Step 1. Binomial model acts similarly to the asset that exists in a risk neutral world.pu+qd = exp(i*∆t) = r, where ∆t = t/nt = term of the optionn= number of periodsIts variance: pu^2 + qd^2 – (exp(i*∆t))^2 =𝜎^2∆tStep 1. Binomial model acts similarly to the asset that exists in a risk neutral world.pu+qd = exp(i*∆t) = r, where ∆t = t/nt = term of the optionn= number of periodsIts variance: pu^2 + qd^2 – (exp(i*∆t))^2 =𝜎^2∆t
Notice that the lattice is symmetrical, that is due to the assumption that d=1/u (ud=1).
Cox, Ross and Rubinstein Binomial Trees Acedo Fabia Reyes Sorbito Vidamo
Report Outline1 • Overview2 • General Assumptions3 • Steps and Formulas4 • Example5 • Summary
Overview• A type of binomial asset pricing model first proposed by John C. Cox, Stephen A. Ross and Mark Rubinstein (1979).• “Simple and efficient numerical procedure for valuing options for which premature exercise may be optional”• “All corporate securities can be interpreted as portfolios of puts and calls on the asset of the firm.”• Uses discrete time model of varying price over time of the underlying financial instrument• Uses binomial tree of possible price of the underlying asset ; each nodes valuation is performed iteratively
Assumptions uS with probability p S dS with probability q = p ‒ 1• Underlying asset price S follows a multiplicative binomial process over discrete period.• Rate of return on the stock over each period can have two possible values.• u and d parameters are constant over the whole tree.
Assumptions• u and d are chosen so that u = 1/d .• Interest rates are assumed constant, d < Rf < u. It means that there is no arbitrage opportunity.• No taxes, transaction cost, or margin requirements• The underlying doesnt pay dividends over the life of the option.
Steps and Formulas Step 1. Compute for the Risk free Return r is the one period rate of returnr = EXP(i*(t/n)) t is term in yearsp = (r-d)/(u-d) n is the number of periodsq=1-p p is the risk-neutral probability up move q is the risk-neutral probability down move Step 2. Generate the price of the tree uxS S is the price of underlying asset,S u is the up move factor with probability p, dxS d is the down move factor with probability q
Steps and FormulasStep 3. Calculation of option value at each final node(Backward Induction) Sn is the computed At Final Node n: underlying asset price If it is a Call Option, then use MAX(0,Sn-K) at node n If it is a Put Option, then use MAX(K-Sn,0) K is the strike priceStep 4. Sequential calculation of the option value at eachpreceding node Cu is the older upper At other Nodes 0 to n-1 option price other nodes = [p * Cu + q * Cd] / r Cd is the older lower option price
Example:Step 1. Compute for the Risk free ReturnStock price [S] $ 60.00 GivenInterest rate [i] 5.00% GivenStrike price [K] $55.00 GivenTerm in years [t] 1 GivenNumber of periods - quarterly [n] 4 GivenUp move factor [u] 1.05 GivenDown move factor [d] 0.9524 d = 1/uOne period rate of return [r] 1.0126 r = EXP(i*(t/n))Risk-neutral probability - up move [p] 61.67% p = (r-d)/(u-d)Risk-neutral probability - down move [q] 38.33% q=1-pNotes: The price of LDI stock is $60/share and the one-year interest rate is0.05. We wish to price one-year call option with a strike price of $55. Using afour-step tree (quarterly) with assumed stock price factor increase of 1.05, wewill compute for the price of the underlying asset and the call option.
Example: Step 2. Generate the price of the tree Formula: CRR Tree: 0 1 … n 0 1 2 3 4 Suuuu 72.93 Suuu 69.46 Suu Suuud 66.15 66.15 Su Suud 63.00 63.00 S Sud Suudd 60.00 60.00 60.00 Sd Sudd 57.14 57.14 Sdd Suddd 54.42 54.42 Sddd 51.83 Sdddd 49.36S is the price of underlying asset, S = $ 60u is the up move factor u = 1.05d is the down move factor d = 0.9524n is the number of periods n=4
Example:Step 3. Calculation of option value at each final node CRR Tree: Binomial Tree for Pricing a $55 Call Option 0 1 2 3 4 0 1 2 3 4 72.93 17.93 69.46 66.15 66.15 11.15 63.00 63.0060.00 60.00 60.00 5.00 57.14 57.14 54.42 54.42 - 51.83 49.36 - Given: K = $ 55At Final Node n: Sample Computation: If it is a Call Option, then use MAX(0,Sn-K) MAX(0, 72.93-55) = 17.93 If it is a Put Option, then use MAX(K-Sn,0) MAX(0, 66.15-55) = 11.15
Example: Step 4. Calculation of the option value at each preceding node Binomial Tree for Pricing a $55 Call Option At other Nodes 0 to n-1 other nodes = [p * Cu + q * Cd] / r 0 1 2 3 4 where Cu is the older upper option price 17.93 Cd is the older lower option price 15.14 12.51 11.15Given: p = 1.05, q = 0.9524, r = 1.0126 10.06 8.68 7.87 6.44 5.00Sample Computation: 4.62 3.04 1.85 -O31 = [1.05*17.93+0.9524*11.15]/1.0126 - = 15.14 -O32 = [1.05*11.15+0.9524*5.00]/1.0126 = 8.68O21 = [1.05*15.14+0.9524*8.68]/1.0126 = 12.51
Summary and Conclusions• Cox-Ross-Rubinstein Model is one of many available binomial options pricing models. It is a simplified alternative numerical method that can be used for practical computations of complex option values. It assumes a constant interest rate (risk free return), absence of arbitrage opportunities and constant probability of underlying assets upward (u) and downward (d) movement.• Options priced derived from Cox-Ross-Rubinstein binomial tree can be used in formulating strategy that will generate/ lock in pure arbitrage profits if the market price of an option differs from the value given by the model.
References:• Cox, J.C., Ross S.A, Rubinstein, M., Option Pricing : A Simplified Approach. (1979). Published in Journal of Finance and Economics• Watsham, Terry J., and Parramore, Keith. Quantitative Methods in Finance. (1997)• http://investexcel.net/736/binomial-option-pricing-excel/• http://www.sitmo.com/article/binomial-and-trinomial-trees/• http://en.wikipedia.org/wiki/Binomial_options_pricing_mode l• http://sfb649.wiwi.hu- berlin.de/fedc_homepage/xplore/tutorials/xlghtmlnode63.ht ml#bin-fig2• http://www.terry.uga.edu/~mayhew/Old/chapter9.pdf