Lecture 3.1: Option Pricing Models: The Binomial Model


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1 Important Concepts Lecture 3.1: Option Pricing Models: The Binomial Model The concept of an option pricing model The one and two period binomial option pricing models Explanation of the establishment and maintenance of a risk free : Risk Management and Financial Instrument Nattawut Jenwittayaroje, Ph.D., CFA Chulalongkorn Business School Chulalongkorn University hedge Illustration of how early exercise can be captured The extension of the binomial model to any number of time periods 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 (e.g., u = = 1.14) d = 1 + return if stock goes down (e.g., d = = 0.91) r = risk free rate C = current call price Value of European call at expiration one period later C u = Max(0,Su X) or C d = Max(0,Sd X) The objective of this model is to derive a formula for the theoretical fair value of the option. See Figure
2 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 (i.e., C) can be inferred from other known values (i.e., u, d, S, r, X). We construct a hedge portfolio of h shares of stock and one short call. Current value of portfolio: V = hs C The objective of the hedge portfolio (i.e., the riskless portfolio of stock and options) is to develop the formula for C. At expiration the hedge portfolio will be worth V u = hsu C u, where C u = Max(0, us X) V d = hsd C d, where C d = Max(0, ds X) If we are hedged, these must be equal. Setting V u = V d and solving for h gives These values are all known so h is easily computed Since the portfolio is riskless, it should earn the risk free rate. Thus V(1+r) = V u (or V d ) Substituting for V and V u (hs C)(1+r) = hsu C u Substituting for h, 5 6 One Period Binomial Model (continued) Thus, the theoretical value of the option is One Period Binomial Model (continued) An Illustrative Example S = 100, X = 100, u = 1.25, d = 0.80, r = 0.07 First find the values of C u,c d, h, and p: C u = Max(0,100(1.25) 100) = 25 C d = 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. 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 the stock goes up (down) in the next period is weighted by the factor p (1 p). The probabilities of the up and down moves were never specified. They are irrelevant to the option price. h = (25 0)/(125 80) = p = ( )/( ) = 0.6 Then insert into the formula for C: 7 8
3 One Period Binomial Model (continued) A Hedged Portfolio Short 1,000 calls and long 1000h = 1000(0.556) = 556 shares. See Figure 4.2. Value of investment: V = 556($100) 1,000($14.02) $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 One Period Binomial Model (continued) An Overpriced Call Let the call be selling for $15.00 Your amount invested is 556($100) 1,000($15.00) = $40,600 You will still end up with $44,500, which is a 9.6% return. Everyone will take advantage of this, forcing the call price to fall to $14.02 OnePeriod Binomial Model (continued) An Underpriced Call Let the call be priced at $13 Sell short 556 shares at $100 and buy 1,000 calls at $13. This will generate a cash inflow of $42,600. At expiration, you will end up paying out $44,500. This is like a loan in which you borrowed $42,600 and paid back $44,500, a rate of 4.46%, which beats the risk free borrowing rate
4 Two Period Binomial Model We now let the stock go up/down another period so that it ends up Su 2, Sud or Sd 2. See Figure 4.3. The option expires after two periods with three possible values: 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
5 Two Period Binomial Model (continued) The price of the call today can again be calculated as a weighted average of the two possible call prices in the next period (even if the call does not expire at the end of the next period); Two Period Binomial Model (continued) An Illustrative Example Input: S = 100, X = 100, u = 1.25, d = 0.80, r = 0.07 Su 2 = 100(1.25) 2 = Sud = 100(1.25)(0.80) = 100 Sd 2 = 100(0.80) 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 next period, discounted at the risk free rate. The two future option prices, in turn, are obtained from the one period binomial model. The call option prices are as follows The hedge ratios are different in the different states: Two Period Binomial Model (continued) The two values of the call at the end of the first period are Two Period Binomial Model (continued) 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
6 Extensions of the Binomial Model American Calls and Early Exercise Pricing Put Options American Puts and Early Exercise Dividends, European Calls, American Calls, and Early Exercise Extending the Binomial Model to n Periods The Behavior of the Binomial Model for Large n and a Fixed Option Life American Calls and Early Exercise The multi period binomial model is an excellent opportunity to illustrate how American options can be exercised early. Consider the American call where S = 100, X = 100, u = 1.25, d = 0.80, r = 0.07 (the same as the previous European call) Now we must consider the possibility of exercising the call early. At time 1 the European call values were C u = when the stock is at 125 C d = 0.0 when the stock is at 80 When the stock is at 125, the call is in the money by $25, but it is still lower than holding value. So not early exercise it. The value of the American call today is now the same at American Calls and Early Exercise Pricing Put Options The binomial model can easily accommodate the early exercise of an American call by simply comparing the computed value (holding value) and intrinsic value (exercise value), and select the greater value. Pricing a put with the binomial model is the same procedure as pricing a call, except that the expiration payoffs are computed by using put payoff formula. Consider a European put where S = 100, X = 100, u = 1.25, d = 0.80, r = 0.07 American Call Path In our example the put prices at expiration are; Exercise or intrinsic value = $0 Exercise or intrinsic value = $
