Binomial Model
The binomial model is a mathematical tool used to evaluate the value of options. It assumes that the price of the underlying asset can move up or down over time in a series of steps. By calculating the expected value at each step, the model helps businesses make decisions about options and other financial instruments.
8 Key excerpts on "Binomial Model"
eBook - ePub- Paul Wilmott(Author)
- 2013(Publication Date)
- Wiley(Publisher)
...CHAPTER 15 the Binomial Model In this Chapter… a simple model for an asset price random walk delta hedging no arbitrage the basics of the binomial method for valuing options risk neutrality 15.1 INTRODUCTION We have seen in Chapter 3 a model for equities and other assets that is based on the mathematical theory of stochastic calculus. There is another, equally popular, approach that leads to the same partial differential equation, the Black–Scholes equation, in a way that some people find more ‘accessible,’ which can be made equally ‘rigorous.’ This approach, via the Binomial Model for equities, is the subject of this chapter. Undoubtedly, one of the reasons for the popularity of this model is that it can be implemented without any higher mathematics (such as differential calculus) and there is actually no need to derive a partial differential equation before this implementation. This is a positive point, however the downside is that it is harder to attain greater levels of sophistication or numerical analysis in this setting. Before I describe this model I want to stress that the Binomial Model may be thought of as being either a genuine model for the behavior of equities, or, alternatively, as a numerical method for the solution of the Black–Scholes equation. 1 Most importantly, we see the ideas of delta hedging, risk elimination and risk-neutral valuation occurring in another setting. The Binomial Model is very important because it shows how to get away from a reliance on closed-form solutions. Indeed, it is extremely important to have a way of valuing options that only relies on a simple model and fast, accurate numerical methods. Often in real life, a contract may contain features that make analytic solution very hard or impossible. Some of these features may be just a minor modification to some other, easily-priced, contract but even minor changes to a contract can have important effects on the value and especially on the method of solution...
- Moorad Choudhry(Author)
- 2001(Publication Date)
- Butterworth-Heinemann(Publisher)
...45 Options III: The Binomial Pricing Model 45.1 The binomial option pricing model We have already introduced the binomial approach to pricing securities, in the chapters on callable bonds and convertibles. The methodology may be applied to options as well, and in fact there are a number of option variants that cannot be priced accurately by the B–S model, so that it is necessary to use the Binomial Model instead. Under certain scenarios the Binomial Model approximates to the B–S model, depending on the number of lattices used in the model and certain other factors. The Binomial Model, which is also known as the lattice approach, is perhaps a more academic approach than the continuous-time models pioneered by Black–Scholes, although it appeared later. However under certain conditions it is preferred as a valuation tool, most commonly for the pricing of options on equities and equity indices, that is, for products where a dividend payment is associated with the underlying asset. The binomial approach was first presented by Cox, Ross and Rubinstein in 1979. Although first applied to equity options, it has since been extended to products such as callable bonds and convertibles and bond options, and to model the term structure of interest rates. A subsequent paper by Ho and Lee (1986) used the binomial approach to value zero-coupon bonds, and therefore can be used to describe the term strucure of interest rates. Other models developed on a similar basis include Black, Derman and Toy (BDT, 1990) and Hull and White (1993). The BDT model incorporates a lattice approach to construct a one-factor model of the short-term interest rate, and the parameters used in deriving the model include the volatility values for all zero-coupon rates along the term structure, in addition to the current zero-coupon term structure...
