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## Valuing Options Using Binomial Trees

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**Binomial Trees**• A binomial tree is designed to represent the possible paths the stock price might follow. • To begin we want to assume that the stock will be held for one period of time with two possible outcomes for the price at the end of the period, either an increase or decrease (following a binomial probability distribution) • The model will then be extended to account for multiple price movements through multiple periods of time.**A simple Example**• Assume that the stock is currently selling for $20 and at the end of three months it will be worth either $22 or $18. • Assume we want to value a European call option with an exercise price of $21 at the end of the three months. • There are two possible outcomes: • Stock Price = $22 and intrinsic value of option = $1 • Stock Price = $18 and intrinsic value of option = $0**Possible Outcomes Shown on Binomial Tree Model**Stock Price = $22 l Option Price = $1 Stock Price =$20 l Stock Price = $18 l Option Price = $0 Time 0 Time 1**A no arbitrage solution**• To price the option we want to assume that we can use it to eliminate risk. • Combining a short position in the call option with a spot position in the underlying stock produces a result where gains on the stock are offset by option. • We want to establish a riskless portfolio so that the value is the same regardless of whether the price of the stock increases to $22 or decreases to $18 in 3 months. • Given that the portfolio is riskless, the portfolio should earn a rate equal to the risk free rate.**Value of Portfolio in 3 months**• Consider a portfolio of D shares of the stock and one option. • If the stock price increases to $22 and the option has a $1 intrinsic value the portfolio is worth: $22D-$1 • If the stock price decreases to $18 and the option has a $0 intrinsic value the portfolio is worth: $18**Determining the number of shares in the portfolio**• The portfolio has two possible outcomes in three months: $22 D - $1 and $18 D • Since the two possible outcomes have equal values it is possible to solve for D $22 D - $1 = $18 D $22 D - $18 D = $1 ($22 - $18) D = $1 D = $1/$4 = .25**Time Value**• If the stock price increases to $22 the portfolio is worth $22 D-$1 = $22(.25) -1 = $4.5 • If the stock price decreases to $18 the portfolio is worth $18 D = $18(.25) = $4.5 • Since the portfolio is risk free it should have a present value at time 0 equal to $4.5 discounted at the risk free rate for 3 months. • Assuming a risk free rate of 12% the PV is: $4.5e-0.12(.25)=$4.367**Pricing the option**• Let the price of the option be f • Given the current share price of $20, the value of the portfolio at time zero should be equal to $20(.25) – f • Which must equal the value of either portfolio 4.367 • Setting the two equal to each other an solving for f finds the value of the call. $20(.25) – f = 4.367 f =.633**An overpriced call**• If the option price was greater than 0.633 you could make a risk free return above the risk free rate. • What if the value of the call was .80 today? • The current value of the portfolio = $20(.25)-.80 = $4.20 • In either outcome the value of the portfolio after three months is 4.5 this implies a risk free return of 4.20er(.25)=4.5 r = .2759 = 27.59%**An overpriced call**• Since you can earn a return greater than the risk free rate the demand for the stock will increase causing its price to increase. Additionally the supply of the option will increase (since everyone will want to write an option to take advantage of the higher price) decreasing its price.**An underpriced call**• What if the price of the call is less than .633? • Assume it is .50, since it is cheap, you buy it and sell the stock short receiving $25(.25) = $5 your total cash flow is $5 - .50 = $4.50 • In three months you need to buy the stock to close the short sale. • If the price increased to $22 you need to pay -$22(.25) + 1 = -$4.50 • If the price decreased to $18 you need to pay -$18(.25) = -$4.50**An underpriced call**• In either case you essentially borrowed $4.50 at time zero and repaid $4.50 in three months. You have borrowed at a rate much less than the risk free rate. • The market will react to this increasing the call price (as demand for the option increases) and decreasing the share price (as there are more people shorting the stock).