# Simple formulas to option pricing and hedging in the Black Scholes model

Save this PDF as:

Size: px
Start display at page:

Download "Simple formulas to option pricing and hedging in the Black Scholes model"

## Transcription

1 Simple formulas to option pricing and hedging in the Black Scholes model Paolo Pianca Department of Applied Mathematics University Ca Foscari of Venice Dorsoduro 385/E, 3013 Venice, Italy pianca/ Abstract. For option whose striking price equals the forward price of the underlying asset, the Black-Scholes pricing formula can be approximated in closed-form. A interesting result is that the derived equation is not only very simple in structure but also that it can be immediately inverted to obtain an explicit formula for implied volatility. In this contribution we present and compare the accuracy of three of such approximation formulas. The numerical analysis shows that the first order approximations are close only for small maturities, Pólya approximations are remarkably accurate for a very large range of parameters, while logistic values are the most accurate only for extreme maturities. Keywords. Option pricing, hedging, Taylor approximation, Pólya approximation, logistic approximation. M.S.C. classification: 91B8, 11K99. J.E.L. classification: G13. 1 Introduction For the special case of at the money European call options, using suitable approximations of the standard normal distribution, the Black-Scholes pricing formula can be written in closed-form. A useful result is that the derived equation is not only very simple in structure but also that it can be immediately inverted to obtain an explicit formula for implied volatility. This approach has been introduced by [3], [4] e [5] where first order Taylor polynomial approximation is suggested and has been taken up again in [8] where the use of Pólya approximation is proposed. In this contribution, we derive a similar closed form invertible equation using an approximation of the standard normal distribution, based on logistic distribution. Obviously, the accuracy of values depends on volatility level and on time to maturity. The analysis carried out highlights that first order polynomial uses very simple forms but presents close approximations only for a limited range of

2 4 parameters. The Pólya and logistic distributions require some additional computations. Pólya approximations are remarkably accurate for a large range of volatility and maturity, while the logistic approximations are the best only for very large maturities. This note is organized as follows. The next Section presents the approximation formulas to standard normal distribution and the extent of tracking errors. Section 3 contains computational evidence of Black-Scholes values obtained with the three approximation functions. Implied volatility and hedging parameters are discussed in Section 4 and 5, respectively. The final Section reports some concluding remarks. First order, logistic and Pólya approximation formulas The well known Black Scholes formula [] for European call option on underlying assets paying no dividends is and where: In the above equations: c BS = S N(d 1 ) X e rτ N(d ) (1) d 1 = ln (S/X) + (r + σ /)τ σ τ d = ln (S/X) + (r σ /)τ σ τ = d 1 σ τ. c BS is the theoretical price of a European call option on a stock, S is the stock price, X is the striking price, r is the continuously compounded rate of interest, τ is the time to option expiration, σ is the volatility (standard deviation of the instantaneous rate of return on the stock), N( ) is the cumulative standard normal density function. We analyze three approximations to the cumulative standard normal distribution N(d) = 1 π d e t / dt = 1 + d 0 e t / π dt = { } d d3 π 6 + d () The first approximation ignores in () all the terms beyond d (i.e. terms of third or higher order are dropped). Therefore A 1 (d) = 1 + d d. (3) π

3 Table 1. Standard normal distribution: exact values, approximate values and relative errors ( 100) 5 d N(d) A 1(d) 100 E r,1 A (d) 100 E r, A 3(d) 100 E r, The first order Taylor polynomial (3) is very close to the standard normal distribution only for d values between about ±0.50, but then starts to diverge. The second approximation is based on logistic distribution F (d) = e kd k > 0. (4) The value k = π/ 3 is often used; with such a constant the maximum value of the differenze F (d) N(d) is about 0.08, attained when d = 0.07 (see [6]). It can be proved that the constant k = π/ 3.41 leads to a better approximation, with a maximum difference of Therefore, as second approximation to N(d), we select A (d) = e kd, k = π/ (5) The third approximation is the distribution suggested by Pólya in [7] to approximate N(d), i.e. [ ] A 3 (d) = e d /π d 0. (6) The Pólya approximation is very accurate for a wide range of possible values for d; the maximum absolute error is 0.003, when d = 1.6. The behavior of the difference A j N(d) (j = 1,, 3) is shown in Figures 1 and, for d values ranging from 0.0 to 3.0 and from 0.0 to 0.5, respectively. The extent of tracking errors is presented in Table 1 for d values ranging from 0.0 to 4.

