Chapter 18

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Chapter 18

  1. 1. <ul><li>The Basics of Options </li></ul><ul><ul><li>A look at options on stocks </li></ul></ul><ul><ul><li>Value of a call option at expiration </li></ul></ul><ul><ul><li>The buyer's position versus the seller's position </li></ul></ul><ul><li>Valuing European Call Options </li></ul><ul><ul><li>Basic determinants of option values </li></ul></ul><ul><ul><li>Valuing call options that have only two possible outcomes </li></ul></ul><ul><ul><li>Valuing options that have many possible outcomes </li></ul></ul><ul><ul><li>Shortcuts for valuing call options </li></ul></ul><ul><li>Put options </li></ul><ul><ul><li>Valuing European put options </li></ul></ul><ul><ul><li>Shortcut for valuing put options </li></ul></ul>Outline: Chapter 18 Options
  2. 2. <ul><li>An option provides its owner with the right, but not the obligation, to buy or sell a particular good for a limited time at a specified price </li></ul><ul><li>Major North American options exchanges and the options trade on them are the following </li></ul><ul><ul><li>Chicago Board Options Exchange </li></ul></ul><ul><ul><ul><li>Individual stocks </li></ul></ul></ul><ul><ul><ul><li>General stock market indexes </li></ul></ul></ul><ul><ul><ul><li>Treasury bonds </li></ul></ul></ul><ul><ul><li>American Exchange </li></ul></ul><ul><ul><ul><li>Individual stocks </li></ul></ul></ul><ul><ul><ul><li>General stock market indices </li></ul></ul></ul><ul><ul><ul><li>Oil and gas index </li></ul></ul></ul>The Basics of Options
  3. 3. <ul><ul><li>Philadelphia Exchange </li></ul></ul><ul><ul><ul><li>Individual stocks </li></ul></ul></ul><ul><ul><ul><li>Foreign currencies </li></ul></ul></ul><ul><ul><ul><li>Gold and silver indexes </li></ul></ul></ul><ul><ul><li>Pacific Stock Exchange </li></ul></ul><ul><ul><ul><li>Individual stock </li></ul></ul></ul><ul><ul><ul><li>Morgan Stanley Emerging Growth Index </li></ul></ul></ul><ul><ul><li>Montreal Exchange </li></ul></ul><ul><ul><ul><li>Individual stocks </li></ul></ul></ul><ul><ul><ul><li>Government of Canada bonds </li></ul></ul></ul><ul><ul><ul><li>Futures </li></ul></ul></ul>The Basics of Options (continued)
  4. 4. <ul><li>Basic Terms </li></ul><ul><ul><li>Call option versus put option </li></ul></ul><ul><ul><ul><li>A call option provides the owner of the option with the right, but not the obligation, to buy the underlying asset </li></ul></ul></ul><ul><ul><ul><li>A put option provides the owner with the right, but not the obligation, to sell the underlying asset </li></ul></ul></ul><ul><ul><li>Exercise price (or strike price) </li></ul></ul><ul><ul><ul><li>The fixed price, stated in the option contract, at which the underlying asset may be purchased or sold is the exercise (or strike) price </li></ul></ul></ul>The Basics of Options (continued)
  5. 5. <ul><ul><li>Expiration date or maturity </li></ul></ul><ul><ul><ul><li>The maturity date is when the option expires. After this date the option is worthless </li></ul></ul></ul><ul><ul><li>Exercising an option </li></ul></ul><ul><ul><ul><li>The act of buying or selling the underlying asset via an option contract is called exercising the option </li></ul></ul></ul><ul><ul><li>American option versus European option </li></ul></ul><ul><ul><ul><li>An American option may be exercised any time up to and including the expiration date </li></ul></ul></ul><ul><ul><ul><li>A European option can be exercised only at the expiration date </li></ul></ul></ul>The Basics of Options (continued)
  6. 6. <ul><ul><li>Long position </li></ul></ul><ul><ul><ul><li>The buyer of an option contract has purchased the option and has a long position, or holds the contract long </li></ul></ul></ul><ul><ul><li>Short position </li></ul></ul><ul><ul><ul><li>The seller, or writer, of an option contract has a short position, or has sold the option </li></ul></ul></ul><ul><ul><li>In-the-money </li></ul></ul><ul><ul><ul><li>A call option is in-the-money if P 0 > X </li></ul></ul></ul><ul><ul><ul><li>A put option is in-the-money if P 0 < X </li></ul></ul></ul><ul><ul><li>Out-of-the-money </li></ul></ul><ul><ul><ul><li>A call option is out-of-the-money if P 0 < X </li></ul></ul></ul><ul><ul><ul><li>A put option is out-of-the-money if P 0 > X </li></ul></ul></ul><ul><ul><ul><li>Where P 0 = Current price of stock and X = Exercise or strike price </li></ul></ul></ul>The Basics of Options (continued)
