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- 1. 1
- 2. Chapter 7 Option valuationMcGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc. All rights reserved.
- 3. Group MembersM.ZEESHAN ANWARMUSHTAQ HASSANRIZWAN ASHRAFSHAHID IQBAL 3
- 4. Financial Options and Their Valuation• Financial options• Valuation to expiration with one period• Binomial option pricing of a hedged volatility• Black-Scholes Option Pricing Model 4
- 5. What is a financial option?“Keep your option open is sound business advice ,andwe are out of option is sure sign of trouble”An option is an agreement/contract which gives itsholder the right, but not the obligation, to buy (or sell) anasset at some predetermined price within a specifiedperiod of time. 5
- 6. Options Contracts: Preliminaries• Calls versus Puts – Call options gives the holder the right, but not the obligation, to buy a given quantity of some asset at some time in the future, at prices agreed upon today. When exercising a call option, you “call in” the asset. – Put options gives the holder the right, but not the obligation, to sell a given quantity of an asset at some time in the future, at prices agreed upon today. When exercising a put, you “put” the asset to someone. 6
- 7. Options Contracts: Preliminaries• Exercising the Option – The act of buying or selling the underlying asset through the option contract.• Strike Price or Exercise Price – Refers to the fixed price in the option contract at which the holder can buy or sell the underlying asset.• Expiry – The maturity date of the option is referred to as the expiration date, or the expiry.• European versus American options – European options can be exercised only at expiry. – American options can be exercised at any time up to expiry. 7
- 8. Options Contracts: Preliminaries• In-the-Money – The exercise price is less than the spot price of the underlying asset.• At-the-Money – The exercise price is equal to the spot price of the underlying asset.• Out-of-the-Money – The exercise price is more than the spot price of the underlying asset. 8
- 9. Options Contracts: Preliminaries• Intrinsic Value – The difference between the exercise price of the option and the spot price of the underlying asset.• Speculative Value – The difference between the option premium and the intrinsic value of the option. OPTION PREMIUM= INTRINSIC VALUE+SPECULATIVE VALUE 9
- 10. Call Options• Call options gives the holder the right, but not the obligation, to buy a given quantity of some asset on or before some time in the future, at prices agreed upon today.• When exercising a call option, you “call in” the asset. 10
- 11. Basic Call Option Pricing Relationships at Expiry• At expiry, an American call option is worth the same as a European option with the same characteristics. – If the call is in-the-money, it is worth ST – E. – If the call is out-of-the-money, it is worthless: Vo= Max[ST – E, 0]Where ST is the value of the stock at expiry (time T) E is the exercise price. C is the value of the call option at expiry 11
- 12. 22-12 CALL OPTION PAYOFFS 60 Option payoffs ($) 40 20 20 40 60 80 100 120 50 Stock price ($) –20 –40 Exercise price = $50 12
- 13. 22-13 CALL OPTION PAYOFFS 60 Option payoffs ($) 40 20 20 40 60 80 100 120 50 Stock price ($) –20 Exercise price = $50 –40 13
- 14. 22-14 CALL OPTION PROFITS 60 Option payoffs ($) 40 Buy a call 20 10 20 40 50 60 80 100 120 –10 Stock price ($) –20 Exercise price = $50; –40 Sell a call option premium = $10 14
- 15. “Stock options are Zero sum Game” • Example: You sell 50 option contracts. You receive $16250 up front, with strike price $20,you will be $16250 ahead. • You will have to sell something for less than its worth, so will lose the difference. • If the stock price is $25 you will have to sell 50x100=5000 shares at $20 per share, so you will be out $25-20=$5 per share, or $25000 total and net loss is $8750. 15
- 16. Exercise price = $20.Ending stock Net profit to option Price seller $15 $16250 17 16250 20 16250 23 -1250 25 -8750 30 -33750 16
- 17. Call Premium DiagramOptionvalue 30 25 20 15 10 Market price 5 Exercise value 5 10 15 20 25 30 35 40 45 50 Stock Price 17
- 18. Notations for option valuationS1 = Stock price at expiration(In one period)S0 = Stock price todayC1 = Value of the call option on the expiration dateC0 = Value of the call option todayE = Exercise price on the option 18
- 19. Case 1 : If the strike price (S1) ends up below the exercise price (E) on the expiration date, then the call option (C1) is worth zero . In other words: C1= 0 if S1 ≤ EOr equivalently: C1= 0 if S1-E ≤ 0Case 2 : If the option finishes in the money then S1 › E ,and the value of the option at expiration is equal to the difference: C1= S1-E if S1 › E C1= S1-E if S1 › 0 19
- 20. OPTION VALUATION WITH ONE PERIOD• We assume a European option with unknown value of stock at expiration date. We assume that we are able to formulate probabilistic belief about its value one period hence. The 450 line represents the theoretical value of the option. It simply the current stock price less the exercise price of the option. When the price of the stock is less than the exercise price of the option, the option has a zero theoretical value; when more, it has a theoretical value on the line. 20
- 21. MARKET VERSUS THEORETICAL VALUE• Suppose the current market price of ABC Corporation`s stock is $10, which is equal to the exercise price. Theoretically, the option has no value; however, if there is some probability that the price of the stock will exceed $10 before expiration. Suppose further the that the option has 30 days to expiration and that there is .3 probability that the stock will have a market price of $5 per share at the end of 30 days, .4 that it will be $10, and .3 that it will be $15. The expected value of the option at the end of 30 days is thus• 0(.3) +0(.4) + ($15-$10) (.3) =$1.50 21
- 22. Chapter7.2 BINOMIAL OPTION PRICING OF HEDGED RATIO By MUSHTAQ HassanMcGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc. All rights reserved.
