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.
8 Key excerpts on "Black Scholes Formula"
eBook - ePubDerivatives
Theory and Practice
- Keith Cuthbertson, Dirk Nitzsche, Niall O'Sullivan(Authors)
- 2019(Publication Date)
- Wiley(Publisher)
...Merton, who had collaborated with Black and Scholes, also produced a paper on options pricing in the 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...
eBook - ePub- Frans de Weert(Author)
- 2011(Publication Date)
- Wiley(Publisher)
...Chapter 2 THE BLACK – SCHOLES FORMULA In the previous chapter the definition of an option was given. Furthermore, some examples showed what the payoff and profit of an option look like. In these examples the price of the option was always given. It is however possible to identify what the fair price of a European option should be. In this perspective ‘fair’ means that the expected profit for both sides of the option contract is 0. The Black – Scholes formula is a good tool for determining the fair price of an option. From the definition of an option it is clear that the price should depend on the strike price, the price of the underlying stock and the time to maturity. It appears that the price of an option also depends on less obvious variables. These other variables are interest rates, the volatility of the underlying stock (the way the stock moves) and the dividends on the stock. By some simple examples it can be clarified that the option price should also depend on the last mentioned variables: • Interest rate. Suppose that the interest rate given on a savings account is 5% per year. Consider a put option with a time to maturity of 1 year, and, given an interest rate of 5%, the price of this option is $10. Since the holder of the short position in this option gets this $10, he can put this money in a savings account, getting a 5% interest rate. By doing so he will have $10.5 (10 1.05) at the expiration date of the option. Since the price of the option is fair, the expected payoff for the holder of the long position in the option will also be $10.5. Now suppose that the interest rate was not 5% but 6%. Higher interest rates cause expected growth rates on stocks to increase, otherwise investors in stocks could be tempted to sell their stocks and put the money in a savings account. This means that the expected payoff of a put option is likely to be less...
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?...
- 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 - 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...
- Paul Wilmott(Author)
- 2010(Publication Date)
- Wiley(Publisher)
...Chapter 9 The Black-Scholes Formulæ and the Greeks In the following formulæ and The formulæ are also valid for time-dependent σ, D and r, just use the relevant ‘average’ as explained in the previous chapter. Warning The greeks which are ‘greyed out’ can sometimes be misleading. They are those greeks which are partial derivatives with respect to a parameter (σ, r or D) as opposed to a variable (S and t) and which are not single signed (i.e. always greater than zero or always less than zero). Differentiating with respect a parameter, which has been assumed to be constant so that we can find a closed-form solution, is internally inconsistent. For example, ∂ V/∂σ is the sensitivity of the option price to volatility, but if volatility is constant, as assumed in the formula, why measure sensitivity to it? This may not matter if the partial derivative with respect to the parameter is of one sign, such as ∂ V/∂σ for calls and puts. But if the partial derivative changes sign then there may be trouble. For example, the binary call has a positive vega for low stock prices and negative vega for high stock prices, in the middle vega is small, and even zero at a point. However, this does not mean that the binary call is insensitive to volatility in the middle. It is precisely in the middle that the binary call value is very sensitive to volatility, but not the level, rather the volatility skew. Table 9.1 : Formulæ for European call. Table 9.2 : Formulæ for European put. Table 9.3 : Formulæ for European binary call. Table 9.4 : Formulæ for European binary put....
