Option Pricing. Chapter 11 Options on Futures. Stefan Ankirchner. University of Bonn. last update: 13/01/2014 at 14:25

Size: px
Start display at page:

Download "Option Pricing. Chapter 11 Options on Futures. Stefan Ankirchner. University of Bonn. last update: 13/01/2014 at 14:25"

Transcription

1 Option Pricing Chapter 11 Options on Futures Stefan Ankirchner University of Bonn last update: 13/01/2014 at 14:25 Stefan Ankirchner Option Pricing 1

2 Agenda Forward contracts Definition Determining forward prices Futures contracts Definition The margining mechanism Options on Futures Pricing PDEs Black76 formula Further reading: Chapter 5 & 16 in J. C. Hull: Options, Futures, and other Derivatives, Prentice-Hall, New York. Stefan Ankirchner Option Pricing 2

3 Forward contracts Definition from Wikipedia: A forward contract, or simply a forward, is a contract between two parties to buy or sell an asset at a specified future time at a price agreed today. In a forward contract one specifies the asset to be delivered, the quantity, the delivery date, the delivery price. The buyer of the contract is said to be long in the asset, and the seller is said to be short. Stefan Ankirchner Option Pricing 3

4 Forward prices The delivery price is usually chosen such that it does not cost anything to enter the forward contract. This particular price is called forward price. This means that at the time where a forward contract is entered, the forward price is equal to the delivery price. Notice that between the two parties making a forward contract there is only a cash resp. asset flow at the delivery date, but not at the contract date. Stefan Ankirchner Option Pricing 4

5 Investment and consumption assets There are many different types of assets underlying forward contracts. We distinguish between the following two: Investment assets stocks bonds gold... Consumption assets commodities (e.g. crude oil, copper,...) orange juice... Stefan Ankirchner Option Pricing 5

6 Forward prices for investment assets The forward price of an investment asset can be determined from its spot price and other observable market factors. Notation S 0 = spot price of the asset T = delivery date r = risk free rate (continuous compounding) F 0 = forward price Theorem Consider an investment asset providing no additional income (e.g. a stock paying no dividends, or a zero-coupon bond). The only arbitrage free forward price for the asset to be delivered at T is F 0 = e rt S 0. (1) Stefan Ankirchner Option Pricing 6

7 Constructive proof of (1) Proof of (1): Suppose first that F 0 > e rt S 0. Set up the following portfolio: forward contract underl. asset cash position S 0 portfolio value at time 0 is zero: V 0 = 0 + S 0 S 0 = 0 portfolio value at time T is positive with probability one: V T = (S T F 0 ) + S T e rt S 0 = F 0 e rt S 0 > 0 Thus the market admits arbitrage. Stefan Ankirchner Option Pricing 7

8 Constructive proof of (1) cont d Next suppose that F 0 < e rt S 0. Set up the following portfolio: forward contract underl. asset cash position S 0 portfolio value at time 0 is zero: V 0 = 0 S 0 + S 0 = 0 portfolio value at time T is positive with probability one: V T = (S T F 0 ) S T + e rt S 0 = e rt S 0 F 0 > 0 Again the market admits arbitrage. Therefore F 0 = e rt S 0 is the unique arbitrage free price. Stefan Ankirchner Option Pricing 8

9 Proof of (1) via risk neutral pricing A forward can be seen as a derivative with payoff S T K, where K is the delivery price. According to the pricing principle, the arbitrage free price is the expected payoff with respect to the risk neutral measure Q. As it costs nothing to enter a forward contract, in an arbitrage free market is must hold true that E Q (S T F 0 ) = 0. (2) Under Q, the discounted asset price is a martingale; hence E Q (e rt S T ) = S 0. Therefore, with (2), which entails (1). F 0 = E Q (S T ) = e rt E Q (e rt S T ) = e rt S 0, Stefan Ankirchner Option Pricing 9

10 Forward price of dividend paying stocks Theorem Consider a dividend paying stock, and let I be the discounted value of all dividends paid up to T. The only arbitrage free forward price with delivery at T is F 0 = e rt (S 0 I ). (3) Sketch of the proof: If F 0 > e rt (S 0 I ), then consider the portfolio forward contract underl. asset cash position S 0 The portfolio value at time 0 is zero, and at T it satisfies V T = (S T F 0) + S T e rt S 0 + e rt I = F 0 e rt (S 0 I ) > 0, and hence the market admits arbitrage. The case F 0 < e rt (S 0 I ) can be treated similarly. Stefan Ankirchner Option Pricing 10

11 The margining mechanism Futures prices Futures contracts Like a forward, a futures contract (or simply a futures) is an agreement to buy/sell an asset at a future time at a price specified already today. The main difference to forwards: futures are exchange-traded, whereas forwards are traded OTC (over-the-counter). Stefan Ankirchner Option Pricing 11

