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Option Valuation

16. Option Valuation. Learning Objectives. Make sure the price is right by making sure that you have a good understanding of: 1. How to price options using the one-period and two-period binomial models. 2. How to price options using the Black-Scholes model.

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Option Valuation

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  1. 16 Option Valuation

  2. Learning Objectives Make sure the price is right by making sure that you have a good understanding of: 1. How to price options using the one-period and two-period binomial models. 2. How to price options using the Black-Scholes model. 3. How to hedge a stock portfolio using options. 4. The workings of employee stock options.

  3. Option Valuation • Our goal in this chapter is to discuss how to calculate stock option prices. • We will discuss many details of the very famous Black-Scholes-Merton option pricing model. • We will discuss "implied volatility," which is the market’s forward-looking uncertainty gauge.

  4. Just What is an Option Worth? • In truth, this is a very difficult question to answer. • At expiration, an option is worth its intrinsic value. • Before expiration, put-call parity allows us to price options. But, • To calculate the price of a call, we need to know the put price. • To calculate the price of a put, we need to know the call price. • So, we want to know the value of a call option: • Before expiration, and • Without knowing the price of the put

  5. A Simple Model to Value Options Before Expiration, I. • Suppose we want to know the price of a call option with • One year to maturity. • A $110 exercise price. • The current stock price is $108. • The one-year risk-free rate, r, is 10 percent. • We know (somehow) that the stock price will be $130 or $115 in one year. • The stock price in one year is still uncertain. • We know that the stock price is going to be $130 or $115 (but no other values). • We do not know the probabilities of these two values. • Therefore, we know the call option value at expiration will be: • $130 – $110 = $20 OR • $115 - $110 = $5 • This call option is certain to finish in the money. • A similar put option is certain to finish out of the money.

  6. A Simple Model to Value Options Before Expiration, II. • If you know the price of a similar put, you can use put-call parity to price a call option before it expires. • The chosen pair of stock prices guarantees that the call option finishes in the money. • Suppose, however, we want to allow the call option to expire in the money OR out of the money. • How do we proceed in this case? Well, we need a different option pricing model.

  7. The One-Period Binomial Option Pricing Model—The Assumptions • Suppose the stock price today is S, and the stock pays no dividends. • We assume that the stock price in one period is either S ×u or S ×d, where: • u (for “up” factor) is bigger than 1 • and d (for “down” factor) is less than 1 • Suppose the stock price today is $100, and u = 1.1 and d = .95. • The stock price in one period will either be • $100 × 1.1 = $110 or • $100 × .95 = $95. • What is the call price today, if: • K = 100 • R = 3%

  8. The One-Period Binomial Option Pricing Model—The Setup • Consider the following portfolio: • Buy a fractional share of the underlying asset--this fraction is represented by the Greek letter, D (Delta) • Sell one call option • Finance the difference by borrowing the amount: DS – C • Key Question: What is the value of this portfolio, today and at option expiration?

  9. Cu and Cd: The intrinsic value of the call if the stock price increases to S×u or decreases to S×d. Portfolio Value Today Portfolio Value At Expiration The Value of this Portfolio(long D Shares and short one call) is: Important: DS is NOT the change in S. Rather, it is a dollar amount, DS. DS×u - Cu DS - C DS×d - Cd

  10. To Calculate Today’s Call Price, C: • A Brilliant Insight: There is one combination of a fractional share and one call that makes this portfolio risk-less. • That is, the portfolio will have the same value when the underlying asset increases as it does when the underlying asset decreases in value. • The portfolio is riskless if: DSu – Cu = DSd – Cd • We know all values in this equation today, except D. • S = $100; Su = $110; Sd = $95 • Cu = MAX(Su – K, 0) = MAX($110 – 100,0) = $10. • Cd = MAX(Sd – K, 0) = MAX($95 – 100,0) = $0.

  11. Therefore, Our First Step is to Calculate D To make the portfolio riskless: DSu – Cu = DSd – Cd DSu – DSd = Cu – Cd D(Su – Sd) = Cu – Cd, Therefore, we can calculate D: D = (Cu – Cd) / (Su – Sd) D = (10 – 0) / 110 – 95 D = 10 / 15 D = 2 / 3.

  12. Sidebar: What is D? • D, delta, is the riskless hedge ratio. • D, delta, is the fractional share amount needed to hedge one call. • Therefore, the number of calls to hedge one share is 1/D.

  13. The One-Period Binomial Option Pricing Model—The Formula • A riskless portfolio today should be worth (DS – C)(1+r) in one period. • So, (DS – C)(1+r) = DSu – Cu (which equals DSd – Cd because we chose the “correct” D). • Solving the equation above for C:

  14. Now We Can Calculate the Call Price, C. What is the price of a similar put? Using Put-Call Parity: Why can we use Put-Call Parity?

  15. The Two-Period Binomial Option Pricing Model • Suppose there are two periods to expiration instead of one. What do we do in this case? • It turns out that we repeat much of the process we used in the one-period binomial option pricing model. • This method can be used to price: • European call options. • European put options. • American calls and puts (with a modification to allow for early exercise). • An exotic array of options (with the appropriate modifications).

