# Pricing of an Exotic Forward Contract

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Pricing of an Exotic Forward Contract Jirô Akahori, Yuji Hishida and Maho Nishida Dept. of Mathematical Sciences, Ritsumeikan University Nojihigashi, Kusatsu, Shiga , Japan {akahori, rp003994, Abstract In this paper we study a pricing problem of an exotic Forward contract. Unlike the standard Forward, the contract is not fair, and like an option, it is compensated by the premium. Using standard arguments in the Black- Scholes economy, an explicit formula for hedging as well as pricing is obtained. This is possible because of an exotic way of settlement, which is another focus of this paper. Contrary to our intuition, simpler ways of settlement do not necessarily imply a simpler formula. 1 Introduction A Forward contract is an agreement between two parties to buy or sell an asset (which can be of any kind) on a pre-agreed future day at a pre-agreed price which is called Forward price. Usually the Forward price is set to make the contract fair, and so no payment takes place on the agreement day. For details of standard Forward contracts, see e.g. [2]. If an extra agreement that benefits one party only is added to the contract, as is the situation that we will study, then an adjustment is needed to keep the contract fair. There are two possibilities: one is by a change of the Forward price, and the other is by payment at the agreement day. In this paper we will study such a contract which we call exotic Forward. We adopt the latter adjustment and study the premium how much should be paid for the contract. The Exotic Forward we will study is defined by the following

2 Extra Agreement 1. A buyer has a right to choose the price-date at the delivery day between N 1 days before the delivery day and N 2 days after, with the Extra Agreement 2 (Settlement of Difference). A settlement day is the delivery day in both cases. If a buyer chooses the N 2 days after, the payment is taken by the way of settlement of difference; she/he is paid a pre-agreed price at the delivery day T. and then is paid the difference at T + N 2. We will give an explicit formula for the premium and the corresponding hedging strategy under a Black-Scholes type model. What interests us most is that the formula is very much alike the Black-Scholes formula for the plain European option. The reduction is caused by the exotic way of settlement (Extra Agreement 2). 2 The Setting and Notations We denote the delivery day by T, T 1 := T N 1, and T 2 = T +N 2. We assume the Black-Scholes Economy; i.e., a spot price S = {S t } 0 t T2 and risk-free bond B = {B t } 0 t T2 satisfy the following stochastic differential equations: ds t = σs t dw t + µs t dt db t = rb t dt (1) where σ > 0, r 0, µ R, and W is a standard 1-dimensional Brownian motion. For simplicity, we assume that B 0 = 1. Under these settings, the Exotic Forward contract in question is restated as follows. 1. The agreement day is time 0, when the Forward price K is quoted in the market. 2. The buyer pays the premium π on the agreement day. 3. On the delivery day T, the buyer chooses the price day; T 1 or T If she/he chooses T 1, she/he is paid S T1 K. 5. If she/he chooses T 2, she/he is paid Y T for the moment. 6. And then on T 2, she/he is paid the difference S T2 K Y T. Though studying general Y T is possible, we concentrate on the case Y T = S T K in this paper. 2

3 Before stating the results, we introduce some notations. Let d ± (t, x) = (r ± 1 2 σ2 )(T t) + log(2 e r(t 2 T) )x/s T1 σ T t (2) for t [T 1, T) and x > 0. We write simply d ± for d ± (T 1, S T1 ). That is, d ± = (r ± 1 2 σ2 )(T T 1 ) + log(2 e r(t 2 T) ) σ T T 1. (3) Let also A := {S T1 (2 e r(t 2 T) )S T }. (4) Finally, we use the following conventions: φ(x) = 1 2π e x2 2 (5) and Φ(x) = x φ(y) dy, (6) the density and the distribution function of the standard Gaussian random variable. 3 The Results The first result which corresponds to the pricing is the following. Theorem 1. Under the hypothesis of No-Arbitrage, the fair premium is given by π = S 0 e r(t T 1) Φ( d ) + S 0 (2 e r(t 2 T) )Φ(d + ) e rt K. (7) From the above result, we notice that the premium of this contract is represented by a mono-integral, even though this contract has several exotic aspects. We stress that when Y T = 0, i.e. no settlement of difference, the premium is no more mono-integral; a little far away from Black-Scholes formula. Further, we can also obtain the explicit hedging strategy for the seller as follows. 3

