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Black-Scholes Model
The Black-Scholes Model is a mathematical formula used to calculate the theoretical price of European-style options. It takes into account factors such as the underlying asset's price, the option's strike price, time to expiration, risk-free interest rate, and volatility. This model has been influential in the field of options pricing and has had a significant impact on financial markets.
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10 Key excerpts on "Black-Scholes Model"
- Bruno Remillard(Author)
- 2016(Publication Date)
- CRC Press(Publisher)
Chapter 1 Black-Scholes Model In this chapter, we introduce a first model for the dynamical behavior of an asset, the so-called Black-Scholes Model. One will learn how to estimate its parameters and how to compute the estimation errors. Then we will state the famous Black-Scholes formula for the price of a European call option, together with the general Black-Scholes equation for the value of a European option, and its representation as an expectation. The concept of implied volatility is then introduced. Finally, we conclude this chapter by estimating the sensitivity parameters of the option value, called “Greeks,” using Monte Carlo methods. One interesting application of greeks is to measure the impact of estimation errors on the value of options. 1.1 The Black-Scholes Model In Black and Scholes [1973], the authors introduced their famous model for the price of a stock, together with a partial differential equation for the value of a European option. To do so, they assumed the following “ideal conditions” on the market: • The short-term interest rate, also called risk-free rate, is known and constant until the maturity of the option. • The log-returns of the stock follows a Brownian motion with drift. • The option is European, i.e., it can only be exercised at maturity. • There are no transaction costs, nor penalties for short selling. • It is possible to buy or borrow any fraction of the security. To make things simpler, we assume that the market is liquid and friction-less, that trading can be done in continuous time, and that the risk-free rate is constant during the life of the option. The latter hypothesis will be weakened in Chapter 3. 1 2 Statistical Methods for Financial Engineering 1.2 Dynamic Model for an Asset Before discussing the model proposed by Black and Scholes [1973] for the distribution of an asset, we examine first some properties of financial data.
eBook - PDFDerivative Instruments
A Guide to Theory and Practice
- Brian Eales, Moorad Choudhry(Authors)
- 2003(Publication Date)
- Butterworth-Heinemann(Publisher)
The Monte Carlo method will be introduced to demonstrate pricing, once again, under a non-standard option payoff scenario. 11.2 The Black and Scholes model The option pricing model developed by Fischer Black and Myron Scholes first appeared in the academic literature in 1973. It now forms the basis for calculating the benchmark premia for regular and many forms of exotic options. The original model that Black and Scholes introduced to the world was based on some quite restrictive assumptions, subsequently researchers have been able to relax some of those underlying assumptions and develop their own less constrained option pricing models. 1 It is not the intention here to derive the B&S option pricing model from first principles and then to compare it algebraically with other models. It is, however, worth getting an intuitive appreciation of the B&S option pricing model, the inputs it requires, and the assumptions which underpin it. 1 For a thorough coverage of recent options pricing models see: Hull, J., Options, Futures, and Other Derivatives , 4th edition, Prentice Hall, 2000. 189 11.2.1 Inputs The inputs required to price an option in a basic B&S world are listed below: 1. The current price of the share; 2. The strike price; 3. The time remaining until expiration; 4. The risk-free rate of interest; 5. A measure of the standard deviation of the continuously compounded annual rate of return on the share ± this input is crucial to the pricing model and is known as volatility. Extensions of the B&S model require additional inputs such as dividend yield, foreign as well as domestic interest rates (currency options), resetting of strike prices, and barriers (exotic options). Some of these variations will be discussed and introduced in the sections that follow. 11.2.2 Assumptions The assumptions on which the basic model is built are, prima facie , formidable: 1. The option is only exercisable at expiration; 2. The market operates continuously; 3.