  16. The Method We can find binomial option prices for two (or more) periods by using the following five steps: • Build a price “tree” for stock prices through time. • Use the intrinsic value formula to calculate the possible option values at expiration. • Calculate the fractional share needed to form each riskless portfolio at the next-to-last date. • Calculate all possible option prices at the next-to-last date. • Repeat this process by working back to today.

  17. The Binomial Option Pricing Model with Many Periods • When there are more than two periods, nothing really changes—we just keep working on back to today.

  18. What Happens When the Number of Periods Gets Really, Really Big? • We can always use a computer to handle this situation. • However, for European options on non-dividend paying stocks, the binomial method converges to the Black-Scholes option pricing formula. • To calculate the prices of many other types of options, however, we still need to use a computer (and methods similar in spirit to the binomial method).

  19. The Black-Scholes Option Pricing Model • The Black-Scholes option pricing model allows us to calculate the price of a call option before maturity (and, no put price is needed). • Dates from the early 1970s • Created by Professors Fischer Black and Myron Scholes • Made option pricing much easier—The CBOE was launched soon after the Black-Scholes model appeared. • The Black-Scholes option pricing model calculates the price of European options on non-dividend paying stocks.

  20. The Black-Scholes Option Pricing Model • The Black-Scholes option pricing model says the value of a stock option is determined by five factors: • S, the current price of the underlying stock. • K, the strike price specified in the option contract. • r, the risk-free interest rate over the life of the option contract. • T, the time remaining until the option contract expires. • , (sigma) which is the price volatility of the underlying stock.

  21. The Black-Scholes Option Pricing Formula • The price of a call option on a single share of common stock is: C = SN(d1) – Ke–rTN(d2) • The price of a put option on a single share of common stock is: P = Ke–rTN(–d2) – SN(–d1) d1 and d2 are calculated using these two formulas:

  22. Formula Details • In the Black-Scholes formula, three common functions are used to price call and put option prices: • e-rt, or exp(-rt), is the natural exponent of the value of –rt (in common terms, it is a discount factor) • ln(S/K) is the natural log of the "moneyness" term, S/K. • N(d1) and N(d2) denotes the standard normal probability for the values of d1 and d2. • In addition, the formula makes use of the fact that: N(-d1) = 1 - N(d1)

  23. Example: Computing Pricesfor Call and Put Options • Suppose you are given the following inputs: S = $50 K = $45 T = 3 months (or 0.25 years) s= 25% (stock volatility) r = 6% • What is the price of a call option and a put option, using the Black-Scholes option pricing formula?

  24. We Begin by Calculating d1 and d2 Now, we must compute N(d1) and N(d2). That is, the standard normal probabilities.

  25. Using the =NORMSDIST(x) Function in Excel • If we use =NORMSDIST(1.02538), we obtain 0.84741. • If we use =NORMSDIST(0.90038), we obtain 0.81604. • Let’s make use of the fact N(-d1) = 1 - N(d1). N(-1.02538) = 1 – N(1.02538) = 1 – 0.84741 = 0.15259. N(-0.90038) = 1 – N(0.90038) = 1 – 0.81604 = 0.18396. • We now have all the information needed to price the call and the put.

  26. The Call Price and the Put Price: • Call Price = SN(d1) – Ke–rTN(d2) = $50 x0.84741 – 45 x e-(0.06)(0.25) x 0.81604 = 50 x 0.84741 – 45 x 0.98511 x 0.81604 = $6.195. • Put Price = Ke–rTN(–d2) – SN(–d1) = $45 x e-(0.06)(0.25) x0.19479 – 50 x 0.15259 = 45 x 0.98511 x 0.18396 – 50 x 0.15259 = $0.525.

  27. We can Verify Our Results Using Put-Call Parity Note: The options must have European-style exercise. Verified.

  28. Valuing the Options Using Excel

  29. Using a Web-based Option Calculator • www.numa.com.

  30. Varying the Option Price Input Values • An important goal of this chapter is to show how an option price changes when only one of the five inputs changes. • The table below summarizes these effects.

  31. Varying the Underlying Stock Price • Changes in the stock price has a big effect on option prices.

  32. Varying the Time Remaining Until Option Expiration

  33. Varying the Volatility of the Stock Price

  34. Varying the Interest Rate

  35. Calculating the Impact of Stock Price Changes on Option Prices • Option traders must know how changes in input prices affect the value of the options that are in their portfolio. • An important effect on option prices is how changes in the stock price affects option prices. • The street name for this effect is “Delta.” • The other inputs also affect the option price, but we will concentrate on Delta.

  36. Calculating Delta • Deltameasures the dollarimpact of a change in the underlying stock price on the value of a stock option. Call option delta = N(d1) > 0 Put option delta = –N(–d1) < 0 • A $1 change in the stock price causes an option price to change by approximately delta dollars.

