecture 21 Options Pricing
Lecture 21 Options Pricing Readings BM, chapter 20 Reader, Lecture 21 M. Spiegel and R. Stanton, 2000 1 Outline Last lecture: Examples of options Derivatives and risk (mis)management Replication and Put-call
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 informationOption Valuation. Chapter 21
Option Valuation Chapter 21 Intrinsic and Time Value intrinsic value of in-the-money options = the payoff that could be obtained from the immediate exercise of the option for a call option: stock price
More informationFinancial Options: Pricing and Hedging
Financial Options: Pricing and Hedging Diagrams Debt Equity Value of Firm s Assets T Value of Firm s Assets T Valuation of distressed debt and equity-linked securities requires an understanding of financial
More informationChapter 11 Options. Main Issues. Introduction to Options. Use of Options. Properties of Option Prices. Valuation Models of Options.
Chapter 11 Options Road Map Part A Introduction to finance. Part B Valuation of assets, given discount rates. Part C Determination of risk-adjusted discount rate. Part D Introduction to derivatives. Forwards
More informationUnderlier Filters Category Data Field Description
Price//Capitalization Market Capitalization The market price of an entire company, calculated by multiplying the number of shares outstanding by the price per share. Market Capitalization is not applicable
More informationVanna-Volga Method for Foreign Exchange Implied Volatility Smile. Copyright Changwei Xiong 2011. January 2011. last update: Nov 27, 2013
Vanna-Volga Method for Foreign Exchange Implied Volatility Smile Copyright Changwei Xiong 011 January 011 last update: Nov 7, 01 TABLE OF CONTENTS TABLE OF CONTENTS...1 1. Trading Strategies of Vanilla
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 informationThe Black-Scholes Formula
FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 The Black-Scholes Formula These notes examine the Black-Scholes formula for European options. The Black-Scholes formula are complex as they are based on the
More informationOverview. Option Basics. Options and Derivatives. Professor Lasse H. Pedersen. Option basics and option strategies
Options and Derivatives Professor Lasse H. Pedersen Prof. Lasse H. Pedersen 1 Overview Option basics and option strategies No-arbitrage bounds on option prices Binomial option pricing Black-Scholes-Merton
More informationDigital barrier option contract with exponential random time
IMA Journal of Applie Mathematics Avance Access publishe June 9, IMA Journal of Applie Mathematics ) Page of 9 oi:.93/imamat/hxs3 Digital barrier option contract with exponential ranom time Doobae Jun
More informationUCLA Anderson School of Management Daniel Andrei, Derivative Markets 237D, Winter 2014. MFE Midterm. February 2014. Date:
UCLA Anderson School of Management Daniel Andrei, Derivative Markets 237D, Winter 2014 MFE Midterm February 2014 Date: Your Name: Your Equiz.me email address: Your Signature: 1 This exam is open book,
More informationVolatility as an indicator of Supply and Demand for the Option. the price of a stock expressed as a decimal or percentage.
Option Greeks - Evaluating Option Price Sensitivity to: Price Changes to the Stock Time to Expiration Alterations in Interest Rates Volatility as an indicator of Supply and Demand for the Option Different
More informationHow To Know Market Risk
Chapter 6 Market Risk for Single Trading Positions Market risk is the risk that the market value of trading positions will be adversely influenced by changes in prices and/or interest rates. For banks,
More informationThe Binomial Option Pricing Model André Farber
1 Solvay Business School Université Libre de Bruxelles The Binomial Option Pricing Model André Farber January 2002 Consider a non-dividend paying stock whose price is initially S 0. Divide time into small
More informationChapters 15. Delta Hedging with Black-Scholes Model. Joel R. Barber. Department of Finance. Florida International University.