12 The margining mechanism Futures prices Some stylized differences between forwards and futures futures forwards exchange traded OTC traded highly standardized tailor-made margin payments no margining no counterparty risk counterparty risk Stefan Ankirchner Option Pricing 12

13 The margining mechanism Futures prices The margining mechanism Anyone trading futures has to set up a margin account at the exchange. At the moment a futures contract is entered, an initial margin has to be deposited. At the end of every trading day the futures position is marked-to-market, and the margin account is adjusted accordingly. The precise mechanism is best explained with an Example: Suppose that on July 4, 2011, an investor buys one gold futures at a price of 1500$/oz. Delivery is December 2011, and the contract size is 100 ounces. Suppose that the exchange requires an initial margin of 2000$. Stefan Ankirchner Option Pricing 13

14 The margining mechanism Futures prices Example cont d Assume that prices fall after the investor has entered the contract, and that the futures closes at 1492$/oz on July 4. The value of the investor s position has declined by ( ) $ 100oz = 800$. oz The margin account is reduced by 800$, and has a new balance of 1200$. Suppose that on July 5 prices soar, and the futures closes at 1520$/oz. Then the exchange transfers ( ) $ 100oz = 2800$ oz to the margin account, having then a new balance of 4000$.... the account is adjusted like this every day up to delivery... The investor can withdraw from the margin account the cash exceeding the initial margin, but has to make sure that a minimum maintenance margin is always set. Stefan Ankirchner Option Pricing 14

15 The margining mechanism Futures prices Do futures prices coincide with forward prices? Definition. futures price = the delivery price of a traded futures contract (note that it does not cost anything to enter a futures) Under the assumptions that the interest rate r is constant, there is no default risk, the futures price, in theory, is equal to the forward price. In particular, the futures price of an investment asset with no additional income is given by e rt S 0. Caution: Under stochastic interest rate r(t), t 0, the futures price Fut and the forward price For, with delivery T, are given by Fut = E Q [S T ] For = S 0 E Q [ T 0 e r(s) ds] (see e.g. Ch.5 in Shreve: Stoch. Calculus for Finance). Stefan Ankirchner Option Pricing 15

16 The margining mechanism Futures prices Futures price dynamcis under the risk neutral measure Theorem Consider a futures on an investment asset without additional income. Then the futures price is a martingale wrt the risk neutral measure Q. Proof under the additional assumption that the interest rate is constant. Remark. If interest rates are stochastic, then in general forward prices are not martingales, whereas futures prices are. Stefan Ankirchner Option Pricing 16

17 Properties Self-financing portfolios Options on Futures Definition. The owner of an option on a futures contract, or simply a futures option, has the right, but not the obligation, to enter a futures contract at a specific future date. A call is the option to buy the futures, and a put is the option to sell it. Option strike price = contract price Examples Treasury bond futures options Crude Oil futures options see CBOT for many further examples... Question: Why options on futures and not on the spot itself? Stefan Ankirchner Option Pricing 17

18 Properties Self-financing portfolios Futures option value at expiration Notation T = expiration date of the option ( the delivery date of the underlying futures) F t = futures price at t K = strike price Payoff resp. value at expiration of a call: (F T K) + of a put: (K F T ) + Stefan Ankirchner Option Pricing 18

19 Properties Self-financing portfolios Self-financing futures portfolios Consider a portfolio consisting of ξ futures contracts and η bonds at time t. Let S 0 t = bond price at t V t = portfolio value at t After any margin payment: futures position has value zero. Therefore, V t = ηs 0 t. Let t + δ be the next trading day. By how much does the portfolio value change? Let F = F t+δ F t be the futures price change. Then the margin account is adjusted by ξ F the bond position earns an interest of (e r δ 1)V t rδv t the new portfolio value is V t+δ ξ F + rδv t Stefan Ankirchner Option Pricing 19

20 Properties Self-financing portfolios Self-financing futures portfolios V t+δ V t ξ F + rδv t Letting δ 0, we get dv t = ξ(t)df t + rv t dt. (4) Equation (4) is the self-financing condition for futures portfolios. Remark. The solution of the SDE (4) is given by t ) V t = e (V rt 0 + ξ(s)e rs df s. 0 Proof. Stefan Ankirchner Option Pricing 20

21 Properties Self-financing portfolios Put-call parity for futures options Theorem Suppose that a European call on a futures, with strike K and expiration T, is traded at a market price of C. Then the only arbitrage free price for a put with same strike K and expiration T is given by P = C e rt F 0 + e rt K, (5) where F 0 is the current futures price. Proof. Stefan Ankirchner Option Pricing 21

22 Deriving pricing PDEs Black76 formula Black s model for futures options In a famous model by Black 1 the futures price is assumed to satisfy the dynamics df t = σf t dw t, where W is a BM with respect to the risk-neutral measure Q. On the following slides we derive the pricing PDE for a futures option with payoff h : R R +. The option is a call if h(x) = (x K) +, and a put if h(x) = (K x) +. 1 F. Black. The pricing of Commodity Contracts, Journal of Financial Economics, 3 (1976), Stefan Ankirchner Option Pricing 22

