1. 1Lec 10 Discover Option Prices Lec 10: How to Discover Option Prices (Hull, Ch. 10) Suppose S 0 = $50 and r = 25%. Q: What might be reasonable prices for C 0 E, C 0 A, or P 0 E, P 0 A (given K=40, T=1 year)?. Intuition, or what questions to think about. ▸ Is the stock price expected to ↑ or ↓? ▸ If call is American, I would pay at least $10. Why? ▸ If call is European, why pay anything? (Exercise ONLY on the Expiration Day!.) ▸ Is it ever possible for C 0 E = C 0 A, or P 0 E = P 0 A ? The purpose of this Lecture is to help you develop “good intuition” about option pricing.

2. 2Lec 10 Discover Option Prices European Call, Stock pays no dividends: C 0 E (p.2) Do these prices make sense? S 0 = $50, C 0 E (K = $40, T=1yr) = $5, and r = 25%(simple interest) Intuition. There are two ways to buy stock: A: Buy the stock right now, CF 0 = -50 Or B: Buy the call and a bond and wait until expiration {+C, +B(FV=$40, T=1yr)} ➟ CF 0 = -5-32 = -$37 At Expiration, for the synthetic stock: if call is in the money (S T > 40) ➟ C T + 40 = S T if call is out of the money (S T < 40) ➟ C T + 40 = 40. Which is the better investment A or B ? Is it possible to make some “free money”?

3. 3Lec 10 Discover Option Prices Yes, try the following strategy: {-S, +C, +B(FV=40, T=1yr)} {Short the stock for $50, Buy the call for $5, Buy a bond for $32 =40/1.25} CF 0 = +50-5-32 = +$13 At Expiration, if S T ≥ 40, call is in the money. Bond matures for $40, use $40 plus call to buy stock. Use stock to cover short position. CF T =0. if S T < 40, call is worthless. Bond matures for $40. Use some of $40, buy stock and cover the short position. CF T = 40 - S T > 0.

4. 4Lec 10 Discover Option Prices This strategy is great! Think about it: Receive $13 now. If stock price ↓, make more money (40 - S T ). If S T ↑, lose nothing! This is known as an ARBITRAGE OPPORTUNITY. The “Arbitrage Profit” = $13. Clearly, Call is mis-priced. To preclude this arbitrage C 0 E must be at least 5+13 = $18. In sum, If S 0 = $50, and r = 25%, then C 0 E (K = $40, T=1yr) ≥ $18 (Compare this answer with initial intuition: “European Call has little value” ).

5. 5Lec 10 Discover Option Prices European Call Price C 0 E, Stock pays a Dividend ( p. 3) Assume stock pays a $5 Dividend (for sure) in 3 months. How will this affect the Call value? Do these prices make sense? S 0 = $50, C 0 E (K = $40, T=1yr)=$6, r=25%, and Div=$5 in 3 months. There are two ways to buy stock: A: Buy the stock right now: CF 0 = -50, Or B: Buy the call. Buy a bond to mimic the dividend, and another bond to cover the $40. Wait until expiration. {+C, +B(FV=5, t=3 months), +B(FV=40, T=1yr) } CF 0 = -6 -4.71 -32 = -$42.71

6. 6Lec 10 Discover Option Prices Cash Flows for the synthetic stock: In 3 months, 1 st bond matures for $5, just like the $5 Dividend from the stock. At Expiration, if call is in the money (S T > 40) C T + 40 = S T if call is out of the money (S T < 40) C T + 40 = 40 Which is the better investment A or B ? (Synthetic is better: it costs less and has better future outcomes)

7. 7Lec 10 Discover Option Prices Arbitrage Strategy: {-S, +C, +B(FV=5, T=3 months), +B(FV=40, T=1yr) } CF 0 = +50 - 6 - 32 - 4.71 = +$7.29 In 3 months, use $5 from the first bond to cover Dividend on short position. At Expiration, if S T ≥ 40, Bond matures, receive $40. Call is in the money; use $40 (from the bond) plus the call to buy stock. Use stock to cover short position. CF T = 0. if S T < 40, Bond matures, receive $40. Call is worthless. Use some of the $40 from the bond to buy stock and cover the short. CF T = 40 - S T > 0.