4 (a) A 1(d) N(d) (b) A (d) N(d) (c) A 3(d) N(d) Fig. 1. Differences A j(d) N(d), d [0, 3] (a) A 1(d) N(d) (b) A (d) N(d) (c) A 3(d) N(d) Fig.. Differences A j(d) N(d), d [0, 0.5] The columns 100 E r,j (j = 1,, 3) report the relative errors E r,j = A j(d) N(d) N(d) (7) of the approximation functions, multiplied by 100. The valued obtained with first order approximation are always lower than true values; the corresponding relative errors are increasing with d and for d 1.0 tend to rise more and more. This is in part due to the fact that the approximation A 1 (d) is not a distribution function. The accuracy of approximation A (d) is inferior than A 1 (d) for d 0.5, while becomes excellent for d in a neighborhood of 1.5 (here the difference A (d) N(d) chances sign). The relative errors related to Pólya approximation are always positive, i.e. the approximation A 3 (d) consistently overestimates N(d); nevertheless, the values obtained with Pólya technique are so precise that we can state that on the whole A 3 (d) is the best approximation. 3 Comparison of Black-Scholes Values In this section we focus on options that are at the money forward, where S = e rτ X. Note that many transactions in the over the counter market are quoted and executed at or near at the money forward. Moreover, empirical evidence shows (see [1]) that the implied volatility of the at the money options is superior to any combination of all the implied volatilities. For at the money options the value of d 1 and d simplify in: d 1 = σ τ d = σ τ (8)

5 and the Black-Scholes formula becomes [ c BS = p BS = S N [ = S N ( σ τ ) ( N σ τ )] = 7 ( ) ] (9) σ τ 1 where p BS is the theoretical price of a European put option at the same maturity and exercise price. In the special case of at the money options, the first order approximation for N(d 1 ) is and substituting in (9) we have The logistic approximation for N(d 1 ) is N A1 (d 1 ) = d 1 = σ τ, (10) c A1 = 0.4 S σ τ. (11) N A (d 1 ) = e kσ τ/, (1) and the logistic approximation for at the money call is ( ) c A = S 1 + e 1. (13) kσ τ/ The Pólya approximation for N(d 1 ) is [ N A3 (d 1 ) = ] 1 e σ τ/(π) (14) and the Pólya approximation for at the money call is = S 1 e σ τ/(π). (15) c A3 Table compares the call option values obtained by the first-order Taylor polynomial, the logistic and the Pólya approximation formulas (respectively relations (11), (13) and (15)) to the Black-Scholes model (1). The volatility is σ = 0.30, the stock price is S = 100 and the expiration data going out to 100 years. To have at the money options, the strike price X is set for each time to option expiration τ so that X = 100 e rτ. First order approximation performs well good for short times to expiration (τ 1), which are the most interesting cases in practice. For τ 3, the logistic approximation presents relative errors greater than 6 %. For example, when τ = 1/1 we have d 1 = 0.3 1/1 0.5 = ; the exact value for is N(d 1 ) = , the logistic approximation is A (d 1 ) =

6 8 Table. At the money call option values for different time to expiration, with S = 100 and σ = 0.3 τ c BS c A1 100 E r,1 c A 100 E r, c A3 100 E r,3 1/ / / / / / , an error of The Black Scholes value is c BS = and the approximate call value is c A = , a relative error of Given the close tracking that the Pólya approximation has to the standard normal distribution it is no surprise that the relative errors of c A3 are at all times very small and are not perceptible in correspondence to short maturities. Only for τ = 75 the logistic approximation presents a lower error, but the case τ = 75 is more an academic curiosity than a real-world phenomenon. 4 Implied standard deviation One of the most widely used application of Black-Scholes formula is the estimation of volatility (instantaneous standard deviation) of the rate of return on the underlying stock, using the market prices of the option and the stock. The implied volatility is the value of standard deviation σ that perfectly explains the option price, given all other variables. It is a widely documented phenomenon that implied volatility is not constant as other parameters varied. For example, the implied volatility from options with different maturities should not combined to provide a single estimate, because they reflect different perception on short-run versus long-run volatility, a kind of term structure of volatility. Computing the implied volatility from Black-Scholes formula requires the solution of non linear equation and hence the practice has been to use a iterative procedure. However, for at the money European option this cumbersome numerical procedure can be avoided: namely, the formulas (11), (13) and (15) can