eBook - PDF- Robert Zipf(Author)
- 2003(Publication Date)
- Academic Press(Publisher)
With the European option, the option may be exercised only on the expiration date. Options may be used for hedging, income production, and speculation. Factors that determine the option price are the security price, the strike price, the time to expiration, the security volatility, and the risk-free interest rate. The Black-Scholes equation is the standard option pricing (evaluation) equation. Some problems arise with Black-Scholes: the security volatility may change, and the risk-free interest rate may change. Chapter Summary 319 Black-Scholes requires assumptions about the institutional structure of the market and human behavior in the securities markets. Black-Scholes has a special problem with bond options because it assumes a probability of bond prices exceeding the total amount of payments due (negative interest rate). Hedging techniques include the put-call parity relation and the use of hedging ratios. The delta is an important hedging ratio. Fractal analysis offers an alternative option analysis model with different human behavior assumptions. C OMPUTER P ROJECTS 1. Using your favorite mathematics computer program, enter the Black-Scholes options pricing model. Enter the parameters for a recently traded, active option to solve for the price. Then vary the five parameters that affect the price. How much does the options price vary as each parameter is varied? How would this affect your options trading system, if you had one? 2. Using your favorite mathematics computer program, enter the Black-Scholes options pricing model. How would you adjust it so that the probability of a price higher than the total of interest and principal to be received is zero? You could lower the variance of the underlying normal distribution so that the tail over the maximum total amount is acceptably low. What would this do to the overall analysis? You could simply lop off the tail of the normal distribution at the maximum total.
eBook - ePubDerivatives
Theory and Practice
- Keith Cuthbertson, Dirk Nitzsche, Niall O'Sullivan(Authors)
- 2019(Publication Date)
- Wiley(Publisher)
Bell Journal of spring 1973.Coincidentally, the Chicago Board Options Exchange (CBOE) began trading options (initially in the large smoking room of the Chicago Board of Trade) in April 1973 and the ‘new’ Black–Scholes formula was soon in use by traders. (For more details of this ‘story’ see the excellent book by Bernstein 2007.) The Ivory Towers of academia produced something of real practical value (as well as aesthetically pleasing). Whether it be ‘Black Holes’ or ‘Black–Scholes’, the power of mathematics to solve problems is impressive – not least in modern finance dealing with derivatives.Source: Adapted from Cuthbertson and Nitzsche (2001).The Black–Scholes (1973) formula for pricing European options (on a stock which pays no dividends) was derived using continuous time finance and stochastic calculus. The assumptions of the Black–Scholes model are:- All risk-free arbitrage opportunities are eliminated.
- No transactions costs or taxes.
- Investors can borrow and lend unlimited amounts at the risk-free interest rate which is constant over the life of the option.
- Stock prices are random, like a ‘coin flip’ – if your first flip gives ‘heads’ this does not help you to predict whether the next flip will give you a head or a tail. The technical term for the random stock price process (over very small time intervals) is a ‘geometric Brownian motion’.
- Stock prices are continuous and do not experience sudden extreme jumps – such as after an announcement of a takeover or other major unexpected firm specific events (e.g. new patents granted).
- The stock pays no dividends.
- The volatility of the stock (return) is known and constant over the life of the option (or is a deterministic function of time).
16.2.1 Call Option
To work out the Black–Scholes equation is rather difficult and at first sight the formula looks rather formidable. For the call premium:
eBook - ePubActuarial Finance
Derivatives, Quantitative Models and Risk Management
- Mathieu Boudreault, Jean-François Renaud(Authors)
- 2019(Publication Date)
- Wiley(Publisher)
16 Introduction to the Black-Scholes-Merton model The year 1973 shall be remembered as the year of breakthroughs in the history of options and derivatives. First of all, the largest U.S. options market, the Chicago Board Options Exchange (CBOE), was created that year. Moreover, the works from Fischer Black, Myron Scholes and Robert C. Merton, gave birth to modern financial mathematics and thus largely contributed to the shape of today’s derivatives markets. More precisely, in the paper The Pricing of Options and Corporate Liabilities, published in 1973, Black and Scholes pioneered rational option pricing by dynamically replicating the option payoff until maturity. During the same period, Robert C. Merton published Theory of Rational Option Pricing. He extended Black and Scholes’ approach in several ways, e.g. by pricing options on dividend-paying stocks and down-and-out barrier options. See [] and []. In his paper, Merton refers to Black and Scholes’ framework as the”Black-Scholes’ theory of option pricing” and even nowadays the model is widely known as the Black-Scholes’ model. Their work was recognized by the Royal Swedish Academy of Sciences (Nobel Prizes) in 1997. 1 The model is still very popular today on the financial markets. In this chapter, our main objective is to lay the foundations of the famous Black-Scholes-Merton market model and its pricing formula. We will provide a heuristic approach to this formula by linking as much as possible the derivations to the binomial model of Part I using a limiting argument