4 Theorem 2. The contract is perfectly hedged by holding η t = e r(t T 1) Φ( d ) + (2 { e r(t2 T) )Φ(d + ) =: η 0, η 0 + (2 e r(t2 T) )φ(d + (t, S t )) B } ts T1 φ( d (t, S t )) /σ T t, B T S t (0 t < T 1 ) (T 1 t < T) 1 A (T t < T 2 ) amount of the risky asset at time t [0, T 2 ), and ν t = K/B T =: ν 0, (0 t < T 1 ) ν 0 + (2 e r(t2 T) ) S tφ(d + (t, S t )) B 1 t σ T t { + S T1 B 1 T Φ( d (t, S t )) ν A S T (B 1 T B 1 φ( d (t, S t )) σ T t T 2 ) + 1 Ω\A S T1 B 1 T }, (T 1 t < T) (T t < T 2 ) (8) (9) amount of the bond. Remark. The hedging strategy is statistical unless t [T 1, T), as is the case with standard Forward contracts; known as cost of carry (see e.g. [2]). The exotic aspect of our contract forces the seller to hedge dynamically, though it is needed only for t [T 1, T). 4 No-Arbitrage Hypothesis Firstly, we review the consequences of the No-Arbitrage hypothesis on the Black-Scholes economy (1). Let (Ω, F, P) be a probability space on which W is defined, and {F t } be the natural filtration of W. Put θ := (µ r)/σ. The P-{F }-martingale Z t := exp(θw t 1 2 θ2 t) defines an equivalent probability measure ˆP by d ˆP dp = Z t, t [0, T 2 ]. (10) Ft The probability measure ˆP is usually called equivalent martingale measure, EMM for short, and plays a central role in the pricing of options and other derivatives. 4

5 The Black-Scholes economy (1) under the EMM is ds t = σs t dŵ t + rs t dt db t = rb t dt, (11) or equivalently S t = S 0 exp{σŵ t σ2 t 2 }, B t = B 0 exp{rt}. (12) where Ŵ t := W t θt is a standard Brownian motion on ({F t } 0 t T2, ˆP), as the Maruyama-Girsanov theorem says. The fair value at time t of a derivative whose pay-off at time T is H, which is the unique one that excludes arbitrage opportunities, is given by C t = e r(t t) E ˆP [H F t ], (0 t T) (13) the discounted conditional expectation of the pay-off random variable with respect to the EMM. The hedging strategy for the derivative is obtained via the stochastic differential expression of C; dc t = η t ds t + ν t db t (14) which means holding η amounts of the asset and ν amounts of the non-risky asset perfectly hedges the derivative. For details of the arguments in this section, see e.g. [1] or [2] and the references therein. 5 Pricing/A Proof of Theorem 1 In this section, we will give the value of the Exotic Forward following the framework of the previous section. To apply (13), we unify the cash flows as an F T2 measurable random variable. Let us define an F T measurable set A(Y T ) := {S T1 K Y T + e r(t2 T) E ˆP [S T2 K Y T F T ]} = {S T1 K (1 e r(t2 T) )Y T + S T e r(t2 T) K} = {S T1 S T (1 e r(t2 T) )(Y T + K)}. (15) 5

6 On A(Y T ), the buyer is sure to choose T 2 as the price-date, while on Ω\A(Y T ) she/he will not fail to choose T 1. The cash flow on Ω \ A(Y T ) is S T1 K at T, and will be e r(t 2 T) (S T1 K) (16) at T 2, if all is invested to the bond. While on A(Y T ), applying the same argument as above, the cash flow at T 2 is e r(t2 T) Y T + S T2 K Y T = (e r(t2 T) 1)Y T + S T2 K. Hence, the pay-off at T 2 of the exotic forward can be seen as H(Y T ) = 1 Ω\A(YT )e r(t2 T) (S T1 K) + 1 A(YT ){(e r(t2 T) 1)Y T + S T2 K}. Therefore, applying (13), we have (17) (18) C t (Y T ) = e r(t 2 t) E ˆP [H(Y T ) F t ] (0 t T 2 ) and if t T 1, = e r(t 2 t) e r(t 2 T) E ˆP [1 Ω\A(YT )(S T1 K) F t ] + e r(t 2 t) E ˆP [1 A(YT ){(e r(t 2 T) 1)Y T } F t ] + e r(t 2 t) E ˆP [1 A(YT )(S T2 K) F t ], = e r(t t) E ˆP [1 Ω\A(YT )(S T1 K) F t ] + e r(t t) E ˆP [1 A(YT ){(1 e r(t 2 T) )Y T } F t ] + E ˆP [1 A(YT )(e rt E ˆP [e rt 2 S T2 F T ] e r(t 2 t) K) F t ] = e r(t t)( E ˆP [1 Ω\A(YT )(S T1 K) F t ] + E ˆP [1 A(YT ){(1 e r(t 2 T) )Y T + S T e r(t 2 T) K} F t ] ) (19) (20) = e r(t t) E ˆP [max ( S T1 K, (1 e r(t2 T) )Y T + S T e r(t2 T) K ) F t ]. In particular, we see that C t (S T K) = e r(t t) E ˆP [max ( S T1, (2 e r(t2 T) ) )S T Ft ] e r(t t) K, (21) 6