7 9 be easily inverted to provide reasonably accurate estimates of implied volatility. With simple algebra we obtain the following analytical estimates of implied standard deviation: ISD A1 =.5 c me S τ ISD A = log [(S c me)/(s + c me )] k τ ISD A3 = π log[1 (c me/s) ] τ (16) (17) (18) where c me is the market price of the call option. To assess the accuracy of these approximations, consider an at the money forward option (X = Se rτ ) with S = 100. The option expires in six months and the market price of the option is c me = 6. The true implied standard deviation, computed by Newton method is ISD = 0.19; the estimates obtained with the approximation formulas are: ISD A1 = 0.11, ISD A = , ISD A3 = Note that, in this example, the approximated volatilities (16) and (18) prove to be very accurate, while the approximation (17) entails a small error. In any case, all three estimates (16), (17) and (18) may also be used to obtain a good starting point for a more accurate numerical procedure in order to compute the exact implied standard deviation, even for very deep out (or in) the money options. 5 Option hedging parameters The key purpose in option portfolios management is hedging to eliminate or to reduce risk. Hedging requires the measurement and the monitoring of the sensitivity of the option value to changes in the parameters. These sensitivities can be compute by partial derivatives which, for at the money options, have not cumbersome expressions. The hedge ratio, or Delta, of an at the money call option is c = c S = N(d 1) = 1 σ τ + 0 e t / dt π. (19) The three approximations for the hedge ratio are immediately obtained. For example, when S = 100, τ = 0.5 and σ = 0.3 the true value is c = , while the approximated values are: c,a1 = , c,a = , c,a3 = Factor Gamma is defined by Γ c = c S = S = N (d 1 ) d 1 S = e d 1 / (0) πsσ τ

8 30 and measures the change in the hedge ratio in correspondence to a small change in the price of the stock. Note that, in most practical circumstances, we have d 1/ 0 and therefore Γ c 0.4 Sσ τ. (1) Factor Gamma indicates the cost of adjusting a hedge. Using the data of the previous example, from approximation (1) we have Γ c = = ; this value is accurate to the third decimal place (the correct value is ). Factor Vega, which is defined as the change in the price of the call for a small change in the volatility parameter, is given by V ega = c σ = N (d 1 )S τ 0.4 S τ. () Factor Omega is a measure of leverage and is given by the product of the Delta and the ratio S/c, i.e. Ω c = c S c. (3) Factor Theta, or time decay, is given by Θ c = c τ = Sσ τ N (d 1 ) rxe rτ N(d ) (4) and for at the money options it becomes [ ] σe ( σ τ/4) Θ c,atm = S + r N(d ) τπ [ ] 0.4σ S τ + r N(d ). (5) The time decay is usually measured over one day. The negative sign indicates that, as the time to expiration is declining, the option value is decaying. Including r = 0.05 in data set of our numerical example, we find that the true value is Θ c = , while the approximated values are: Θ c,a1 = , Θ c,a = , Θ c,a3 = Concluding remarks Nowadays computing an exact Black-Scholes European call option value is really easy: the large use of computer programs and the great availability of free or low cost software to price options, undoubtedly, reduces the need for approximate formulas, apart from the accuracy of proposed formulas. So, one could wonder which is the usefulness of formulas derived from a first order polynomial or logistic and Pólya approximations to standard normal distributions.