- Lishang Jiang(Author)
- 2005(Publication Date)
- WSPC(Publisher)
Chapter 5 European Option Pricing Black-Scholes Formula In this chapter, we will describe the price movement of an underlying asset by a continuous model — geometrical Brownian motion. Then we will set up a math-ematical model for the option pricing (Black-Scholes partial differential equation) and find the pricing formula (Black-Scholes formula). We will also discuss how to manage risky assets using the Black-Scholes formula and hedging technique. 5.1 History Option pricing is an old problem in finance. In 1900, Louis Bachelier published his doctoral thesis Theorie de la Speculation, now reckoned a milestone of the modern financial theory. In his thesis, Bachelier made the first attempt to model the stock price movement as a random walk. Option pricing problem was also addressed in his thesis. In 1964, Paul Samuelson, a Nobel Economics Prize winner, modified Bache-lier's model, using return instead of stock price in the original model. Let St be the stock price, then ^-p is its return. The stochastic differential equation proprosed by P. Samuelson is: ^p-= lidt + adWt. (5.1.1) This correction eliminates the unrealistic negative value of stock price in the original model. Using this model, P. Samuelson studied the call option pricing problem (C. Sprenkle(1965) and J. Baness(1964) also studied this problem at the same time). The result is given in the following. Let V be the premium of a call option, S be the asset price, K be the strike price, and T the expiration date, then V = e-acT [Se QsT iV(di) -KN(d 2 )}, (5.1.2) 73
eBook - ePub- Abdol S. Soofi, Yuqin Zhang(Authors)
- 2018(Publication Date)
- Business Expert Press(Publisher)
CHAPTER 10 Valuation of the Target Company Using the Black–Scholes Model In the framework of a merger and acquisition (M&A), a real call option gives the acquiring company the right, not the obligation, to acquire the target during the time the call option is valid.As discussed in Chapter 9 , one could use a real call option in valuing a target company. Here, we consider the Black–Scholes model for option pricing by first using an example of an option involving stocks and then by another example that illustrates valuation of a potential target firm.Black and Scholes (1973) developed a model for pricing the fair value of European options. The model is a partial differential equation, which describes the dynamics of option price adjustment within the time to expiration of the options. The derivation of Black–Scholes option pricing formulas is beyond the scope of the study. It suffices to say that the Black–Scholes stochastic differential equation has the following form:where V is the investor’s long position in one option, σ is the standard deviation or a measure of the volatility of the underlying asset of the option, S is the asset price, t is time, and r is the risk-free interest rate (interest rate on U.S. Treasury bills). The term measures the variation of the investor’s long position over time. The term shows the variation of the long position with respect to variation in the price of the underlying asset for the option.Pricing the option requires the solution of the above differential equation. However, an analytical closed-form solution1 for some of the stochastic differential equations modeling the dynamics of the option may not exist. For example, American and Asian options2 do not have analytical, closed-form solutions (Richardson 2009). In such cases, one must use one of the numerical methods in solving the differential equations. These numerical solutions include finite difference, binomial,3
- Fabrice D. Rouah, Gregory Vainberg(Authors)
- 2012(Publication Date)
- Wiley(Publisher)
Chapter 4
The Black-Scholes, Practitioner Black-Scholes, and Gram-Charlier Models
INTRODUCTION
In this chapter we review the Black-Scholes option pricing model and present the VBA code to implement it. We do not derive this model nor spend too much time explaining it since so much has been written about it already, in textbooks such as those by Hull (2006), Haug (1998), and Chriss (1997). We review implied volatility and the moneyness and maturity biases that give rise to volatility smiles. We then review the Practitioner Black-Scholes Model, which uses the Deterministic Volatility Function of Dumas, Fleming, and Whaley (1998). Finally, we discuss the model of Backus, Foresi, and Wu (2004), which introduces skewness and excess kurtosis into the Black-Scholes Model to account for moneyness and maturity biases.THE Black-Scholes Model
