Option Pricing and Strategies

1. Option Pricing and Strategies Lecture Notes for FIN 353 Yea-Mow Chen Department of Finance San Francisco State University

2. I. Option Market Structure 1. A call option gives the holder the right to buy an asset by a certain date for a certain price. A put option gives the holder the right to sell an asset by a certain date for a certain price. � Options can be either American or European. American Options are options that can be exercised at any time up to the expiration date, whereas European options are options that can only be exercised on the expiration date.

3. I. Option Market Structure There are two sides to every option contract. The writer of an option receives cash up front but has potential liabilities later. His or her profit/loss is the reverse of that for the purchaser of the option. Buy (Long) Sell (Short) _____________________________________ Call Right to buy Obligation to sell Put Right to sell Obligation to buy _____________________________________

4. I. Option Market Structure 2. Underlying Assets Stock Options: Foreign Currency Options: Philadelphia Exchange is the major exchange for foreign currency options trading. Index Options: Settlement is in cash rather than by delivering the portfolio underlying the index. Futures Options - When the holder of a call option exercises, he or she acquires from the writer a long position in the underlying futures contract plus a cash amount equal to the excess of the futures price over the strike price.

5. I. Option Market Structure 3. Specification of Stock Options � Expiration Dates Strike Prices Underlying Stock

6. I. Option Market Structure 4. Dividends and Stock Splits The early over-the-counter options were dividend protected. If a company declared a cash dividend, the strike price for options on the company's stock was reduced on the ex-dividend day by the amount of dividend. Exchange-traded options are not generally adjusted for cash dividends. Exchange-traded options are adjusted for stock splits. In general, an n-for-m stock split should cause the stock price to go down to m/n of its previous value. � Stock options are adjusted for stock dividends.

7. I. Option Market Structure 5. Position Limits and Exercise Limits: A position limit defines the maximum number of option contract than an investor can hold on one side of the market. For this purpose, long calls and short puts are considered to be on the same side of the market. Also short calls and long puts are considered to be on the same side of the market. The exercise limit equals the position limit. It defines the maximum number of contracts that can be exercised by any individual in any period of 5 consecutive business days.

8. I. Option Market Structure 6. Trading: � Market Makers The Floor Broker The Order Book Official Offsetting Orders

9. I. Option Market Structure 7. Margins: When call and put options are purchased, the option price must be paid in full. Investors are not allowed to buy options on margin. When naked call options are written, an initial margin requirement is the maximum of either: (1) the call premium plus 20% of the market value of the stock, less an amount equal to the difference in the exercise value and the stock value if the call is out of the money, or (2) the call premium plus 10% of the market value of the stock. �

10. I. Option Market Structure Margin = ( Max{[c + .20S - Max(E-S, 0)], [c +.10S]}) *N Example: If a writer sells an ABC 50 call contract for $3 when ABC is selling for $48, then the initial margin requirement would be $1,060. Margin = (Max{[$3 + .20 *$48 - Max($50 - $48, 0)], [$3 + .10 * $48]} * 100 = $1,060. � For a naked put on a stock, the initial margin requirement is: � Margin = ( Max{[p + .20S - Max(S-E, 0)], [ p + .10S]}) * N

11. II. Cross-Sectional Characteristics of Option Prices � Option prices on November 1, 1995 � Call Put _______________________________________________________ Nov Dec Mar Nov Dec Mar BrMSq 75 r r r r 1/4 1/2 86 5/8 80 6 3/4 7 3/4 9 3/8 1/8 3/8 1 1/2 86 5/8 85 2 3/8 3 3/4 5 7/8 5/8 1 1/2 3 1/4 86 5/8 90 1/4 1 1/4 3 1/4 3 1/4 4 1/4 5 3/4 _________________________________________________________

12. II. Cross-Sectional Characteristics of Option Prices Two Observations on Option Pricing 1. The price of a call (and a put) option will be greater the more distant the expiration date of the option, everything else being the same. � 2. The price of a call option will be lower the greater the exercise price of that option, everything else being the same; while the price of a put option becomes higher the greater the exercise price of that option.

