2. Kim, Gyutai Dept. of Industrial Engineering, Chosun University 1 Properties of Stock Options

3. Kim, Gyutai Dept. of Industrial Engineering, Chosun University 2 Options Option contracts give the holders the right, but not the obligation, to buy/sell an asset at a predetermined time for a predetermined price. Call options give the holder the right to buy an asset, whereas put options give the holder the right to sell an asset. The premium, or option value, is the cost the holder must pay to enter into the contract. The date in the contract is known as the expiration date, and the predetermined price is known as the strike price or exercise price. European options can only be exercised on the expiration date, whereas American options can be exercised any time up to and including the expiration date.

4. Kim, Gyutai Dept. of Industrial Engineering, Chosun University 3 Option Terminology Example You are the holder of a European call option to buy Microsoft stock. The contract gives you the right,but not the obligation, to buy Microsoft stock T = 1year from now (expiration date) for X = $50(exercise price). The payoff to an option is similar to a forward contract – except you have the option not to buy the underlying asset (i.e. Microsoft stock).

5. Kim, Gyutai Dept. of Industrial Engineering, Chosun University 4 Let X = $50 and T = 1 year. Assume that Microsoft stock at T = 1 year is $55 per share – would you exercise the option and buy the stock? Assume that Microsoft stock at T = 1 year is $45 per share – would you exercise the option and buy the stock? Option Example

6. Kim, Gyutai Dept. of Industrial Engineering, Chosun University 5 Option Payoff Let S T = Stock price at expiration X = Exercise price The payoff of an option at expiration is summarized below: Long Call: Max[(S T – X), 0] Short Call: -Max[(S T – X), 0] Long Put: Max[(X – S T ), 0] Short Put: -Max[(X – S T ), 0] Note: From the outset of the option contract, S T is a random variable, with the payoff dependent upon the magnitude of S T.

7. Kim, Gyutai Dept. of Industrial Engineering, Chosun University 6 Option Payoff Diagrams

8. Kim, Gyutai Dept. of Industrial Engineering, Chosun University 7 Option Payoff Example

9. Kim, Gyutai Dept. of Industrial Engineering, Chosun University 8 Option Attributes Options can be considered in-the-money, out-of-the- money, or at-the-money. For a call option, in-the-money implies S t > X,out-of- the-money implies S t < X, and at-the-money implies S t = X. For a put option, in-the-money implies S t X, and at-the-money implies S t = X.

10. Kim, Gyutai Dept. of Industrial Engineering, Chosun University 9 Option Attributes Assume stock on company ABC is currently trading for S 0 = $103per share. A call option expiring in T = 3 months with an exercise price of X = $100 is selling for a call option premium, c = $5. The value of an option is comprised of two components - the intrinsic value and time premium. The intrinsic value is the value of the option if it was exercised today, or S 0 - X = $3. (Note: The intrinsic value can never be negative, therefore it is the max[S 0 - X, 0]). The time premium represents the probability the underlying asset will move in the favorable direction by the maturity date. The time premium is the option premium minus the intrinsic value or $5 - 3 = $2. At expiration, the time premium is zero because no time remains for the underlying security to move in a favorable direction. Therefore, at expiration the value of the option is the max(S T - X, 0).

11. Kim, Gyutai Dept. of Industrial Engineering, Chosun University 10 Why Use Financial Options? Financial options provide three general benefits: (1) Leverage (2) Insurance (3) Acting on prior beliefs

12. Kim, Gyutai Dept. of Industrial Engineering, Chosun University 11 Leverage Options contracts allow investors to magnify gains/losses. Take for example the following situation for a stock where an investor has $3,000 to invest. The investor plans on either buying: 1. 100 shares of stock at S 0 = $30 2. Entering into a call option contract with an X = $30 and call option premium c = $3. (Note: Most option contracts require that you purchase 100 shares, and thus this investor intends on entering into 10 option contracts.)

