Options pricing ( calculation of option premium)
4 likes1,521 views
The document discusses various pricing models for options including: 1. Upper and lower bounds for call option value based on the stock price. 2. Factors that determine option value such as exercise price, expiration date, stock price, stock price variability, and interest rate. 3. The binomial model and risk-neutral valuation method for pricing options using a single period binomial tree framework. 4. The Black-Scholes model for pricing European options using a continuous time framework based on the stock's lognormal distribution. 5. Assumptions of the Black-Scholes model and adjustments needed for dividends, short-term options, and long-term options.
![Factors Determining The Option Value
• EXERCISE PRICE
• EXPIRATION DATE
• STOCK PRICE
• STOCK PRICE VARIABILITY
• INTEREST RATE
C0 = f [S0 , E, 2, t , rf ]
+ - + + +](https://image.slidesharecdn.com/optionspricing-151104043804-lva1-app6891/85/Options-pricing-calculation-of-option-premium-3-320.jpg)
![Pioneer Stock
1. PROBABILITY OF RISE IN A RISK-NEUTRAL WORLD
RISE 40% TO 280
FALL 10% TO 180
EXPECTED
RETURN = [PROB OF RISE x 40%] + [(1 - PROB OF RISE) x - 10%]
= 10% p = 0.4
2. EXPECTED FUTURE VALUE OF THE OPTION
STOCK PRICE Cu = RS. 60
STOCK PRICE Cd = RS. 0
0.4 x RS. 60 + 0.6 x RS. 0 = RS. 24
3. PRESENT VALUE OF THE OPTION
RS. 24
= RS. 21.82
1.10](https://image.slidesharecdn.com/optionspricing-151104043804-lva1-app6891/85/Options-pricing-calculation-of-option-premium-8-320.jpg)
Options pricing ( calculation of option premium)
- 2. Option Value: Bounds UPPER AND LOWER BOUNDS FOR THE VALUE OF CALL OPTION VALUE OF UPPER LOWER CALL OPTION BOUND (S0) BOUND ( S0 – E) STOCK PRICE 0 E
- 3. Factors Determining The Option Value • EXERCISE PRICE • EXPIRATION DATE • STOCK PRICE • STOCK PRICE VARIABILITY • INTEREST RATE C0 = f [S0 , E, 2, t , rf ] + - + + +
- 4. Binomial Model Option Equivalent Method - 1 A SINGLE PERIOD BINOMIAL (OR 2 - STATE) MODEL • S CAN TAKE TWO POSSIBLE VALUES NEXT YEAR, uS OR dS (uS > dS) • B CAN BE BORROWED .. OR LENT AT A RATE OF r, THE RISK-FREE RATE .. (1 + r) = R • d < R > u • E IS THE EXERCISE PRICE Cu = MAX (u S - E, 0) Cd = MAX (dS - E, 0)
- 5. Binomial Model Option Equivalent Method - 1 PORTFOLIO SHARES OF THE STOCK AND B RUPEES OF BORROWING STOCK PRICE RISES : uS - RB = Cu STOCK PRICE FALLS : dS - RB = Cd Cu - Cd SPREAD OF POSSIBLE OPTION PRICE = = S (u - d) SPREAD OF POSSIBLE SHARE PRICES dCu - uCd B = (u - d) R SINCE THE PORTFOLIO (CONSISTING OF SHARES AND B DEBT) HAS THE SAME PAYOFFAS THAT OFA CALL OPTION, THE VALUE OF THE CALL OPTION IS C = S - B