  7. 7. The Basics of Options A Look at Options on Stocks
  8. 8. <ul><li>The value of an European call, V c , at maturity is </li></ul><ul><ul><li> </li></ul></ul>The Basics of Options Value of a Call Option at Expiration Value of call option at expiration V c = Max(0, P 0 - X) 0 Market price of stock is less than the exercise price Market price-Exercise price of the option Market price of stock is greater than the exercise price Value of Call Option Condition
  9. 9. The Basics of Options Value of a Call Option at Expiration (concluded) <ul><li>The value of a call option at expiration </li></ul>Value of call, V c ($) 45 o 0 X Market price of common stock, P 0 ($) } V c = P 0 – X if P 0 is greater than X
  10. 10. <ul><li>Although the potential benefit to the purchaser of a call option may be evident, why would anyone want to write or sell the option? </li></ul><ul><ul><li>The writer of the call option receives the premium and realizes a gain as long as the value of the stock at the expiration date is less than the exercise price plus the premium </li></ul></ul><ul><ul><ul><li>The expiration date gain or loss to the buyer and to the writer are mirror images of each other. It is a zero-sum game </li></ul></ul></ul><ul><ul><ul><li>Because only 10 to 15 percent of all stock options written end up being in-the-money at expiration, there are sufficient incentives for some individuals or investment dealers to write options </li></ul></ul></ul>The Basics of Options Buyer's Position Versus Seller's Position
  11. 11. The Basics of Options Buyer's Position Versus Seller's Position (concluded) <ul><li>Profit opportunities for a buyer and a seller of a call option </li></ul>Market price of common stock, P 0 ($) Profit ($) V c Seller X X + V c 0 0 Premium Premium Market price of common stock, P 0 ($) Profit ($) - V c Buyer X X + V c
  12. 12. <ul><li>Basic determinants of option values </li></ul><ul><ul><li>Price of the underlying asset, P 0 </li></ul></ul><ul><ul><li>Exercise price, X </li></ul></ul><ul><ul><li>Time to expiration, t </li></ul></ul><ul><ul><li>Risk-free rate, k RF </li></ul></ul><ul><ul><li>Variability of the underlying asset,  </li></ul></ul><ul><li>The value of a call option, V c , on a nondividend-paying stock, or asset, is </li></ul>Valuing European Call Options Basic Determinants of Option Values
  13. 13. Valuing European Call Options Basic Determinants of Option Values (continued) <ul><li>The value of a call option before expiration </li></ul>45 o } Share price minus present value of the exercise price Market price of common stock, P 0 ($) Present value of exercise price Value of call, V c ($) Value of call option, V c Premium
  14. 14. <ul><ul><li>Effect of an Increase of </li></ul></ul><ul><ul><li>Variable each Factor on V c </li></ul></ul><ul><ul><li>Asset price, P 0 + </li></ul></ul><ul><ul><li>Exercise price, X - </li></ul></ul><ul><ul><li>Time to expiration, t + </li></ul></ul><ul><ul><li>Risk-free rate, k RF + </li></ul></ul><ul><ul><li>Variability of asset's return, + </li></ul></ul>Valuing European Call Options Basic Determinants of Option Values (continued)
  15. 15. <ul><li>The value of a call option as the time to maturity decreases </li></ul>Valuing European Call Options Basic Determinants of Option Values (concluded) Market price of common stock, P 0 ($) Present value of exercise price Value of call, V c ($) V c with 3 months to maturity V c with 1 year to maturity
  16. 16. <ul><li>A replicating portfolio of stock and borrowing </li></ul><ul><ul><li>Assume you want to value a call option that is good for 1 year to buy a share of stock of Yale Ltd. The current price of Yale is $35, the exercise price is $40, and the risk-free rate is 10 percent. There are only two possible outcomes. The price of Yale will increase to $50 at the end of the year or it will decrease to $20. The probability of each of the two outcomes is identical. The possible payoffs are </li></ul></ul><ul><ul><li>Stock price = $50 Stock price = $20 </li></ul></ul><ul><ul><li>One call is worth $10 $0 </li></ul></ul>Valuing European Call Options Valuing Call Options That Have Only Two Possible Outcomes
