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Options

Options. Option Basics Option strategies Put-call parity Binomial option pricing Black-Scholes Model. 1. Option Basics: Terminologies. Buy - Long Sell - Short Call Put Key Elements Exercise or Strike Price Premium or Price Maturity or Expiration.

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Options

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  1. Options • Option Basics • Option strategies • Put-call parity • Binomial option pricing • Black-Scholes Model

  2. 1. Option Basics: Terminologies • Buy - Long • Sell - Short • Call • Put • Key Elements • Exercise or Strike Price • Premium or Price • Maturity or Expiration

  3. Market and Exercise Price Relationship In the Money - exercise of the option would be profitable. Call: market price>exercise price Put: exercise price>market price Out of the Money - exercise of the option would not be profitable. Call: market price<exercise price Put: exercise price<market price At the Money - exercise price and asset price are equal.

  4. Examples • Tell if it is in or out of the money • For a put option on a stock, the current stock price=$50 while the exercise price is $47. • For a call option on a stock, the current stock price is 55 while exercise price is 45.

  5. American vs. European Options American - the option can be exercised at any time before expiration or maturity. European - the option can only be exercised on the expiration or maturity date. Should we early exercise American options?

  6. Different Types of Options • Stock Options • Index Options • Futures Options • Foreign Currency Options • Interest Rate Options

  7. Payoffs and Profits at Expiration - Calls Notation Stock Price = ST Exercise Price = X Payoff to Call Holder (ST - X) if ST >X 0 if ST < X Profit to Call Holder Payoff - Purchase Price

  8. Payoffs and Profits at Expiration - Calls Payoff to Call Writer - (ST - X) if ST >X 0 if ST < X Profit to Call Writer Payoff + Premium (pages 678 – 680)

  9. Payoff and Profit to Call Option at Expiration

  10. Payoff and Profit to Call Writers at Expiration

  11. Payoffs and Profits at Expiration - Puts Payoffs to Put Holder 0 if ST> X (X - ST) if ST < X Profit to Put Holder Payoff - Premium

  12. Payoffs and Profits at Expiration - Puts Payoffs to Put Writer 0 if ST > X -(X - ST) if ST < X Profits to Put Writer Payoff + Premium

  13. Payoff and Profit to Put Option at Expiration

  14. 2. Option Strategies Straddle (Same Exercise Price) – page 685 Long Call and Long Put Spreads - A combination of two or more call options or put options on the same asset with differing exercise prices or times to expiration -- page 686. Vertical or money spread: Same maturity Different exercise price Horizontal or time spread: Different maturity dates

  15. Value of a Protective Put Position at Expiration

  16. Protective Put vs Stock (at-the-money) Page 684

  17. Value of a Covered Call Position at Expiration

  18. Covered call example • Go to yahoo.finance.com, click on “investing”. Select the Citi Corp. (ticker symbol: C). Choose “options”. Suppose you are interested in the July 2013 options (i.e., option expiration date is July 13, 2013) and decided to buy 100 shares of Citi Corp on 4/24/2013. In the meantime, you shorted 1 contract (100 shares) of Citi’s stock option with an exercise price of $____/share. • What is the profit from the portfolio at the expiration date when the price of Citi on July 13, 2013 is $35?

  19. Option Values • Intrinsic value - profit that could be made if the option was immediately exercised. • Call: stock price - exercise price • Put: exercise price - stock price • Time value - the difference between the option price and the intrinsic value.