7 and π = C 0 (S T K) = e rt E ˆP [max ( S T1, (2 e r(t 2 T) )S T ) ] e rt K. (22) We will calculate the expectation in (22), to complete the proof of Theorem 1. The most remarkable property is that A(S T K) is independent of F t for any t T 1, since A(S T K) ={S T1 K (2 e r(t 2 T) )S T K} ={S T1 (2 e r(t 2 T) )S T } = A ={log S 0 + σŵ T1 + (r 1 2 σ2 )T 1 log S 0 + σŵ T + (r 1 2 σ2 )T + log(2 e r(t 2 T) )} (23) = { Ŵ T Ŵ T1 (r 1 2 σ2 )(T T 1 ) + log(2 e r(t 2 T) ) σ ={Ŵ T Ŵ T1 d T T1 }. Therefore, π = e rt E ˆP [S T1 ; Ω \ A] + (2 e r(t2 T) )e rt E ˆP [S T ; A] Ke rt = e rt E ˆP [S T1 ]{ ˆP(Ω \ A) + (2 e r(t2 T) )E ˆP [S T /S T1 ; A]} = e r(t T1) S 0 {Φ( d ) + (2 e r(t2 T) )E ˆP [S T /S T1 ; A]}. (24) Noting that S T /S T1 = exp{σ(ŵ T Ŵ T1 ) σ2 2 (T T 1)}, (25) 7

8 we have E ˆP [S T /S T1 ; A] σ2 (r = e 2 )(T T 1) = e r(t T 1) y d = e r(t T 1) Φ(d + ). By (24) and (26), we obtain (7). y d 1 2π e y2 2 e yσ T T 1 dy 1 2π e 1 2 (y σ T T 1 ) 2 dy (26) Remark. For the cases of Y T = 0, the set A(0) defined in (15) is no longer independent of F T1, and therefore the pricing formula is somehow more complicated than (7). On the other hand, if we have Y T = cs T K (27) for any deterministic c, then A(Y T ) is again independent of F T1. Further generalizations of (27) are possible and will be studied elsewhere. 6 Hedging/A Proof of Theorem 2 As was mentioned in section 4, the expression (14) gives the hedging strategy. Let us start with (19), the expression for all t [0, T 2 ]: C t C t (S T K) = e r(t t) E ˆP [1 Ω\A (S T1 K) F t ] + e r(t2 t) E ˆP [1 A {(e r(t2 T) 1)(S T K)} F t ] + e r(t2 t) E ˆP [1 A (S T2 K) F t ] = e r(t t) E ˆP [1 Ω\A S T1 F t ] e r(t t) K + e r(t2 t) E ˆP [1 A {S T2 + (e r(t2 T) 1)S T } F t ]. (28) First, let us consider the case when t [T, T 2 ). Since all terms except 8