23 Deriving pricing PDEs Black76 formula Letting the 4 steps roll 1) Assume that the futures option is replicable and that its value at time t [0, T ] is equal to v(t, F t ), where v C 1,2. 2) v(t, F t ) is an Ito process with 2 decompositions: From the self-financing condition we have dv(t, F t ) = ξ(t)df t + r v(t, F t )dt and from Ito s formula = ξ(t)σf t dw t + r v(t, F t )dt, dv(t, F t ) = v t (t, F t )dt + v f (t, F t )df t v ff (t, F t )df t df t = v t (t, F t )dt + v f (t, F t )σf t dw t v ff (t, F t )σ 2 F 2 t dt. Stefan Ankirchner Option Pricing 23

24 4 steps cont d Deriving pricing PDEs Black76 formula 3) Matching the coefficients yields first ξ(t) = v f (t, F t ), and then that v(t, f ) has to solve the PDE v t (t, f ) σ2 f 2 v ff (t, f ) r v(t, f ) = 0, (6) with terminal condition v(t, x) = h(x). Stefan Ankirchner Option Pricing 24

25 4 steps cont d Deriving pricing PDEs Black76 formula 4) Discounted Feynman-Kac shows that the solution of the PDE (6) is given by v(t, x) = e r(t t) E t,x [h(f T )]. Stefan Ankirchner Option Pricing 25

26 Deriving pricing PDEs Black76 formula Price of a futures call: the Black76 formula Assume a futures price of F t = f at time t. Then the time t arbitrage free price of a European call futures option, with strike K e and expiration date T, is given by where Black76 call(f, K, T t, σ X, r) = e r(t t) f Φ(d 1 ) e r(t t) KΦ(d 2 ), Proof. d 1 = log ( ) f σ K + X 2 2 (T t) σ, T t d 2 = d 1 σ T t. Stefan Ankirchner Option Pricing 26

27 Deriving pricing PDEs Black76 formula futures put and link to Black Scholes formula The arbitrage free price of a European put futures option is given by Black76 put(f, K, T t, σ, r) = e r(t t) KΦ( d 2 ) e r(t t) f Φ( d 1 ). Remark. Note that Black76 call(f, K, τ, σ, r) = BS call(fe rτ, K, τ, σ, r). Stefan Ankirchner Option Pricing 27

Option Pricing. Chapter 4 Including dividends in the BS model. Stefan Ankirchner. University of Bonn. last update: 6th November 2013

Option Pricing. Chapter 4 Including dividends in the BS model. Stefan Ankirchner. University of Bonn. last update: 6th November 2013 Option Pricing Chapter 4 Including dividends in the BS model Stefan Ankirchner University of Bonn last update: 6th November 2013 Stefan Ankirchner Option Pricing 1 Dividend payments So far: we assumed

More information

Option Pricing. Chapter 9 - Barrier Options - Stefan Ankirchner. University of Bonn. last update: 9th December 2013

Option Pricing. Chapter 9 - Barrier Options - Stefan Ankirchner. University of Bonn. last update: 9th December 2013 Option Pricing Chapter 9 - Barrier Options - Stefan Ankirchner University of Bonn last update: 9th December 2013 Stefan Ankirchner Option Pricing 1 Standard barrier option Agenda What is a barrier option?

More information

Lecture 6 Black-Scholes PDE

Lecture 6 Black-Scholes PDE Lecture 6 Black-Scholes PDE Lecture Notes by Andrzej Palczewski Computational Finance p. 1 Pricing function Let the dynamics of underlining S t be given in the risk-neutral measure Q by If the contingent

More information

Mathematical Finance

Mathematical Finance Mathematical Finance Option Pricing under the Risk-Neutral Measure Cory Barnes Department of Mathematics University of Washington June 11, 2013 Outline 1 Probability Background 2 Black Scholes for European

More information

Option Valuation. Chapter 21

Option 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 information

On Black-Scholes Equation, Black- Scholes Formula and Binary Option Price

On 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 information

Lecture 15. Sergei Fedotov. 20912 - Introduction to Financial Mathematics. Sergei Fedotov (University of Manchester) 20912 2010 1 / 6

Lecture 15. Sergei Fedotov. 20912 - Introduction to Financial Mathematics. Sergei Fedotov (University of Manchester) 20912 2010 1 / 6 Lecture 15 Sergei Fedotov 20912 - Introduction to Financial Mathematics Sergei Fedotov (University of Manchester) 20912 2010 1 / 6 Lecture 15 1 Black-Scholes Equation and Replicating Portfolio 2 Static

More information

Chapter 1: Financial Markets and Financial Derivatives

Chapter 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 information

American Options. An Undergraduate Introduction to Financial Mathematics. J. Robert Buchanan. J. Robert Buchanan American Options