  37. Example: Calculating Delta

  38. The "Delta" Prediction: • The call delta value of 0.8474 predicts that if the stock price decreases by $1, the call option price will decrease by $0.85. • If the stock price is $49, the call option value is $5.368—an actual decrease of about $0.83. • How well does Delta predict if the stock price changes by $0.25? • The put delta value of -0.1526 predicts that if the stock price decreases by $1, the put option price will increase by $0.15. • If the stock price is $49, the put option value is $0.698—an actual increase of about $0.17. • How well does Delta predict if the stock price changes by $0.25?

  39. Hedging with Stock Options • You own 1,000 shares of XYZ stock AND you want protection from a price decline. • Let’s use stock and option information from before—in particular, the “delta prediction” to help us hedge. • Here you want changes in the value of your XYZ shares to be offset by the value of your options position. That is:

  40. Hedging Using Call Options—The Prediction • Using a Delta of 0.8474 and a stock price decline of $1: You should write 12 call options with a $45 strike to hedge your stock.

  41. Hedging Using Call Options—The Results • XYZ Shares fall by $1—so, you lose $1,000. • What about the value of your option position? • At the new XYZ stock price of $49, each call option is now worth $5.37—a decrease of $.83 for each call ($83 per contract). • Because you wrote 12 call option contracts at $6.20 (rounded), your call option gain was $996 = ($6.20 - $5.37) ×12 ×100. • Your call option gain nearly offsets your loss of $1,000. • Why is it not exact? • Call Delta falls when the stock price falls. • Therefore, you did not sell quite enough call options.

  42. Hedging Using Put Options—The Prediction • Using a Delta of -0.1526 and a stock price decline of $1: You should buy 66 put options with a strike of $45 to hedge your stock.

  43. Hedging Using Put Options—The Results • XYZ Shares fall by $1—so, you lose $1,000. • What about the value of your option position? • At the new XYZ stock price of $49, each put option is now worth $.70—an increase of $.17 for each put ($17 per contract). • Because you bought 62 put option contracts at $.53 (rounded), your put option gain was $1,122 = ($.70 - $.53) × 66 ×100. • Your put option gain more than offsets your loss of $1,000. • Why is it not exact? • Put Delta also falls (gets more negative) when the stock price falls. • Therefore, you bought too many put options—this error is more severe the lower the value of the put delta. • To get closer: Use a put with a strike closer to at-the-money.

  44. Hedging a Portfolio with Index Options • Many institutional money managers use stock index options to hedge the equity portfolios they manage. • To form an effective hedge, the number of option contracts needed can be calculated with this formula: • Note that regular rebalancing is needed to maintain an effective hedge over time. Why? Well, over time: • Underlying Value Changes • Option Delta Changes • Portfolio Value Changes • Portfolio Beta Changes

  45. Example: Calculating the Number of Option Contracts Needed to Hedge an Equity Portfolio • Your $10,000,000 portfolio has a beta of 1.00. • You decide to hedge the value of this portfolio with the sale of call options. • The call options have a delta of 0.579 • The value of the index is 1508. So, you sell 115 call options.

  46. Implied Standard Deviations • Of the five input factors for the Black-Scholes option pricing model, only the stock price volatility is not directly observable. • A stock price volatility estimated from an option price is called an implied standard deviation (ISD) or implied volatility (IVOL). • Calculating an implied volatility requires: • All other input factors, and • Either a call or put option price

  47. CBOE Implied Volatilities for Stock Indexes • The CBOE publishes data for three implied volatility indexes: • S&P 500 Index Option Volatility, ticker symbol VIX • S&P 100 Index Option Volatility, ticker symbol VXO • NASDAQ 100 Index Option Volatility, ticker symbol VXN • The VIX, VXO, and VXN indexes are estimates of expected market volatility. • The VIX was once known as the “investor fear gauge.” • This name stems from the belief that the VIX reflects investors’ collective prediction of near-term market volatility, or risk. • Generally, the VIX increases during times of high financial stress and decreases during times of low financial stress. • Some investors use the VIX as a buy-sell indicator. • The market saying is: “When the VIX is high, it’s time to buy; when the VIX is low, it’s time to go!”

  48. Employee Stock Options, ESOs • Essentially, an employee stock option is a calloption that a firm grants (i.e., gives) to employees. • ESOs allow employees to buy shares of stock in the company. • Giving stock options to employees is a widespread practice. • Because you might soon be an ESO holder, an understanding of ESOs is important.

  49. Features of ESOs • ESOs have features that ordinary call options do not. • The details vary by firm, but: • The life of the ESO is generally 10 years. • ESOs cannot be sold. • ESOs have a “vesting” period of about 3 years. • Employees cannot exercise their ESOs until they have worked for the company for this vesting period. • If an employee leaves the company before the ESOs are “vested," the employees lose the ESOs. • If an employee stays for the vesting period, the ESOs can be exercised any time over the remaining life of the ESO.

  50. Why are ESOs Granted? • Owners of a corporation (i.e., the stockholders) have a basic problem. How do they get their employees to make decisions that help the stock price increase? • ESOs are a powerful motivator, because payoffs to options can be large. • High stock prices: ESO holders gain and shareholders gain. • ESOs have no upfront costs to the company. • ESOs can be viewed as a substitute for ordinary wages. • Therefore, ESOs are helpful in recruiting employees.

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