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.
7 Key excerpts on "Black-Scholes Model"
eBook - ePubDerivatives
Theory and Practice
- Keith Cuthbertson, Dirk Nitzsche, Niall O'Sullivan(Authors)
- 2019(Publication Date)
- Wiley(Publisher)
...CHAPTER 16 Black–Scholes Model Aims To demonstrate how option prices change with changes in the price of the underlying asset, its volatility, interest rates and time to maturity. To establish upper and lower bounds for the price of European calls and puts. To show how the Black–Scholes formula is used to price European calls and puts. To show how options and the underlying asset (e.g. stock-Z) can be combined into a portfolio which does not change in value when there is a small change in the price of the underlying asset – this is delta hedging. To explain implied volatility and its use in options trading. 16.1 DETERMINANTS OF OPTION PRICES It can be shown that the option premium varies second-by-second as the stock price, the risk-free interest rate and the volatility of the stock change, over time. Let us develop some intuitive arguments which give some insight into the determination of European option prices. We consider each factor in turn, holding all the other factors constant. This intuitive approach will help us understand the mathematical formulas for option prices which we present later. In each case we assume the investor has a long options position (i.e. has purchased a call or a put) and we only consider European stock options (where the stock pays no dividends). The results are summarised in Table 16.1. TABLE 16.1 Factors affecting the option premia Long European call option Long European put option Time to expiration, T + + Current stock price, + – Strike price, K – + Stock return volatility, + + Risk-free rate, r + – Notes: We only consider options on stocks which pay no dividends. ‘+’ indicates a positive relationship between the option price and the variable chosen. That is, a rise (fall) in the variable is accompanied by a rise (fall) in the option premium. A ‘–’ indicates a negative relationship between the option price and the variable chosen...
- 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) where S t = time t price of the stock σ = annual stock return volatility (assumed constant) r = annual risk-free interest rate T = time to maturity in years Some 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) or C BS can be substituted into (4.2) to get The 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...
- Mario Cerrato(Author)
- 2012(Publication Date)
- Wiley(Publisher)
...5 The Black and Scholes Model 5.1 INTRODUCTION In this chapter we introduce the Black and Scholes model and discuss its practical use. The main assumptions underlying the model are: a. it is possible to short-sell the underlying stock; b. trading stocks is continuous; c. there are no arbitrage opportunities; d. there are no transaction costs or taxes; e. all securities are perfectly divisible (i.e. you can buy or sell a fraction); f. it is possible to borrow or lend cash at a constant risk-free rate; g. the stock does not pay a dividend. 5.2 THE BLACK AND SCHOLES MODEL Define a standard Brownian motion W (t), on 0 ≤ t ≤ T, and a probability space (Ω, F, P). The process driving the stock price is a geometric Brownian motion: (5.1) Suppose that C (t)= C (S (t), t) is the value at t of an option on the stock S (t). Changes in the value of the option on small intervals, d C (t), are 1 (5.2) The term is stochastic. However, we can eliminate it from (5.2). Suppose we can construct the following portfolio P consisting of a long-position on the call option and a short-position on n units of stock: We can differentiate P to obtain (5.3) Using (5.2) in combination with (5.3) we obtain (5.4) Thus, setting n in (5.4) equal to, we have (5.5) The portfolio in (5.5) is risk free. Thus, it should earn the risk-free rate of return. 2 Since, it follows that 3 (5.6) This result is known as the Black and Scholes partial differential equation. In the absence of arbitrage any derivative security having as an underlying stock S should satisfy (5.6). 5.3 THE BLACK AND SCHOLES FORMULA For simplicity, we now set T =1. We already know that if the market is arbitrage free, the value of an option can be written as the expected value of its cash flows (see Sections 4.3 and 4.5): (5.7) where S (T) is the stock price at T, K is the strike of the option and the expectation is taken under the measure Q...
eBook - ePub- Paul Wilmott(Author)
- 2013(Publication Date)
- Wiley(Publisher)
...This is market completeness. The risk and reward on an option and on its underlying are related and the Black–Scholes equation follows. 5.13 SUMMARY This was an important but not too difficult chapter. In it I introduced some very powerful and beautiful concepts such as delta hedging and no arbitrage. These two fundamental principles led to the Black–Scholes option pricing equation. Everything from this point on is based on, or is inspired by, these ideas. FURTHER READING The history of option theory, leading up to Black–Scholes is described in Briys, Mai, Bellalah & de Varenne (1998). The story of the derivation of the Black–Scholes equation, written by Bob Whaley, can be found in the 10th anniversary issue of Risk magazine, published in December 1997. Of course, you must read the original work, Black & Scholes (1973) and Merton (1973). See Black (1976) for the details of the pricing of options on futures, and Garman & Kohlhagen (1983) for the pricing of FX options. For details of other ways to derive the Black–Scholes equation, see Andreasen, Jensen & Poulson (1998). 1 Actually. I’m lying. One of these parameters does not affect the option value. 2 The pricing formulae were being used even earlier by Ed Thorp to make money. 3 Life, and everything in it, is based on arbitrage opportunities and their exploitation. I always wonder how, after learning about the absence of free lunches, students can join a bank and then expect to get six or seven figure bonuses. Where do they expect this money to come from?...