9 31 Actually, the first order formulas are very simple, easy to remember and can be used to give immediately operational information about the option price and the implied volatility. But this simplicity shows marks of weakness in correspondence of high volatilities and times to maturity. The formulas based on logistic and Pólya approximations without any doubt are more complicating looking, but can be easily programmed onto a pocket calculator. In particular the Pólya formula presents an impressive accuracy even for long-lived options; the approximations obtained with the logistic distribution present larger relative errors, nevertheless are sufficiently accurate for most practical circumstances. References 1. Beckers, S.: Standard deviation implied in option pricing as predictors of futures stock price variabily. Journal of Banking and Finance 5 (1981) Black, F., Scholes, M.: The pricing of options and corporate liabilities. Journal of Political Economy 81, May/June (1973) Brenner, M., Subrahmanyam, M.: A simple formula to compute the implied standard deviation. Financial Analysts Journal 44, September/October (1988) Brenner, M., Subrahmanyam, M.: A simple approach to option valuation and hedging in the Black-Scholes model. Financial Analysts Journal 50, March/April (1994) Fienstein, S.: A source of unbiased implied standard deviation. Working Paper 88-9, Federal Reserve of Atlanta, Atlanta (1988) 6. Johnson, N.L., Kotz, S., Balakrishna, N. Continuous univariate distribution. J. Wiley & Sons, Inc., New York (1994) 7. Pólya, G.: Remarks on computing the probability integral in one and two dimension. Proceedings of 1st Berkeley Symposium on Mathematics Statistics and Probabilities (1945) Smith, D.J.: Comparing at-the-money Black-Scholes call option values. Derivatives Quarterly 7 (001) 51-57

10 3

### An Improved Approach to Computing Implied Volatility. Donald R. Chambers* Lafayette College. Sanjay K. Nawalkha University of Massachusetts at Amherst

EFA Eastern Finance Association The Financial Review 38 (2001) 89-100 Financial Review An Improved Approach to Computing Implied Volatility Donald R. Chambers* Lafayette College Sanjay K. Nawalkha University

### 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:

### 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

### 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

### 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

### 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

### 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

### FINANCIAL ECONOMICS OPTION PRICING

OPTION PRICING Options are contingency contracts that specify payoffs if stock prices reach specified levels. A call option is the right to buy a stock at a specified price, X, called the strike price.

### CHAPTER 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

### Journal Of Financial And Strategic Decisions Volume 10 Number 2 Summer 1997

Journal Of Financial And Strategic Decisions Volume 10 Number 2 Summer 1997 AN EMPIRICAL INVESTIGATION OF PUT OPTION PRICING: A SPECIFICATION TEST OF AT-THE-MONEY OPTION IMPLIED VOLATILITY Hongshik Kim,

### BINOMIAL OPTIONS PRICING MODEL. Mark Ioffe. Abstract

BINOMIAL OPTIONS PRICING MODEL Mark Ioffe Abstract Binomial option pricing model is a widespread numerical method of calculating price of American options. In terms of applied mathematics this is simple

### Volatility 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

### 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

### The equilibrium price of the call option (C; European on a non-dividend paying stock) is shown by Black and Scholes to be: are given by. p T t.

THE GREEKS BLACK AND SCHOLES (BS) FORMULA The equilibrium price of the call option (C; European on a non-dividend paying stock) is shown by Black and Scholes to be: C t = S t N(d 1 ) Xe r(t t) N(d ); Moreover

### Hedging 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

### DETERMINING THE VALUE OF EMPLOYEE STOCK OPTIONS. Report Produced for the Ontario Teachers Pension Plan John Hull and Alan White August 2002

DETERMINING THE VALUE OF EMPLOYEE STOCK OPTIONS 1. Background Report Produced for the Ontario Teachers Pension Plan John Hull and Alan White August 2002 It is now becoming increasingly accepted that companies

### 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

### 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

OPTIONS CALCULATOR QUICK GUIDE Reshaping Canada s Equities Trading Landscape OCTOBER 2014 Table of Contents Introduction 3 Valuing options 4 Examples 6 Valuing an American style non-dividend paying stock

### 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

### The 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

### Option Values. Option Valuation. Call Option Value before Expiration. Determinants of Call Option Values

Option Values Option Valuation Intrinsic value profit that could be made if the option was immediately exercised Call: stock price exercise price : S T X i i k i X S Put: exercise price stock price : X

### 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-Merton approach merits and shortcomings

Black-Scholes-Merton approach merits and shortcomings Emilia Matei 1005056 EC372 Term Paper. Topic 3 1. Introduction The Black-Scholes and Merton method of modelling derivatives prices was first introduced