9 S T2 are F t measurable, we have C t = e r(t t) 1 Ω\A S T1 e r(t t) K + (e r(t t) e r(t 2 t) )1 A S T + 1 A e rt E ˆP [e rt 2 S T2 F t ] (29) = e r(t t) 1 Ω\A S T1 e r(t t) K + (e r(t t) e r(t 2 t) )1 A S T + 1 A S t. Therefore, we have dc t = 1 A ds t + {1 A S T (e rt e rt 2 ) + 1 Ω\A S T1 e rt e rt K}dB t. (30) Next, when t [T 1, T), C t = e r(t t) S T1 E ˆP [1 Ω\A F t ] e r(t t) K + e r(t t) (2 e r(t 2 T) )S t E ˆP [1 A S T /S t F t ]. (31) Since we have A = {S T1 (2 e r(t 2 T) )S T } = {S T1 /S t (2 e r(t 2 T) )S T /S t } = { log(s T /S t ) log(2 e r(t 2 T) ) + log(s t /S T1 )} (32) = {Ŵ T Ŵ t (r 1 2 σ2 )(T t) + log(2 e r(t2 T) )S t /S T1 } σ from (23), the conditional expectations in (31) are explicitly calculated as and E ˆP [1 Ω\A F t ] = Φ ( d (t, S t )), (33) E ˆP [1 A S T /S t F t ] = e r(t t) Φ (d + (t, S t )). (34) Noting that Φ(±d ± (t, x)) C 1,2 ([T 1, T) R), we can apply Itô s formula to get dφ(±d ± (t, S t )) = x Φ(±d ± (t, x)) x=s t ds t + ( t σ 2 x 2 xx )Φ(±d ± (t, x)) x=s t dt = ± φ(d ±(t, S t )) σ T t [ S 1 t ds t (r σ 2 ± 2 1 σ 2 ) ] dt. (35) 9

10 Hence dc t = e rt {S T1 Φ( d (t, S t )) K} db t + e rt B t S T1 dφ( d (t, S t )) + (2 e r(t 2 t) ) [ Φ(d + (t, S t ))ds t + S t dφ(d + (t, S t )) + d Φ(d + ), S t ] = e rt {S T1 Φ( d (t, S t )) K} db t e rt φ(d (t, S t )) B t S T1 σ (S t 1 ds t rdt) T t + (2 e r(t2 t) ) [ Φ(d + (t, S t ))ds t + S t φ(d + (t, S t )) σ {S t 1 ds t (r + σ 2 ) dt} T t φ(d + (t, S t )) + σs t dt ] T t = [ ( (2 e r(t2 t) ) Φ(d + (t, S t )) + φ(d ) +(t, S t )) σ T t B ts T1 φ( d (t, S t )) B T S t σ ] ds t T t [ ( S T1 + Φ( d (t, S t )) + φ(d (t, S t )) B T σ T t (2 e r(t 2 t) ) S tφ(d + (t, S t )) B t σ T t ) K B T ] db t (36) Finally, consider the cases when t [0, T 1 ). Since A is independent of F t, from (28) we have C t = e r(t T1) S t ˆP(Ω \ A) e r(t t) K + e r(t t) (2 e r(t2 T) )E ˆP [S T1 F t ]E ˆP [1 A S T /S T1 ] = e r(t T1) S t Φ( d ) e r(t t) K + (2 e r(t2 T) )Φ(d + )S t. (37) Therefore, dc t = Ke rt db t + {e r(t T 1) Φ( d ) + (2 e r(t 2 T) )Φ(d + )}ds t. By (30), (36) and (38), we get (8) and (9). (38) 10

11 References [1] J. Akahori: Stochastic Analysis in Mathematical Finance, (in Japanese) Systems, Control and Information, 44, pp , [2] J. C. Hull Options, Futures, and Other Derivatives, (5th Edition), Prentice Hall,

### 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

### 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

### 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

### 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

### Martingale 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

### 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

### τ θ What is the proper price at time t =0of this 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

### The Black-Scholes Model

The Black-Scholes Model Liuren Wu Zicklin School of Business, Baruch College Options Markets (Hull chapter: 12, 13, 14) Liuren Wu The Black-Scholes Model Options Markets 1 / 19 The Black-Scholes-Merton

### 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

### where N is the standard normal distribution function,

The Black-Scholes-Merton formula (Hull 13.5 13.8) Assume S t is a geometric Brownian motion w/drift. Want market value at t = 0 of call option. European call option with expiration at time T. Payout at

### 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

### 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

### Valuation of the Surrender Option Embedded in Equity-Linked Life Insurance. Brennan Schwartz (1976,1979) Brennan Schwartz

Valuation of the Surrender Option Embedded in Equity-Linked Life Insurance Brennan Schwartz (976,979) Brennan Schwartz 04 2005 6. Introduction Compared to traditional insurance products, one distinguishing

### Chapter 2: Binomial Methods and the Black-Scholes Formula

Chapter 2: Binomial Methods and the Black-Scholes Formula 2.1 Binomial Trees We consider a financial market consisting of a bond B t = B(t), a stock S t = S(t), and a call-option C t = C(t), where the

### 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

### 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

### Two-State Option Pricing

Rendleman and Bartter [1] present a simple two-state model of option pricing. The states of the world evolve like the branches of a tree. Given the current state, there are two possible states next period.