  17. 17. <ul><ul><li>If instead of buying the option you had purchased the stock directly and borrowed against it at 10 percent. The amount of the borrowing is based on the lower stock price outcome, $20. The size of the loan would have been $20/(1.10) = $18.18. Thus, you borrowed $18.18 and will repay $20 in 1 year. The possible payoffs from this strategy are </li></ul></ul><ul><ul><li>Stock price = $50 Stock price = $20 </li></ul></ul><ul><ul><li>One share of </li></ul></ul><ul><ul><li>stock is worth $50 $20 </li></ul></ul><ul><ul><li>Repay loan -20 -20 </li></ul></ul><ul><ul><li>Total payoff $30 0 </li></ul></ul>Valuing European Call Options Valuing Call Options That Have Only Two Possible Outcomes (continued)
  18. 18. <ul><ul><li>The payoff from the replicating portfolio is three times that of the payoff if the call is purchased. Therefore, </li></ul></ul><ul><ul><li>Value of 3 calls = Original stock price - Loan </li></ul></ul><ul><ul><li>= $35 - $18.18 = $16.82 </li></ul></ul><ul><ul><li>Value of 1 call, V c , = $16.82/3 = $5.61 </li></ul></ul><ul><ul><li>This is the binomial option pricing model </li></ul></ul>Valuing European Call Options Valuing Call Options That Have Only Two Possible Outcomes (continued)
  19. 19. <ul><li>Hedge ratio (Delta) </li></ul><ul><ul><li>The number of shares of stock that are needed to replicate one option is called the hedge ratio </li></ul></ul><ul><ul><li>For Yale </li></ul></ul><ul><ul><li>So, to replicate one call option, we need only 1/3 of the loan of $18.18 which is $6.06 </li></ul></ul>Valuing European Call Options Valuing Call Options That Have Only Two Possible Outcomes (continued)
  20. 20. <ul><ul><li>The value of the call option can be expressed as follows </li></ul></ul><ul><ul><li>where the present value of the loan is the present value of the loan from before adjusted by the hedge ratio, i.e., $18.18(1/3)=$6.06 </li></ul></ul><ul><ul><li>V c = $35(1/3) - $6.06 = $5.61 </li></ul></ul>Valuing European Call Options Valuing Call Options That Have Only Two Possible Outcomes (continued) <ul><ul><li>V c = (Stock price)(Hedge ratio) - Present value of loan </li></ul></ul>
  21. 21. <ul><li>An alternative way to value options </li></ul><ul><ul><li>What if the option did not sell for $5.61? Suppose, for example, it sold for $8? </li></ul></ul><ul><ul><ul><li>Then you can make a guaranteed profit with no risk simply by purchasing the stock, selling 3 call options, and borrowing $18.18 </li></ul></ul></ul><ul><ul><ul><li>Likewise, if the call option sells for less than $5.61, you can make a guaranteed profit with no risk by selling the stock, buying 3 call options, and lending the balance of $18.18 </li></ul></ul></ul><ul><ul><ul><li>This ability to profit is independent of your preference for risk. </li></ul></ul></ul><ul><ul><ul><li>The valuation of options does not depend on the risk preferences of individuals; therefore, the simple assumption that all investors are risk-neutral can be made </li></ul></ul></ul><ul><ul><li>This leads to the risk-neutral approach to option valuation </li></ul></ul>Valuing European Call Options Valuing Call Options That Have Only Two Possible Outcomes (continued)
  22. 22. <ul><li>Risk neutral approach </li></ul><ul><ul><li>If investors are risk-neutral, the expected return from investing in Yale stock is equal to the risk-free rate of interest, k RF = 10 percent </li></ul></ul><ul><ul><ul><li>Knowing this we can determine the probability of an upward or downward movement in the price of the stock </li></ul></ul></ul><ul><ul><ul><li>An increase to $50 is a 43 % increase in value [i.e., ($50/$35) - 1 = 0.43 = 43 %] from the current market price of $35 </li></ul></ul></ul><ul><ul><ul><li>A fall in price to $20 is a decrease of 43% [i.e., ($20/$35) - 1 = -0.43 = -43%] </li></ul></ul></ul>Valuing European Call Options Valuing Call Options That Have Only Two Possible Outcomes (continued)
  23. 23. <ul><ul><li>Solving for W, the probability of an upward movement in the price of Yale, we have </li></ul></ul><ul><ul><li>W = 0.616 </li></ul></ul>Valuing European Call Options Valuing Call Options That Have Only Two Possible Outcomes (continued)
  24. 24. <ul><ul><li>The general formula for determining the probability of an upward movement, W, is </li></ul></ul><ul><ul><li>The probability of an upward movement in the price to $50 is 0.616, while the probability of a downward movement to $20 is 1 - 0.616 = 0.384 </li></ul></ul>Valuing European Call Options Valuing Call Options That Have Only Two Possible Outcomes (continued)
