OptionsIQ

1. OptionsIQ Montgomery Investment Technology, Inc. Financial Modeling Software and Consulting

2. BASIC TOPICS: Option Contracts Defined Intrinsic and Time Value Type of Options Theoretical or Fair Value Volatility The Black-Scholes Option Pricing Model The Binomial Method of Pricing Options Risk Sensitivities Delta Hedging Trading Strategies Involving Options Exotic Options ADVANCED TOPICS: The Markov Process and the Efficient Market Hurst Exponent to Measure Trend in the Time Series Normality of the Return Rates Log-Normality of the Stock Prices Autocorrelation of the Return Rates Ito and Stratonovich Interpretations The Risk-Free Interest Rate vs. Return Rate Contents:

3. Option Contracts Defined • An option is a derivative security which gives the holder the right, but not the obligation, to buy or sell an underlying asset by a certain date for a certain price. • A call option gives the holder the right to buy, while a put option gives the right to sell the underlying asset. The price designated in the contract is known as the exercise or strike price. The time to expiration is calculated based on the time from the value date to the expiration date. • American-style options can be exercised at anytime up to expiration, while European-style options can only be exercised at expiration.

4. Option Contracts Defined Question 1: An option gives the holder... A. the obligation to buy the underlying asset. B. the right to sell the underlying asset. C. the obligation to buy or sell the underlying asset. D. the right to buy or sell the underlying asset.

5. Option Contracts Defined Question 1: Answer D An option gives the holder... A. the obligation to buy the underlying asset. B. the right to sell the underlying asset. C. the obligation to buy or sell the underlying asset. D. the right to buy or sell the underlying asset.

6. Option Contracts Defined Question 2: What is the difference between an American option and a European option? A. An American option is traded on American exchanges, while European options are traded on European exchanges. B. An American option is written on an the assets of an American company, while a European option is written on the assets of a European company. C. An American option can only be exercised at expiration, while a European option can be exercised at anytime up to expiration. D. An American option can be exercised at anytime up to expiration, while a European option can be exercised only at expiration.

7. Option Contracts Defined Question 2: Answer D What is the difference between an American option and a European option? A. An American option is traded on American exchanges, while European options are traded on European exchanges. B. An American option is written on an the assets of an American company, while a European option is written on the assets of a European company. C. An American option can only be exercised at expiration, while a European option can be exercised at anytime up to expiration. D. An American option can be exercised at anytime up to expiration, while a European option can be exercised only at expiration.

8. Intrinsic and Time Value • The premium, or price of an option, can be divided into two components, intrinsic and time value. Intrinsic value is the payoff if the option were to be exercised immediately. Intrinsic value is always greater than or equal to zero. For example, a certain asset is trading for $30. The intrinsic value of a $25 call is therefore $5. • Usually the price of an option in the marketplace will be greater than its intrinsic value. The difference between the market value of an option and its intrinsic value is called the time value (or extrinsic value) of an option. An option is trading at parity when the price of the option is equal to its intrinsic value (the time value is zero).

9. Intrinsic and Time Value • An option which has a positive intrinsic value is considered to be in-the-money by the amount of the intrinsic value. If a stock is trading at $50, a $40 call is $10 in-the-money. A $50 call on the same stock would be considered to be at-the-money, and a $55 call would be considered to be out-of-the-money.

10. Intrinsic and Time Value Question 1: What is the intrinsic value of a $30 put if the underlying asset is trading at $23? A. zero B. $7 C. -$7 D. There is not enough information to determine the intrinsic value.

11. Intrinsic and Time Value Question 1: Answer B What is the intrinsic value of a $30 put if the underlying asset is trading at $23? A. zero B. $7 C. -$7 D. There is not enough information to determine the intrinsic value.

12. Intrinsic and Time Value Question 2: The underlying asset of a $55 call is trading at $48. This call would be considered... A. In-the-money. B. At-the-money. C. Out-of-the-money.

13. Intrinsic and Time Value Question 2: Answer C The underlying asset of a $55 call is trading at $48. This call would be considered... A. In-the-money. B. At-the-money. C. Out-of-the-money.

14. Intrinsic and Time Value Question 3: A certain underlying asset is trading at $50. A $42 call is trading in the marketplace for $10. What is the time value of the option? A. zero B. $8 C. $2 D. $10

15. Intrinsic and Time Value Question 3: Answer C A certain underlying asset is trading at $50. A $42 call is trading in the marketplace for $10. What is the time value of the option? A. zero B. $8 C. $2 D. $10

16. Types of Options • A person who has purchased an option is considered to be long the option (owner). A person who has sold the option is considered short the option (seller). However, these terms (long and short) are also used to refer to a position in the market. A person who thinks that the market will rise (or is bullish about the market) will make a long market position (buy a call or sell a put), while a person who thinks that the market will decline (or is bearish about the market) will make a short market position (sell a call or buy a put).