  20. Figure 21.1 Call Option Value before Expiration

  21. Restrictions on Option Value: Call • Value cannot be negative • Value cannot exceed the stock value • Value of the call must be greater than the value of levered equity C > S0 - ( X + D ) / ( 1 + Rf )T C > S0 - PV ( X ) - PV ( D )

  22. If the prices are not equal arbitrage will be possible Put-call Parity

  23. Stock Price = 110 Call Price = 17 Put Price = 5 Risk Free = 5% Maturity = 1 yr X = 105 117 > 115 Since the leveraged equity is less expensive, acquire the low cost alternative and sell the high cost alternative Put Call Parity - Disequilibrium Example

  24. 4. Binomial Option Pricing: Text Example 120 10 100 C 90 0 Call Option Value X = 110 Stock Price

  25. Binomial Option Pricing: Text Example 30 Alternative Portfolio Buy 1 share of stock at $100 Borrow $81.82 (10% Rate) Net outlay $18.18 Payoff Value of Stock 90 120 Repay loan - 90 - 90 Net Payoff 0 30 18.18 0 Payoff Structure is exactly 3 times the Call

  26. Binomial Option Pricing: Text Example 30 30 18.18 C 0 0 3C = $18.18 C = $6.06

  27. Generalizing the Two-State Approach See page 725, get hedge ratio, then compute call price.

  28. 5. Black-Scholes Option Valuation Co = SoN(d1) - Xe-rTN(d2) d1 = [ln(So/X) + (r + 2/2)T] / (T1/2) d2 = d1 + (T1/2) where Co = Current call option value. So = Current stock price N(d) = probability that a random draw from a normal dist. will be less than d.

  29. Black-Scholes Option Valuation X = Exercise price e = 2.71828, the base of the natural log r = Risk-free interest rate (annualizes continuously compounded with the same maturity as the option) T = time to maturity of the option in years ln = Natural log function Standard deviation of annualized cont. compounded rate of return on the stock

  30. Call Option Example So = 100 X = 95 r = .10 T = .25 (quarter) = .50 d1 = [ln(100/95) + (.10+(5 2/2))] / (5.251/2) = .43 d2 = .43 + ((5.251/2) = .18

  31. Call Option Value Co = SoN(d1) - Xe-rTN(d2) Co = 100 X .6664 - 95 e- .10 X .25 X .5714 Co = 13.70 Implied Volatility Using Black-Scholes and the actual price of the option, solve for volatility. Is the implied volatility consistent with the stock?

  32. Black-Scholes Model with Dividends • The call option formula applies to stocks that pay dividends. • One approach is to replace the stock price with a dividend adjusted stock price. Replace S0 with S0 - PV (Dividends)

  33. Put Value Using Black-Scholes P = Xe-rT [1-N(d2)] - S0 [1-N(d1)] Using the sample call data S = 100 r = .10 X = 95 g = .5 T = .25 95e-10x.25(1-.5714)-100(1-.6664) = 6.35

  34. Determinants of Call Option Values

  35. Using the Black-Scholes Formula Hedging: Hedge ratio or delta The number of stocks required to hedge against the price risk of holding one option. Call = N (d1) Put = N (d1) - 1 Option Elasticity Percentage change in the option’s value given a 1% change in the value of the underlying stock.

  36. Figure 21.9 Call Option Value and Hedge Ratio

  37. Portfolio Insurance • Buying Puts - results in downside protection with unlimited upside potential. • Limitations • Tracking errors if indexes are used for the puts. • Maturity of puts may be too short. • Hedge ratios or deltas change as stock values change.

  38. Solution • Synthetic Put • Selling a proportion of shares equal to the put option’s delta and placing the proceeds in risk-free T-bills (page 763-764) • Additional problem • Delta changes with stock prices • Solution: delta hedging • But increase market volatilities • October 19, 1987

  39. Hedge Ratios Change with Stock Prices

  40. Hedging On Mispriced Options Option value is positively related to volatility: • If an investor believes that the volatility that is implied in an option’s price is too low, a profitable trade is possible. • Profit must be hedged against a decline in the value of the stock. • Performance depends on option price relative to the implied volatility.

  41. Hedging and Delta The appropriate hedge will depend on the delta. Recall the delta is the change in the value of the option relative to the change in the value of the stock. • Example (page 767): • Implied volatility = 33% • Investor believes volatility should = 35% • Option maturity = 60 days • Put price P = $4.495 • Exercise price and stock price = $90 • Risk-free rate r = 4% • Delta = -.453

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