Options Pricing. This is sometimes referred to as the intrinsic value of the option.

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1 Options Pricing We will use the example of a call option in discussing the pricing issue. Later, we will turn our attention to the Put-Call Parity Relationship. I. Preliminary Material Recall the payoff (not profit) from a call option. Payoff S 0 X where S 0 is the stock price if exercised now and X is the exercise (strike) price. This is sometimes referred to as the intrinsic value of the option. Intrinsic Value S 0 X The above is true for in-the-money options. Otherwise, the value is zero. Thus, as the stock price exceeds the exercise price, the intrinsic value will increase. Value X S 0 Time Value of the Option the difference between the actual call price and the intrinsic value. Thus, an option will have value (even if currently out-of-the-money ) when it still has time left to expire. For example, even if the current stock price were less than the exercise price, the option would still have value if it had time left till expiration. 1

2 Value Value of option Time value X S 0 At the stock price increases, then it will become more likely that the option will be exercised and the time value will decline --- causing the value of the option to converge on the adjusted intrinsic value. Adjusted Intrinsic Value S 0 PV(X) If you know that you will exercise the option, then you will pay X amount when the option is exercised. Thus, the payment of X in the future has a present value now (note, you could put PV(X) into the bank now). The straightline now represents the adjusted intrinsic value. [we ignore dividend payments throughout] Basic Determinants: 1. Stock Price: as stock price increases, the value of the option should increase. 2. Exercise Price: as exercise price increases, the value of the option decreases 3. Volatility of Stock Price: as volatility increase, the value of the option increases (remember, the downside is limited to zero payoff) 4. Time to expiration: as time to expiration increases, the value of the option will increase (closely related to volatility; also the longer time to expiration means that the present value of the exercise price declines) 5. Interest rate: as the interest rate increases, the value of the option will increase (because the present value of the exercise price decreases) 6. Dividend payments: a higher dividend payment (or, dividend yield) will decrease the value of the option because it will dampen the capital gains. 2

3 II. Binomial Options Pricing Model Example 1 Current Price $100 Exercise Price of Call Option $100 Two Possible Stock Prices at expiration: $150 or $75 Two Possible Values of Option at Expiration: $50 ($150-$100) or $0 Now, suppose you purchased 2/3 share of the stock and wrote (sold) one call. What would be the payoffs? Suppose for simplicity we re talking about 1 year. If Price at expiration is $150 The option will be exercised and you receive $100 You will need to go out and buy 1/3 share of a stock at $150, thus you must spend $50 Your net payoff is $50 If Price at expiration is $75 The option will not be exercised You own 2/3 of a stock worth $75, thus $50. Same payoff for each case! Example 2 Current Price $100 Exercise Price of Call Option $100 Two Possible Stock Prices at expiration: $110 and $90 Two Possible Values of Option at Expiration: $10 ($110-$100) or $0 Now, suppose you purchased 1/2 share of the stock and wrote (sold) one call. What would be the payoffs? Suppose for simplicity we re talking about 1 year. If Price at expiration is $110 The option will be exercised and you receive $100 You will need to go out and buy 1/2 share of a stock at $110, thus you must spend $55 Your net payoff is $45 3

4 If Price at expiration is $90 The option will not be exercised You own 1/2 of a stock worth $90, thus $45. Same payoff for each case! What is so interesting about this? You get the same payoff regardless thus, the risky business of investing has now turned risk-free! But, what was your profit? That depends on your investment. In this case we know that you bought half a share of stock valued at $100, thus you have invested $50 in the stock. But, you re total investment is less than that. Why? Because you wrote a call option. Thus, Investment $50 C (where C is the price of the call option you received) Now, we know that there is a certain payoff of $ that is, it is a risk-free payoff. If we assume markets are efficient at making sure there are not arbitrage opportunities, then this strategy should earn the risk-free rate. Suppose the risk-free interest rate is 10% and we ll assume 1 year to make things simple. [$50 C] x (1+.10) $45 $50 C C $50 $45 $45 $50 $40.91 $9.09 To recap, I write a call option and receive $9.09 then purchase half a share for $50 making my total investment $ At the end of one year, I receive $45 for a 10% return. [notice that a higher interest rate would have caused the value of the option to increase; same for a longer time to maturity]. Suppose now, to take an extreme case, the value of the option was actually $20. Now, I conduct the same strategy. I write one call option and receive $20, then purchase half a share for $ my investment is then $30. Still, though, I am certain to have something of value of $45 just as before. Thus, my $30 investment would have grown to $45 for a 50% return. If this were the case, then lots of people would be doing it --- increasing the supply of the call options (writing more of them) and driving down their price until it reached $