### Options/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

### 14 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

### Financial 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

### 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

### VALUATION 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

### Options, pre-black Scholes

Options, pre-black Scholes Modern finance seems to believe that the option pricing theory starts with the foundation articles of Black, Scholes (973) and Merton (973). This is far from being true. Numerous

### 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

### Underlier 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

### IMPLIED VOLATILITY SKEWS AND STOCK INDEX SKEWNESS AND KURTOSIS IMPLIED BY S&P 500 INDEX OPTION PRICES

IMPLIED VOLATILITY SKEWS AND STOCK INDEX SKEWNESS AND KURTOSIS IMPLIED BY S&P 500 INDEX OPTION PRICES Charles J. Corrado Department of Finance 14 Middlebush Hall University of Missouri Columbia, MO 6511

### Return to Risk Limited website: www.risklimited.com. Overview of Options An Introduction

Return to Risk Limited website: www.risklimited.com Overview of Options An Introduction Options Definition The right, but not the obligation, to enter into a transaction [buy or sell] at a pre-agreed price,

### Journal of Financial and Economic Practice

Analyzing Investment Data Using Conditional Probabilities: The Implications for Investment Forecasts, Stock Option Pricing, Risk Premia, and CAPM Beta Calculations By Richard R. Joss, Ph.D. Resource Actuary,

### Options Pricing. This is sometimes referred to as the intrinsic value of the option.

Options Pricing We will use the example of a call option in discussing the pricing issue. Later, we will turn our attention to the Put-Call Parity Relationship. I. Preliminary Material Recall the payoff

### Sensex Realized Volatility Index

Sensex Realized Volatility Index Introduction: Volatility modelling has traditionally relied on complex econometric procedures in order to accommodate the inherent latent character of volatility. Realized

### 1 The Black-Scholes Formula

1 The Black-Scholes Formula In 1973 Fischer Black and Myron Scholes published a formula - the Black-Scholes formula - for computing the theoretical price of a European call option on a stock. Their paper,

### 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

### IMPLIED VOLATILITY SKEWS AND STOCK INDEX SKEWNESS AND KURTOSIS IMPLIED BY S&P 500 INDEX OPTION PRICES

IMPLIED VOLATILITY SKEWS AND STOCK INDEX SKEWNESS AND KURTOSIS IMPLIED BY S&P 500 INDEX OPTION PRICES Charles J. Corrado Department of Finance University of Missouri - Columbia Tie Su Department of Finance

### 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

### OPTIONS and FUTURES Lecture 2: Binomial Option Pricing and Call Options

OPTIONS and FUTURES Lecture 2: Binomial Option Pricing and Call Options Philip H. Dybvig Washington University in Saint Louis binomial model replicating portfolio single period artificial (risk-neutral)

### CAPM Option Pricing. Sven Husmann a, Neda Todorova b

CAPM Option Pricing Sven Husmann a, Neda Todorova b a Department of Business Administration, European University Viadrina, Große Scharrnstraße 59, D-15230 Frankfurt (Oder, Germany, Email: husmann@europa-uni.de,

### Option pricing in detail

Course #: Title Module 2 Option pricing in detail Topic 1: Influences on option prices - recap... 3 Which stock to buy?... 3 Intrinsic value and time value... 3 Influences on option premiums... 4 Option

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

### Simplified Option Selection Method

Simplified Option Selection Method Geoffrey VanderPal Webster University Thailand Options traders and investors utilize methods to price and select call and put options. The models and tools range from

### Use the option quote information shown below to answer the following questions. The underlying stock is currently selling for \$83.

Problems on the Basics of Options used in Finance 2. Understanding Option Quotes Use the option quote information shown below to answer the following questions. The underlying stock is currently selling

### Journal Of Financial And Strategic Decisions Volume 11 Number 1 Spring 1998

Journal Of Financial And Strategic Decisions Volume Number Spring 998 TRANSACTIONS DATA EXAMINATION OF THE EFFECTIVENESS OF THE BLAC MODEL FOR PRICING OPTIONS ON NIEI INDEX FUTURES Mahendra Raj * and David

### 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

### Valuing Coca-Cola and Pepsi Options Using the Black-Scholes Option Pricing Model

Valuing Coca-Cola and Pepsi Options Using the Black-Scholes Option Pricing Model John C. Gardner, University of New Orleans Carl B. McGowan, Jr., CFA, Norfolk State University ABSTRACT In this paper, we