8. 8Lec 10 Discover Option Prices Thus, we have an arbitrage opportunity. Receive a CF 0 = $7.29 now. If S↑, lose nothing! If S ↓, make even more money (40 - S T ). To preclude the arbitrage C 0 E must be at least 6 + 7.29 = $13.29. (Exercise: Assume a $10 Div. in 3 months. Show that C 0 E ≥ $8.59). In sum: if S 0 = $50, C 0 E ($40, T=1yr), r = 25%, plus a dividend in 3 months No Div $5 Div$10 Div C 0 E ≥$18 $13.29 $8.59

9. 9Lec 10 Discover Option Prices American Call Price C 0 A, no Dividends ( p. 4) Do these prices make sense? S 0 = $50, C 0 A (K = $40, T=1yr) = $5 There are two ways to buy the stock: 1.Pay $50 and buy the stock immediately. Or 2.Buy the Call for $5, exercise immediately, Pay only $45 Smell Arbitrage? Buy the Call for $5, pay $40 to exercise call, sell the stock for + $50, CF 0 = -5 - 40 + 50 =+5. ➟ Arb. profit = $5 To preclude arbitrage we must have: C 0 A + 40 > 50; i.e., C 0 A > $10;

10. 10Lec 10 Discover Option Prices American Call C 0 A, Stock Pays a SMALL Dividend ( p. 5) If the stock paid a dividend, you would want to exercise in order to collect the dividend. Yes or No? Suppose: S 0 = $50, C 0 A (K = $40, T=1yr) = $11 Div = $5 (for sure) in 3 months(t*), r = 25%/yr ⇒ r for 3 months = 25%/4 = 6¼%. Must consider: 1) exercise before dividend is paid out 2) forgo dividend, wait till expiration

11. 11Lec 10 Discover Option Prices 1. Plan to exercise just before dividend is paid. Strategy: {+C, +B(FV = $40, t* = 3 months), -S}. Exercise just before dividend is paid. CF 0 = - C 0 - PV(K) + S 0 = -11 - 40/1.0625 + 50 = $1.35 > 0 ☺ At t* = 3 months, just before ex-dividend day if S 3 > $40 Call is in the money. Use $40 from bond to exercise call, receive stock, use it to cover short position before dividend is paid. CF 3 = 0. If S 3 < $40 Call out money ∴ do not exercise. Bond matures for $40; use some of it to buy stock and cover short. CF 3 =40-S 3 > 0, and you still own the call! Clearly, this is an Arb. opp. ⇒ C 0 must be > $11. C 0 A must be > S 0 - PV(K,t*) = $12.35 = 11 + 1.35.

12. 12Lec 10 Discover Option Prices 2. Create a Synthetic Stock for 1 year. Synthetic Stock={+C, +B (FV =$5, t*=3 months), +B(FV =$40, T =1yr)}, Synthetic Stock Price = { C 0 + D/(1+r/4) + 40/(1+r) } = 11 + 4.71 + 32 = $47.71 Actual Stock Price = S 0 = $50 ⇒ Arb. opp. Set up an arbitrage: {Buy Synthetic Stock, Short the actual (i.e., physical) stock} {-S, +C, +Bond(FV=5, t*=3 months), +Bond(FV=40, T=1) }. ➟ Net CF 0 = +$50 - (11 + 4.71 +32) = $2.29 Will it work?

13. 13Lec 10 Discover Option Prices In 3 months, use the 1 st bond (=$5) to cover dividend on short stock. At T = 1 yr (= Expiration) if S T > 40 Call in the money, receive $40 from 2 nd bond, use it to exercise call, receive stock, cover short. Net CF T = 0. if S T < 40 Call out of the money, throw it away. Receive $40 from bond, use some of it to buy back stock and cover short. CF T = 40 - S T > 0! To preclude Arb.C 0 A > $13.29 (=11+2.29)

14. 14Lec 10 Discover Option Prices What do we learn? Go back to original question: “Is it a good idea to exercise just to receive the dividend?” If you exercise right before the dividend payment, C 0 A = $12.35, If you DO NOT plan to exercise in 3 months, C 0 A = $13.29, It seems that the option is worth more if we forgo the dividend.

15. 15Lec 10 Discover Option Prices American Call C 0 A, Stock Pays a LARGE Dividend ( p. 6) Stock pays a $10 Dividend (for sure) in 3 months (time t*). Again, consider: 1) exercise before dividend is paid out or 2) wait till expiration 1 ) Plan to exercise just before dividend is paid. The synthetic position in the stock for 3 months consists of: {+C, +B (FV = $40, t = 3months)} = { C 0 + 40/(1.0625) } = C 0 A + $37.65 The real stock costs $50. ➟ C 0 A > $12.35 ( = 50-37.65)