7 Pricing Put Options P=( )/( ) Pricing Put Options The two values of the put at the end of the first period are Therefore, the value of the put today is American Puts and Early Exercise The multi period binomial model is an excellent opportunity to illustrate how American options can be exercised early. Consider the American put where S = 100, X = 100, u = 1.25, d = 0.80, r = 0.07 (the same as the previous European put) Now we must consider the possibility of exercising the put early. At time 1 the European put values were P u = 0.00 when the stock is at 125 P d = when the stock is at 80 When the stock is at 80, the put is in the money by $20 so exercise it early. Replace P u = with P u = 20. The value of the American put today is higher at American Puts and Early Exercise The binomial model can easily accommodate the early exercise of an American put by simply comparing the computed value (holding value) and intrinsic value (exercise value), and select the greater value. 27 Exercise or intrinsic value 28
8 Dividends, European Calls, American Calls, and Early Exercise One way to incorporate dividends is to assume a constant yield,, per period. The stock moves up, then drops by the rate. See Figure 4.5, p. 109 for example with a 10% yield The call prices at expiration are The European call prices after one period are Dividends, European Calls, American Calls, and Early Exercise If the call is American, when the stock is at 125, it pays a dividend of $12.50 and then falls to $ We can exercise it, paying $100, and receive a stock worth $125. The stock goes ex dividend, falling to $ but we get the $12.50 dividend. So at that point, the option is worth $25. We replace the binomial value of C u = $22.78 with C u = $25. At time 0 the value of the American call is The European call value at time 0 is Extending the Binomial Model to n Periods With n periods to go, the binomial model can be easily extended. The basic procedure is the same. See Figure 4.9, p. 114 in which we see below the stock prices, the prices of European, and American puts. This illustrates the early exercise possibilities for American puts, which can occur in multiple time periods. Stock price path with 10% dividend yield at Time 1 The effect of dividend on the early exercise decision of an American call, and hence its value 31 At each step, we must check for early exercise by comparing the value if exercised to the value if not exercised and use the higher value of the two. 32
9 S = 100, X = 100, u = 1.25, d = 0.80, r = 0.07 The Behavior of the Binomial Model for Large n and a Fixed Option Life The risk free rate is adjusted to (1 + r) T/n 1 The up and down parameters are adjusted to Early exercise of American put where is the annualized volatility The Behavior of the Binomial Model for Large n and a Fixed Option Life Let us price the DCRB June 125 call with one period. The parameters are as follows; the stock price is , the option has 35 days remaining, the risk free rate is 4.56 percent per year, and the (annual) DCRB volatility is 83%. The Behavior of the Binomial Model for Large n and a Fixed Option Life The new option prices would be C u = Max(0, ) = C d = Max(0, ) = 0.0 p would be ( )/( ) =.444; 1 p =.556. The price of the option at time 0 is, therefore, The new stock prices are Su = ( ) = Sd = ( ) =
10 Summary If we make n large enough, we obtain a very accurate depiction of what happens to the stock over the option s life. Therefore, we can be confident that our estimated option value is a quite accurate reflection of the true value of the option. 37 An option is priced by combining the stock and option in a riskfree hedge portfolio such that the option price can be inferred from other known values. The one period binomial option pricing formula provides the option price as a weighted average of the two possible option prices at expiration, discounted at the risk free rate. The two period binomial option pricing formula provides the option price as a weighted average of the two possible option prices the next period, discounted at the risk free rate, where the two future option prices are obtained from the one period binomial model. 38 Summary Pricing a put with the binomial model is the same procedure as pricing a call, except that the expiration payoffs reflect the put payoff. The binomial model can easily accommodate the early exercise of an American option by simply replacing the computed value with the intrinsic value if the latter is greater. The binomial model converges to a specific value of the option as the number of time periods increases. 39
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