eBook - ePubAn Introduction to Financial Mathematics
Option Valuation
- Hugo D. Junghenn(Author)
- 2019(Publication Date)
- Chapman and Hall/CRC(Publisher)
...Chapter 12 The Black-Scholes-Merton Model With the methods of Chapter 11 at our disposal, we are now able to derive the celebrated Black-Scholes formula for the price of a call option. The formula is based on the solution of a partial differential equation arising from a stochastic differential equation that governs the price of the underlying stock S. We assume throughout that the market is arbitrage-free. 12.1 The Stock Price SDE Let W be a Brownian motion on a probability space (Ω, ℱ, ℙ). The price S of a single share of S is assumed to satisfy the SDE d S S = σ d W + μ d t, (12.1) where μ and σ are constants called, respectively, the drift and volatility of the stock. Equation (12.1) asserts that the relative change in the stock price has two components: a deterministic part μ dt, which accounts for the general trend of the stock, and a random component σ dW, which reflects the unpredictable nature of S. The volatility is a measure of the riskiness of the stock and its sensitivity to changes in the market. If σ = 0, then (12.1) is an ODE with solution S t = S 0 e μt. Equation (12.1) may be written in standard form as d S = σ S d W + μ S d t, (12.2) which is the SDE of Example 11.5.2. The solution there was found to be S t = S 0 exp [ σ W t + (μ − 1 2 σ 2) t ]. (12.3) The integral version of (12.2) is S t = S 0 + ∫ 0 t σ S (s) d W (s) + ∫ 0 t μ S (s) d s. (12.4) Taking expectations in (12.4) and using Theorem 11.3.3, we see. that E S t = S 0 + μ ∫ 0 t E S (s) d s. The function x (t) ≔ E S t therefore satisfies the ODE x ′ = μx, hence E S t = S 0 e μ t. This is the solution of (12.1) for the case σ = 0 and represents the return on a risk-free investment. Thus taking expectations in (12.4) “removes” the random component of (12.1). Both the drift μ and the volatility σ may be stochastic processes, in which case the solution to (12.1) is given by S t = S 0 exp { ∫ 0 t σ (s) d W (s) + ∫ 0 t [ μ (s) − 1 2 σ 2 (s) ] d s }, as was shown in Example 11.5.2...
eBook - ePubFX Barrier Options
A Comprehensive Guide for Industry Quants
- Zareer Dadachanji(Author)
- 2016(Publication Date)
- Palgrave Macmillan(Publisher)
...[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...
eBook - ePub- Antonio Castagna(Author)
- 2010(Publication Date)
- Wiley(Publisher)
...2 Pricing Models for FX Options 2.1 PRINCIPLES OF OPTION PRICING THEORY We will shortly review the theory of option pricing with a strict reference to the FX world. First, we introduce a (slightly extended) BS economy, then we relax one of the basic assumptions: we will allow the volatility of the FX rate process to be stochastic. These principles will pave the way to the analysis of some well-known models employed in practice to price FX options. 2.1.1 The Black-Scholes economy We work in continuous time and assume that W t is a standard Brownian motion, and a martingale with respect to a filtered probability space (Ω, Ƒ, F, P) for the time set [0, ∞). We assume also that the filtration F satisfies the usual conditions, 8 and that we have a perfect frictionless market, with one domestic and foreign interest rate (at which interest accrues continuously). In the economy, one risky asset is traded: an FX pair whose price process is the following stochastic differential equation (SDE): (2.1) where µ t and ς t are time-dependent parameters. A second traded asset is a riskless (domestic) deposit, 9 whose price changes according to the following differential equation: (2.2) An FX pair can be considered as an asset yielding a continuous cash flow equal to the foreign interest rate. In fact, when a trader buys one unit of a given pair, they sell S t quantity of domestic currency: this quantity can be invested in a money market deposit and earn accrued interest at the rate r t f. We have described a BS economy, since it is basically the same economy assumed by Black and Scholes [9] (apart from time-dependent parameters). The two assets can be employed in a trading strategy, and we are interested in designing a strategy with the following specific feature: Definition 2.1.1. Self-financing strategy. Assume that we establish a trading strategy by holding, at time 0, the quantity α 0 of the risky asset (i.e., the FX pair) and the quantity β 0 of the deposit...