13. II. Cross-Sectional Characteristics of Option Prices Option Pricing At Expiration �On the expiration date: C[S,t*] = max [0, S-E] for a call P[S,t*] = max [0, E-S] for a put. � The term max[0, S-E] is commonly referred to as the intrinsic value of a call option. If S<E, the option is said to be out-of-the-money and will have zero intrinsic value; If S>E, the option is in-the-money and has a positive value equating to (S-E).

14. II. Cross-Sectional Characteristics of Option Prices Intrinsic Value of the BrMSq if November 1's price of 86 5/8 were the expiration price: � call option put option ------------------------ ------------------------------- Nov Dec March Nov Dec March ----------------------------------------------------------------- 75 11 5/8 11 5/8 11 5/8 0 0 0 80 6 5/8 6 5/8 6 5/8 0 0 0 85 1 5/8 1 5/8 1 5/8 0 0 0 90 0 0 0 3 3/8 3 3/8 3 3/8 ------------------------------------------------------------------

15. II. Cross-Sectional Characteristics of Option Prices Option Pricing Before Expiration Time Value: Market participants are usually willing to pay more than the intrinsic value for an option, because they expect the market price of the stock to increase before the option expires. The amount by which the market price of an option exceeds its intrinsic value is its time value.

16. II. Cross-Sectional Characteristics of Option Prices The market price, or the premium, of an unexpired option will nearly always be equal to or greater than its intrinsic value. If the option price falls below the intrinsic value, net of transaction costs, arbitrageurs will buy the options, exercise them, and immediately sell the stock. Such riskless arbitrage prevents the option price from falling substantially below the intrinsic value of the option.

17. II. Cross-Sectional Characteristics of Option Prices Time value of the BrMSq on November 1, 1991 � call option put option ------------------------- ----------------------------- Nov Dec March Nov Dec March --------------------------------------------------------------- 75 r r r r 1/4 1/2 80 1/8 1 1/8 3 1/8 1/8 3/8 1 1/2 85 3/4 2 1/8 4 1/4 5/8 1 1/2 3 1/4 90 1/4 1 1/4 3 1/4 -1/8 7/8 2 3/8 -----------------------------------------------------------------

18. II. Cross-Sectional Characteristics of Option Prices Total Value: On its expiration date, the price of an option will lie on the intrinsic value line. Prior to that date, the value of an option varies with the price of the underlying stock. The option price/stock price curve shifts closer to the intrinsic value line as the expiration date approaches. This downward shifting shows why market participants sometimes refer to an option as a wasting asset. If the price of the stock does not rise, the value of an option declines as it approaches the expiration date.

19. III. Determinants of Option Pricing �Option pricing using the Black-Scholes Model: � c = S*N(d1) - (Ee-rt)*N(d2) � where � d1 = [ln(S/E) + (r + ?2/2)*t]/ ?? t � d2 = [ln(S/E) + (r -(?2 /2)*t]/ ?? t.

20. III. Determinants of Option Pricing Key Option Pricing Determinants and their Impacts on Option Prices: --------------------------------------------------------------------------------------------------------------------- European American European American Calls Calls Puts Puts ---------------------------------------------------------------------------------------------------------------------- 1. Exercise Price - - + + 2. Time to Maturity NA + NA + 3. Underlying security price + + - - 4. Underlying security price + + + + Volatility 5. Dividend policy - - + + 6. The risk-free interest rate + + - - --------------------------------------------------------------------------------------------------------------------- �

21. IV. OPTION DERIVATIVES 1. Delta: Defined as the rate of change of an option price with respect to the price of the underlying asset. It is the slope of the curve that relates the option price to the underlying asset price. Delta = ?c/?S = N(d1) where ?S = a small change in the stock price; ?c = the corresponding change in the call price.

22. IV. OPTION DERIVATIVES For example: If Eurodollar futures advanced 10 ticks, a call option on the futures whose delta is .30 would increase by only 3 ticks. Similarly, a call option whose delta is .11 would increase in value approximately 1 tick. The delta for a European call on a non-dividend-paying stock is N(d1), and for a European put is N(d1) -1. The delta for a call is positive, ranging in value from approximately 0 for deep out-of-the-money calls to approximately 1 for deep in-the-money ones. In contrast, the delta for a put is negative, ranging from approximately 0 to -1.