13. Kim, Gyutai Dept. of Industrial Engineering, Chosun University 12 Leverage

14. Kim, Gyutai Dept. of Industrial Engineering, Chosun University 13 Insurance Put options provide a form of insurance for an investment. Consider an investor intent on holding onto Exxon stock. In order to protect herself from a downside price movement, she purchases a put options contract. Assume the stock is trading for S 0 = $100. She enters into a put option with an X = $100 and put premium of p = $3. – If S T = $105, then her put option expires worthless and she loses p = $3. – If S T = $90, then the net profit of the option is max(100 – 90, 0) – 3 = $7. – Even though the stock has fallen $10, her stock has only been exposed to a loss of the premium amount or $3.

15. Kim, Gyutai Dept. of Industrial Engineering, Chosun University 14 Acting on Prior Beliefs Any desired payoff can be constructed by holding combinations of call options, put options, and long/short underlying asset positions. Some names of the most popular combinations are bull spreads, bear spreads, butterfly spreads,calendar spreads, straddles, strips, straps, and strangles. Each of these combinations addresses a different prior belief about the movement of the underlying asset.

16. Kim, Gyutai Dept. of Industrial Engineering, Chosun University 15 Option Notation The following notation will be used throughout the remainder of this section: S 0 = Underlying asset price today S T = Underlying asset price at expiration X = Exercise price T = Time to expiration r = Risk-free rate q = Continuous dividend yield

17. Kim, Gyutai Dept. of Industrial Engineering, Chosun University 16 Option Six key variable inputs determine the value of an option: (1). the value of the underlying asset, (2) the exercise price, (3) the variability of the underlying asset (or σ),(4)the time to maturity, (5) the risk-free rate, and (6) dividends. Each of these inputs affects the change in the value of the option directly. Table 1 summarizes the effect on the price of a European call and put option by assuming an increase in one input and keeping all the other inputs fixed.

18. Kim, Gyutai Dept. of Industrial Engineering, Chosun University 17 Option Attributes

19. Kim, Gyutai Dept. of Industrial Engineering, Chosun University 18 Option Attributes When the underlying asset value increases, the payoff of the option at expiration increases; thus, increasing the value of the option. When the exercise price increases the payoff amount decreases; thus, decreasing the value of the option. Due to the asymmetric payoff of an option, when the volatility increases, so does the option value. An increase in the volatility translates to a greater probability of both a larger positive and negative return. But, since the option will not be exercised if there is a large negative return, the option capitalizes only on the larger positive return.

20. Kim, Gyutai Dept. of Industrial Engineering, Chosun University 19 Option Attributes As the time to make a decision increases, so does the option value.Increased time until a decision needs to be made allows for more time for information gathering and/or observation of critical events. In addition, an increase in time allows the value of the underlying asset to fluctuate to larger values, thus increasing the value of the option. An increase in the risk-free asset will increase the value of the call option. Although less intuitive, an increase in the risk-free rate reduces the amount of money that needs to be set aside and invested in order to exercise the option. Put another way, the increase in the risk-free rate increases the discount rate, which decreases the investment cost: as the investment cost decreases, the call option value increases. Dividends (or cash payouts) from the underlying asset reduce the value of the underlying asset by the dividend amount. Thus a decrease in the underlying asset’s value leads to a decrease in the payoff of the option at expiration which implies a decrease in option price.

21. Kim, Gyutai Dept. of Industrial Engineering, Chosun University 20 Option Bounds Upper Bound Call Option Price < S 0 If this did not hold, then a risk-less (or arbitrage) profit could be made by shorting the call and buying the stock. Justification: No matter what the payoff of the call option is at expiration, it will always differ by the amount X from the stock price.

22. Kim, Gyutai Dept. of Industrial Engineering, Chosun University 21 Option Bounds Upper Bound Put Option Price < Xe -rT If this did not hold, then an arbitrage profit could be made by shorting the put and investing the proceeds at the risk-free rate. Justification: The put option payoff, max(X – S T, 0), can never exceed X because S T can’t be less than zero.