Chapters 15 Delta Hedging with Black-Scholes Model Joel R. Barber Department of Finance Florida International University Miami, FL 33199 1 Hedging Example A bank has sold for $300,000 a European call option
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 informationVALUATION IN DERIVATIVES MARKETS
VALUATION IN DERIVATIVES MARKETS September 2005 Rawle Parris ABN AMRO Property Derivatives What is a Derivative? A contract that specifies the rights and obligations between two parties to receive or deliver
More informationExam MFE Spring 2007 FINAL ANSWER KEY 1 B 2 A 3 C 4 E 5 D 6 C 7 E 8 C 9 A 10 B 11 D 12 A 13 E 14 E 15 C 16 D 17 B 18 A 19 D
Exam MFE Spring 2007 FINAL ANSWER KEY Question # Answer 1 B 2 A 3 C 4 E 5 D 6 C 7 E 8 C 9 A 10 B 11 D 12 A 13 E 14 E 15 C 16 D 17 B 18 A 19 D **BEGINNING OF EXAMINATION** ACTUARIAL MODELS FINANCIAL ECONOMICS
More informationOption Values. Determinants of Call Option Values. CHAPTER 16 Option Valuation. Figure 16.1 Call Option Value Before Expiration
CHAPTER 16 Option Valuation 16.1 OPTION VALUATION: INTRODUCTION Option Values Intrinsic value - profit that could be made if the option was immediately exercised Call: stock price - exercise price Put:
More informationTABLE OF CONTENTS. Introduction Delta Delta as Hedge Ratio Gamma Other Letters Appendix
GLOBAL TABLE OF CONTENTS Introduction Delta Delta as Hedge Ratio Gamma Other Letters Appendix 3 4 5 7 9 10 HIGH RISK WARNING: Before you decide to trade either foreign currency ( Forex ) or options, carefully
More informationChapter 21 Valuing Options
Chapter 21 Valuing Options Multiple Choice Questions 1. Relative to the underlying stock, a call option always has: A) A higher beta and a higher standard deviation of return B) A lower beta and a higher
More informationIntroduction to Options. Derivatives
Introduction to Options Econ 422: Investment, Capital & Finance University of Washington Summer 2010 August 18, 2010 Derivatives A derivative is a security whose payoff or value depends on (is derived
More informationUnderlying (S) The asset, which the option buyer has the right to buy or sell. Notation: S or S t = S(t)
INTRODUCTION TO OPTIONS Readings: Hull, Chapters 8, 9, and 10 Part I. Options Basics Options Lexicon Options Payoffs (Payoff diagrams) Calls and Puts as two halves of a forward contract: the Put-Call-Forward
More informationChapter 13 The Black-Scholes-Merton Model
Chapter 13 The Black-Scholes-Merton Model March 3, 009 13.1. The Black-Scholes option pricing model assumes that the probability distribution of the stock price in one year(or at any other future time)
More informationFinance 436 Futures and Options Review Notes for Final Exam. Chapter 9
Finance 436 Futures and Options Review Notes for Final Exam Chapter 9 1. Options: call options vs. put options, American options vs. European options 2. Characteristics: option premium, option type, underlying
More informationGAMMA.0279 THETA 8.9173 VEGA 9.9144 RHO 3.5985
14 Option Sensitivities and Option Hedging Answers to Questions and Problems 1. Consider Call A, with: X $70; r 0.06; T t 90 days; 0.4; and S $60. Compute the price, DELTA, GAMMA, THETA, VEGA, and RHO
More informationWeek 12. Options on Stock Indices and Currencies: Hull, Ch. 15. Employee Stock Options: Hull, Ch. 14.
Week 12 Options on Stock Indices and Currencies: Hull, Ch. 15. Employee Stock Options: Hull, Ch. 14. 1 Options on Stock Indices and Currencies Objective: To explain the basic asset pricing techniques used
More informationThe Black-Scholes Model
Chapter 4 The Black-Scholes Model 4. Introduction Easily the best known model of option pricing, the Black-Scholes model is also one of the most widely used models in practice. It forms the benchmark model
More informationStock. Call. Put. Bond. Option Fundamentals
Option Fundamentals Payoff Diagrams hese are the basic building blocks of financial engineering. hey represent the payoffs or terminal values of various investment choices. We shall assume that the maturity
More informationJorge Cruz Lopez - Bus 316: Derivative Securities. Week 11. The Black-Scholes Model: Hull, Ch. 13.
Week 11 The Black-Scholes Model: Hull, Ch. 13. 1 The Black-Scholes Model Objective: To show how the Black-Scholes formula is derived and how it can be used to value options. 2 The Black-Scholes Model 1.
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 informationDIGITAL FOREX OPTIONS
DIGITAL FOREX OPTIONS OPENGAMMA QUANTITATIVE RESEARCH Abstract. Some pricing methods for forex digital options are described. The price in the Garhman-Kohlhagen model is first described, more for completeness
More informationDigital Options. and d 1 = d 2 + σ τ, P int = e rτ[ KN( d 2) FN( d 1) ], with d 2 = ln(f/k) σ2 τ/2
Digital Options The manager of a proprietary hedge fund studied the German yield curve and noticed that it used to be quite steep. At the time of the study, the overnight rate was approximately 3%. The
More informationLecture 7: Bounds on Options Prices Steven Skiena. http://www.cs.sunysb.edu/ skiena
Lecture 7: Bounds on Options Prices Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena Option Price Quotes Reading the
More informationJorge Cruz Lopez - Bus 316: Derivative Securities. Week 9. Binomial Trees : Hull, Ch. 12.
