Week_5_Binomial_option_pricing

Binomial TreesDr David Liu Department of Mathematical Sciences XJTLU1Is Stock Market Predictable?2Asset Price ModelOur basic assumption: We don’t know and cannot predict tomorrow’s values of asset prices But, we can set up some statistical asset models based upon historical or market data, i.e. Asset price models.3Asset Price ModelEfficient Market Hypothesis The past history is fully reflected in the present price, which don’t hold any further information Markets respond immediately to any new information about an asset.1)2)4Asset Price ModelWith the two assumptions above, the changes in the asset prices are a Markov Process. Stock (share) prices S can be treated as Random Walks. This leads to two different models 1) Binary tree (Binomial tree) model (this week’s topic) 2) Stochastic differential model (we will cover this topic in the future)5Binomial Trees6A one-step binomial model and a no-arbitrage argumentA Simple Binomial Model A stock price is currently $20 In three months it will be either $22 or $18Stock Price = $22 Stock price = $20 Stock Price = $187Binomial TreesBinomial Tree representing different possible paths that might be followed by the stock price In each time step, it has a certain probability of moving up and a certain probability of moving down8Use Binomial Trees to Value Options9A Call OptionA 3-month call option on the stock has a strike price of 21.Stock Price = $22 Option Price = $1 Stock price = $20 Option Price=? Stock Price = $18 Option Price = $010Consider the Portfolio:long ∆sharesshort 1 call option ∏= ∆S -C22∆ –1Setting Up a Riskless Portfolio 11Portfolio is riskless when 22∆ –1 = 18∆=> ∆= 0.2518∆Valuing the Portfolio(If risk-Free Rate is 0%)The riskless portfolio is:long 0.25 shares short 1 call option 12 The value of the portfolio in 3 months is 22 ×0.25 –1 = 4.50 The value of the portfolio today is ∏= ∆S -C =4.500.25x20-C=4.50C = 0.50Valuing the Portfolio(Risk-Free Rate is 12%)The riskless portfolio is:long 0.25 shares short 1 call option 13 The value of the portfolio in 3 months is 22 ×0.25 –1 = 4.50 The value of the portfolio today is 4.5e –0.12×0.25 = 4.3670Valuing the OptionThe portfolio that islong 0.25 sharesshort 1 option; is worth 4.367 14The value of the shares is 5.00 (= 0.25 ×20 )The value of the option is thereforeƒ = 5.000 –4.367=0.633GeneralizationA derivative lasts for time T and isdependent on a stock (e.g. u=1.1,d=0.9 in the above example)15S0uƒuS0dƒdS 0ƒp as a ProbabilityIt is natural to interpret p and 1-p as probabilities of up and down movements The value of a derivative is then its expected payoff discounted at the risk-free rate. So it is in a risk-19neutral world.S 0u ƒu S 0dƒd S 0ƒIrrelevance of Stock ’s Expected Return When we are valuing an option in terms of the price of the underlying 20asset , the probability of up and down movements in the real world are irrelevantOriginal Example RevisitedS 0u = 22ƒu = 1S 0d = 18S 0ƒ22Now calculate the probability that gives a return on the stock equal to the risk-free rate. We can find it from 20e 0.12 ×0.25 = 22p + 18(1 –p )which gives p = 0.6523ƒd = 0Risk-neutral ValuationWhen the probability of an up and down movements are p and 1-p the expected stock price at time T is E(S T )=S 0e rT23This shows that the stock price earns the risk-free rate, so it is in a risk-neutral world This method is known as using risk-neutral valuationValuing the Option Using Risk-Neutral ValuationS 0u = 22ƒu = 1S 24The value of the option isf=e –0.12×0.25 (0.6523×1 + 0.3477×0)= 0.633S 0d = 18ƒd = 00ƒTwo-Step Binomial Trees2224.225Each time step is 3 months E=21, r=12%201819.816.2Valuing a Call Option201.2823221824.23.219.80.02.0257ABCD E26Value at node B= e –0.12×0.25(0.6523×3.2 + 0.3477×0) = 2.0257Value at node A= e –0.12×0.25(0.6523×2.0257 + 0.3477×0)= 1.282316.20.00.0FGeneralizationS S 0u2ƒuuThe length of time step is ∆t years270u ƒu S 0d ƒdS 0ƒS 0d2ƒddS 0ud ƒudA Put Example as an exerciseE = 52, time steps = 1yr r = 5%72D29506040048432201.41479.4636ABCEFf =4.1923?American OptionsAmerican options can be valued using a binomial treeThe procedure is to work back through the 30tree from the end to the beginning, testing at each node to see whether early exercise is optimalAmerican OptionsAmerican Put option (E=52, r =5%)72D 31505.0894604048432201.4147Early exercise 12.0A BCEF 9.4634DeltaDelta (∆) is the ratio of the change in the price of a stock option to the change in the price of the 32underlying stockThe value of ∆varies from node to node~End~Thanks for listening 34。