**Generalizing the Model**• Let the stock price at time 0 be S0 with two possible outcomes: • The stock price increases by a factor greater than 1, u The resulting stock price is then S0u • The stock price decrease by a factor less than 1, d. The resulting stock price is then S0d • If the stock price increases the value of the option is fu • If the stock price decreases the value of the option is fd**Binomial Tree Model**S0u fu l S0 f l S0d fd l Time 0 Time 1**The generalization**• Setting the value of the two portfolios at time 1 equal produces S0uD-fu= S0dD-fd • Solving for D produces: • Delta is thus the change in the option price divided by the change in the stock price**The PV of the portfolio**• Since the value of the portfolio at time 1 is the same regardless of whether the price increased or decreased either price can be used to find the value of the portfolio at time 0 • Using a stock price increase the value of the portfolio at time 0 is (S0uD-fu)e-rT**The value of the option**• The PV of the portfolio has to equal the cost of setting up the option at time 0: S0 D-f = (S0u D-fu)e-rT • The goal is to solving for f, the current value of the option. f = S0 D- (S0u D -fue-rT f = S0 D- S0u D e-rT-fue-rT f = S0 D(1- ue-rT)-fue-rT**Solving for the price of the option, f**substitute f = S0 D(1- ue-rT)-fue-rT Simplifying produces f = e-rT[pfu+(1-p)fd] Where :**Returning to the Example:**• In the previous example we had a current price of $20 with the possibility of an increase to $22 or $20u=$22 solving for u, u =1.1 • Similarly $20d=$18, solving for d, d=.9 • Given the inputs from before u = 1.1 and d=.9 r=.12, T = .25, fu=1, and fd=0 • Plugging into the equations on the previous slide p = .6523 and f = .6333 Which is the same as before**A Quick Question**• Does the probability of an increase in the stock price impact the value of the option? • NO! regardless of the probability the value of the option is the same – the probability of an increase or decrease is already incorporated in the price of the stock. Since the option price is based on the stock price, the probability does not impact the option price.**Risk Preferences Review**• Risk Loving vs. Risk Neutral vs. Risk Adverse • Assume you are faced with two equally likely outcomes: A gain of $10 and a loss of $5. How much would you be willing to pay to accept the risk of the possible loss?**Risk Preferences Review**• Risk Neutral: If you are willing to pay $2.50, you are willing to pay a “fair price” to accept the risk. (If you repeated the event over and over on average you would receive your $2.50) • Risk Averse: If you are willing to pay less than 2.50 lets say 2.00, you are risk averse. The $.50 represents a risk premium, the additional return you expect to earn for accepting the risk. The lower the amount you are willing to pay the more risk averse you are.**Risk - Neutral Valuation**• Now assume that p is the probability of an increase in the stock price and 1-p is the probability of a decrease in the price of the stock. • This would imply that the option price f = e-rT[pfu+(1-p)fd] is the expected payoff from owning the option discounted at the risk free rate.**Risk Neutral Valuation**• Using p in the same way produces the Expected value of the stock at time t E(ST)=pS0u+(1-p)S0d E(ST)=pS0(u-d)+S0d E(ST)=S0erT The stock price grows on average at the risk free rate! Substituting**Risk Neutral Valuation**• In a risk neutral world investors do not get compensated for risk. Implying the expected return on all securities is the risk free rate of interest. • In the case of assuming p = the probability of an increase in the stock price, the option is equal to its expected payoff discounted at the risk free rate. • This is based upon our portfolio assumptions that both return the risk free return. The risk neutral valuation is the same as the no arbitrage valuation.**No Arbitrage**Stock Price = $22 l Option Price = $1 Stock Price =$20 l Stock Price = $18 l Option Price = $0 Time 0 Time 1**No arbitrage**• Previously we found that the no arbitrage value of the option was .633 assuming that the we used the risk free rate of 12% in the valuation.**No arbitrage vs. Risk Neutral**• In the risk neutral world the expected value of the stock price in one period would be also equal to the risk free return of 12% • Therefore, if p is the probability of a stock price increase, the expected value of the stock in the future must equal the value of the stock today assuming it grows at the risk free rate or: 22p+18(1-p) = 20e.12(.25) 22p-18p = 20e.12(.25)-18 4p=2.6092 p=0.6523**Expected Value of Option in 3 months**• Given the probability of a stock price increase is 0.6523, if the price increases to $22 in our example the option is worth $1. If the price decreased to $18 the option is worth $0 • The expected value of the option in three months in our example would then be: 0.6523($1)+(1-0.6523)$0 =0.6523**Value of Option at Time 0**• In a risk neutral setting the value of the option should also increase at the risk free rate of 12% • By discounting the value of the option at time = 3 months back to time =0 at the risk free rate we can find the value of the option at time 0 0.6523e-(.12)(.25)= 0.633 Which is the same value of the option in the No Arbitrage solution**Risk Neutral vs. “Real World”**• In the “real world” the probability of an increase in the stock price is not equal to p. This occurs when the expected payoff on the stock is equal to the risk free rate, which is not usually the case. • What if expected return on the stock is 16%? • Let q = the probability of an increase and solve for q assuming a 16% expected return on the stock. 