23. IV. OPTION DERIVATIVES Deltas change in response not only to stock price changes, but also to the time to expiration. As the time to expiration decreases, the delta of an in-the-money call or put increases, while an out-of-the-money call or put tends to decrease. � Delta also can be used to measure the probability that the option will be in the money at expiration. Thus, the call with a delta = N(d1) = .40 has an approximately 40% chance that its stock price will exceed the option�s exercise price at expiration.

24. IV. OPTION DERIVATIVES 2. Gamma: The gamma is the second derivative of the option premium with respect to the stock price. It tells you how much the delta will change when the stock price increases or decreases. If an option has a small gamma value, the option� s delta value is relatively stable and thus can hedge a large price change in the underlying stock better than if the option has a larger gamma value. � ?2 C N?(d1) ? = ------------ = ---------------- ? S2 S0 ?? T for a call option. The gamma values for European puts are the same as those for calls.

25. IV. OPTION DERIVATIVES Ex: Suppose delta = 50% and gamma = 5%; if the stock price increases by 1.00 then the delta will increase by 5 percentage points to 55% (50% + 5%). In other words, the option premium will increase or decrease in value at the rate of 50% of the stock price before the +1.00 point move, and 55% after the +1.00 point move. The gamma of a call or put varies with respect to the stock price and time to maturity. It can increase dramatically as the time to expiration decreases. Gamma values are largest for at -the-money options and smallest for deep-in-the-money and deep-out-of-the-money options.

26. IV. OPTION DERIVATIVES 3. Theta: The theta is the first derivative of the option premium with respect to time. It measures time decay - the amount of premium lost as another day passes. � ?C ?c = - ---------- ?T S0N(d1)? = ---------------- - r E e-rT N(d2) 2? T �

27. IV. OPTION DERIVATIVES Stock options with large negative theta values can lose their time premium rapidly. The value changes the most as maturity approaches. The theta value for call options on nondividend stocks is always negative. Put options usually have negative thetas as well. However, deep-in-the-money European puts could have positive thetas. EX: Assume a premium of 1.00 and a theta of 0.04. You would expect the premium to lose 4 points by tomorrow - to 0.96, assuming that no other variables have changed.

28. IV. OPTION DERIVATIVES 4. Vega: The vega is the first derivative of the option premium with respect to volatility. It measures the dollar change in the value of option when the underlying implied volatility increases by one percentage point.� ?C ? = ----------- = S ?T N(d1) ?? European puts with the same terms have the same vega values. A change in volatility will give the greatest total dollar effect on at-the-money options and the greatest percentage effect on out-of-the-money options.

29. IV. OPTION DERIVATIVES EX: If the implied volatility is 20%, the call premium is 2.00, and the vega is 0.12, then you would expect the premium to increase to 2.12 (2.00 + 0.12) when implied volatility moves up to 21%. Vega gives you an idea of how sensitive the option premium is to perceived changes in market value.

30. IV. OPTION DERIVATIVES Derivative Exercise: Option Value Delta Gamma Theta Vega -------------------------------------------------------------------------------------- Deutsche mark 58 call 2.29 60 14 -.04 .05 Eurodollar 92 put .24 -50 2 001 .03 Japanese Yen 75 call 1.15 20 3 -.012 .22 S&P 500 250 put .70 -30 9 -.007 .13 Swiss Franc 65 call 6.20 90 2 -.002 .08 --------------------------------------------------------------------------------------

31. IV. OPTION DERIVATIVES Derivative Exercise: �1. If Deutsche Mark futures rally one full point, the 58 call will advance from 2.29 to 2.89 �2. If volatility in the Yen futures contract increases from 12% to 13%, the Yen 75 call will advance from 1.15 to 1.37 3. If the S&P 500 futures decline 1.00 point, the delta on the 250 put will move from -30 to -39

32. IV. OPTION DERIVATIVES 4. If six days pass with the Japanese Yen futures contract remaining unchanged (and all other parameters remain unchanged), how much value will the Yen 75 call lose? 1.32 5. If implied volatility in Eurodollar futures drops from 9% to 7%, the 92 put will decline from .24 to .18 6. If a trader sells 10 Deutsche Mark 58 calls, how much futures contracts will he have to buy/sell in order to establish a delta neutral position? buy 6 futures contracts