5 Example 3 Current Price $100 Exercise Price of Call Option $125 Two Possible Stock Prices at expiration: $200 or $50 Two Possible Values of Option at Expiration: $75 ($200-$125) or $0 Now, suppose that you purchased one share of the stock and wrote (sold) two call options. If Price at expiration were $200 Call option will be exercised and you get $250 You need to buy one more share for $200 Net payoff $50 If Price at expiration were $50 Call option would not be exercised Net Payoff $50 Again, let s see how the option would be priced. You purchase one share for $100 but sell (write) 2 calls. Investment $100 2C We know that this will lead to a certain payoff of $50. Assume again that the risk-free interest rate were 10% and 1 year. $100 2C 2C $100 C $27.27 $50 $50 $ Thus, you write 2 calls receiving $54.55 then purchase one share for $100, your investment is the $45.45 which grows to $50 without risk for a 10% return. Now, here let s take things a step further and actually show that you do not need money now. Suppose you can borrow at the risk-free rate as well. So, you write 2 calls and receive $54.55, then borrow the $45.45 to combine it with the money from the calls to purchase the 1 share. At the end of the one year will have $50 which will just be enough to payback your loan. So why do it? No reason now. But 5

6 Suppose the price of the option was $30. So, you sell (write) 2 options and get $60. You borrow $40 at the risk-free rate to buy the 1 share. At the end of the 1 year you have $50 regardless of which price the stock ends up at. You will need to repay $40 plus $4 interest. Notice, though, you have made $6 without any initial investment and without taking on any risk!!!! This is the kind of profit that should not exist if markets work efficiently. Hedge Ratio H C S + C S III. Black-Scholes C 0 S 0 e δt N( d ) Xe 1 rt N( d ) 2 d 1 ln( S 0 2 / X ) + ( r δ + σ / 2) T σ T d 2 d 1 σ T N(d) is the probability that a random draw from a standard normal distribution will be less than d. this equals the area under the normal curve up to d. Delta is the annual dividend yield of the stock. T is the time remaining until maturity of the option (in years) Sigma is the standard deviation of the annualized continuously compounded rate of return of the stock, expressed as a decimal. To see the intuition, assume no dividend payment (thus, delta 0) S δt 0e d N( d1) S0 N( 1) Now, think of the N(d) s as the risk-adjust probablities that the call option will expire in the money. Suppose that the probability that the option will expire in the money is one. C 0 S 0 Xe rt But, this is just the adjusted intrinsic value of the option discussed earlier. 6

7 Extended Example of Binomial Option Pricing used in Class 1. Start with the simplest case Current Price of the Stock $100 Exercise Price of Call Option $100 Two Possible Stock Prices at expiration: $110 and $90 (we ll assume expiration is one year from now. Not because this assumption is realistic, but rather because it makes calculations easier.) Two Possible Values of Option at Expiration: $10 ($110-$100) or $0 Now, suppose you purchased 1/2 share of the stock and wrote (sold) one call. What would be the payoffs? If Price at expiration is $110 The option will be exercised and you receive $100 by whomever purchased the call option from you. You will need to go out and buy 1/2 share of a stock at $110, thus you must spend $55 Your net payoff is $45 If Price at expiration is $90 The option will not be exercised You own 1/2 of a stock worth $90, thus your net payoff is $45. Same payoff for each case! Notice we re not talking profits here, just payoffs! 2. You don t like the idea of purchasing half shares. Given the same assumptions as above, you could purchase 1 share of the stock and write (sell) 2 call options. If Price at expiration is $110 The 2 options you wrote will be exercised and you will receive $200. You own 1 share already, so you ll need to purchase another share for $110 (i.e., current stock price). Thus, you now have 2 shares to hand over to whomever purchases the 2 call from you. What is the payoff? $200 - $110 $90 If Price at expiration is $90 The options will not be exercised. You own 1 share worth $90. Thus, your payoff is $90. Again, the idea is that in either case, you get the same payoff. 7