22q+18(1-q) = 20e.16(.25) q = .7041**Real World return on the Option**• If q = .7041 the expected payoff on the option is .7041(1)+(1-.7041)0 = .7041 • Given a risk neutral value of the call at time 0 of 0.633 the implied discount rate on the option is .633=.7041e-r(.25) R = .4258 = 42.58%**Risk Neutral**• While the implied discount rate for the option could be found it is based on the assumption that the value of the option at time 0 is based on the risk neutral value of the option. • Without the risk neutral value of the option it would not be possible to compare how much extra return the option should pay based on its increased risk compared to the stocks expected return of 16%.**Extending the tree**• Now assume that we want to look another 6 months out. • In the previous example it was assumed that the price increased by10% or decreased by 10%. • Assume that at time t = 3 months this is again the case. You can build the tree out from the possible payoffs at time t= 3 months.**Possible Outcomes Shown on Binomial Tree Model**Stock Price = $22 l Stock Price =$20 l Stock Price = $18 l Time 1 (3 months) Time 0**Possible Outcomes after an initial price increase**l Stock Price = $24.2 Stock Price =$22 l Stock Price = $19.8 l Time 1 (3 months) Time 2 (6 months)**Possible Outcomes after an initial price decrease**l Stock Price = $19.8 Stock Price =$18 l Stock Price = $16.2 l Time 1 (3 months) Time 2 (6 months)**Combining the possible outcomes**Stock Price = $24.2 l Stock Price =$22 l Stock Price = $19.8 l l $20 Stock Price = $18 l Stock Price = $16.2 l Time 1 (3 months) Time 2 (6 months) Time 0**Calculating the Option price**• The option price at time zero now depends upon two different time periods. • Assume we want to look at an option that has a strike price of $21 at time t = 6 months. • The price of the option at time 6 months will equal its intrinsic value as before, but the price at time t = 3 months will need to be calculated based upon the possible outcomes at time 6 months.**Possible Outcomes after an initial price increase**Stock Price = $24.2 Option Price = $3.2 l Stock Price =$22 l Stock Price = $19.8 Option Price = $0 l Time 1 (3 months) Time 2 (6 months)**Option price**• Using the option price equation from before the price of the option assuming an initial increase to time t=3 can be found. f = e-rT[pfu+(1-p)fd] f = e-0.12(.25)[(.6523)(3.2)+(.3477)(0)] =2.2057**Possible Outcomes after an initial price decrease**Stock Price = $19.8 Option Price = $0 l Stock Price =$18 l Stock Price = $16.2 Option Price = $0 l Time 1 (3 months) Time 2 (6 months)**Option price**• Using the option price equation from before the price of the option assuming an initial decrease to time t-3 can be found. f = e-rT[pfu+(1-p)fd] f = e-0.12(.25)[(.6523)(0)+(.3477)(0)] =0**Combining the possible outcomes**Stock Price = $24.2 Option Price = $3.2 l Stock Price =$22 Option Price =$2.0257 l Stock Price=$19.8 Option Price =$0 l l $20 l Stock Price = $18 Option Price =$0 Stock Price = $16.2 Option Price =$0 l Time 1 (3 months) Time 2 (6 months) Time 0**Option price at time 0**• Using the option price equation from before the price of the option at time 0 can be found using the option prices at time t= 3 months f = e-rT[pfu+(1-p)fd] f = e-0.12(.25)[(.6523)(2.0257)+(.3477)(0)] =1.2823**Note:**• In our example the time period was fixed for each step of the process and the proportional up and down movement were the same for each step. • Therefore the risk neutral probability p is the same at each step.**Generalizing the possible outcomes**Stock Price = S0u2 Option Price = fuu l Stock Price =S0u Option Price =fu l Stock Price=S0ud Option Price =fud l l f l Stock Price = S0d Option Price =fd Stock Price = S0d2 Option Price =fdd l Time 1 (3 months) Time 2 (6 months) Time 0**Substituting for generalization of f**The original pricing equation gave us f = e-rDT[pfu+(1-p)fd] When applied to time t=3 it becomes fu = e-rDT[pfuu+(1-p)fud] and fd = e-rDT[pfud+(1-p)fdd] f = e-rDT[pfu+(1-p)fd] f = e-2rDT[p2fuu+2p(1-p)fud+(1-p)2fdd] Substituting**Put Options**• The same procedure can be used to calculate the value of a put option. Using the same problem as before but to find the price of a put option we would need to look at the tree to find the value of the option at time t = 6 months then work back.