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Option Valuation

16. Option Valuation. What is an Option Worth?. 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.

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Option Valuation

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  1. 16 Option Valuation

  2. What is an Option Worth? • 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

  3. 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 (T = 1.0) • A $110 exercise price (K = 110) • The current stock price is $108 (S0 =108) • The one-year risk-free rate is 10 percent (r = 10%)

  4. A Simple Model to Value Options Before Expiration, I. • We know what the stock price will be in one year. • S1 = $130 or $115 (but no other values). • S1 is still uncertain. • We do not know the probabilities of these two values. • Therefore, 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.

  5. 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.

  6. The One-Period Binomial Option Pricing Model—The Assumptions • S = the stock price today; the stock pays no dividends. • Assume that the stock price in one period is eitherS ×u or S ×d, where: • u (for “up” factor) is > 1 • and d (for “down” factor) < 1 • Suppose S = $100, u = 1.1 and d = .95 • S1 = $100 × 1.1 = $110 or S1 = $100 × .95 = $95 • What is the call price today, if: • K = 100 • R = 3%

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

  8. 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 = Δ * S. DS×u - Cu DS - C DS×d - Cd

  9. To Calculate Today’s Call Price, C: • There is one combination of a fractional share and one call that makes this portfolio risk-less. • 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, 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

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

  11. 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

  12. 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). • Solve the equation above for C.

  13. The One-Period Binomial Option Pricing Model—The Formula • Solving the equation above for C:

  14. Calculate the Call Price, C.

  15. Now We Can Calculate the Call Price, C. If C= $5.18, what is the price of a similar put? Using Put-Call Parity:

  16. The Two-Period Binomial Option Pricing Model • Suppose there are two periods to expiration instead of one. • Repeat much of the process 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).

  17. 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.

  18. The Binomial Option Pricing Model with Many Periods

  19. What Happens When the Number of Periods Gets Really, Really Big? • 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).

  20. The Black-Scholes Option Pricing Model • The BSOPM calculates the price of a call option before maturity • Dates from the early 1970s • Professors Fischer Black and Myron Scholes • Facilitated option pricing • CBOE was launched soon after BSOPM appeared • 1997 Nobel Prize in Economics • Important contributions by professor Robert Merton • The Black-Scholes-Merton option pricing model

  21. Black-Scholes Option ValuationVariables C= Current call option value P = Current put option value S= Current stock price K = Option strike or exercise price r = Risk-free interest rate  = Stock price volatility T = time to maturity of the option in years ln = Natural log function; ln(x) e = 2.71828, the base of the natural log; exp() N(d) = probability that a random draw from a normal distribution will be less than d Solution Variables Input Variables Functions

  22. Black-Scholes Option Valuation

  23. Formula Functions • e-rt = exp(-rt) = natural exponent of the value of –rt (a discount factor) • ln(S/K) = 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. • Formula makes use of the fact that: N(-d1) = 1 - N(d1)

  24. 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%

  25. Step 1: Calculating d1 and d2

  26. Step 2: Excel’s “=NORMSDIST(x)” Function =NORMSDIST(1.02538) = 0.84741 = N(d1) =NORMSDIST(0.90038) = 0.81604 = N(d2) N(-d1) = 1 - N(d1): N(-1.02538) = 1 – N(1.02538) = 1 – 0.84741 = 0.15259 = N(-d1) N(-0.90038) = 1 – N(0.90038) = 1 – 0.81604 = 0.18396 = N(-d2)

  27. Step 3a: The Call Price C = SN(d1) – Ke–rTN(d2) = $50(0.84741) – 45(e-(.06)(.25))(0.81604) = 50(0.84741) – 45(0.98511)(0.81604) = $6.195

  28. Step 3b: The Put Price: P = Ke–rTN(–d2) – SN(–d1) = $45(e-(.06)(.25))(0.18396) – 50(0.15259) = 45(0.98511)(0.18396) – 50(0.15259) = $0.525

  29. Verify Our Results Using Put-Call Parity Note: Options must be European-style

  30. Valuing the Options Using Excel

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

  32. 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.

  33. Factors Influencing Option Values

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

  35. Varying the Time Remaining Until Option Expiration

  36. Varying the Volatility of the Stock Price

  37. Varying the Interest Rate

  38. 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.

  39. Example: Calculating Delta

  40. The Call "Delta" Prediction: • Call delta = 0.84741 • if the stock price increasesby $1, the call option price will increase by about $0.85 • From the previous example: • Stock price = $50 • Call option price = $6.195 • If the stock price is $51: • Call option value = $7.060 • Increase of about $0.868

  41. The Put "Delta" Prediction: • Put delta value = -0.15259 • if the stock price increases by $1, the put option price will decrease by $0.15. • From the previous example: • Stock price = $50 • Put option price = $0.525 • If the stock price is $51: • Put option value = $0.390 • Decrease of about $0.14

  42. 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. • You want changes in the value of your XYZ shares to be offset by the value of your options position. That is:

  43. Hedging Using Call Options—The Prediction • Delta = 0.8474; stock price declines by $1: Write 12 call options with a $45 strike to hedge your stock.

  44. Hedging with Calls - Results • 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.

  45. Hedging Using Put Options—The Prediction • Delta = -0.1526; stock price declines by $1: Buy 66 put options with a strike of $45 to hedge your stock.

  46. Hedging Using Put Options: Results • Put option gain more than offsets $1,000 loss • 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.

  47. Hedging a Portfolio with Index Options • Many institutional money managers use stock index options to hedge equity portfolios • Regular rebalancing needed to maintain an effective hedge • Underlying Value Changes • Option Delta Changes • Portfolio Value Changes • Portfolio Beta Changes

  48. Hedging with Stock Index Options You manage a $10 million stock portfolio. You attempt to maintain a portfolio beta of 1.00 You decide to hedge your position by buying index put options with a contract value of $125,000 and a delta of 0.579. How many contracts do you need to buy?

  49. Hedging with Stock Index Options Portfolio value = $10 million Portfolio beta = 1.00 Index contract value = $150,800 Option delta = 0.579 Sell 115 call options

  50. Implied Standard Deviations • Implied standard deviation (ISD) • = Implied volatility (IVOL) • Stock price volatility estimated from an option price • Of the six BSOPM input factors, only stock price volatility is not directly observable • Calculating an implied volatility requires: • All 5 other input factors • Either a call or put option price

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