Week 9 Binomial Trees : Hull, Ch. 12. 1 Binomial Trees Objective: To explain how the binomial model can be used to price options. 2 Binomial Trees 1. Introduction. 2. One Step Binomial Model. 3. Risk Neutral
More informationImplied Volatility Surface
Implied Volatility Surface Liuren Wu Zicklin School of Business, Baruch College Options Markets (Hull chapter: 16) Liuren Wu Implied Volatility Surface Options Markets 1 / 18 Implied volatility Recall
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 informationOptions on an Asset that Yields Continuous Dividends
Finance 400 A. Penati - G. Pennacchi Options on an Asset that Yields Continuous Dividends I. Risk-Neutral Price Appreciation in the Presence of Dividends Options are often written on what can be interpreted
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 informationValuing Stock Options: The Black-Scholes-Merton Model. Chapter 13
Valuing Stock Options: The Black-Scholes-Merton Model Chapter 13 Fundamentals of Futures and Options Markets, 8th Ed, Ch 13, Copyright John C. Hull 2013 1 The Black-Scholes-Merton Random Walk Assumption
More informationOptions, Derivatives, Risk Management
1/1 Options, Derivatives, Risk Management (Welch, Chapter 27) Ivo Welch UCLA Anderson School, Corporate Finance, Winter 2014 January 13, 2015 Did you bring your calculator? Did you read these notes and
More informationOPTIONS. FINANCE TRAINER International Options / Page 1 of 38
OPTIONS 1. FX Options... 3 1.1 Terminology... 4 1.2 The Four Basic Positions... 5 1.3 Standard Options... 7 1.4 Exotic Options... 7 1.4.1 Asian Option (Average Rate Option, ARO)... 7 1.4.2 Compound Option...
More informationSteve Meizinger. FX Options Pricing, what does it Mean?
Steve Meizinger FX Options Pricing, what does it Mean? For the sake of simplicity, the examples that follow do not take into consideration commissions and other transaction fees, tax considerations, or
More informationOption Portfolio Modeling
Value of Option (Total=Intrinsic+Time Euro) Option Portfolio Modeling Harry van Breen www.besttheindex.com E-mail: h.j.vanbreen@besttheindex.com Introduction The goal of this white paper is to provide
More information第 9 讲 : 股 票 期 权 定 价 : B-S 模 型 Valuing Stock Options: The Black-Scholes Model
1 第 9 讲 : 股 票 期 权 定 价 : B-S 模 型 Valuing Stock Options: The Black-Scholes Model Outline 有 关 股 价 的 假 设 The B-S Model 隐 性 波 动 性 Implied Volatility 红 利 与 期 权 定 价 Dividends and Option Pricing 美 式 期 权 定 价 American
More informationHedging of Financial Derivatives and Portfolio Insurance
Hedging of Financial Derivatives and Portfolio Insurance Gasper Godson Mwanga African Institute for Mathematical Sciences 6, Melrose Road, 7945 Muizenberg, Cape Town South Africa. e-mail: gasper@aims.ac.za,
More informationBINOMIAL OPTION PRICING
Darden Graduate School of Business Administration University of Virginia BINOMIAL OPTION PRICING Binomial option pricing is a simple but powerful technique that can be used to solve many complex option-pricing
More informationLecture 4: The Black-Scholes model
OPTIONS and FUTURES Lecture 4: The Black-Scholes model Philip H. Dybvig Washington University in Saint Louis Black-Scholes option pricing model Lognormal price process Call price Put price Using Black-Scholes
More informationOption Properties. Liuren Wu. Zicklin School of Business, Baruch College. Options Markets. (Hull chapter: 9)
Option Properties Liuren Wu Zicklin School of Business, Baruch College Options Markets (Hull chapter: 9) Liuren Wu (Baruch) Option Properties Options Markets 1 / 17 Notation c: European call option price.