17. Types of Options • The owner of a call (put) benefits from price increases (decreases) in the stock with limited downside risk. The most that can be lost is the price or premium of the option. However, the seller of a call (put) benefits from price decreases (increases) with unlimited loss potential and limited gains. The most that can be gained is the premium of the option.

18. Types of Options Question 1: An investor is bullish about the market for a particular underlying asset. Which of the following strategies might this investor pursue? A. Long a put. B. Short a put. C. Short a call. D. None of the above.

19. Types of Options Question 1: Answer B An investor is bullish about the market for a particular underlying asset. Which of the following strategies might this investor pursue? A. Long a put. B. Short a put. C. Short a call. D. None of the above.

20. Types of Options Question 2: A trader is bearish about the market for a certain underlying asset. Which of the following strategies might this trader pursue? A. Long a put. B. Short a put. C. Long a call. D. Short a call. E. A & D.

21. Types of Options Question 2: Answer E A trader is bearish about the market for a certain underlying asset. Which of the following strategies might this trader pursue? A. Long a put. B. Short a put. C. Long a call. D. Short a call. E. A & D.

22. Theoretical or Fair Value • The theoretical or fair value of an option is the price one would expect to pay in order to just break even in the long run. Theoretical value can be thought of as the "production cost" of the option. • The cost of purchasing an option in the marketplace is called the option premium. This amount is often different from the theoretical value. • There are several characteristics involved in the pricing of an option. They include: underlying price, exercise price, amount of time remaining until expiration, the volatility of the underlying asset, the risk-free rate of interest over the life of the option, and the dividend yield rate of the asset. • There are several models available to price options using these characteristics. The two most commonly used models are the Black-Scholes and the Binomial models.

23. Theoretical or Fair Value Question 1: What is the theoretical value of an option? A. An option value generated by a mathematical model given certain assumptions B. The price one would expect to pay for an option to just break even in the long run. C. The option value determined by inputting values into the Black-Scholes model. D. All of the above.

24. Theoretical or Fair Value Question 1: Answer D What is the theoretical value of an option? A. An option value generated by a mathematical model given certain assumptions B. The price one would expect to pay for an option to just break even in the long run. C. The option value determined by inputting values into the Black-Scholes model. D. All of the above.

25. Volatility • Volatility is one of the key inputs to an option pricing model. Volatility is the degree to which the price of an underlying asset tends to fluctuate over time. More generally, it is a measure of how uncertain we are about future stock price movements. If an underlying asset has a small volatility or price variability, then an option on that asset would not have much value to the holder.

26. Volatility • There are several different types of volatility. Future or projected volatility is based on the expected future distribution of prices for a particular underlying asset. Implied volatility is calculated based on the option price traded in the marketplace. It is the volatility which would have to be input into a theoretical pricing model in order to yield a theoretical value equal to the market value of the option. Historical volatility is calculated based on a range of historical prices. At lease 20 observations are usually desirable to ensure statistical significance. Seasonal volatility comes into affect with certain commodities, for example as a consequence of changes in weather conditions or demand.

27. Volatility • n order to estimate the volatility of an underlying asset using historical data (e.g. daily, weekly, monthly), the following formula can be used: • Definitions: n : number of observations S(i) : stock price at the end of the ith interval (i=0,1,2,...,n) T : length of time interval in years s* : the standard deviation of s (volatility) u(i)= ln (S(i)/S(i-1) for i= 0,1,2,...,n n n s = [ {1/(n-1)} E [ u(i)]^2 - {1/n(n-1)} {E u(i) }^2 ]^1/2 i=1 i=1 s*= s/(T^1/2)

28. Volatility Question 1: What is volatility? A. It is the degree to which an asset price fluctuates over time. B. It is a measure of speed of the market. C. It is a measure of uncertainty of future stock price distributions. D. All of the above.

29. Volatility Question 1: Answer D What is volatility? A. It is the degree to which an asset price fluctuates over time. B. It is a measure of speed of the market. C. It is a measure of uncertainty of future stock price distributions. D. All of the above.

30. Volatility Question 2: Given E u(i) = 0.09531 and E [u(i)]^2 =0.00333 for a twenty trading day period, assume that there are 252 trading days per year. What is the approximate volatility per year? A. 10% B. 20% C. 30% D. 40%

31. Volatility Question 2: Answer B Given E u(i) = 0.09531 and E [u(i)]^2 =0.00333 for a twenty trading day period, assume that there are 252 trading days per year. What is the approximate volatility per year? A. 10% B. 20% C. 30% D. 40%

32. The Black-Scholes Option Pricing Model • The Black-Scholes model is one of the most basic pricing models used. It is designed for use with European options on non-dividend paying stocks. The following is the mathematical equation for the Black-Scholes model. C = theoretical value of a call P = theoretical value of a put U = price of the underlying asset E = exercise price t = time to expiration in years v = annual volatility expressed as a decimal fraction r = risk-free interest expressed as a decimal fraction e = base of the natural logarithm ln = natural logarithm N'(x) = the normal distribution curve N(x) = the cumulative normal density function