American Options. An Undergraduate Introduction to Financial Mathematics. J. Robert Buchanan. J. Robert Buchanan American Options American Options An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2010 Early Exercise Since American style options give the holder the same rights as European style options plus

More information

The Black-Scholes Formula

The 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 information

How To Price A Call Option

How To Price A Call Option Now by Itô s formula But Mu f and u g in Ū. Hence τ θ u(x) =E( Mu(X) ds + u(x(τ θ))) 0 τ θ u(x) E( f(x) ds + g(x(τ θ))) = J x (θ). 0 But since u(x) =J x (θ ), we consequently have u(x) =J x (θ ) = min

More information

24. Pricing Fixed Income Derivatives. through Black s Formula. MA6622, Ernesto Mordecki, CityU, HK, 2006. References for this Lecture:

24. Pricing Fixed Income Derivatives. through Black s Formula. MA6622, Ernesto Mordecki, CityU, HK, 2006. References for this Lecture: 24. Pricing Fixed Income Derivatives through Black s Formula MA6622, Ernesto Mordecki, CityU, HK, 2006. References for this Lecture: John C. Hull, Options, Futures & other Derivatives (Fourth Edition),

More information

Lecture. S t = S t δ[s t ].

Lecture. S t = S t δ[s t ]. Lecture In real life the vast majority of all traded options are written on stocks having at least one dividend left before the date of expiration of the option. Thus the study of dividends is important

More information

Manual for SOA Exam FM/CAS Exam 2.

Manual for SOA Exam FM/CAS Exam 2. Manual for SOA Exam FM/CAS Exam 2. Chapter 7. Derivatives markets. c 2009. Miguel A. Arcones. All rights reserved. Extract from: Arcones Manual for the SOA Exam FM/CAS Exam 2, Financial Mathematics. Fall

More information

FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008

FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 Options These notes consider the way put and call options and the underlying can be combined to create hedges, spreads and combinations. We will consider the

More information

Finance 400 A. Penati - G. Pennacchi. Option Pricing

Finance 400 A. Penati - G. Pennacchi. Option Pricing Finance 400 A. Penati - G. Pennacchi Option Pricing Earlier we derived general pricing relationships for contingent claims in terms of an equilibrium stochastic discount factor or in terms of elementary

More information

CS 522 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options

CS 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 information

Introduction to Mathematical Finance

Introduction to Mathematical Finance Introduction to Mathematical Finance Martin Baxter Barcelona 11 December 2007 1 Contents Financial markets and derivatives Basic derivative pricing and hedging Advanced derivatives 2 Banking Retail banking

More information

Trading Strategies Involving Options. Chapter 11

Trading Strategies Involving Options. Chapter 11 Trading Strategies Involving Options Chapter 11 1 Strategies to be Considered A risk-free bond and an option to create a principal-protected note A stock and an option Two or more options of the same type

More information

Lecture 21 Options Pricing

Lecture 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 information

Introduction, Forwards and Futures

Introduction, Forwards and Futures Introduction, Forwards and Futures Liuren Wu Zicklin School of Business, Baruch College Fall, 2007 (Hull chapters: 1,2,3,5) Liuren Wu Introduction, Forwards & Futures Option Pricing, Fall, 2007 1 / 35

More information

Monte Carlo Methods in Finance

Monte Carlo Methods in Finance Author: Yiyang Yang Advisor: Pr. Xiaolin Li, Pr. Zari Rachev Department of Applied Mathematics and Statistics State University of New York at Stony Brook October 2, 2012 Outline Introduction 1 Introduction

More information

Call and Put. Options. American and European Options. Option Terminology. Payoffs of European Options. Different Types of Options

Call and Put. Options. American and European Options. Option Terminology. Payoffs of European Options. Different Types of Options Call and Put Options A call option gives its holder the right to purchase an asset for a specified price, called the strike price, on or before some specified expiration date. A put option gives its holder

More information

Forward Price. The payoff of a forward contract at maturity is S T X. Forward contracts do not involve any initial cash flow.

Forward Price. The payoff of a forward contract at maturity is S T X. Forward contracts do not involve any initial cash flow. Forward Price The payoff of a forward contract at maturity is S T X. Forward contracts do not involve any initial cash flow. The forward price is the delivery price which makes the forward contract zero

More information

Lectures. Sergei Fedotov. 20912 - Introduction to Financial Mathematics. No tutorials in the first week

Lectures. Sergei Fedotov. 20912 - Introduction to Financial Mathematics. No tutorials in the first week Lectures Sergei Fedotov 20912 - Introduction to Financial Mathematics No tutorials in the first week Sergei Fedotov (University of Manchester) 20912 2010 1 / 1 Lecture 1 1 Introduction Elementary economics

More information

Determination of Forward and Futures Prices. Chapter 5

Determination of Forward and Futures Prices. Chapter 5 Determination of Forward and Futures Prices Chapter 5 Fundamentals of Futures and Options Markets, 8th Ed, Ch 5, Copyright John C. Hull 2013 1 Consumption vs Investment Assets Investment assets are assets