This model hardly needs introduction. It is the most popular option pricing model, due to its simplicity, closed-form solution, and ease of implementation. The Black and Scholes (1973) price at time t for a European call option with maturity at time t + T with strike price K on a stock paying no dividends is(4.1)whereσ = annual stock return volatility (assumed constant)St= time t price of the stockr = annual risk-free interest rateT = time to maturity in yearsSome authors use the notation and n (x ) = φ(x ).To price a put option, the put-call parity relation can be used, which produces(4.2)orCBScan be substituted into (4.2 ) to getThe Excel file Chapter4BlackScholes contains VBA code for implementing the Black-Scholes Model. The VBA function BS_call() produces the Black-Scholes price of a European call given by Equation (4.1 ), while function BS_put() uses the put-call parity relationship (4.2 ) to produce the price of a European put. Both functions require as inputs the spot price of the asset (S ), the strike price (K ), the risk-free rate of interest (r ), the time to maturity (T
eBook - PDFThe Rise of the Quants
Marschak, Sharpe, Black, Scholes and Merton
- C. Read(Author)
- 2012(Publication Date)
- Palgrave Macmillan(Publisher)
Even Black and Scholes found this result odd, at least until their colleague Merton explained that the return is included implicitly in the price of the stock, which is part of their equation. Disseminating and marketing a new financial tool However, practitioners found the equation liberating and were willing to put academic reservations aside in pragmatic return for the overwhelm- ing usefulness of a pricing equation that is relatively simple to calculate. In addition, the assumptions of the equation seemed reasonable and were already broadly accepted in other applications in finance, statistics, economics, and the decision sciences. Finally, the equation could act as a bright light in a previously unilluminated new frontier. In this respect, it provided a focal point from which speculation could deviate. The champions of the CBOE knew intuitively just how valuable the Black-Scholes options pricing equation could be. Up to that point, the finance literature was nascent and qualitative, well before developments from economic theory had become mainstream and well before the quantitative wave had swept over Wall Street and the literature. Black and Scholes had aimed for the economic theory literature as a result of the perceived lack of rigor within finance, but met with the same resist- ance in getting their work published as had William Sharpe and his CAPM model almost a decade earlier. However, once their article was accepted for publication in the prestigious Journal of Political Economy for its 1973 volume, 2 Black’s star was in its ascendancy, especially at the University of Chicago, where the Journal of Political Economy was published. Word had already got out about the paper and caught the ear of the University of Chicago’s influential finance professor James H. Lorie (1922–2005) and the CBOE’s champion and first Vice Chairman, Edmund O’Connor.
eBook - ePubFX Barrier Options
A Comprehensive Guide for Industry Quants
- Zareer Dadachanji(Author)
- 2016(Publication Date)
- Palgrave Macmillan(Publisher)
8.When we lend cash, we are sure to get it back. [Interest-paying counterparties have zero credit risk.]The assumptions in the above list, whilst significant, are present in many models that are more sophisticated, including those models which on the whole yield prices close to the market. We should in no way lose confidence in the Black–Scholes model due to the assumptions above. However, the most serious assumption of the Black–Scholes model is one that lies not in the list above, but rather in the assumed model process itself: the assumption that spot moves in a geometric Brownian Motion with constant volatility σ . We will discuss the ramifications of this assumption in Chapter 4 , just before we embark on the modelling adventure that is Smile Pricing. For the moment, however, we will accept the Black–Scholes model for what it is, and continue to deepen our analysis and understanding of it.2.3.4 Interpretation of the Black–Scholes PDE
It is instructive to examine each of the terms in the Black–Scholes PDE (Equation 2.21 ) and interpret it intuitively. We split the equation up as follows:2.3.4.1 Term 1: theta term
Term 1 is relatively straightforward to interpret: the partial derivative , being taken at constant spot, measures the rate at which the option value changes due to the passage of time alone. Even if spot does not move, the option changes in value. In general, the rate of change in option value with respect to time is termed option theta , and denoted by the Greek upper-case letter Θ, as discussed in Section 3.4 . We may therefore call Term 1 the theta term .2.3.4.2 Term 2: carry term
Term 2 readily admits interpretation if we recall that the delta hedge specified by the model consists of a position in Foreign cash of amount . Given a Foreign interest rate of r f , we expect to earn Foreign interest on this cash position at the rate , which when converted to Domestic gives . In order to buy a Foreign cash position of , we must have had to borrow an amount of Domestic cash equal to , which will be costing us interest at the rate , or equivalently earning us interest at the rate
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