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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.
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6 Key excerpts on "Binomial Model"
eBook - PDF- Douglas Kennedy, M.A.H. Dempster, Dilip B. Madan, Rama Cont(Authors)
- 2016(Publication Date)
- Chapman and Hall/CRC(Publisher)
Chapter 2 THE Binomial Model 2.1 One-period model 2.1.1 Introduction We begin our study of the ideas underlying the elimination or minimization of risk and the pricing of assets in financial markets by considering the Binomial Model. While at first sight it might seem too simple mathematically for any practical appli-cation, we will see that it is rich enough to motivate many of the ideas that we will develop in subsequent chapters; also it is an important tool for approximating results in the more realistic models which come later. The first case to consider is the one-period model operating from time 0 to time 1 . There are just two assets: the first may be thought of as a bank account (or bond) which is riskless in that 1 unit of wealth at time 0 held in the bank account becomes 1 C with certainty at time 1 , where > 0 is a constant and may be interpreted as the interest rate on the bank account; the second asset is a stock for which the price at time 0 is S 0 , where S 0 > 0 is constant, and its price at time 1 is a random variable S 1 taking just two possible values uS 0 and dS 0 where u > 0 and d > 0 are given constants. The random variable S 1 is defined on the underlying probability sample space D f ! 1 ; ! 2 g with S 1 .! 1 / D uS 0 and S 1 .! 2 / D dS 0 . We will assume that u and d are distinct values and take u > d . We denote the underlying probabilities for the two outcomes ! 1 and ! 2 by p 1 D P . f ! 1 g / D P .S 1 D uS 0 / and p 2 D P . f ! 2 g / D P .S 1 D dS 0 / ; where p i > 0 , i D 1; 2 and, of course, p 1 C p 2 D 1 . A useful way to represent the change in the stock price between time 0 and time 1 is by a binomial tree (or binary tree) S 0 S 1 .! 1 / D uS 0 S 1 .! 2 / D dS 0 . . . . p 1 p 2 with u and d being proportional changes to the stock price; u is often regarded as being an ‘up’ jump of the stock and d as a ‘down’ jump although we do not assume 25 26 The Binomial Model necessarily that u > 1 and d < 1 .
eBook - PDF- Alexander Melnikov(Author)
- 2011(Publication Date)
- Chapman and Hall/CRC(Publisher)
Chapter 2 Financial Risk Management in the Binomial Model 2.1 The Binomial Model of a financial market. Absence of arbitrage, uniqueness of a risk-neutral probability measure, martingale representation. The Binomial Model of a ( B, S )-market was introduced in the previous chapter. Sometimes this model is also referred to as the Cox-Ross-Rubinstein model. Recall that the dynamics of the market are represented by equations Δ B n = rB n − 1 , B 0 = 1 , (2.1) Δ S n = ρ n S n − 1 , S 0 > 0 , where r ≥ 0 is a constant rate of interest with − 1 < a < r < b , and profitabil-ities or risky asset returns ρ n = b with probability p ∈ [0 , 1] a with probability q = 1 − p , n = 1 , . . . , N , form a sequence of independent identically distributed random variables. The stochastic basis in this model consists of Ω = { a, b } N , the space of sequences ω = ( ω 1 , . . . , ω N ) of length N whose elements are equal to either a or b ; F = 2 Ω , the set of all subsets of Ω. The probability P has Bernoulli probability distribution with p ∈ [0 , 1], so that P ( { ω } ) = p P N i =1 I { b } ( ω i ) (1 − p ) P N i =1 I { a } ( ω i ) . The filtration F is generated by the sequence ( ρ n ) n ≤ N : F n = σ ( ρ 1 , . . . , ρ n ). Note that many authors use different parameters u (up) and d (down) in the Binomial Model setting, which can be interpreted as possible values of stock returns, and they can be expressed in terms of our parameters a and b : u = 1 + b and d = 1 + a . In the framework of model (2.1), we can specify the following notions. A predictable sequence π = ( π n ) n ≤ N ≡ ( β n , γ n ) n ≤ N is an investment strategy 33 34 Risk Analysis in Finance and Insurance (portfolio). A contingent claim f N is a random variable on the stochastic basis (Ω , F , F , P ). Hedge for a contingent claim f N is a self-financing portfolio with the terminal value X π n ≥ f N . A hedge π ∗ with the value X π ∗ N ≤ X π N for any other hedge π is called the minimal hedge .