More information

Institutional Finance 08: Dynamic Arbitrage to Replicate Non-linear Payoffs. Binomial Option Pricing: Basics (Chapter 10 of McDonald)

Institutional 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 information

Four Derivations of the Black Scholes PDE by Fabrice Douglas Rouah www.frouah.com www.volopta.com

Four Derivations of the Black Scholes PDE by Fabrice Douglas Rouah www.frouah.com www.volopta.com Four Derivations of the Black Scholes PDE by Fabrice Douglas Rouah www.frouah.com www.volopta.com In this Note we derive the Black Scholes PDE for an option V, given by @t + 1 + rs @S2 @S We derive the

More information

Lecture 4: Properties of stock options

Lecture 4: Properties of stock options Lecture 4: Properties of stock options Reading: J.C.Hull, Chapter 9 An European call option is an agreement between two parties giving the holder the right to buy a certain asset (e.g. one stock unit)

More information

EC372 Bond and Derivatives Markets Topic #5: Options Markets I: fundamentals

EC372 Bond and Derivatives Markets Topic #5: Options Markets I: fundamentals EC372 Bond and Derivatives Markets Topic #5: Options Markets I: fundamentals R. E. Bailey Department of Economics University of Essex Outline Contents 1 Call options and put options 1 2 Payoffs on options

More information

1 The Black-Scholes model: extensions and hedging

1 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 information

Convenient Conventions

Convenient Conventions C: call value. P : put value. X: strike price. S: stock price. D: dividend. Convenient Conventions c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 168 Payoff, Mathematically Speaking The payoff

More information

The Black-Scholes pricing formulas

The Black-Scholes pricing formulas The Black-Scholes pricing formulas Moty Katzman September 19, 2014 The Black-Scholes differential equation Aim: Find a formula for the price of European options on stock. Lemma 6.1: Assume that a stock

More information

Option pricing. Vinod Kothari

Option 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 information

Options. Moty Katzman. September 19, 2014

Options. Moty Katzman. September 19, 2014 Options Moty Katzman September 19, 2014 What are options? Options are contracts conferring certain rights regarding the buying or selling of assets. A European call option gives the owner the right to

More information

Chapter 1 - Introduction

Chapter 1 - Introduction Chapter 1 - Introduction Derivative securities Futures contracts Forward contracts Futures and forward markets Comparison of futures and forward contracts Options contracts Options markets Comparison of

More information

Options pricing in discrete systems

Options pricing in discrete systems UNIVERZA V LJUBLJANI, FAKULTETA ZA MATEMATIKO IN FIZIKO Options pricing in discrete systems Seminar II Mentor: prof. Dr. Mihael Perman Author: Gorazd Gotovac //2008 Abstract This paper is a basic introduction

More information

Moreover, under the risk neutral measure, it must be the case that (5) r t = µ t.

Moreover, under the risk neutral measure, it must be the case that (5) r t = µ t. LECTURE 7: BLACK SCHOLES THEORY 1. Introduction: The Black Scholes Model In 1973 Fisher Black and Myron Scholes ushered in the modern era of derivative securities with a seminal paper 1 on the pricing

More information

Introduction to Futures Contracts

Introduction to Futures Contracts Introduction to Futures Contracts September 2010 PREPARED BY Eric Przybylinski Research Analyst Gregory J. Leonberger, FSA Director of Research Abstract Futures contracts are widely utilized throughout

More information

Lecture 5: Put - Call Parity

Lecture 5: Put - Call Parity Lecture 5: Put - Call Parity Reading: J.C.Hull, Chapter 9 Reminder: basic assumptions 1. There are no arbitrage opportunities, i.e. no party can get a riskless profit. 2. Borrowing and lending are possible

More information

1 Introduction to Option Pricing

1 Introduction to Option Pricing ESTM 60202: Financial Mathematics Alex Himonas 03 Lecture Notes 1 October 7, 2009 1 Introduction to Option Pricing We begin by defining the needed finance terms. Stock is a certificate of ownership of

More information

Consider a European call option maturing at time T

Consider a European call option maturing at time T Lecture 10: Multi-period Model Options Black-Scholes-Merton model Prof. Markus K. Brunnermeier 1 Binomial Option Pricing Consider a European call option maturing at time T with ihstrike K: C T =max(s T

More information

BUSM 411: Derivatives and Fixed Income

BUSM 411: Derivatives and Fixed Income BUSM 411: Derivatives and Fixed Income 2. Forwards, Options, and Hedging This lecture covers the basic derivatives contracts: forwards (and futures), and call and put options. These basic contracts are

More information

Chapter 3: Commodity Forwards and Futures

Chapter 3: Commodity Forwards and Futures Chapter 3: Commodity Forwards and Futures In the previous chapter we study financial forward and futures contracts and we concluded that are all alike. Each commodity forward, however, has some unique