eBook - ePubAn Introduction to Financial Mathematics
Option Valuation
- Hugo D. Junghenn(Author)
- 2019(Publication Date)
- Chapman and Hall/CRC(Publisher)
...Chapter 7 The Binomial Model In this chapter, we construct the geometric Binomial Model for stock price movement and use the model to determine the value of a general European claim. An important consequence is the Cox-Ross-Rubinstein formula for the price of a call option. Formulas for path-dependent claims are given in the last section. The valuation techniques in this chapter are based on the notion of self-financing portfolio described in Chapter 5. 7.1 Construction of the Binomial Model Consider a (non-dividend paying) stock S with initial price S 0 such that during each discrete time period the price changes either by a factor u with probability p or by a factor d with probability q ≔ 1 − p, where 0 < d < u. The symbols u and d are meant to suggest the words “up” and “down” and we shall use these terms to describe the price movement from one period to the next, even though u may be less than one (prices drift downward) or d may be greater than one (prices drift upward). The model has essentially three components: the stock price formula, the probability law that governs the price, and the flow of information in the form of a filtration. The Stock Price The stock’s movements are modeled as follows: For a fixed positive integer N, the number of periods over which the stock price is observed, let Ω be the set of all sequences ω = (ω 1, …, ω N), where ω n = u if the stock moved up during the n th time period and ω n = d if the stock moved down. We may express Ω as a Cartesian product Ω = Ω 1 × Ω 2 × ⋯ × Ω N, where Ω n = { u, d } represents the possible movements of the stock at time n. A member of Ω represents a particular market scenario. For example the sequence (u, d, *, …, *) represents all market scenarios with the property that during the time interval [0, 1] the stock value changed from S 0 to uS 0, and during the time interval [1, 2] the stock value changed from uS 0 to duS 0...
eBook - ePubFinancial Simulation Modeling in Excel
A Step-by-Step Guide
- Keith A. Allman, Josh Laurito, Michael Loh(Authors)
- 2011(Publication Date)
- Wiley(Publisher)
...CHAPTER 4 Option Pricing With a set of basic skills developed, we can now move on to practical application. The first financial realm that we will apply simulation techniques to is option pricing. As we work our way through option pricing, you will see that simulation techniques can take on many forms and be intertwined with multiple financial disciplines. For instance, in the derivatives or options industry, one routinely encounters discussions about bonds, which then leads to interest rates. One may wonder why, if the purpose of a stock option is to purchase stocks, that one would need to know anything about the bond market? The quick answer to this question is that to price an option, the price must in some way be enforced by a market driven asset that represents the cost of borrowing as viewed by the entire financial industry. That asset is a bond, and it is the anchor all the derivative and option pricing models adhere to. We will discuss this concept in a little more detail further on, but to begin with we will start off with the most basic pricing model, the binomial tree. To demonstrate why an enforcer is necessary, we will use the binomial tree to model our most basic asset, the stock. A note: This text is focused on practical application and will not formally explicate binomial trees. It also assumes that the reader is familiar with options and bonds as financial instruments. Binomial trees will be discussed only in a general sense to present ideas and concepts necessary to understand how to implement the Hull-White Trinomial tree later on. If the reader is interested in learning more about the theory of binomial trees, there are many good texts that cover this topic in more detail. BINOMIAL TREES The most basic method of presenting stock price movement is a binomial tree...
eBook - ePubOption Trading
Pricing and Volatility Strategies and Techniques
- Euan Sinclair(Author)
- 2010(Publication Date)
- Wiley(Publisher)
...Actually the underlying price is a random function that has a certain distribution. So anything else that may plausibly influence option prices through the underlying and its risk will enter the analysis here. For example, stock prices are influenced by earnings, seasonality effects, prices of raw materials, and currency prices, but these can all be viewed as contributors to the shape of the stock’s return distribution and do not need to be modeled separately. With this preamble behind us, we can look at two specific option-pricing models. THE Binomial Model Examining the Binomial Model (or binomial tree) is the easiest way to get intuition about option pricing. It is also a very good way to price general options with more complex expiration conditions or payoff structures than the standard exchange traded vanilla options. First we will look at a very simple “toy” version of the model. Then we will show how it can be extended into a more realistic form. FIGURE 4.1 The Evolution of the Stock Price in the One-Step Binomial Tree Imagine that we have a stock that is trading $100. It can either move up $1 with probability, p, or move down $1 with probability, 1 − p. This situation is illustrated in Figure 4.1. Now let’s imagine that we own the at-the-money call option. This has a strike of 100. If the stock rises we will make a profit of $1. If it drops we make nothing. We will try to hedge this position by selling a quantity, h, of the underlying. When we have done this, if the stock rises we will make money on our call but lose money on our hedge. Our portfolio will be worth (4.1) If the stock drops we lose money on our option, but our hedge will be profitable. In this case our portfolio will be worth (4.2) For this to actually be a hedge we chose the hedge ratio, h, so that these two expressions are equal. This requires that h = 0.5. Now we are indifferent to the stock’s movement...