### 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

### Introduction 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

### The 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

### Valuation, 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

### 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

### Understanding Options Gamma to Boost Returns. Maximizing Gamma. The Optimum Options for Accelerated Growth

Understanding Options Gamma to Boost Returns Maximizing Gamma The Optimum Options for Accelerated Growth Enhance Your Option Trading Returns by Maximizing Gamma Select the Optimum Options for Accelerated

### ON DETERMINANTS AND SENSITIVITIES OF OPTION PRICES IN DELAYED BLACK-SCHOLES MODEL

ON DETERMINANTS AND SENSITIVITIES OF OPTION PRICES IN DELAYED BLACK-SCHOLES MODEL A. B. M. Shahadat Hossain, Sharif Mozumder ABSTRACT This paper investigates determinant-wise effect of option prices when

### CHAPTER 21: OPTION VALUATION

CHAPTER 21: OPTION VALUATION 1. Put values also must increase as the volatility of the underlying stock increases. We see this from the parity relation as follows: P = C + PV(X) S 0 + PV(Dividends). Given

### FX 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

### GAMMA.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

### Or part of or all the risk is dynamically hedged trading regularly, with a. frequency that needs to be appropriate for the trade.

Option position (risk) management Correct risk management of option position is the core of the derivatives business industry. Option books bear huge amount of risk with substantial leverage in the position.

### Discussions of Monte Carlo Simulation in Option Pricing TIANYI SHI, Y LAURENT LIU PROF. RENATO FERES MATH 350 RESEARCH PAPER

Discussions of Monte Carlo Simulation in Option Pricing TIANYI SHI, Y LAURENT LIU PROF. RENATO FERES MATH 350 RESEARCH PAPER INTRODUCTION Having been exposed to a variety of applications of Monte Carlo

### The Valuation of Currency Options

The Valuation of Currency Options Nahum Biger and John Hull Both Nahum Biger and John Hull are Associate Professors of Finance in the Faculty of Administrative Studies, York University, Canada. Introduction

### 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

### Information Content of Right Option Tails: Evidence from S&P 500 Index Options

Information Content of Right Option Tails: Evidence from S&P 500 Index Options Greg Orosi* October 17, 2015 Abstract In this study, we investigate how useful the information content of out-of-the-money

Pair Trading with Options Jeff Donaldson, Ph.D., CFA University of Tampa Donald Flagg, Ph.D. University of Tampa Ashley Northrup University of Tampa Student Type of Research: Pedagogy Disciplines of Interest:

### American Index Put Options Early Exercise Premium Estimation

American Index Put Options Early Exercise Premium Estimation Ako Doffou: Sacred Heart University, Fairfield, United States of America CONTACT: Ako Doffou, Sacred Heart University, John F Welch College

### Steve 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

Swing Trade Warrior Chapter 1. Introduction to swing trading and how to understand and use options How does Swing Trading Work? The idea behind swing trading is to capitalize on short term moves of stocks

### FTS Real Time System Project: Using Options to Manage Price Risk

FTS Real Time System Project: Using Options to Manage Price Risk Question: How can you manage price risk using options? Introduction The option Greeks provide measures of sensitivity to price and volatility

### Study 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

### Model-Free Boundaries of Option Time Value and Early Exercise Premium

Model-Free Boundaries of Option Time Value and Early Exercise Premium Tie Su* Department of Finance University of Miami P.O. Box 248094 Coral Gables, FL 33124-6552 Phone: 305-284-1885 Fax: 305-284-4800

### 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.

### THE OHANA GROUP AT MORGAN STANLEY EMPLOYEE STOCK OPTION VALUATION

THE OHANA GROUP AT MORGAN STANLEY EMPLOYEE STOCK OPTION VALUATION BRYAN GOULD, FINANCIAL ADVISOR, PORTFOLIO MANAGER TELEPHONE: (858) 643-5004 4350 La Jolla Village Drive, Suite 1000, San Diego, CA 92122

### Option Pricing Beyond Black-Scholes Dan O Rourke

Option Pricing Beyond Black-Scholes Dan O Rourke January 2005 1 Black-Scholes Formula (Historical Context) Produced a usable model where all inputs were easily observed Coincided with the introduction

