1. Ch8. Financial Options

2. 1. Def: a contract that gives its holder the right to buy or sell an asset at predetermined price within a specific period of time. E.g) Currently a stock price is $53.50 at 1/9/2006. A call option provides a right to buy the stock at $55 at 5/14/2006. Here, $55 is a strike (exercise) price. 5/14/2006 is an expiration date. The seller of the option is called “option writer.”

3. 2. Types of options and Terminology Call option: right to buy an asset at fixed price. Put option: right to sell an asset at fixed price. Covered option: options sold with stocks (or assets). Naked option: option sold without stocks (or assets). Out of money: when an option provides an unfavorable difference between exercise price and market price. In the money: when an option provides an favorable difference.

4. Usually options are written for six months or less. But LEAPS (long term equity anticipation security) have maturity of up to 2 and ½ years. 3. Exercise value versus option price. Exercise value = Max[market (stock)price – Exercise (strike) price, 0] Option market value > Exercise value Option market value > 0 even when it is out of money due to maturity or volatility.

5. (source: Intermediate Financial Management by Brigham and Daves, 9 th edition)

8. The time value decreases as the stock price increases. Due to the sure profit, the time value would approaches to zero at higher price. It is due to the declining degree of leverage provided by options as the underlying stock price increases (that is, option price change is more volatile than stock price change). Considerable upside potential but limited downward risk.

9. 4. Option pricing model: binomial approach. Concept of riskless (hedge) portfolio: generating a portfolio producing the same return as a riskless security. That is, stock + a call option = riskless asset/security. E.g) assumptions Current price of stock is $40. At the end of a year, the price would change to either $50 or $32. A call option with the stock has an exercise price of $35 with one year maturity. A riskless security had 8% return.

10. (1) stock price change and exercise value change Stock price: 50 -32 =18 Exercise value: 15(=50-35) or 0 (2) # of stock to be held to cover option Exercise value change / stock price change = 15/18 =0.8333 (3) creating a riskless portfolio Buying 0.8333 share and sell one call option. After one year, stock price would be 41.67 (=0.8333*50) or 26.67 (=0.8333*32)

11. A value of a portfolio composed of 0.833 stock and a sold call option after one year: At $50, 41.67(stock)-15(call option)=26.67 At $32, 26.67(stock) -0 (call option)=26.67 Stock – call option = portfolio Thus after one year, the portfolio composed of 0.833 stock and a sold call option generates the same return regardless of the stock price. It is riskless. If we calculate PV of the portfolio (daily compounding with riskless rate) is 26.67/(1+0.08/365)^365 =24.67 It implies that 1) PV of portfolio = PV of stock – PV of call option. 2) PV of call option =PV of stock – PV of portfolio = 0.8333*40 – 24.62 =8.71

12. 5. Black sholes’ model

13. (source: Intermediate Financial Management by Brigham and Daves, 9 th edition) E.g) Assume: P = $27, X = $25, r RF = 6%, t = 0.5 years and σ 2 = 0.11. d 1 = {ln($27/$25) + [(0.06 + 0.11/2)](0.5)} ÷ {(0.3317)(0.7071)} d 1 = 0.5736. d 2 = d 1 - (0.3317)(0.7071) d 2 = 0.5736 - 0.2345 = 0.3391

14. N(d 1 ) = N(0.5736) = 0.7168. N(d 2 ) = N(0.3391) = 0.6327. Note: Values obtained from Excel using NORMSDIST function. For example: N(d 1 ) = NORMSDIST(0.5736) V = $27(0.7168) - $25e -(0.06)(0.5) (0.6327) = $19.3536 - $25(0.97045)(0.6327) = $4.0036.

15. 5-1) factors affecting the call option price Current stock price: + Exercise price: - Maturity: + Risk-free rate: + Variance:+

16. 6. Put option Valuation Def: right to sell an asset Put-call parity: two portfolios generate the same returns. Put option + stock = call option +PV of exercise price

18. Put option = call option –stock + PV of exercise price