4.1 Option Prices: numerical approach Lecture 4

4.2 Pricing: 1.Binomial Trees

Binomial Trees Binomial trees are frequently used to approximate the movements in the price of a stock or other asset In each small interval of time the stock price is assumed to move up by a proportional amount u or to move down by a proportional amount d

4.4 A Simple Binomial Model A stock price is currently $20 In three months it will be either $22 or $18 Stock Price = $22 Stock Price = $18 Stock price = $20 probabilities can’t be 50%- 50%, unless you are risk- neutral

4.5 Stock Price = $22 Option Price = $1 Stock Price = $18 Option Price = $0 Stock price = $20 Option Price=? A Call Option A 3-month call option on the stock has a strike price of 21. if you were risk-neutral (and r=0), you could say that the option is worth: 0.5=50%*1$+50%*$0

U(18) U(22) U(20) Expcted U (50% prob) prob(22) has to be > prob (18), because otherwise Utility function is linear Then, in order to know prob we need to know the Utility function. But this is an impossible task, and we have to find a shortcut.... i.e. we have to find a way of “linearizing” the world

4.7 Consider the Portfolio:long  shares short 1 call option Portfolio is riskless when 22  – 1 = 18  or  =  – 1 18  Setting Up a Riskless Portfolio

4.8 Valuing the Portfolio risk-fre rate=12% p.a. ---> 3% quarterly ---> disc. factor=exp(-0.12*0.25)= The riskless portfolio is: long 0.25 shares short 1 call option The value of the portfolio in 3 months is 22  0.25 – 1 = 4.50 Note that this pay-off is deterministic, so its PV is obtained by simple discounting

4.9 Valuing the Option The value of the portfolio today is 4.5e – 0.12  0.25 = The portfolio that is long 0.25 shares short 1 option is worth The value of the shares is (= 0.25  20 ) The value of the option is therefore (= – )

4.10 Valuing the Option note that the value of the option has been obtained without knowing the shape of the utility function but if the solution is independent of preferences functional form, then it is valid also for all utility function Then, it is valid also for risk-neutral preferences eureka !!! let’s imagine a risk-neutral world ---> derive risk-neutral probabilities

Summing up... Movements in Time  t Su Sd S p 1 – p

4.12 Risk-neutral Evaluation Since p is a risk-neutral probability 20e 0.12  0.25 = 22p + 18(1 – p ); p = p is called the risk-neutral probability show simple_example.xls Su = 22 ƒ u = 1 Sd = 18 ƒ d = 0 S ƒS ƒ p (1  – p ) hyp: risk-free rate=12% p.a.; t = 3m

Tree Parameters for a Nondividend Paying Stock We choose the tree parameters p, u, and d so that the tree gives correct values for the mean & standard deviation of the stock price changes in a risk-neutral world e r  t = pu + (1– p )d  2  t = pu 2 + (1– p )d 2 – [pu + (1– p )d ] 2 A further condition often imposed is u = 1/ d

Tree Parameters for a Nondividend Paying Stock When  t is small a solution to the equations is

The Complete Tree S0S0 S0uS0u S0dS0d S0S0 S0S0 S0u 2S0u 2 S0d 2S0d 2 S0u 2S0u 2 S0u 3S0u 3 S0u 4S0u 4 S 0 d 2 S0uS0u S0dS0d S0d 4S0d 4 S0d 3S0d 3

Backwards Induction We know the value of the option at the final nodes We work back through the tree using risk-neutral valuation to calculate the value of the option at each node, testing for early exercise when appropriate

4.17 Valuing the Option The value of the option is e – 0.12  0.25 [   0] = Su = 22 ƒ u = 1 Sd = 18 ƒ d = 0 SƒSƒ

4.18 A Two-Step Example Each time step is 3 months

4.19 Valuing a Call Option Value at node B = e –0.12  0.25 (   0) = Value at node C =0 Value at node A = e –0.12  0.25 (   0) = A B C D E F

4.20 Pricing: 2.Monte Carlo

An Ito Process for Stock Prices (See pages 225-6) where  is the expected return  is the volatility. The discrete time equivalent is

Monte Carlo Simulation We can sample random paths for the stock price by sampling values for  Suppose  = 0.14,  = 0.20, and  t = 0.01, then see simple_example.xls

Monte Carlo Simulation – One Path (continued. See Table 10.1)

Monte Carlo Simulation When used to value European stock options, this involves the following steps: 1.Simulate 1 path for the stock price in a risk neutral world 2.Calculate the payoff from the stock option 3.Repeat steps 1 and 2 many times to get many sample payoff 4.Calculate mean payoff 5.Discount mean payoff at risk free rate to get an estimate of the value of the option

A More Accurate Approach (Equation 16.15, page 407)

Extensions When a derivative depends on several underlying variables we can simulate paths for each of them in a risk-neutral world tocalculate the values for the derivative

To Obtain 2 Correlated Normal Samples

Standard Errors The standard error of the estimate of the option price is the standard deviation of the discounted payoffs given by the simulation trials divided by the square root of the number of observations.

Application of Monte Carlo Simulation Monte Carlo simulation can deal with path dependent options, options dependent on several underlying state variables, & options with complex payoffs It cannot easily deal with American- style options