- 23. Hedged Position• Tow related financial assets – Stock – Option on that Stock• In this way prices of one financial assets off set by opposite price movements.• To maintain the risk free position 23
- 24. • Return on option and stock, opportunity cost is important to maintain hedged position.• The opportunity cost is equal to risk free rate of return to establishing hedged position. 24
- 25. Binomial OptionMaps probabilities as a branching process. 25
- 26. Problem for solutionCurrent value = 50Probability of Occurrence 2/3 for increase by 20% 1/3 for decrease by 10%Calculate (a) Stock Value at the End of Period (b) Expected Value of Stock Value at the End of Period (c) Option Value at the End of Period (d) Expected Value of Option Value at the End of Period 26
- 27. uVs = One value higher than current valuedVs = One value lower than current valueVs = Current valueu = One plus percentage increase in value from beginning to endd = One minus percentage decrease in value from beginning to endq = Probability of upward movement of stock1 – q = Probability of downward movement of stock 27
- 28. Delta Option• A hedged position ascertained by long position and short position.• This is also called Hedged Ratio of Stock to Options. 28
- 29. Delta Option = Spread of possible option prices Spread of possible stock pricesWhereSpread of possible option prices means uVo – dVoSpread of possible stock prices means uVs – dVs 29
- 30. Stock Prices at Value of long Value of Short Value of end of Period Position in Stock Position in Combined (Out flow) Option (Inflow) Hedged Position 60 2(60) = 120 -3(10) = -30 120 – 3 = 90 45 2(45) = 90 -3(0) = 0 90 – 0 = 90 30
- 31. Determination of Option Value at Beginning Period Equation to solve for Vo B [Long position – Short Position(Vo B)]1.05=Value of Hedged Position 31
- 32. Chapter Black scholes option pricing model7.3 By Rizwan AshrafMcGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc. All rights reserved.
- 33. OBJECTIVEOur main objective is to find the current price of a derivative. • Derivatives are securities that do not convey ownership, but rather a promise to convey ownership. 33
- 34. The concepts behind black-scholes• The option price and the stock price depend on the same underlying source of uncertainty• We can form a portfolio consisting of the stock and the option which eliminates this source of uncertainty• The portfolio is instantaneously riskless and must instantaneously earn the risk-free rate• This leads to the Black-Scholes differential equation 34
- 35. BSOPM• The Black-Scholes OPM: rt C S N (d1 ) Ke N (d 2 ) 2 ln ( S / K ) R ( / 2) t d1 t d2 d1 t 35
- 36. Black-Scholes Option Pricing Model (cont’d)• Variable definitions: ► C = theoretical call premium ► S = current stock price ► t = time in years until option expiration ► K = option striking price ► R = risk-free interest rate ► = standard deviation of stock returns ► N(x) = cumulative standard normal distribution ► functions ► ln = natural logarithm ► e = base of natural logarithm (2.7183) 36
- 37. Assumptions of the Model The stock pays no dividends during the option’s life European exercise terms Markets are efficient No commissions Constant interest rates Lognormal returns 37
- 38. The Stock Pays no Dividends During the Option’s Life• The OPM assumes that the underlying security pays no dividends• If you apply the BSOPM to two securities, one with no dividends and the other with a dividend yield, the model will predict the same call premium• Valuing securities with different dividend yields using the OPM will result in the same price 38
- 39. European Exercise Terms• The OPM assumes that the option is European• Not a major consideration since very few calls are ever exercised prior to expiration 39
- 40. Markets Are Efficient• The OPM assumes markets are informational efficient – People cannot predict the direction of the market or of an individual stock 40
- 41. No Commissions• The OPM assumes market participants do not have to pay any commissions to buy or sell• Commissions paid by individual can significantly affect the true cost of an option – Trading fee differentials cause slightly different effective option prices for different market participants 41
- 42. Constant Interest Rates• The OPM assumes that the interest rate R in the model is known and constant• It is common use to use the discount rate on a U.S. Treasury bill that has a maturity approximately equal to the remaining life of the option – This interest rate can change 42
- 43. Lognormal Returns• The OPM assumes that the logarithms of returns of the underlying security are normally distributed• A reasonable assumption for most assets on which options are available 43