  25. 25. <ul><ul><li>The expected value of the call option 1 year from now is </li></ul></ul><ul><ul><li>The value of the call option today,V c , is </li></ul></ul>Valuing European Call Options Valuing Call Options That Have Only Two Possible Outcomes (continued)
  26. 26. <ul><li>There are two equivalent methods for valuing a call option </li></ul><ul><ul><li>Determine the combination of the asset and borrowing that replicates the call option. Because the call option and the levered position in the asset must produce the same return, the call option and the replicating portfolio sell for the same price </li></ul></ul><ul><ul><li>Determine the expected future value of the option and then discount it back to the present </li></ul></ul>Valuing European Call Options Valuing Call Options That Have Only Two Possible Outcomes (concluded)
  27. 27. <ul><li>Black-Scholes OPM </li></ul><ul><ul><li>The Black-Scholes option pricing model gives the correct expression for the value of European options on nondividend-paying stocks assuming continuous compounding </li></ul></ul>Valuing European Call Options Valuing Options That Have Many Possible Outcomes
  28. 28. Valuing European Call Options Valuing Options That Have Many Possible Outcomes (continued)
  29. 29. <ul><ul><li>Two subsidiary equations are </li></ul></ul>Valuing European Call Options Valuing Options That Have Many Possible Outcomes (continued)
  30. 30. <ul><li>B-S OPM and replicating portfolios </li></ul><ul><ul><li>Although the B-S OPM looks complicated, it is simply a restated version of our replicating portfolio approach to valuing a two-outcome call option </li></ul></ul>Valuing European Call Options Valuing Options That Have Many Possible Outcomes (continued) Stock price Hedge ratio Present value of loan
  31. 31. <ul><li>Using the Black-Scholes model </li></ul><ul><ul><li>Black and Scholes simply employed the knowledge that the value of a call option has to be equal to an equivalent portfolio where N(d 1 ) shares of stock are purchased and then borrowed against </li></ul></ul>Valuing European Call Options Valuing Options That Have Many Possible Outcomes (continued)
  32. 32. <ul><ul><li>An example </li></ul></ul><ul><ul><li>Assume the following data </li></ul></ul>Valuing European Call Options Valuing Options That Have Many Possible Outcomes (continued)
  33. 33. <ul><ul><li>Step 1 </li></ul></ul><ul><ul><ul><li>Calculate d 1 and d 2 , rounding the answers to three decimal places </li></ul></ul></ul>Valuing European Call Options Valuing Options That Have Many Possible Outcomes (continued)
  34. 34. <ul><ul><li>Step 2 </li></ul></ul><ul><ul><ul><li>Compute N(d 1 ) and N(d 2 ) using the cumulative normal distribution function table, Table B.5 </li></ul></ul></ul><ul><ul><ul><ul><li>In our case d 1 = -1.035, the closest tabled value is –1.04, which gives a value for N(d 1 ) of 0.149 </li></ul></ul></ul></ul><ul><ul><ul><ul><li>Similarly, d 2 = -1.135 and the closest tabled N(d 2 ) value is 0.127 </li></ul></ul></ul></ul>Valuing European Call Options Valuing Options That Have Many Possible Outcomes (continued)
  35. 35. <ul><ul><li>Step 3 </li></ul></ul><ul><ul><ul><li>Determine the value of the call option, V c </li></ul></ul></ul>Valuing European Call Options Valuing Options That Have Many Possible Outcomes (continued)
  36. 36. <ul><ul><li>The most difficult part to understand is that risk is captured in the calculation of d 1 and d 2 </li></ul></ul><ul><ul><li>is simply the present value of the exercise price when continuous discounting is employed </li></ul></ul>Valuing European Call Options Valuing Options That Have Many Possible Outcomes (continued)
  37. 37. <ul><ul><li>V c = P 0 N(d 1 ) - (Present value of X) N(d 2 ) </li></ul></ul><ul><ul><li>If the stock had little or no risk (i.e., a very small ), the calculated values for d 1 and d 2 would be large, and the probabilities would both approach the value of 1. If N(d 1 ) and N(d 2 ) both equal 1, </li></ul></ul><ul><ul><li>V c = P 0 - Present value of X </li></ul></ul><ul><ul><li>which is the lower bound on the value of a call option before the expiration date </li></ul></ul>Valuing European Call Options Valuing Options That Have Many Possible Outcomes (continued)