8 3. What is so interesting about all this? Pricing of the Call You get the same payoff regardless thus, the risky business of investing has now turned riskfree! But, what was your profit? That depends on your investment. We ll return to the original case where you purchased half a share and wrote one call. In this case we know that you bought half a share of stock valued at $100, thus you have invested $50 in the stock. But, you re total investment is less than that. Why? Because you wrote a call option. Thus, Investment $50 C (where C is the price of the call option you received) Now, we know that there is a certain payoff of $ that is, it is a risk-free payoff. If we assume markets are efficient at making sure there are not arbitrage opportunities, then this strategy should earn the risk-free rate. Suppose the risk-free interest rate is 10% and we ll assume 1 year to make things simple. [$50 C] x (1+.10) $45 $50 C C $50 $45 $45 $50 $40.91 $9.09 To recap, I write a call option and receive $9.09 then purchase half a share for $50 making my total investment $ At the end of one year, I receive $45 for a 10% return. [notice that a higher interest rate would have caused the value of the option to increase; same for a longer time to maturity]. 4. What if Pricing is Not Efficient Suppose now, to take an extreme case, the value of the option was actually $20. Now, I conduct the same strategy. I write one call option and receive $20, then purchase half a share for $ my investment is then $30. Still, though, I am certain to have something of value of $45 just as before. Thus, my $30 investment would have grown to $45 for a 50% return. If this were the case, then lots of people would be doing it --- increasing the supply of the call options (writing more of them) and driving down their price until it reached $9.09. Suppose now the price of the option was only $5. If I do the same thing, then my investment will be $45 ($50 - $5) in order to get a certain payoff of $45. Hence, 0% rate of return. I d be better off just investing in the risk-free asset. Thus, noone would want to write call options, the supply would decline until the price of the option was driven up to $

9 5. Where does the Hedge Ratio come from? How do we know how many shares to buy and how many calls to write? The hedge ratio is stated in terms of how many shares to own per unit of calls written. Recall, the main idea here is to find the hedge ratio that will make the payoffs exactly the same in each case. Thus, returning to our example, suppose you did not know the perfect hedge ratio (H). Let s state the payoffs with the unknown hedge ratio (H). If the stock price ends up being $110. The value of the call option (with a strike price of $100) will be $10 ($110-$100). Your payoff will be the following: (H x $110) - $10 The term in parentheses is the value of the shares you own at the end. Notice, in the prior example, we purchased half a share and the payoff was $ thus, replace H with ½ and you get the payoff we had before. If the stock price ends up being $90. The value of the call option is zero. Your payoff will be the following. (H x $90) - $0 Again, using ½ for H would get us our $45 payoff. Now, if we want to know the hedge ratio that will provide the same payoff in either case, we simply set the payoffs equal and solve for H. ( H 110) 10 ( H 90) 0 ( H 110) ( H 90) 10 0 H (110 90) 10 0 H C S + + C S This is how I had arrived at the correct number of shares to own per call written. Thus, when we purchased 1 share we had to write 2 calls. We could have purchased 50 shares and written 100 calls to arrive at the same thing. It s the ratio that matters. 9