eBook - ePubFoundations of the Pricing of Financial Derivatives
Theory and Analysis
- Robert E. Brooks, Don M. Chance(Authors)
- 2024(Publication Date)
- Wiley(Publisher)
PART II Discrete Time Derivatives Pricing Theory
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CHAPTER 7 The Binomial Model
To this point we have used a simple model in which the asset price can either move up to one level or down to another. We called this model the two‐state or Binomial Model. In this chapter, we derive the model more formally. This work is attributed to Cox, Ross, and Rubinstein (1979 ) and Rendleman and Bartter (1979 ). An excellent book on Binomial Models in finance is by van der Hoek and Elliott (2006 ).7.1 THE ONE‐PERIOD Binomial Model FOR CALLS
Recalling from the previous chapter, we specified an asset priced at S that can go up to Su or down to Sd. Given the historical decision to assume a lognormal distribution, early authors pursued a multiplicative binomial approach.1 We shall refer to these two states as the up state and the down state, respectively. The basic idea is communicated in Figure 7.1 . At the initial point in time, there is only one node, whereas at the next point in time, there are only two nodes. Also, at the initial point in time, there are two arcs emanating from the initial node.This type of figure is sometimes called a binomial tree, although it will look a little more like a tree when we expand it later.The risk‐free rate is r. Consider a call option with exercise price X that expires in one period. The two possible values of the option at expiration are(7.1)Of course, our objective is to determine the value of the option today, c. The basic layout with the corresponding option prices inserted at each node is in Figure 7.2 .Appealing to the arbitrage principle, we shall create a portfolio consisting of the option and the underlying asset. We shall set this portfolio up so that it is hedged, meaning that its future value is known for certain. A hedged portfolio must, therefore, earn the risk‐free rate. We can then solve for the price of the option that is consistent with a risk‐free return on this portfolio.Let us buy hc
eBook - ePubQuantitative Modeling of Derivative Securities
From Theory To Practice
- Peter Laurence, Marco Avellaneda(Authors)
- 2017(Publication Date)
- Routledge(Publisher)
Chapter 4Refinements of the Binomial ModelThe Binomial Model discussed in Chapter 2 used two input parameters: the interest rate and the volatility. Until now, we assumed implicitly that these parameters were constant. In this chapter, we remove these assumptions, allowing for variations of these parameters with time.A term-structure of interest rates is introduced to model a (more realistic) economy in which rates can vary with the duration of loans. We will also study time-inhomogeneity of the volatility process by introducing a term-structure of volatilities . Time-dependent volatilities are useful to incorporate into the pricing model the market’s expectations about risk across time. Information about the temporal behavior of volatility is contained in the prices of liquid option instruments with different maturities written on a given underlying asset.We will also discuss refinements of the Binomial Model that will permit us to extend the theory to derivative contracts of practical interest. These include derivatives contingent on underlying assets that pay dividends, options on futures and “structured” derivative instruments providing a stream of uncertain cash-flows across time.4.1 Term-Structure of Interest Rates
We incorporate into the model different interest rates for different trading periods. Usually, interest rates are not constant in time. For example, the following table gives market values for interbank dollar deposits on August 23, 1995:1Maturity Bid Offer 1 month 5.8700 6.0000 2 months 5.7800 5.9000 3 months 5.7800 5.9000 6 months 5.8100 5.9500 9 months 5.8100 5.9300 Implied forward interest rates can be obtained from such a “strip” of deposit rates, or from the markets in Eurodollar futures or Treasury bill futures.2In the case of Eurodollar futures, the December 1997 Eurodollar 90-day futures contract gives the expected London Inter-Bank Offered Rate (LIBOR) for the period of January 1998 through March 1998. The March 1998 contract gives the expected 90-day LIBOR from April 1998 through June 1998, and so forth. These values can then be input in the model at different time periods.3
eBook - ePubOption Trading
Pricing and Volatility Strategies and Techniques
- Euan Sinclair(Author)
- 2010(Publication Date)
- Wiley(Publisher)
With this preamble behind us, we can look at two specific option-pricing models.• The underlying price is a function of volatility and time. 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.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 4.1The Evolution of the Stock Price in the One-Step Binomial TreeImagine 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. The position has been transformed into one with no risk (this means that we can legitimately discount the portfolio value at the risk-free interest rate, but we will ignore this for now).
- P. C. G. Vassiliou(Author)
- 2013(Publication Date)
- Wiley-ISTE(Publisher)
fair price of the option.(b) To find a hedging strategy for the writer of the contract, given an appropriate value of the premium V (0). This strategy is based on the writer investing on the underlying asset through the bank account.The present chapter serves more as a pedagogical instrument for introducing the ideas and concepts of stochastic finance. Such a way of introducing the main theme of mathematical finance could be found in many works. Two of the more successful are those by Jarrow and Tunbull (2000) and Shreve (2004, 2004a) and which the reader may use as accompanying sources.4.2. Binomial Model
As we do in the entire book, time will be discrete. We refresh the reader’s memory by repeating the definition of a European call option as defined in Chapter 2 .A European call option is a contract giving the owner the right, but not the obligation, at time 0 to buy a number of units of the underlying asset at the maturity time or expiration time T , at a fixed price called the strike price, say K . It is termed call because the owner of the contract has the option to buy, that is, to call the asset away from someone.We will slightly alter the notation of Chapter 2 for reasons which will be apparent in what follows. In this respect, let be the stochastic process in discrete time, which represents the price of the underlying asset at time t. In order to find solutions to the two basic problems we described, we need a model for the stochastic process . We will now describe what is well known as the Binomial Model, which serves as an approximation to the evolution of the asset price in discrete time. From what the reader will see, it is apparent that the more correct term for the model would have been the random walk
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