Lecture 3.1: Option Pricing Models: The Binomial Model

1 Important Concepts The concept of an Option Pricing Model Lecture : Option Pricing The one and two period Binomial Option Pricing models Models: The Binomial Model Explanation of the establishment and maintenance of a risk free hedge Illustration of how early exercise can be captured The extension of the Binomial Model to any number of time periods 01135531: Risk Management and Nattawut Jenwittayaroje, , CFA. Financial Instrument Chulalongkorn Business School Chulalongkorn University 1 2. one period Binomial Model Conditions and assumptions one period , two outcomes (states). S = current stock price u = 1 + return if stock goes up ( , u = 1 + = ).

2 D = 1 + return if stock goes down ( , d = 1 + = ). r = risk free rate C = current call price Value of European call at expiration one period later Cu = Max(0,Su X) or Cd = Max(0,Sd X). The objective of this Model is to derive a formula for the theoretical fair value of the Option . See Figure 3 4. one period Binomial Model (continued) one period Binomial Model (continued). The Option is priced by combining the stock and Option in a risk free hedge portfolio such that the Option price ( , C) can be inferred from other known values ( , u, d, S, r, X). We construct a hedge portfolio of h shares of stock and one short call.

3 These values are all known so h is easily computed Current value of portfolio: V = hS C. Since the portfolio is riskless, it should earn the risk free The objective of the hedge portfolio ( , the riskless portfolio of stock rate. Thus and options) is to develop the formula for C. V(1+r) = Vu (or Vd). At expiration the hedge portfolio will be worth Substituting for V and Vu Vu = hSu Cu , where Cu = Max(0, uS X) (hS C)(1+r) = hSu Cu Vd = hSd Cd , where Cd = Max(0, dS X). Substituting for h, If we are hedged, these must be equal. Setting Vu = Vd and solving for h gives 5 6. one period Binomial Model (continued) one period Binomial Model (continued).

4 Thus, the theoretical value of the Option is An Illustrative Example S = 100, X = 100, u = , d = , r = First find the values of Cu, Cd, h, and p: Cu = Max(0,100( ) 100) = 25. Cd = Max(0,100(.80) 100) = 0. This is the theoretical value of the call as determined by the stock price, exercise price, risk free rate, and up and down factors. h = (25 0)/(125 80) = Note how the call price is a weighted average of the two possible call prices the next period, discounted at the risk free rate. The call's value if p = ( )/( ) = the stock goes up (down) in the next period is weighted by the factor p (1 p). Then insert into the formula for C: The probabilities of the up and down moves were never specified.

5 They are irrelevant to the Option price. 7 8. one period Binomial Model (continued). A Hedged Portfolio Short 1,000 calls and long 1000h = 1000( ) = 556 shares. See Figure Value of investment: V = 556($100) 1,000($ ). $41,580. (This is how much money you must put up.). Stock goes up to $125. Value of investment = 556($125) 1,000($25) = $44,500. Stock goes down to $80. Value of investment = 556($80) 1,000($0) = $44,480. You invested $41,580 and got back $44,500, a 7 % return, which is the risk free rate. 9 10. one period Binomial Model (continued) One-Period Binomial Model (continued). An Overpriced Call An Underpriced Call Let the call be selling for $ Let the call be priced at $13.

6 Your amount invested is 556($100) 1,000($ ). Sell short 556 shares at $100 and buy 1,000 calls at $13. = $40,600. This will generate a cash inflow of $42,600. You will still end up with $44,500, which is a At expiration, you will end up paying out $44,500. return. Everyone will take advantage of this, forcing the call This is like a loan in which you borrowed $42,600 and price to fall to $ paid back $44,500, a rate of , which beats the risk free borrowing rate. 11 12. Two Period Binomial Model We now let the stock go up/down another period so that it ends up Su2, Sud or Sd2. See Figure The Option expires after two periods with three possible values: 13 14.

7 Two Period Binomial Model (continued). After one period the call will have one period to go before expiration. Thus, using a single period Model , it will worth either of the following two values In a single period world, a call Option 's value is a weighted average of the Option 's two possible values at the end of the next period. 15 16. Two Period Binomial Model (continued) Two Period Binomial Model (continued). The price of the call today can again be calculated as a weighted average An Illustrative Example of the two possible call prices in the next period (even if the call does not expire at the end of the next period); Input: S = 100, X = 100, u = , d = , r = Su2 = 100( )2 = Sud = 100( )( ) = 100.

8 Sd2 = 100( )2 = 64. In summary, the two period Binomial Option Pricing formula provides the Option price as a weighted average of the two possible Option prices the The call Option prices are as next period, discounted at the risk free rate. The two future Option prices, follows in turn, are obtained from the one period Binomial Model . The hedge ratios are different in the different states: 17 18. Two Period Binomial Model (continued) Two Period Binomial Model (continued). The two values of the call at the end of the first period are Therefore, the value of the call today is The value of p is the same, (1+r d) / (u d), regardless of the number of periods in the Model .

9 19 20. Extensions of the Binomial Model American Calls and Early Exercise The multi period Binomial Model is an excellent opportunity to American Calls and Early Exercise illustrate how American options can be exercised early. Pricing Put Options Consider the American call where S = 100, X = 100, u = , d = , r = (the same as the American Puts and Early Exercise previous European call). Dividends, European Calls, American Calls, and Early Now we must consider the possibility of exercising the call early. Exercise At time 1 the European call values were Extending the Binomial Model to n periods Cu = when the stock is at 125.

10 Cd = when the stock is at 80. The Behavior of the Binomial Model for Large n and a When the stock is at 125, the call is in the money by $25, but it Fixed Option Life is still lower than holding value. So not early exercise it. The value of the American call today is now the same at 21 22. American Calls and Early Exercise Pricing Put Options The Binomial Model can easily Pricing a put with the Binomial Model is the same accommodate the early exercise of procedure as Pricing a call, except that the expiration an American call by simply payoffs are computed by using put payoff formula. comparing the computed value (holding value) and intrinsic value (exercise value), and select the Consider a European put where greater value.