- 44. Black-Scholes Option Pricing Model ExampleStock ABC currently trades for $30. A call option onABC stock has a striking price of $25 and expires inthree months. The current risk-free rate is 5%, and ABCstock has a standard deviation of 0.45.According to the Black-Scholes OPM, but should be thecall option premium for this option? 44
- 45. • S = CURRENT STOCK PRICE = $30• K = STRIKE PRICE = $25• t = time = 3 month• R =5%=0.05• =standard deviation=0.45 45
- 46. Black-Scholes Option Pricing Model (cont’d) Example (cont’d)Solution: We must first determine d1 and d2: 2 ln( S / K ) R ( / 2) t d1 t 2 ln(30 / 25) 0.05 (0.45 / 2) 0.25 0.45 0.25 0.1823 0.0378 0.978 0.225 46
- 47. Black-Scholes Option Pricing Model (cont’d) Example (cont’d)Solution (cont’d): d2 d1 t 0.978 (0.45) 0.25 0.978 0.225 0.753 47
- 48. Black-Scholes Option Pricing Model (cont’d) Example (cont’d)Solution (cont’d): The next step is to find the normalprobability values for d1 and d2. Using Excel’sNORMSDIST function yields: N (d1 ) 0 .8 3 6 N (d 2 ) 0 .7 7 4 48
- 49. Using Excel’s NORMSDIST Function• The Excel portion below shows the input and the result of the function: 49
- 50. Black-Scholes Option Pricing Model (cont’d) Example (cont’d)Solution (cont’d): The final step is to calculate theoption premium: rt C S N (d1 ) Ke N (d 2 ) ( 0.05 )( 0.25 ) $30 0.836 $25 e 0.774 $25.08 $19.11 $5.97 50
- 51. Insights Into the Black-Scholes Model• Divide the OPM into two parts: rt C S N (d1 ) Ke N (d 2 ) Part A Part B 51
- 52. Insights Into the Black-Scholes Model (cont’d)• Part A is the expected benefit from acquiring the stock: – S is the current stock price and the discounted value of the expected stock price at any future point – N(d1) is a pseudo-probability • It is the probability of the option being in the money at expiration, adjusted for the depth the option is in the money 52
- 53. Insights Into the Black-Scholes Model (cont’d)• Part B is the present value of the exercise price on the expiration day: – N(d2) is the actual probability the option will be in the money on expiration day 53
- 54. Insights Into the Black-Scholes Model (cont’d)• The value of a call option is the difference between the expected benefit from acquiring the stock and paying the exercise price on expiration day 54
- 55. Fischer Black Born: 1938 Died: 1995 1959 -- earned bachelors degree in physics 1964 -- earned PhD. from Harvard in applied math 1971 -- joined faculty of University of Chicago Graduate School of Business 1973 -- Published "The Pricing of Options and Corporate Liabilities“ 19XX -- Left the University of Chicago to teach at MIT 1984 -- left MIT to work for Goldman Sachs & Co. Myron Scholes Born: 1941 1973 -- Published "The Pricing of Options and Corporate Liabilities“ Currently works in the derivatives trading group at Salomon Brothers. 55
- 56. Chapter Other measures or7.4 parameters of sensitivity By SHAHID IQBALMcGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc. All rights reserved.
- 57. Other parameters measuring the risk • Gamma ┌ • Theta θ • Rho p • Vega 57
- 58. Option Gamma• The gamma of an option indicates how the delta of an option will change relative to a 1 point move in the underlying asset.• The Gamma shows the option deltas sensitivity to market price changes. 58
- 59. 59
- 60. Other parameters measuring the risk • Theta: measure of option price sensitiveness to a change in time to expiration. • Rho:measure of option price sensitiveness to a change in the interest rate • Vega: of an option indicates how much, theoretically at least, the price of the option will change as the volatility of the underlying asset changes. 60
- 61. Parameters measuring the risk• Gamma(stock price, strike price)• Theta(time until to expiration)• Rho (risk free rate)• Vega(volitality) 61
- 62. volatility• volatility is a measure for variation of price of a financial instrument over time.• Volatility can be measure by using the standard deviation or variance. Commonly the higher the volatility the riskier the security. 62
- 63. Implied volatility• Implied volatility tells a trader what level of volatility to expect from the asset given the current share price and current option price. 63
- 64. Debt & Other Options• Debt option may be on the actual debt instrument or on an interest- rate future contract.• Debt option provides a means for protection against adverse- rate movements. 64
- 65. Foreign currency options• Fx options(foreign-exchange option)• is written on the number of units of a foreign currency that a U.S dollar will buy. 65
- 66. Thanks for your listening!! 66

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