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Black Scholes Formula
The Black-Scholes formula is a mathematical model 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. The formula 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 Formula"
eBook - PDF- Mondher Bellalah(Author)
- 2009(Publication Date)
- World Scientific(Publisher)
The story began in 1900, when the French mathematician, Louis Bachelier, obtained an option pricing formula. His model is based on the assumption that stock prices follow a Brownian motion. Since then, numerous studies on option valuation have blossomed. The proposed formulas involve one or more arbi-trary parameters. They were developed by Sprenkle (1961), Ayres (1963), Boness (1964), Samuelson (1965), Thorp and Kassouf (1967), and Chen (1970) among others. The Black and Scholes (1973) formulation, hereafter called as B–S, solved a problem, which has occupied the economists for at least three-quarters of a century. This formulation represented a significant breakthrough in attacking the option pricing problem. In fact, the Black– Scholes theory is attractive since it delivers a closed-form solution to the pricing of European options. Assuming that the option is a function of a single source of uncertainty, namely the underlying asset price, and using a portfolio which combines options and the underlying asset, Black and Scholes constructed a risk-less hedge, which allowed them to derive an analytical formula. This model provides a no-arbitrage value for European options on shares. It is a function of the share price S , the strike price K , the time to maturity T , the risk-free interest rate r , and the volatility of the stock price, σ . This model involves only observable variables to the excep-tion of volatility and it has become the benchmark for traders and market makers. It also contributed to the rapid growth of the options markets by making a brand new pricing technology available to market players. About the same time, the necessary conditions to be satisfied by any rational option pricing theory were summarized in Merton’s (1973) theorems. The post-Black–Scholes period has seen many theoretical devel-opments.
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:
- Thomas Mikosch(Author)
- 1998(Publication Date)
- WSPC(Publisher)
4 Applications of Stochastic Calculus in Finance Since the celebrated papers by Black and Scholes (1973) and Merton (1973) the idea of using stochastic calculus for modelling prices of risky assets (share prices of stock, stock indices such as the Dow Jones, Nikkei or DAX, foreign exchange rates, interest rates, etc.) has been generally accepted. This led to a new branch of applied probability theory, the field of mathematical finance. It is a symbiosis of stochastic modelling, economic reasoning and practical financial engineering. In this chapter we consider the Black-Scholes model for pricing a Euro-pean call option. Do not worry if you do not have much prior knowledge about economic and financial processes. As a minimum, you will need some words of economic language, and as a maximum, the ltd lemma. In Section 4.1 we will explain the basic terminology of finance: bond, stock, option, portfolio, volatil-ity, trading strategy, hedging, maturity of a contract, self-financing, arbitrage will be the catchy words. The Black-Scholes formula for pricing a European call option will be obtained as the solution of a particular partial differential equation. Since notions like equivalent martingale measure and change of measure have become predominant in the financial literature, we give in Section 4.2 a short introduction to this area. This requires some knowledge of measure-theoretic tools, but, as usual, we will keep the theory on a low level. We hope that the key ideas will become transparent, and we give a second derivation of the Black-Scholes formula by using the change of measure ideology. 168 CHAPTER 4. 4.1 The Black-Scholes Option Pricing Formula 4.1.1 A Short Excursion into Finance We assume that the price X t of a risky asset (called stock) at time t is given by geometric Brownian motion of the form X t = f{t,B t )=X Q ^-™^ t + ° B > , (4.1) where, as usual, B = {B u t > 0) is Brownian motion, and X 0 is assumed to be independent of B.