eBook - ePubQuantitative Finance
A Simulation-Based Introduction Using Excel
- Matt Davison(Author)
- 2014(Publication Date)
- Chapman and Hall/CRC(Publisher)
...Chapter 18 Pricing Options Using Binomial Trees 18.1 CHAPTER SUMMARY In this chapter, we discuss a European option written on a very simple stock price model: one in which, at each sequence of time steps, the stock price can either rise or fall. Such models are termed binomial trees. We begin by considering a European call option written on a stock, which either goes up or down over a single period. We show how to hedge such an option to remove all risks from the resulting portfolio, and hence to have a price that is the same for everyone, whatever their risk preference. In the next section, we show that this argument could work, not just for a call but for any option written on the stock with this simple model. This allows us to introduce a more sophisticated tree model in which the stock rises or falls over several time steps, allowing more possible final stock price outcomes. Here too, we show that a sequence of hedging operations can remove risk from the portfolio. The path of a price through a tree resembles a random walk along a stock price number line. In Chapter 20, we deepen this analogy and introduce some new mathematics, which we follow in Chapter 21 with a brief overview of the subject of stochastic calculus...
eBook - ePub- Andreas Binder, Michael Aichinger(Authors)
- 2013(Publication Date)
- Wiley(Publisher)
...If the portfolio is to replicate the option value, the unknowns a 1, a 2 must satisfy with the solutions Hence, if we choose the weights a 1 and a 2 accordingly, P replicates the option and therefore the option value equals the portfolio value also at time 0. Hence, we obtain (2.3) Note that the formula for the option value does not depend on the probability p of the outcomes s 1, s 2 ! We would have obtained the same result by discounting the expected outcome (at time T) with the expectation value taken under a probability q for s 1 and 1 − q for s 2 with (2.4) The measure implied from this change of probability is called the risk neutral measure of the Binomial Model, whereas the physical measure is the one with the probability p. For a more detailed discussion concerning the measure theoretic foundation of risk neutral valuation, see, e.g., Delbaen and Schachermayer (2006). In the risk neutral measure, the expected value of the share price grows with the risk free rate r independent of the physical growth rate implied by s 1 and s 2 and their physical probabilities. In order to obtain a probability q in the interval (0, 1), it is required that the risk-free forward value S 0 e rT lies between s 1 and s 2. If S 0 e rT was outside this interval, arbitrage (a guaranteed profit at a higher rate than the risk-free rate) would be possible. 2.3 THE MULTIPERIOD Binomial Model The assumption of only two possible states for the price of the equity at the expiry of the option is not a very realistic one. However at least in theory, it might be a reasonable assumption if the time interval under consideration is sufficiently small. Therefore, we recursively build an N -level tree as indicated in the following figure. Note that the random variables which choose the up or the down branch are assumed to be identical for all time steps...
eBook - ePub- Peter Christoffersen(Author)
- 2011(Publication Date)
- Academic Press(Publisher)
...10. Option Pricing This chapter is devoted to the pricing of options. An option derives its value from an underlying asset, but its payoff is a nonlinear function of the underlying asset price, and so the option price is also a nonlinear function of the underlying asset price. This nonlinearity adds complications to pricing and risk management. In this chapter we will first introduce the binomial tree approach and the Black-Scholes-Merton (BSM) approach to option pricing. We then extend the BSM model by allowing for skewness and kurtosis in returns as well as for time-varying volatility. We will also introduce the ad hoc implied volatility function (IVF) approach to option pricing. The IVF method is not derived from any coherent theory but it works well in practice. Keywords: Binomial trees, Black-Scholes-Merton model, Gram-Charlier approximation, implied volatility functions. 1. Chapter Overview The previous chapters have established a framework for constructing the distribution of a portfolio of assets with simple linear payoffs—for example, stocks, bonds, foreign exchange, forwards, futures, and commodities. This chapter is devoted to the pricing of options. An option derives its value from an underlying asset, but its payoff is not a linear function of the underlying asset price, and so the option price is not a linear function of the underlying asset price either. This nonlinearity adds complications to pricing and risk management. In this chapter we will do the following: • Provide some basic definitions and derive a no-arbitrage relationship between put and call prices on the same underlying asset. • Briefly summarize the binomial tree approach to option pricing. • Establish an option pricing formula under the simplistic assumption that daily returns on the underlying asset follow a normal distribution with constant variance. We will refer to this as the Black-Scholes-Merton (BSM) formula...
