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Binomial Option Pricing Model

Chris Hebert 10/27/10. Binomial Option Pricing Model. History of The Binomial Option Pricing Model. Cox-Ross-Rubinstein 1979 After Black- Scholes (1973) After Monte Carlo Methods . What is it?.

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Binomial Option Pricing Model

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  1. Chris Hebert 10/27/10 Binomial Option Pricing Model

  2. History of The Binomial Option Pricing Model • Cox-Ross-Rubinstein • 1979 • After Black-Scholes (1973) • After Monte Carlo Methods

  3. What is it? • The Binomial Option Pricing Model is essentially a tree that is constructed to show possible values that an underlying asset can take and the resulting value of the option at these values.

  4. One Simplifying Assumption • The Binomial Option Pricing Model makes an assumption that over a certain time period, the underlying can only do one of two things: go up, or go down.

  5. Up and down by how much? • Let’s call the upward movement u, and the downward movement d. • Intuitively, the magnitude of this movement should be based on volatility and the size of the time interval. • Cox-Ross-Rubinstein found u and d to be:

  6. How can we assume only up or down? • We can assume that the stock can only go up or down because we use a large number of short time periods. • With a small number of time intervals, the model only considers a few possible final outcomes. • As we increase the number of time intervals, more possible final outcomes are considered and accuracy increases.

  7. Example • S=90.00 • K=100.0 • T= 1 • σ= .3 • r= .01 • Δt= .1

  8. How did we construct that? • Step 1: Create the Binomial Price Tree • Step 2: Calculate the Payoff at time T for all nodes • Step 3: Pull back calculation to calculate value of option at each node from time T to time 0.

  9. Important Observation • Because of the nature of stock movements, the binomial lattice is recombining. • Going up then down is the same as going down then up. • This enables us to make fewer node calculations, expediting process.

  10. Step 1: Create the Binomial Price Tree • We know the current stock price (S0) • Multiply (S0) by u to find the stock price at time t=1 for an upward movement • Multiply (S0) by d to find the stock price at time t=1 for a downward movement • To find S at any node multiply (S0)(u)n(d)t-n where n= the number of upward movements

  11. Step 2: Calculate the Payoff at time T for all nodes • Plug in the stock price at each final node into the payoff formula to calculate the value of the option at maturity • These are all of the possible payoffs of an option.

  12. Step 3: Pull back calculation • Work from maturity backwards to present • Fair price of option today is equal to the expected payoff of the option discounted by the riskless interest rate • Take the values of two possibilities the option can attain at time t (up or down) and pull back the value for the option at time t-1

  13. Step 3 (continued) • Let C be the value of the option at time t and node i • Factor in riskless interest rate to the probability that the option goes up or down • For p we use .5 to assume random movements of the stock, up and down

  14. Inputs needed • S0= Stock price at time t=0 • K= Strike Price (for payoff) • T= Time to expiry • σ= volatility • r= riskless interest rate • Δt= change in time; size of each step in tree

  15. Example • S=90.00 • K=100.0 • T= 1 • σ= .3 • r= .01 • Δt= .1

  16. Connection to Black-Scholes • The Binomial Options Pricing Method is a discrete time approximation to the continuous Black-Scholes equation for European options.

  17. Advantages • Primary advantage of Binomial trees is that American options become much easier to price. • Other advantages • Can factor in dividends • No calculus needed • Easy to make a spreadsheet to find option values

  18. American Option Pricing • For an American Option, the value of the option at any given point is the maximum of two values: • The payoff if the option were exercised now • The pullback formula’s output • Note: the pullback formula takes into account American option values at future points.

  19. Disadvantages/ Limitations • Speed • In order to have accurate results, we want to maximize the number of time periods, but this means creating a lot of nodes which can be slow to calculate even with today’s computers. • For some extreme options, like a cash or nothing, pricing can be inaccurate until a very large number of time intervals is used.

  20. Extensions • Changing the value of p • Trinomial trees • Up • Down • Constant • Dividends

  21. Works Cited • Albanese, Claudio., and Giuseppe Campolieti. Advanced Derivatives Pricing and Risk Management: Theory, Tools and Hands-on Programming Application. Amsterdam ; Boston: Elsevier Academic Press, 2006. • Berg, Imme van den. Principles of Infinitesimal Stochastic and Financial Analysis: . Singapore ; River Edge, N.J.: World Scientific, 2000. • Daigler, Robert T. Advanced Options Trading: The Analysis and Evaluation of Trading Strategies, Hedging Tactics, and Pricing Models. Chicago, Ill.: Probus Pub. Co., 1994. • http://en.wikipedia.org/wiki/Binomial_options_pricing_model • http://www.hoadley.net/options/bs.htm • http://www.excel-modeling.com/examples/example_007.htm • http://fedc.wiwi.hu-berlin.de/xplore/tutorials/sfehtmlnode36.html

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