### 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

### Finite Differences Schemes for Pricing of European and American Options

Finite Differences Schemes for Pricing of European and American Options Margarida Mirador Fernandes IST Technical University of Lisbon Lisbon, Portugal November 009 Abstract Starting with the Black-Scholes

### 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

### 7: The CRR Market Model

Ben Goldys and Marek Rutkowski School of Mathematics and Statistics University of Sydney MATH3075/3975 Financial Mathematics Semester 2, 2015 Outline We will examine the following issues: 1 The Cox-Ross-Rubinstein

### 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

### European Options Pricing Using Monte Carlo Simulation

European Options Pricing Using Monte Carlo Simulation Alexandros Kyrtsos Division of Materials Science and Engineering, Boston University akyrtsos@bu.edu European options can be priced using the analytical

### 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

### Black-Scholes Option Pricing Model

Black-Scholes Option Pricing Model Nathan Coelen June 6, 22 1 Introduction Finance is one of the most rapidly changing and fastest growing areas in the corporate business world. Because of this rapid change,

### Lecture 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

### MATH3075/3975 Financial Mathematics

MATH3075/3975 Financial Mathematics Week 11: Solutions Exercise 1 We consider the Black-Scholes model M = B, S with the initial stock price S 0 = 9, the continuously compounded interest rate r = 0.01 per

### Arbitrage-Free Pricing Models

Arbitrage-Free Pricing Models Leonid Kogan MIT, Sloan 15.450, Fall 2010 c Leonid Kogan ( MIT, Sloan ) Arbitrage-Free Pricing Models 15.450, Fall 2010 1 / 48 Outline 1 Introduction 2 Arbitrage and SPD 3

### 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

### Computational Finance Options

1 Options 1 1 Options Computational Finance Options An option gives the holder of the option the right, but not the obligation to do something. Conversely, if you sell an option, you may be obliged to

### Hedging Options In The Incomplete Market With Stochastic Volatility. Rituparna Sen Sunday, Nov 15

Hedging Options In The Incomplete Market With Stochastic Volatility Rituparna Sen Sunday, Nov 15 1. Motivation This is a pure jump model and hence avoids the theoretical drawbacks of continuous path models.

### A SNOWBALL CURRENCY OPTION

J. KSIAM Vol.15, No.1, 31 41, 011 A SNOWBALL CURRENCY OPTION GYOOCHEOL SHIM 1 1 GRADUATE DEPARTMENT OF FINANCIAL ENGINEERING, AJOU UNIVERSITY, SOUTH KOREA E-mail address: gshim@ajou.ac.kr ABSTRACT. I introduce

### Understanding Options and Their Role in Hedging via the Greeks

Understanding Options and Their Role in Hedging via the Greeks Bradley J. Wogsland Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996-1200 Options are priced assuming that

### 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

### Bond Options, Caps and the Black Model

Bond Options, Caps and the Black Model Black formula Recall the Black formula for pricing options on futures: C(F, K, σ, r, T, r) = Fe rt N(d 1 ) Ke rt N(d 2 ) where d 1 = 1 [ σ ln( F T K ) + 1 ] 2 σ2

### 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

### The interest volatility surface

The interest volatility surface David Kohlberg Kandidatuppsats i matematisk statistik Bachelor Thesis in Mathematical Statistics Kandidatuppsats 2011:7 Matematisk statistik Juni 2011 www.math.su.se Matematisk

### Options 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

### Session X: Lecturer: Dr. Jose Olmo. Module: Economics of Financial Markets. MSc. Financial Economics. Department of Economics, City University, London

Session X: Options: Hedging, Insurance and Trading Strategies Lecturer: Dr. Jose Olmo Module: Economics of Financial Markets MSc. Financial Economics Department of Economics, City University, London Option

### 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

### 1.1 Some General Relations (for the no dividend case)

1 American Options Most traded stock options and futures options are of American-type while most index options are of European-type. The central issue is when to exercise? From the holder point of view,

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

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 Agenda Forward contracts Definition Determining forward