More information

Option Pricing. Chapter 12 - Local volatility models - Stefan Ankirchner. University of Bonn. last update: 13th January 2014

Option Pricing. Chapter 12 - Local volatility models - Stefan Ankirchner. University of Bonn. last update: 13th January 2014 Option Pricing Chapter 12 - Local volatility models - Stefan Ankirchner University of Bonn last update: 13th January 2014 Stefan Ankirchner Option Pricing 1 Agenda The volatility surface Local volatility

More information

Black-Scholes Equation for Option Pricing

Black-Scholes Equation for Option Pricing Black-Scholes Equation for Option Pricing By Ivan Karmazin, Jiacong Li 1. Introduction In early 1970s, Black, Scholes and Merton achieved a major breakthrough in pricing of European stock options and there

More information

Futures Price d,f $ 0.65 = (1.05) (1.04)

Futures Price d,f $ 0.65 = (1.05) (1.04) 24 e. Currency Futures In a currency futures contract, you enter into a contract to buy a foreign currency at a price fixed today. To see how spot and futures currency prices are related, note that holding

More information

Pricing of an Exotic Forward Contract

Pricing of an Exotic Forward Contract Pricing of an Exotic Forward Contract Jirô Akahori, Yuji Hishida and Maho Nishida Dept. of Mathematical Sciences, Ritsumeikan University 1-1-1 Nojihigashi, Kusatsu, Shiga 525-8577, Japan E-mail: {akahori,

More information

Does 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 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 information

CHAPTER 22: FUTURES MARKETS

CHAPTER 22: FUTURES MARKETS CHAPTER 22: FUTURES MARKETS PROBLEM SETS 1. There is little hedging or speculative demand for cement futures, since cement prices are fairly stable and predictable. The trading activity necessary to support

More information

Hedging Strategies Using Futures. Chapter 3

Hedging Strategies Using Futures. Chapter 3 Hedging Strategies Using Futures Chapter 3 Fundamentals of Futures and Options Markets, 8th Ed, Ch3, Copyright John C. Hull 2013 1 The Nature of Derivatives A derivative is an instrument whose value depends

More information

Determination of Forward and Futures Prices

Determination of Forward and Futures Prices Determination of Forward and Futures Prices Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012 Short selling A popular trading (arbitrage) strategy is the shortselling or

More information

2 Stock Price. Figure S1.1 Profit from long position in Problem 1.13

2 Stock Price. Figure S1.1 Profit from long position in Problem 1.13 Problem 1.11. A cattle farmer expects to have 12, pounds of live cattle to sell in three months. The livecattle futures contract on the Chicago Mercantile Exchange is for the delivery of 4, pounds of cattle.

More information

On Market-Making and Delta-Hedging

On Market-Making and Delta-Hedging On Market-Making and Delta-Hedging 1 Market Makers 2 Market-Making and Bond-Pricing On Market-Making and Delta-Hedging 1 Market Makers 2 Market-Making and Bond-Pricing What to market makers do? Provide

More information

Underlying (S) The asset, which the option buyer has the right to buy or sell. Notation: S or S t = S(t)

Underlying (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 information

Forwards, Swaps and Futures

Forwards, Swaps and Futures IEOR E4706: Financial Engineering: Discrete-Time Models c 2010 by Martin Haugh Forwards, Swaps and Futures These notes 1 introduce forwards, swaps and futures, and the basic mechanics of their associated

More information

Hedging with Futures and Options: Supplementary Material. Global Financial Management

Hedging with Futures and Options: Supplementary Material. Global Financial Management Hedging with Futures and Options: Supplementary Material Global Financial Management Fuqua School of Business Duke University 1 Hedging Stock Market Risk: S&P500 Futures Contract A futures contract on

More information

Finance 436 Futures and Options Review Notes for Final Exam. Chapter 9

Finance 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 information

Lecture 7: Bounds on Options Prices Steven Skiena. http://www.cs.sunysb.edu/ skiena

Lecture 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 information

Hedging. An Undergraduate Introduction to Financial Mathematics. J. Robert Buchanan. J. Robert Buchanan Hedging

Hedging. 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 information

Jung-Soon Hyun and Young-Hee Kim

Jung-Soon Hyun and Young-Hee Kim J. Korean Math. Soc. 43 (2006), No. 4, pp. 845 858 TWO APPROACHES FOR STOCHASTIC INTEREST RATE OPTION MODEL Jung-Soon Hyun and Young-Hee Kim Abstract. We present two approaches of the stochastic interest

More information

Option 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) 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 information

Treasury Bond Futures

Treasury Bond Futures Treasury Bond Futures Concepts and Buzzwords Basic Futures Contract Futures vs. Forward Delivery Options Reading Veronesi, Chapters 6 and 11 Tuckman, Chapter 14 Underlying asset, marking-to-market, convergence

More information

Fundamentals of Futures and Options (a summary)

Fundamentals 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 information

CRUDE OIL HEDGING STRATEGIES An Application of Currency Translated Options

CRUDE OIL HEDGING STRATEGIES An Application of Currency Translated Options CRUDE OIL HEDGING STRATEGIES An Application of Currency Translated Options Paul Obour Supervisor: Dr. Antony Ware University of Calgary PRMIA Luncheon - Bankers Hall, Calgary May 8, 2012 Outline 1 Introductory

More information

Jorge Cruz Lopez - Bus 316: Derivative Securities. Week 9. Binomial Trees : Hull, Ch. 12.