  38. 38. <ul><li>Assumptions of the B-S OPM </li></ul><ul><ul><li>There are no transactions costs or taxes </li></ul></ul><ul><ul><li>The risk-free rate is constant over the life of the option </li></ul></ul><ul><ul><li>The stock market operates continuously (both day and night) </li></ul></ul><ul><ul><li>The stock price is continuous; that is, there are no sudden jumps in price </li></ul></ul><ul><ul><li>The stock pays no cash dividends </li></ul></ul><ul><ul><li>The option can be exercised only at the expiration date (i.e., it is a European option) </li></ul></ul><ul><ul><li>The underlying stock can be sold short without penalty </li></ul></ul><ul><ul><li>The distribution of returns on the underlying stock is lognormal </li></ul></ul>Valuing European Call Options Valuing Options That Have Many Possible Outcomes (continued)
  39. 39. <ul><li>Shortcuts for valuing call options </li></ul><ul><ul><li>Consider the same example used earlier in which </li></ul></ul><ul><ul><li>P 0 = $35 k RF = 0.10 X = $40 = 0.20 t = 0.25 </li></ul></ul><ul><ul><li>Step 1 </li></ul></ul><ul><ul><ul><li>Calculate the standard deviation times the square root of time </li></ul></ul></ul><ul><ul><li>(t) 0.5 = (0.20)(0.25) 0.5 = 0.10 </li></ul></ul><ul><ul><li>Step 2 </li></ul></ul><ul><ul><ul><li>Calculate the market price divided by the present value of the exercise price </li></ul></ul></ul>Valuing European Call Options Valuing Options That Have Many Possible Outcomes (continued)
  40. 40. <ul><ul><li>Step 3 </li></ul></ul><ul><ul><ul><li>Using the two values from steps 1 and 2, determine the tabled factor and multiply it by the share price. </li></ul></ul></ul><ul><ul><ul><ul><li>Table B.6 provides a value of 0.008. Multiplying the stock price of $35 by 0.008, we have the price of the call option, V c , which is $0.28 </li></ul></ul></ul></ul>Valuing European Call Options Valuing Options That Have Many Possible Outcomes (concluded)
  41. 41. <ul><li>The value of an European put, V p , at maturity is </li></ul>Put Options Value of put option at expiration V p = Max(0, X-P 0 ) Exercise price of option-Market price Market price of stock is less than the exercise price 0 Market price of stock is greater than the exercise price Value of Put Option Condition
  42. 42. <ul><ul><li>Effect of an Increase of </li></ul></ul><ul><ul><li>Variable each Factor on V p </li></ul></ul><ul><ul><li>Asset price, P 0 - </li></ul></ul><ul><ul><li>Exercise price, X + </li></ul></ul><ul><ul><li>Time to expiration, t either </li></ul></ul><ul><ul><li>Risk-free rate, k RF - </li></ul></ul><ul><ul><li>Variability of asset's return, + </li></ul></ul>Put Options (continued)
  43. 43. Put Options (continued) <ul><li>The value of a put option at expiration </li></ul>Value of put, V p ($) 45 o Market price of common stock, P 0 ($) V p = X - P 0 if P 0 is less than X { X X
  44. 44. <ul><li>Once you know the value of a call option with a specific exercise price, determining the value of a put option, V p , is easy </li></ul><ul><li>The principle of the put-call parity states that the value of a call option, plus the present value of the exercise price, equals the value of the put option plus the market price of the underlying asset, or </li></ul>Put Options Valuing European Put Options
  45. 45. Put Options Valuing European Put Options (continued)
  46. 46. Put Options Valuing European Put Options (concluded) <ul><ul><li>Example </li></ul></ul><ul><ul><li>With a call option value of $0.26, the value of a put option (with everything else the same) is </li></ul></ul>
  47. 47. <ul><ul><li>From our earlier calculations, (t) 0.5 = 0.10 while </li></ul></ul><ul><ul><ul><li>Going to Table B.7, we find the tabled value is 0.119 </li></ul></ul></ul><ul><ul><ul><li>Multiplying by the market price of $35 produces $4.17 which compares closely with the $4.27 calculated earlier </li></ul></ul></ul><ul><ul><li>We can also value European options on dividend-paying stocks. If only one known cash dividend is expected to be paid before the expiration of the option, the equations are </li></ul></ul>Put Options Shortcuts for Valuing Put Options
  48. 48. Put Options Shortcuts for Valuing Put Options (concluded)

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