### 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

### Overview. 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

### The Behavior of Bonds and Interest Rates. An Impossible Bond Pricing Model. 780 w Interest Rate Models

780 w Interest Rate Models The Behavior of Bonds and Interest Rates Before discussing how a bond market-maker would delta-hedge, we first need to specify how bonds behave. Suppose we try to model a zero-coupon

### Valuing double barrier options with time-dependent parameters by Fourier series expansion

IAENG International Journal of Applied Mathematics, 36:1, IJAM_36_1_1 Valuing double barrier options with time-dependent parameters by Fourier series ansion C.F. Lo Institute of Theoretical Physics and

### UCLA 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,

### Black Scholes Merton Approach To Modelling Financial Derivatives Prices Tomas Sinkariovas 0802869. Words: 3441

Black Scholes Merton Approach To Modelling Financial Derivatives Prices Tomas Sinkariovas 0802869 Words: 3441 1 1. Introduction In this paper I present Black, Scholes (1973) and Merton (1973) (BSM) general

### American and European. Put Option

American and European Put Option Analytical Finance I Kinda Sumlaji 1 Table of Contents: 1. Introduction... 3 2. Option Style... 4 3. Put Option 4 3.1 Definition 4 3.2 Payoff at Maturity... 4 3.3 Example

### Chapter 1. Theory of Interest 1. The measurement of interest 1.1 Introduction

Chapter 1. Theory of Interest 1. The measurement of interest 1.1 Introduction Interest may be defined as the compensation that a borrower of capital pays to lender of capital for its use. Thus, interest

### Implied Volatility for FixedResets

Implied Volatility for FixedResets In this essay I explain the calculation of Implied Volatility for FixedResets, in accordance with a spreadsheet I have developed which is publicly available at http://www.prefblog.com/xls/impliedvolatility.xls.

### Pricing Dual Spread Options by the Lie-Trotter Operator Splitting Method

Pricing Dual Spread Options by the Lie-Trotter Operator Splitting Method C.F. Lo Abstract In this paper, based upon the Lie- Trotter operator splitting method proposed by Lo 04, we present a simple closed-form

### 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

### Hedging 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,

### b. June expiration: 95-23 = 95 + 23/32 % = 95.71875% or.9571875.9571875 X \$100,000 = \$95,718.75.

ANSWERS FOR FINANCIAL RISK MANAGEMENT A. 2-4 Value of T-bond Futures Contracts a. March expiration: The settle price is stated as a percentage of the face value of the bond with the final "27" being read

### 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

### Put-Call Parity and Synthetics

Courtesy of Market Taker Mentoring LLC TM Excerpt from Trading Option Greeks, by Dan Passarelli Chapter 6 Put-Call Parity and Synthetics In order to understand more-complex spread strategies involving

### Summary of Interview Questions. 1. Does it matter if a company uses forwards, futures or other derivatives when hedging FX risk?

Summary of Interview Questions 1. Does it matter if a company uses forwards, futures or other derivatives when hedging FX risk? 2. Give me an example of how a company can use derivative instruments to

### 16. Time dependence in Black Scholes. MA6622, Ernesto Mordecki, CityU, HK, 2006. References for this Lecture:

16. Time dependence in Black Scholes MA6622, Ernesto Mordecki, CityU, HK, 2006. References for this Lecture: Marco Avellaneda and Peter Laurence, Quantitative Modelling of Derivative Securities, Chapman&Hall

### Conn Valuation Services Ltd.

CONVERTIBLE BONDS: Black Scholes vs. Binomial Models By Richard R. Conn CMA, MBA, CPA, ABV, CFFA, ERP For years I have believed that the Black-Scholes (B-S) option pricing model should not be used in the

### CHAPTER 7 FUTURES AND OPTIONS ON FOREIGN EXCHANGE SUGGESTED ANSWERS AND SOLUTIONS TO END-OF-CHAPTER QUESTIONS AND PROBLEMS

CHAPTER 7 FUTURES AND OPTIONS ON FOREIGN EXCHANGE SUGGESTED ANSWERS AND SOLUTIONS TO END-OF-CHAPTER QUESTIONS AND PROBLEMS QUESTIONS 1. Explain the basic differences between the operation of a currency

### 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

### 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