Jorge 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 information

Week 13 Introduction to the Greeks and Portfolio Management:

Week 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 information

Lecture 09: Multi-period Model Fixed Income, Futures, Swaps

Lecture 09: Multi-period Model Fixed Income, Futures, Swaps Lecture 09: Multi-period Model Fixed Income, Futures, Swaps Prof. Markus K. Brunnermeier Slide 09-1 Overview 1. Bond basics 2. Duration 3. Term structure of the real interest rate 4. Forwards and futures

More information

Review of Basic Options Concepts and Terminology

Review of Basic Options Concepts and Terminology Review of Basic Options Concepts and Terminology March 24, 2005 1 Introduction The purchase of an options contract gives the buyer the right to buy call options contract or sell put options contract some

More information

2. How is a fund manager motivated to behave with this type of renumeration package?

2. 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 information

CHAPTER 20. Financial Options. Chapter Synopsis

CHAPTER 20. Financial Options. Chapter Synopsis CHAPTER 20 Financial Options Chapter Synopsis 20.1 Option Basics A financial option gives its owner the right, but not the obligation, to buy or sell a financial asset at a fixed price on or until a specified

More information

SOCIETY OF ACTUARIES FINANCIAL MATHEMATICS. EXAM FM SAMPLE QUESTIONS Financial Economics

SOCIETY OF ACTUARIES FINANCIAL MATHEMATICS. EXAM FM SAMPLE QUESTIONS Financial Economics SOCIETY OF ACTUARIES EXAM FM FINANCIAL MATHEMATICS EXAM FM SAMPLE QUESTIONS Financial Economics June 2014 changes Questions 1-30 are from the prior version of this document. They have been edited to conform

More information

OPTIONS MARKETS AND VALUATIONS (CHAPTERS 16 & 17)

OPTIONS MARKETS AND VALUATIONS (CHAPTERS 16 & 17) OPTIONS MARKETS AND VALUATIONS (CHAPTERS 16 & 17) WHAT ARE OPTIONS? Derivative securities whose values are derived from the values of the underlying securities. Stock options quotations from WSJ. A call

More information

Options: Valuation and (No) Arbitrage

Options: 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 information

CHAPTER 22: FUTURES MARKETS

CHAPTER 22: FUTURES MARKETS CHAPTER 22: FUTURES MARKETS 1. a. The closing price for the spot index was 1329.78. The dollar value of stocks is thus $250 1329.78 = $332,445. The closing futures price for the March contract was 1364.00,

More information

Call Price as a Function of the Stock Price

Call Price as a Function of the Stock Price Call Price as a Function of the Stock Price Intuitively, the call price should be an increasing function of the stock price. This relationship allows one to develop a theory of option pricing, derived

More information

Lecture 12. Options Strategies

Lecture 12. Options Strategies Lecture 12. Options Strategies Introduction to Options Strategies Options, Futures, Derivatives 10/15/07 back to start 1 Solutions Problem 6:23: Assume that a bank can borrow or lend money at the same

More information

Option Values. Determinants of Call Option Values. CHAPTER 16 Option Valuation. Figure 16.1 Call Option Value Before Expiration

Option 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 information

Caps and Floors. John Crosby

Caps and Floors. John Crosby Caps and Floors John Crosby Glasgow University My website is: http://www.john-crosby.co.uk If you spot any typos or errors, please email me. My email address is on my website Lecture given 19th February

More information

Chapter 8 Financial Options and Applications in Corporate Finance ANSWERS TO END-OF-CHAPTER QUESTIONS

Chapter 8 Financial Options and Applications in Corporate Finance ANSWERS TO END-OF-CHAPTER QUESTIONS Chapter 8 Financial Options and Applications in Corporate Finance ANSWERS TO END-OF-CHAPTER QUESTIONS 8-1 a. An option is a contract which gives its holder the right to buy or sell an asset at some predetermined

More information

A) 1.8% B) 1.9% C) 2.0% D) 2.1% E) 2.2%

A) 1.8% B) 1.9% C) 2.0% D) 2.1% E) 2.2% 1 Exam FM Questions Practice Exam 1 1. Consider the following yield curve: Year Spot Rate 1 5.5% 2 5.0% 3 5.0% 4 4.5% 5 4.0% Find the four year forward rate. A) 1.8% B) 1.9% C) 2.0% D) 2.1% E) 2.2% 2.