More informationOptions 1 OPTIONS. Introduction
Options 1 OPTIONS Introduction A derivative is a financial instrument whose value is derived from the value of some underlying asset. A call option gives one the right to buy an asset at the exercise or
More informationFactors Affecting Option Prices. Ron Shonkwiler (shonkwiler@math.gatech.edu) www.math.gatech.edu/ shenk
1 Factors Affecting Option Prices Ron Shonkwiler (shonkwiler@math.gatech.edu) www.math.gatech.edu/ shenk 1 Factors Affecting Option Prices Ron Shonkwiler (shonkwiler@math.gatech.edu) www.math.gatech.edu/
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 informationStudy on the Volatility Smile of EUR/USD Currency Options and Trading Strategies
Prof. Joseph Fung, FDS Study on the Volatility Smile of EUR/USD Currency Options and Trading Strategies BY CHEN Duyi 11050098 Finance Concentration LI Ronggang 11050527 Finance Concentration An Honors
More informationOptions. Pricing. Binomial models. Black-Scholes model. Greeks
Options. Priing. Binomial moels. Blak-Sholes moel. Greeks 1. Binomial moel,. Blak-Sholes moel, assmptions, moifiations (iviens, rreny options, options on ftres 3. Implie volatility 4. Sensitivity measres
More informationFundamentals of Futures and Options (a summary)
Fundamentals of Futures and Options (a summary) Roger G. Clarke, Harindra de Silva, CFA, and Steven Thorley, CFA Published 2013 by the Research Foundation of CFA Institute Summary prepared by Roger G.
More informationChapter 1: Financial Markets and Financial Derivatives
Chapter 1: Financial Markets and Financial Derivatives 1.1 Financial Markets Financial markets are markets for financial instruments, in which buyers and sellers find each other and create or exchange
More informationBlack-Scholes. 3.1 Digital Options
3 Black-Scholes In this chapter, we will study the value of European digital and share digital options and standard European puts and calls under the Black-Scholes assumptions. We will also explain how
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 informationMarket s gamma hedging absorption capability for barrier options
Market s gamma hedging absorption capability for barrier options Alexandre Andriot, Pierre Nirascou Supervisor: Lecturer Mr. Hamel, Paris Dauphine University, Master 272 05/12/2013 Table of contents I
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 informationForeign Exchange Symmetries
Foreign Exchange Symmetries Uwe Wystup MathFinance AG Waldems, Germany www.mathfinance.com 8 September 2008 Contents 1 Foreign Exchange Symmetries 2 1.1 Motivation.................................... 2
More informationLecture 6: Option Pricing Using a One-step Binomial Tree. Friday, September 14, 12
Lecture 6: Option Pricing Using a One-step Binomial Tree An over-simplified model with surprisingly general extensions a single time step from 0 to T two types of traded securities: stock S and a bond
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 informationTHETA AS AN ASSET CLASS By Dominik von Eynern, Swiss Alpha Asset Management GMBH
THETA AS AN ASSET CLASS By Dominik von Eynern, Swiss Alpha Asset Management GMBH It is essential to find new ways or asset classes to deliver consistent and uncorrelated returns within the asset management
More informationRisk Adjustment for Poker Players
Risk Ajustment for Poker Players William Chin DePaul University, Chicago, Illinois Marc Ingenoso Conger Asset Management LLC, Chicago, Illinois September, 2006 Introuction In this article we consier risk
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 informationMore on Market-Making and Delta-Hedging
More on Market-Making and Delta-Hedging What do market makers do to delta-hedge? Recall that the delta-hedging strategy consists of selling one option, and buying a certain number shares An example of
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 informationOption Premium = Intrinsic. Speculative Value. Value
Chapters 4/ Part Options: Basic Concepts Options Call Options Put Options Selling Options Reading The Wall Street Journal Combinations of Options Valuing Options An Option-Pricing Formula Investment in
More informationOne Period Binomial Model
FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 One Period Binomial Model These notes consider the one period binomial model to exactly price an option. We will consider three different methods of pricing
More informationIntroduction to Binomial Trees
11 C H A P T E R Introduction to Binomial Trees A useful and very popular technique for pricing an option involves constructing a binomial tree. This is a diagram that represents di erent possible paths
More information2. How is a fund manager motivated to behave with this type of renumeration package?
MØA 155 PROBLEM SET: Options Exercise 1. Arbitrage [2] In the discussions of some of the models in this course, we relied on the following type of argument: If two investment strategies have the same payoff
More informationCHAPTER 15. Option Valuation
CHAPTER 15 Option Valuation Just what is an option worth? Actually, this is one of the more difficult questions in finance. Option valuation is an esoteric area of finance since it often involves complex
More informationHow to use the Options/Warrants Calculator?
How to use the Options/Warrants Calculator? 1. Introduction Options/Warrants Calculator is a tool for users to estimate the theoretical prices of options/warrants in various market conditions by inputting
More information