10 6. What happens when the stock price is more volatile? Current Price of the Stock $100 Exercise Price of Call Option $100 Two Possible Stock Prices at expiration: $140 and $80 Two Possible Values of Option at Expiration: $40 ($140-$100) or $0 What is the proper hedge ratio? ( H 140) 40 ( H 80) 0 ( H 140) ( H 80) 40 0 H (140 80) 40 0 H C S + + C S Thus, we purchase 2/3 of the stock now and write one call. If the price at expiration is $140, then we ll get $100 from the call buyer. We ll need to go out and purchase 1/3 of a share at a price of $ for a total of $ Our payoff will be $53.33 $ If the price at expiration is $80, then our payoff will be $53.33 $80 x (2/3) What is the value of the option at the beginning? The 2/3 of the stock cost us $66.67 $66.67 C $53.33 $53.33 C $66.67 $66.67 $48.48 $18.18 So, the volatility increased (i.e., wider range of possible prices at expiration) and the value of the call option increased!!! This is a general result (assuming other things constant). 10

11 7. Time Let s return to the original set-up. Current Price of the Stock $100 Exercise Price of Call Option $100 Two Possible Stock Prices at expiration: $110 and $90 Two Possible Values of Option at Expiration: $10 ($110-$100) or $0 Now, suppose you purchased 1/2 share of the stock and wrote (sold) one call. Let s suppose that the option will expire 10 years from today. What is the value of this option? $45 $50 C () 10 $45 C $50 () 10 $50 $17.35 $32.65 As the time to expiration increased, the value of the option increased!!! Another general result. 8. Risk-free Interest Rate Return to the original example and suppose the interest rate on the risk-free asset falls to 5% from 10%. $50 C $ C $50 $ $50 $42.86 $7.14 Thus, a decrease in the interest rate decreased the value of the option. If the interest rate had increased, then the value of the option would increase!!! A general result. 11

12 9. Two-State Approach --- Generalizing the Binomial Model Return to the Original Set-up Current Price of the Stock $100 Exercise Price of Call Option $100 But, now assume the option lasts for 2 years (we re using years for simplicity here), and we ll assume the stock price can change at the end of the first year, then again at the end of the second year. Thus, from our original set-up we change things in the following way. Two Possible Stock Prices at end of year 1: $110 and $90 Three Possible Stock Prices at expiration (end of year 2): $115, $95, $ Now, we apply the same procedure as before, just more times. We work backwards. Current Price of Stock 110 Exercise Price of Call Option 100 Two possible prices at end: 115, 95 Two possible values of the call option: 15 ( ), 0 (if stock price is 95) Hedge ratio (15 0)/ (115-95).75 So, we purchase.75 shares ( 110 x ) of the stock and write 1 call. If the price turns out to be 115, then the call will be exercised. We get $100 (exercise price) and spend $28.75 to purchase the.25 share of the stock at $115. We end up with a payoff of $71.25 If the price turns out to be 95, then the call will not be exercised. We own.75 of a share that is now worth $95. The payoff is $

13 Assuming the risk-free interest rate is 10%, we get a value of the call option at this point of $71.25 $82.50 C $71.25 C $82.50 $82.50 $64.77 $17.73 Now we take the lower price at the end of year 1. Current Price of Stock 90 Exercise Price of Call Option 100 Two possible prices at end: 95, 70 Two possible values of the call option: 0 (if stock price is 95), 0 (if stock price is 70) So here, the call option is worthless in either outcome. Hence, the hedge ratio would be 0. Let s move back in time to the first year in order to see the implications. Current Price of Stock 100 Exercise Price of Call Option 100 Two possible prices at end: 110, 90 Two possible values of the call option: $17.73 (if stock price is 110), 0 (if stock price is 90) Hedge Ratio ( )/(110-90).8865 Now, let s determine the value of things at the end of year 1. Price 90 Price 110 Value of Stock (90*.8865) (110*.8865) Value of Option Total So, the payoff is the same. The initial purchase is 100*.8865 $88.65 $88.65 C $ $ C $88.65 $88.65 $72.53 $