33. Microhedge with Options A. Asset/Liability Management: Support that on March 2, 2000, a bank funded $75 million in loans that reprice every six months with three-month Eurodollar CDs at an annual rate of 9.30%. For each 100 basis points increase in interest rates, the bank would have to pay additional $187,500. To hedge, the bank writes 30 June 2000 Eurodollar futures call options at a strike price of 89.50. Since the Eurodollar futures settled at 89.78, the calls are in-the-money and priced at $14.50 each. If by June 1, 2000, Eurodollar CD rate dropped to 7.6% and Eurodollar futures price settled at 92.44. What is the net result of the this hedging strategy? If Eurodollar CD rate increased to 10.30% instead and futures price settled at 88.00, what is the net result?

34. Microhedge with Options Cash Market Futures Options Market ______________________________________________________________ Today: 6-month $75m loans March: Write 30 June 2000 3-matched with 3-month Eurodollar Eurodollar futures call options CDs at 9.3%. at a strike price of 89.50. (If rates rise by 1%, the bank will have Since the Eurodollar futures to pay an additional $187,500) settled at 89.78, calls earn a ($75M * 1% * 3/12). premium of $14.50 each. June: �If 3-month Eurodollar CD rate June: Eurodollar futures dropped to 7.6% price settled at 92.44. The calls are in-the-money and will be exercised by holders. _________________________________________________________ Gain: $318,750 Loss: (92.44 - 89.50) * = (=$75m * (9.3% - 7.6%) * 3/12) 2500 * 30= $7,350 * 30 savings in financing. = $220,500 Net Gain = $318,750 + $43,500 - $220,500 = $141,750

35. Microhedge with Options June: If 3-month Eurodollar June: Eurodollar CD rate had risen 1% futures price settled at 88.00. Calls are expired out-of money. �__________________________________________ Additional cost: $187,000. gain: premium $14.5 *100 * 30 = $43,500 � Net Loss = $187,000 - $43,500 = $143,500

36. Microhedge with Options B. Mortgage Prepayment Protection: � The prepayment option of fixed-rate mortgage contracts essentially gives borrowers a call option written by banks over the life of the mortgages. It will be exercised when it is in-the-money, i.e., when mortgage rates fall below the contractual rate minus any prepayment penalties or new loan origination costs. To manage the risk of mortgage prepayment if rates should fall, S&Ls should buy interest rate call options.

37. Microhedge with Options A S&L has five mortgage loans on its books, each earning a fixed rate of 14.25% with 20 years to maturity on an outstanding principal of $100,000. These loans are funded with three-month CDs. On Nov. 5, 1999, the three-month CD rate was 9.2%. The S&L imposes a 2.5% fees on new loan origination. To hedge the risk of a fall in mortgage rates and mortgage prepayment, management decides to buy five March 2000 T-bond futures call options at a strike price of 70. On November 5, 1999, each T-bond futures call option has a premium of $851 (March 2000 T-bond futures are priced at 69.78).

38. Microhedge with Options On Feb. 15, 2000, mortgage rates have fallen to 11.7% and 3-month CDs earn 8.7% interest, while the T-bond futures price rose to 72.11, what is the net result of the hedging strategy? Cash Market Future Options Market _______________________________________________________ Today: $500,000 mortgage loans at Today: Buy five March 14.25 fixed, with 20 year maturity 2000 T-bond futures call financed with 3-month CDs at 9.2%. options at a strike price Want to hedge against falling interest of 70. On this day, T-bond rates futures were at 69.78 (at-the- money) and T-bond futures call option has a premium $851 per contract Profit =$500,000 * (14.25% -9.2%) Cost = $851 * 5 *3/12 =$6,313/Quarter = $4,255

39. Microhedge with Options On Feb. 15, 2000, mortgage rates have fallen to 11.7% and 3-month CDs earn 8.7% interest, while the T-bond futures price rose to 72.11, what is the net result of the hedging strategy? (Case I: Falling Interest Rates: Mortgages are refinanced) � Feb. 15 Mortgage rates have fallen Feb. 15 T-bond future price 2000: to 11.7% and 3-month CDs 2000: rises to 72.11 earn 8.7% interest The five futures call options can be offset to return $2,110 per option _________________________________________________________ Profit = $500,000 * (11.7% - 8.7%) Profit = $2,110 * 5 * 3/12 = $3,750/Quarter = $10,550 Loss of profit = $6,313 - $3,750 = $2,563/Quarter Net Result = $10,550 - $4,255 - $2,563 = $3,732.