16. 16Lec 10 Discover Option Prices 2) Create a Synthetic Stock for 1 year (i.e., Plan to wait until expiration). Synthetic stock for 1 yr consists of: {+C, +B(FV= $ 10, t=3months), +B(FV= $ 40, T=1)} = { C 0 + $10/(1+r/4) + 40/(1+r) } = C 0 + $41.41 ➟ C 0 A > 50 - 41.41 = $8.59 What is the math telling us? If you plan to exercise in 3 months, C 0 A = $12.35, If you plan to hold call for 1 yr, C 0 A = $8.59. Implication: If the dividend is large, then we should Exercise right before dividend is paid

17. 17Lec 10 Discover Option Prices The right to exercise at any time: How much is it worth? (p. 7) Asume S 0 = $50, K=$40, T=1 year, Dividend in 3 months, and r = 25%. No Div $5 Div$10 Div C 0 E ≥ $18 $13.29 $ 8.59 C 0 A ≥ $18 $13.29$12.35 $0 $0 $ 3.76 Right to Early exercise

18. 18Lec 10 Discover Option Prices Put Option Prices European Puts on stocks that pay NO dividends. (p. 7) Do these prices make sense? S 0 = $75, P 0 E (K = $100, T=1yr) = $4, and r = 25% There are two ways to buy a bond: A: {Buy the stock and the put} and wait until expiration or B: {Buy the bond right now}, {+S, +P} ➟ CF 0 = -75-4 = -$79 {+B(FV=$100, T=1yr)} ➟ CF 0 = -$80 Arb Strategy: {+S, +P, -B(FV=100, T=1yr)}. CF 0 = -75-4+80 = +$1

19. 19Lec 10 Discover Option Prices At Expiration, if S T ≥ 100 Put is worthless. Sell stock, use some of this cash to pay loan. CF T = S T - K > 0. S T < 100 Put is in the money, exercise it. Hand over stock; receive $100, cover loan. CF T =0 To preclude arbitrage P E 0 > $5 (=4+1). In general, P E 0 > max(0, PV(K) - S 0 )

20. 20Lec 10 Discover Option Prices European PUT prices for stocks that pay dividends (p. 7) Assume a $5 Dividend in 3 months. Do these prices make sense? S 0 = $75, $5 Div, P 0 E (K = $100, T=1yr) = $6, and r = 25% A synthetic position in a 1-year bond consists of: {+S, +P, -B(FV=$5, t=3 months) } ➟ Synthetic bond costs: $76.29 (= -6 - 75 + 5/1.0625) The actual bond costs: 100/1.25 = $80 Is this possible? ➟ There must be an arb. opp.

21. 21Lec 10 Discover Option Prices Buy cheap: synthetic bond, and Sell the expensive one, (actual or physical). Arb Portfolio S T 100CF 0 + Put -(S T -100) 0-6 + Stock + S T + S T -75 - B for Div* -5* -5*+4.71 - B for K -100 -100 +80 0 +$3.71 Arbitrage-free Price: P E 0 > 6 + 3.71 = $9.71 *in 3 months, receive $5 div, pay off first bond.

22. 22Lec 10 Discover Option Prices If the Dividend is $10 Dividend in 3 months, then P E 0 ≥ -75 + 10/1.0625 + 100/1.25 = $14.41 Summary: If S 0 = $75, K = $100, T=1 year, and r = 25% and Dividend in 3 months. Then, No Div$5 Div $10 Div P E 0 ≥ $5$9.71 $14.41 P 0 A ≥$25$25 $28.53 $20$15.29 $14.12 Right to Early exercise ▸ For an American PUT, the right to early exercise is worth quite a bit. ▸ For an American CALL if S 0 = $50, K=$40, T=1 year, r = 25%. No Div$5 Div $10 Div C 0 E ≥$18$13.29 $8.59 C 0 A ≥$18$13.29 $12.35 $0$0 $3.76 Right to Early Exercise

23. 23Lec 10 Discover Option Prices Put-Call Parity (p. 11) European Options on stocks that pay no dividends Proposition: For European Options on a stock that pays no dividends (Call and Put with same K and T), +S, +P = +C, +B(FV=K,T) And By the law of one price: +C 0 = + S 0 + P 0 - B(FV=K,T) - C 0 = - S 0 - P 0 + B(FV=K,T) +P 0 = - S 0 + C 0 + B(FV=K,T) - P 0 = + S 0 - C 0 - B(FV=K,T), etc.

24. 24Lec 10 Discover Option Prices Put-Call Parity (p. 11) European Options on stocks that pay dividends +C 0 + B(FV=K,T) + B(FV=Dividend, t*) = +S 0 + P 0 American Options on stocks with/without Dividends +C 0 + B 0 (FV=K,T) ≤ +S 0 + P 0

25. 25Lec 10 Discover Option Prices Thank You (A Favara) Thank You (A Favara)