14 9. Two-State Approach --- Generalizing the Binomial Model Let s try a new example. Suppose the current price of a stock is $100. A 2-year call option exists with an exercise price of $110. Assume the interest rate is 5%. Also, assume the stock price can change twice at the end of year 1, and at the end of year 2. The possible stock prices are the following Now, we will conduct the same procedure as we ve previously done. This time though, we ll interpret results in terms of the value of things at various stages. We begin at the end of the time period and work our way backwards. Take the branch in which the stock price at the end of year 1 is $110. Now, it will take only two possible values for the end of year 2: $121 or $ What would be the appropriate Hedge ratio at this time? H {you can think of this as own 2 shares and selling 3 calls, or owning.67 shares and selling 1 call} Let s look at values assuming ownership of.67 shares and writing 1 call. Values at end of Year 2 Price Price 121 Value of Stock (104.50*.67) (121 *.67) Value of Call Sold 0-11 ( ) Total Thus, the value of option would be $73.33 C $ C $73.33 $ $73.33 $66.35 $

15 Take the branch in which the stock price at the end of year 1 is $95. Now, this is actually the easy branch. In this case, the stock price can either end up at $ or $ In either case, the value of the option is 0 (remember the exercise price is $110 in this example). Thus, the hedge ratio is holding zero shares per call would give the same payoff as the call. At this point, we go backward in time to the first year. The stock price at the end of the first year can be either 110 or 95. A spread of 15 (for the hedge denominator). The value of the option at the end of the first year is either $6.984 (what we just calculated for the $110 branch) or 0 (that was the case for the $95 branch). Thus, a spread of $6.984 for the hedge numerator. What is the hedge ratio? H Now, let s calculate values at end of year 1. Values at end of Year 1 Price 95 Price 110 Value of Stock (95*.4656) (110 *.4656) Value of Call Sold Total Thus, the payoffs (or, better values ) at the end of year 1 are the same. The value of the call option today is therefore: {note, the initial stock purchase is 46.56) $46.56 C $ $ C $46.56 $46.56 $42.13 $

16 IV. Put-Call Parity Relationship Suppose you buy a call option and write a put option. Assume that each has the same exercise price. What would be the payoffs (not profits)? If S T < X Payoff of call held 0 + Payoff of put written - (X-S T ) Total S T - X If S T > X Payoff of call held S T -X + Payoff of put written 0 Total S T -X Now suppose you formed a portfolio by purchasing a share and borrowing an amount that will grow to X in the future. What would be the payoff? S T -X Thus, the put-call strategy and the levered equity position provide the exact same payoffs. Thus, the investment in the two must be the same. Net Cash Outlay of Option Position C P Where C is the cost of the call and P is the price received from writing the put. The net cash outlay of the levered equity position S 0 X (1 + r) T The put-call parity condition becomes the following: C P S 0 X (1 + r) T P C S 0 X + (1 + r) T P C S 0 + PV ( X ) What if this condition is not satisfied? Stock Price $110 Call Price (X 105) 17 Put Price (X105) 5 Risk-free interest rate 5% 16

17 C P S-PV(X) Now, you can buy the cheap portfolio (right hand side) and sell the expensive portfolio (left hand side). Immediate Cash Flows in one year St <105 St>105 Position Cash Flow St95 St 120 Buy Stock Borrow X/(1+r) Sell call (rather than buy) Buy put (rather than sell) Total Notice, you make an immediate $2 risk-free without any offsetting outflows. Thus, an arbitrage opportunity has arisen. Notice what happens if the put price is driven up to say $6 and the call price is driven down to $16. Immediate Cash Flows in one year St <105 St>105 Position Cash Flow St95 St 120 Buy Stock Borrow X/(1+r) Sell call (rather than buy) Buy put (rather than sell) Total

18 What if the call price was $12 and the put price had been $8. Now, the cheap portfolio is the option portfolio (buy the call and sell the put). Now, you would want to buy the option portfolio and sell the levered equity position. Immediate Cash Flows in one year St <105 St>105 Position Cash Flow St95 St 120 Sell stock Lend money Buy Call Sell put Total

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