23. Kim, Gyutai Dept. of Industrial Engineering, Chosun University 22 Option Bounds Lower Bounds Call Option Price > max (S 0 – Xe -rT, 0) Recall that the value of a forward contract is f = S 0 – Xe -rT. However, the forward contract is an obligation to buy the underlying asset at expiration. The call option is not an obligation to buy the underlying asset. Hence, the added value of the option’s flexibility is a premium over and above the forward’s value. Put Option Price > max(Xe -rT – S 0, 0) The value of a short forward position (or the obligation to sell the asset) is equal to f = Xe -rT – S 0. Just like the call option, the put option is not an obligation to sell the underlying asset.

24. Kim, Gyutai Dept. of Industrial Engineering, Chosun University 23 Option Bounds American > European Options Recall that American options can be exercised any time up to its expiration date, whereas, a European option can only be exercised on its expiration date. In general, the value of a European option will be less than the value of its corresponding American option. An American option is valued higher because the investor must pay the additional premium to have the right to exercise the option earlier if it is deemed worthwhile. Note: It is never optimal to exercise an American call option on a non-dividend-paying stock early.

25. Kim, Gyutai Dept. of Industrial Engineering, Chosun University 24 Option Bounds Put-Call Parity Because call and put option prices will be valued in identical no-arbitrage environment there exists a relationship. This relationship is called the put-call parity. It states that: c + Xe -rT = p + S 0 This relationship holds for European call and put option values with the same option attributes (i.e. same expiration date, volatility, exercise price, etc.).

26. Kim, Gyutai Dept. of Industrial Engineering, Chosun University 25 Payoffs Portfolio P: One European put option plus one share Portfolio C: One European call option plus an amount of cash equal to Xe -rT Position if ST $100 Stock S T S T Pure Dis. Bond with $100 $100 $100 Put $100-S T 0 Call 0 S T -$100 Stock+Put $100 S T B+Call $100 S T Payoff Structure for Portfolio “P ” Payoff Structure for Portfolio “C ” $100 S+P S P $100 B+C S C

27. Kim, Gyutai Dept. of Industrial Engineering, Chosun University 26 Arbitrage with Put-Call Parity If a put-call parity does not hold, there exists an arbitrate opportunity. S 0 =$31, X=$30, r f =10%, C=$3, P=$2.25 C+X e -rT = 3 + 30e -0.1*3/12 =$32.26 p + S 0 = 2.25 + 31 = $33.25 Portfolio “P” is overpriced relative to portfolio “C”. So, buy the call and short the put and stock. Cash Inflow: -3+2.25+31 = $30.25 Invest the proceed into the risk-free interest rate: 30.25*e -0.1*0.25 =$31.02 The Final Payoff: if the stock price at expiration of the option is greater than $30, the call option is exercised. If it is less than $30, the put option is exercised. In either case, one share of the stock can be purchased for $30. $31.25 - $30 =$1.25

28. Kim, Gyutai Dept. of Industrial Engineering, Chosun University 27 Arbitrage with Put-Call Parity If a put-call parity does not hold, there exists an arbitrate opportunity. S 0 =$31, X=$30, r f =10%, C=$3, P=$1 C+X e -rT = 3 + 30e -0.1*3/12 =$32.26 p + S 0 = 1 + 31 = $32 Portfolio “C” is overpriced relative to portfolio “P”. So, short the call and buy the put and stock. Net Initial Investment: 1 + 31 – 2 =$29 Finance $29 at the risk-free interest rate and repay 29e 01.*0.25 = $29.73 The Final Payoff: if the stock price at expiration of the option is greater than $30, the call option is exercised. If it is less than $30, the put option is exercised. In either case, one share of the stock can be sold for $30. $30 - $29.73 = $0.27