eBook - PDFThe Social Life of Financial Derivatives
Markets, Risk, and Time
- Edward LiPuma(Author)
- 2017(Publication Date)
- Duke University Press Books(Publisher)
And so it came to pass that the refinement and publi-cation of the Black-Scholes formula for options pricing (Black and Scholes 1973) corresponded almost to the day with the celebratory opening of the Chicago Board Options Exchange. As we will see, the bs model is a model of simplicity and elegance at the surface level that sits atop an underlying complexity gener-ated by the countervailing forces that course through real markets. But first, to get things rolling, an annotated version of the formulation. The Black-Scholes equation has the following form: 11 ∂ V ∂ t + 1 2 σ 2 S 2 ∂ 2 V ∂ S 2 + rS ∂ V ∂ S − rV = 0 ∂ V ∂ t All derivatives have expiration dates. This part of the equation models how much the option’s value (v) changes over time (t) if the price of the underlying asset remains constant. In other words, its role is to determine how much the premium built into the derivative wanes as the contract moves toward expiration. 1 2 σ 2 S 2 ∂ 2 V ∂ S 2 The formula is founded on a financial argument about hedging. This part of the equation defines how much a hedged position changes when the under- lying asset changes. It specifies the sensitivity of delta to the underlier, thereby al-lowing the model to account for the volatility of the underlying asset. It is a convexity term in that it measures how changes in the underlier are reflected in the derivative. Doing this entails specifying that if the point of expiration and point of pricing are in a space of events then so are all the points on the line segment joining them. rS ∂ V ∂ S All derivatives posit and calculate from a risk-free rate (of return). This part of the formula allows for increases in the value of the underlying asset as the risk free rate. − rV Since derivatives are valued at some time point prior to the payout which (hy-pothetically) takes place at the derivative’s expiration, it is necessary to include a discounting term.
- 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 - 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 - 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 - PDFGreen's Functions
Construction and Applications
- Yuri A. Melnikov, Max Y. Melnikov(Authors)
- 2012(Publication Date)
- De Gruyter(Publisher)
However, the equation is becoming increasingly popular in its field where it is widely used for qualitative as well as quantitative analysis of stock option pricing problems. This attracts numerous researchers working in this rapidly developing area of financial engineering, that is slowly but surely becoming an integral part of applied mathematics. Our approach to the Black–Scholes equation and its treatment are very similar to those of the diffusion equation that we described in detail in Chapter 7: the use of the Laplace transform will be combined with the method based on eigenfunction ex-pansion, in order to treat a number of terminal-boundary-value problems for the gov-erning equation. Some necessary preparatory work is presented in Section 8.1, where we will show that a special solution of the Black–Scholes equation, which is, quite customarily in the field, referred to as its Green’s function , represents, in fact, its fun-damental solution . In Section 8.2 the emphasis will be on the construction of those Green’s functions that were never before exposed in books. In Section 8.3 our focus Section 8.1 The Fundamental Solution 363 will be on several methodological issues related to our approach to the construction of Green’s functions for parabolic type equations. Section 8.4 changes our view an-gle, as we focus on numerical results, illustrating the computational potential of the procedures, based on Green’s functions. 8.1 The Fundamental Solution The intention in this section is to establish a terminological basis for our work on Green’s functions for the Black–Scholes equation @V.S;t/ @t C 2 S 2 2 @ 2 V.S;t/ @S 2 C rS @V.S;t/ @S rV.S;t/ D 0 (8.1) which represents a linear, backward in time, parabolic type partial differential equa-tion with variable coefficients. In shorthand form, it will in this book, frequently be referred to as BSE.
eBook - ePub- Paul Wilmott(Author)
- 2013(Publication Date)
- Wiley(Publisher)
If we can see the price at which the option is trading, we can ask ‘What volatility must I use to get the correct market price?’ This is called the implied volatility. The implied volatility is the volatility of the underlying which when substituted into the Black–Scholes formula gives a theoretical price equal to the market price. In a sense it is the market’s view of volatility over the life of the option. Assuming that we are using call prices to estimate the implied volatility then provided the option price is less than the asset and greater than zero then we can find a unique value for the implied volatility. (If the option price is outside these bounds then there’s a very extreme arbitrage opportunity.) Because there is no simple formula for the implied volatility as a function of the option value we must solve the equation for σ, where V BS is the Black–Scholes formula. Today’s asset price is S 0, the date is t 0 and everything is known in this equation except for σ. Below is an algorithm for finding the implied volatility from the market price of a call option to any required degree of accuracy. The method used is Newton–Raphson which uses the derivative of the option price with respect to the volatility (the vega) in the calculation
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