More information

INTRODUCTION TO OPTIONS MARKETS QUESTIONS

INTRODUCTION TO OPTIONS MARKETS QUESTIONS INTRODUCTION TO OPTIONS MARKETS QUESTIONS 1. What is the difference between a put option and a call option? 2. What is the difference between an American option and a European option? 3. Why does an option

More information

Introduction to Arbitrage-Free Pricing: Fundamental Theorems

Introduction to Arbitrage-Free Pricing: Fundamental Theorems Introduction to Arbitrage-Free Pricing: Fundamental Theorems Dmitry Kramkov Carnegie Mellon University Workshop on Interdisciplinary Mathematics, Penn State, May 8-10, 2015 1 / 24 Outline Financial market

More information

Chapter 11 Options. Main Issues. Introduction to Options. Use of Options. Properties of Option Prices. Valuation Models of Options.

Chapter 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 information

Lecture 12: The Black-Scholes Model Steven Skiena. http://www.cs.sunysb.edu/ skiena

Lecture 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 information

Lecture 3: Put Options and Distribution-Free Results

Lecture 3: Put Options and Distribution-Free Results OPTIONS and FUTURES Lecture 3: Put Options and Distribution-Free Results Philip H. Dybvig Washington University in Saint Louis put options binomial valuation what are distribution-free results? option

More information

Option Pricing. Stefan Ankirchner. January 20, 2014. 2 Brownian motion and Stochastic Calculus

Option Pricing. Stefan Ankirchner. January 20, 2014. 2 Brownian motion and Stochastic Calculus Option Pricing Stefan Ankirchner January 2, 214 1 The Binomial Model 2 Brownian motion and Stochastic Calculus We next recall some basic results from Stochastic Calculus. We do not prove most of the results.

More information

Assumptions: No transaction cost, same rate for borrowing/lending, no default/counterparty risk

Assumptions: No transaction cost, same rate for borrowing/lending, no default/counterparty risk Derivatives Why? Allow easier methods to short sell a stock without a broker lending it. Facilitates hedging easily Allows the ability to take long/short position on less available commodities (Rice, Cotton,

More information

Invesco Great Wall Fund Management Co. Shenzhen: June 14, 2008

Invesco 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 information

Option Pricing. 1 Introduction. Mrinal K. Ghosh

Option 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 information

Online Appendix: Payoff Diagrams for Futures and Options

Online Appendix: Payoff Diagrams for Futures and Options Online Appendix: Diagrams for Futures and Options As we have seen, derivatives provide a set of future payoffs based on the price of the underlying asset. We discussed how derivatives can be mixed and

More information

第 9 讲 : 股 票 期 权 定 价 : B-S 模 型 Valuing Stock Options: The Black-Scholes Model

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 information

9 Basics of options, including trading strategies

9 Basics of options, including trading strategies ECG590I Asset Pricing. Lecture 9: Basics of options, including trading strategies 1 9 Basics of options, including trading strategies Option: The option of buying (call) or selling (put) an asset. European

More information

Lecture Notes: Basic Concepts in Option Pricing - The Black and Scholes Model

Lecture Notes: Basic Concepts in Option Pricing - The Black and Scholes Model Brunel University Msc., EC5504, Financial Engineering Prof Menelaos Karanasos Lecture Notes: Basic Concepts in Option Pricing - The Black and Scholes Model Recall that the price of an option is equal to

More information

Other variables as arguments besides S. Want those other variables to be observables.

Other variables as arguments besides S. Want those other variables to be observables. Valuation of options before expiration Need to distinguish between American and European options. Consider European options with time t until expiration. Value now of receiving c T at expiration? (Value

More information

or enters into a Futures contract (either on the IPE or the NYMEX) with delivery date September and pay every day up to maturity the margin

or enters into a Futures contract (either on the IPE or the NYMEX) with delivery date September and pay every day up to maturity the margin Cash-Futures arbitrage processes Cash futures arbitrage consisting in taking position between the cash and the futures markets to make an arbitrage. An arbitrage is a trade that gives in the future some

More information

Valuing Stock Options: The Black-Scholes-Merton Model. Chapter 13

Valuing 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 information

How To Value Options In Black-Scholes Model

How To Value Options In Black-Scholes Model Option Pricing Basics Aswath Damodaran Aswath Damodaran 1 What is an option? An option provides the holder with the right to buy or sell a specified quantity of an underlying asset at a fixed price (called

More information

Black-Scholes and the Volatility Surface

Black-Scholes and the Volatility Surface IEOR E4707: Financial Engineering: Continuous-Time Models Fall 2009 c 2009 by Martin Haugh Black-Scholes and the Volatility Surface When we studied discrete-time models we used martingale pricing to derive

More information

Swiss Risk Disclosure - Characteristics and Risks of Options

Swiss Risk Disclosure - Characteristics and Risks of Options This is a sample form and will not submit any information. Swiss Risk Disclosure for Options Print Swiss Risk Disclosure - Characteristics and Risks of Options 1. Characteristics 1.1 Definitions 1.1.1

More information