40. Microhedge with Options (Case II: Falling Interest Rates: Mortgages are not refinanced) � Feb. 15 Mortgage rates have fallen Feb. 15 T-bond futures 2000: to 13.25% and 3-month 2000: price rises to 70.70. CDs earn 9.0% interest The five futures call options can be offset to return $700 per option _________________________________________________________ Profit = $500,000 * (14.25% - 9.0%) Profit = $700 * 5 * 3/12 = $6,562.50/Quarter = $3,500 Loss of Profit = $6,313 - $6,562.50 = -$249.50/Quarter Net Result = $249.50 + 3,500 - $4,255 = -$505.50.

41. Microhedge with Options (Rising Interest Rates: Options are not exercised) � Feb. 15 Mortgage rates have been rising Feb. 15 T-bond futures 2000: to 15% and 3-month CDs 2000: price falls to 69 earn 9.8% interest The five futures call options are not exercised _________________________________________________________ Profit = $500,000 * (14.25 - 9.8%) * 3/1 Loss of premiums = $4,255 = $5,562.5/Quarter Loss = $6,313 - $5,562.5 = $750.5 Net Result = -$750.5 - $4,255 = -$5,005.5

42. Microhedge with Options (Rising Interest Rates: Options are not exercised) � Feb. 15 Mortgage rates have been rising Feb. 15 T-bond futures 2000: to 15% and 3-month CDs 2000: price falls to 69 earn 9.8% interest The five futures call options are not exercised _________________________________________________________ Profit = $500,000 * (14.25 - 9.8%) * 3/1 Loss of premiums = $4,255 = $5,562.5/Quarter Loss = $6,313 - $5,562.5 = $750.5 Net Result = -$750.5 - $4,255 = -$5,005.5

43. Macrohedge with Options Using Options to Hedge the Interest Rate Risk of the Balance Sheet An FI's net worth exposure to an interest rate shock could be represented as: � ?R ?E = -(DA - kDL) * A * --------- (1+R) We want to adopt a put option position to generate profits that just offset the loss in net worth due to a rate shock , given a positive duration gap for the FI. �

44. Macrohedge with Options Let ?P be the total change in the value of the put position in T-bonds. This can be decomposed into: � ?P = (Np x ? p) (1) Where Np is the number of $ 100,000 put option on T-bond contracts to be purchased (the number for which we are solving) and ? p is the change in the dollar value for each $ 100,000 face value T-bond put option contract.

45. Macrohedge with Options �The change in dollar value for each contract (? p) can be further decomposed into: � ? p = dp x dB x ? R (2) dB dR 1+R

46. Macrohedge with Options The value of a basis point can be linked to duration: dB = - MD x dR (3) B �Equation (3) can be arranged by cross multiplying as: dB = - MD x B (4) dR The first term (dp/dB) shows how the value of a put option change for each $ 1 dollar change in the underlying bond. This is called the delta of an option (? ) and lies between 0 and 1. The second term (dB/dR) shows how the market value of a bond changes if interest rates rise by one basis point.

47. Macrohedge with Options As a result, we can rewrite Equation (2) as: � ? p = (-?) x MD x B x ? R (5) 1+R �Thus the change in the total value of a put option ? P is: � ? P = Np * [(-?) x MD x B x ? R ] (6) 1+R The term in square brackets is the change in the value of one $100,000 face value T-bond put option as rates change and Np is the number of put option contracts.

48. Macrohedge with Options To hedge net worth exposure, we require the profit on the off-balance sheet put option to just offset the loss of on balance sheet net worth when rates rise (or bond prices fall). That is: ? P = ? E Np * (-?) * MD * B * = (DA-kDL) * A * � Solving for Np the number of put option to buy- we have: � Np = (DA-kDL) x A (-?) x MD x B