### Stocks paying discrete dividends: modelling and option pricing

Stocks paying discrete dividends: modelling and option pricing Ralf Korn 1 and L. C. G. Rogers 2 Abstract In the Black-Scholes model, any dividends on stocks are paid continuously, but in reality dividends

### QUANTIZED INTEREST RATE AT THE MONEY FOR AMERICAN OPTIONS

QUANTIZED INTEREST RATE AT THE MONEY FOR AMERICAN OPTIONS L. M. Dieng ( Department of Physics, CUNY/BCC, New York, New York) Abstract: In this work, we expand the idea of Samuelson[3] and Shepp[,5,6] for

### Additional 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

### OPTION PRICING FOR WEIGHTED AVERAGE OF ASSET PRICES

OPTION PRICING FOR WEIGHTED AVERAGE OF ASSET PRICES Hiroshi Inoue 1, Masatoshi Miyake 2, Satoru Takahashi 1 1 School of Management, T okyo University of Science, Kuki-shi Saitama 346-8512, Japan 2 Department

### The Effect of Management Discretion on Hedging and Fair Valuation of Participating Policies with Maturity Guarantees

The Effect of Management Discretion on Hedging and Fair Valuation of Participating Policies with Maturity Guarantees Torsten Kleinow Heriot-Watt University, Edinburgh (joint work with Mark Willder) Market-consistent

### 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

### THE BLACK-SCHOLES MODEL AND EXTENSIONS

THE BLAC-SCHOLES MODEL AND EXTENSIONS EVAN TURNER Abstract. This paper will derive the Black-Scholes pricing model of a European option by calculating the expected value of the option. We will assume that

### LECTURE 9: A MODEL FOR FOREIGN EXCHANGE

LECTURE 9: A MODEL FOR FOREIGN EXCHANGE 1. Foreign Exchange Contracts There was a time, not so long ago, when a U. S. dollar would buy you precisely.4 British pounds sterling 1, and a British pound sterling

### The British Put Option

Appl. Math. Finance, Vol. 18, No. 6, 211, (537-563) Research Report No. 1, 28, Probab. Statist. Group Manchester (25 pp) The British Put Option G. Peskir & F. Samee We present a new put option where the

### Numerical methods for American options

Lecture 9 Numerical methods for American options Lecture Notes by Andrzej Palczewski Computational Finance p. 1 American options The holder of an American option has the right to exercise it at any moment

### The 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 Exotic Options. 1 Barrier Options. 2 Compound Options. 3 Gap Options

More Exotic Options 1 Barrier Options 2 Compound Options 3 Gap Options More Exotic Options 1 Barrier Options 2 Compound Options 3 Gap Options Definition; Some types The payoff of a Barrier option is path

### On Quantile Hedging and Its Applications to the Pricing of Equity-Linked Life Insurance Contracts 1

On Quantile Hedging and Its Applications to the Pricing of Equity-Linked Life Insurance Contracts 1 Alexander Melnikov Steklov Mathematical Institute of Russian Academy Sciences and Department of Mathematical

### 1 Geometric Brownian motion

Copyright c 006 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

### Options and Derivative Pricing. U. Naik-Nimbalkar, Department of Statistics, Savitribai Phule Pune University.

Options and Derivative Pricing U. Naik-Nimbalkar, Department of Statistics, Savitribai Phule Pune University. e-mail: uvnaik@gmail.com The slides are based on the following: References 1. J. Hull. Options,

### Barrier Options. Peter Carr

Barrier Options Peter Carr Head of Quantitative Financial Research, Bloomberg LP, New York Director of the Masters Program in Math Finance, Courant Institute, NYU March 14th, 2008 What are Barrier Options?

### 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

### Introduction to Stochastic Differential Equations (SDEs) for Finance

Introduction to Stochastic Differential Equations (SDEs) for Finance Andrew Papanicolaou January, 013 Contents 1 Financial Introduction 3 1.1 A Market in Discrete Time and Space..................... 3

### 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

### 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

### 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

### Lecture 11. Sergei Fedotov. 20912 - Introduction to Financial Mathematics. Sergei Fedotov (University of Manchester) 20912 2010 1 / 7

Lecture 11 Sergei Fedotov 20912 - Introduction to Financial Mathematics Sergei Fedotov (University of Manchester) 20912 2010 1 / 7 Lecture 11 1 American Put Option Pricing on Binomial Tree 2 Replicating