- 6. Illustration S = 200, u = 1.4, d = 0.9 E = 220, r = 0.10, R = 1.10 Cu = MAX (u S - E, 0) = MAX (280 - 220, 0) = 60 Cd = MAX (dS - E, 0) = MAX (180 - 220, 0) = 0 Cu - Cd 60 = = = 0.6 (u - d) S 0.5 (200) dCu - uCd 0.9 (60) B = = = 98.18 (u - d) R 0.5 (1.10) 0.6 OF A SHARE + 98.18 BORROWING … 98.18 (1.10) = 108 REPAYT PORTFOLIO CALL OPTION WHEN u OCCURS 1.4 x 200 x 0.6 - 108 = 60 Cu = 60 WHEN d OCCURS 0.9 x 200 x 0.6 - 108 = 0 Cd = 0 C = S - B = 0.6 x 200 - 98.18 = 21.82
- 7. Binomial Model Risk-Neutral Method WE ESTABLISHED THE EQUILIBRUIM PRICE OF THE CALL OPTION WITHOUT KNOWING ANYTHING ABOUT THE ATTITUDE OF INVESTORS TOWARD RISK. THIS SUGGESTS … ALTERNATIVE METHOD … RISK-NEUTRAL VALUATION METHOD 1. CALCULATE THE PROBABILITY OF RISE IN A RISK NEUTRAL WORLD 2. CALCULATE THE EXPECTED FUTURE VALUE .. OPTION 3. CONVERT .. IT INTO ITS PRESENT VALUE USING THE RISK-FREE RATE
- 8. Pioneer Stock 1. PROBABILITY OF RISE IN A RISK-NEUTRAL WORLD RISE 40% TO 280 FALL 10% TO 180 EXPECTED RETURN = [PROB OF RISE x 40%] + [(1 - PROB OF RISE) x - 10%] = 10% p = 0.4 2. EXPECTED FUTURE VALUE OF THE OPTION STOCK PRICE Cu = RS. 60 STOCK PRICE Cd = RS. 0 0.4 x RS. 60 + 0.6 x RS. 0 = RS. 24 3. PRESENT VALUE OF THE OPTION RS. 24 = RS. 21.82 1.10
- 9. Black-Scholes Model E C0 = S0 N (d1) - N (d2) ert N (d) = VALUE OF THE CUMULATIVE NORMAL DENSITY FUNCTION S0 1 ln E + r + 2 2 t d1 = t d2 = d1 - t r = CONTINUOUSLY COMPOUNDED RISK - FREE ANNUAL INTEREST RATE = STANDARD DEVIATION OF THE CONTINUOUSLY COMPOUNDED ANNUAL RATE OF RETURN ON THE STOCK
- 10. Black-Scholes Model Illustration S0 = RS.60 E = RS.56 = 0.30 t = 0.5 r = 0.14 STEP 1 : CALCULATE d1 AND d2 S0 2 ln E + r + 2 t d1 = t .068 993 + 0.0925 = = 0.7614 0.2121 d2 = d1 - t = 0.7614 - 0.2121 = 0.5493 STEP 2 : N (d1) = N (0.7614) = 0.7768 N (d2) = N (0.5493) = 0.7086 STEP 3 : E 56 = = RS. 52.21 ert e0.14 x 0.5 STEP 4 : C0 = RS. 60 x 0.7768 - RS. 52.21 x 0.7086 = 46.61 - 37.00 = 9.61
- 11. Assumptions • THE CALL OPTION IS THE EUROPEAN OPTION • THE STOCK PRICE IS CONTINUOUS AND IS DISTRIBUTED LOGNORMALLY • THERE ARE NO TRANSACTION COSTS AND TAXES • THERE ARE NO RESTRICTIONS ON OR PENALTIES FOR SHORT SELLING • THE STOCK PAYS NO DIVIDEND • THE RISK-FREE INTEREST RATE IS KNOWN AND CONSTANT
- 12. Adjustment For Dividends Short-Term Options Divt ADJUSTED STOCK PRICE = S = S - (1 + r)t E VALUE OF CALL = S N (d1) - N (d2) ert S 2 ln E + r + 2 t d1 = t
- 13. Adjustment For Dividends – 2 Long-Term Options C = S e -yt N (d1) - E e -rt N (d2) S 2 ln E + r - y + 2 t d1 = t d2 = d1 - t y - dividend yield THE ADJUSTMENT • DISCOUNTS THE VALUE OF THE STOCK TO THE PRESENT AT THE DIVIDEND YIELD TO REFLECT THE EXPECTED DROP IN VALUE ON ACCOUNT OF THE DIVIDEND PAYMENTS • OFFSETS THE INTEREST RATE BY THE DIVIDEND YIELD TO REFLECT THE LOWER COST OF CARRYING THE STOCK
- 14. Put – Call Parity - Revisited JUST BEFORE EXPIRATION C1 = S1 + P1 - E IF THERE IS SOME TIME LEFT C0 = S0 + P0 - E e -rt THE ABOVE EQUATION CAN BE USED TO ESTABLISH THE PRICE OF A PUT OPTION & DETERMINE WHETHER THE PUT - CALL PARITY IS WORKING