### 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

### LECTURE 15: AMERICAN OPTIONS

LECTURE 15: AMERICAN OPTIONS 1. Introduction All of the options that we have considered thus far have been of the European variety: exercise is permitted only at the termination of the contract. These

### 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

### Simulating Stochastic Differential Equations

Monte Carlo Simulation: IEOR E473 Fall 24 c 24 by Martin Haugh Simulating Stochastic Differential Equations 1 Brief Review of Stochastic Calculus and Itô s Lemma Let S t be the time t price of a particular

### 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

### FORWARDS AND EUROPEAN OPTIONS ON CDO TRANCHES. John Hull and Alan White. First Draft: December, 2006 This Draft: March 2007

FORWARDS AND EUROPEAN OPTIONS ON CDO TRANCHES John Hull and Alan White First Draft: December, 006 This Draft: March 007 Joseph L. Rotman School of Management University of Toronto 105 St George Street

### 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

### COMPLETE MARKETS DO NOT ALLOW FREE CASH FLOW STREAMS

COMPLETE MARKETS DO NOT ALLOW FREE CASH FLOW STREAMS NICOLE BÄUERLE AND STEFANIE GRETHER Abstract. In this short note we prove a conjecture posed in Cui et al. 2012): Dynamic mean-variance problems in

### Jorge 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.

### 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

### Some stability results of parameter identification in a jump diffusion model

Some stability results of parameter identification in a jump diffusion model D. Düvelmeyer Technische Universität Chemnitz, Fakultät für Mathematik, 09107 Chemnitz, Germany Abstract In this paper we discuss

### 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.

### 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

### 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

### 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

### 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

### Put-Call Parity. chris bemis

Put-Call Parity chris bemis May 22, 2006 Recall that a replicating portfolio of a contingent claim determines the claim s price. This was justified by the no arbitrage principle. Using this idea, we obtain

### 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

### ARBITRAGE-FREE OPTION PRICING MODELS. Denis Bell. University of North Florida

ARBITRAGE-FREE OPTION PRICING MODELS Denis Bell University of North Florida Modelling Stock Prices Example American Express In mathematical finance, it is customary to model a stock price by an (Ito) stochatic

### Foreign 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

### 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

### A Martingale System Theorem for Stock Investments

A Martingale System Theorem for Stock Investments Robert J. Vanderbei April 26, 1999 DIMACS New Market Models Workshop 1 Beginning Middle End Controversial Remarks Outline DIMACS New Market Models Workshop

### Chapter 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)

### Notes on Black-Scholes Option Pricing Formula

. Notes on Black-Scholes Option Pricing Formula by De-Xing Guan March 2006 These notes are a brief introduction to the Black-Scholes formula, which prices the European call options. The essential reading

### Pricing Barrier Option Using Finite Difference Method and MonteCarlo Simulation

Pricing Barrier Option Using Finite Difference Method and MonteCarlo Simulation Yoon W. Kwon CIMS 1, Math. Finance Suzanne A. Lewis CIMS, Math. Finance May 9, 000 1 Courant Institue of Mathematical Science,

### 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

### ASimpleMarketModel. 2.1 Model Assumptions. Assumption 2.1 (Two trading dates)

2 ASimpleMarketModel In the simplest possible market model there are two assets (one stock and one bond), one time step and just two possible future scenarios. Many of the basic ideas of mathematical finance

### ENGINEERING AND HEDGING OF CORRIDOR PRODUCTS - with focus on FX linked instruments -

AARHUS SCHOOL OF BUSINESS AARHUS UNIVERSITY MASTER THESIS ENGINEERING AND HEDGING OF CORRIDOR PRODUCTS - with focus on FX linked instruments - AUTHORS: DANIELA ZABRE GEORGE RARES RADU SIMIAN SUPERVISOR:

### Pricing Options with Discrete Dividends by High Order Finite Differences and Grid Stretching

Pricing Options with Discrete Dividends by High Order Finite Differences and Grid Stretching Kees Oosterlee Numerical analysis group, Delft University of Technology Joint work with Coen Leentvaar, Ariel

### 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

### 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

### MTH6120 Further Topics in Mathematical Finance Lesson 2

MTH6120 Further Topics in Mathematical Finance Lesson 2 Contents 1.2.3 Non-constant interest rates....................... 15 1.3 Arbitrage and Black-Scholes Theory....................... 16 1.3.1 Informal