33. The Black-Scholes Option Pricing Model C = UN(h) - Ee^(-rt) N (h-v(t^1/2)) N(x) = 1 - N'(x) (a1*k + a2*k^2 + a3*k^3) x>0 1 - N (-x) x<0 P = -UN(-h) + Ee(-rt) N (v(t^1/2) -h) k = 1 / (1 + yx) h = [ ln(U/E) + (r + (v^2)/2) t] y = 0.33267 [ v(t^1/2)] a1 = 0.4361836 a2 = -0.1201676 N'(x) = [e^(-(x^2)/2)] / (2 Pi)^1/2 a3 = 0.9372980

34. The Black-Scholes Option Pricing Model Question 1: What is the approximate theoretical value (using the Black-Scholes pricing model) of a call if U=50, E=45,t=1, v=30%, and r=5%? A. $10. B. $12 C. $15 D. None of the above.

35. The Black-Scholes Option Pricing Model Question 1: Answer A What is the approximate theoretical value (using the Black-Scholes pricing model) of a call if U=50, E=45,t=1, v=30%, and r=5%? A. $10. B. $12 C. $15 D. None of the above.

36. The Black-Scholes Option Pricing Model Question 2: What is the approximate theoretical value (using the Black-Scholes pricing model) of a call if U=80, E=65,t=.5, v=25%, and r=6%? A. $15. B. $17 C. $20 D. $25

37. The Black-Scholes Option Pricing Model Question 2: Answer B What is the approximate theoretical value (using the Black-Scholes pricing model) of a call if U=80, E=65,t=.5, v=25%, and r=6%? A. $15. B. $17 C. $20 D. $25

38. The Binomial Method of Pricing Options • The Binomial method, detailed in a 1979 journal article by Cox, Ross and Rubinstein, can be used to accurately value American-style options. The "open architecture" allow flexibility in relaxing some of the constraints that are imposed by the Black-Scholes model. The binomial method is used extensively to price both standard and non-standard option contracts. The degree of accuracy can be specified by selecting a desired number of nodes or "iterations". Discrete cash flows or dividend yields may be incorporated into the option calculation when using the binomial or lattice approach.

39. The Binomial Method of Pricing Options • In order to provide the binomial tree which will approximate the lognormal distribution, the following is defined: • u = e^{v(t/n)^1/2} d = 1/u where: n = the number of periods to expiration (number of branches of the binomial tree) v = the annual volatility of the underlying asset t = the time to expiration in years j = the underlying price (from 0 - n) i = the period (from 0 - n) rr = the risk free rate of interest over the life of the option defined as rr = 1 + (rt / n) p = the probability defined as p = (rr-d) / (u-d) E = the exercise price U = the underlying asset price

40. The Binomial Method of Pricing Options • C(i , j) = max [ pC (i + 1, j) + (1 - p) C (i + 1, j + 1) ] [ rr ' U (i , j) – E ] • P(i , j) = max [ pP (i + 1, j) + (1 - p) P (i + 1, j + 1) ] [ rr ' E - U (i , j) ]

41. The Binomial Method of Pricing Options Question 1: The binomial method is often used to calculate option values when: A. the exercise style is American. B. discrete cash flows are generated by the underlying asset. C. speed of recalculation is not the overriding factor. D. all of the above.

42. The Binomial Method of Pricing Options Question 1: Answer D The binomial method is often used to calculate option values when: A. the exercise style is American. B. discrete cash flows are generated by the underlying asset. C. speed of recalculation is not the overriding factor. D. all of the above.

43. The Binomial Method of Pricing Options Question 2: What is the approximate theoretical value of a put option with the following properties: U=85, E=82, t=1,v=35%, n=6 and ri=7%? A. $8 B. $12 C. $16 D. $20

44. The Binomial Method of Pricing Options Question 2: Answer A What is the approximate theoretical value of a put option with the following properties: U=85, E=82, t=1,v=35%, n=6 and ri=7%? A. $8 B. $12 C. $16 D. $20

45. Risk Sensitivities (Greeks) • Delta (also known as the hedge ratio) is the sensitivity of an option's theoretical value to a change in the price of the underlying contract. Calls have deltas ranging from zero to one hundred. Puts have deltas ranging from zero to negative one hundred. An underlying contract always has a delta of one hundred delta = change in the option price change in the stock price

46. Risk Sensitivities (Greeks) • The Gamma of a portfolio of derivatives on an underlying asset is the rate of change of the portfolio's delta with respect to the price of the underlying asset. If gamma is large, delta is highly sensitive to the price of the underlying asset. gamma = change in the value of the portfolio * change in time change in stock price • The Theta of a portfolio of derivatives is the rate of change of the value of the portfolio with respect to time with all else remaining the same. It is sometimes referred to as the time decay of the portfolio. theta = change in the value of the portfolio change in time

47. Risk Sensitivities (Greeks) • The Vega of a portfolio of derivatives is the rate of change of the value of the portfolio with respect to the volatility of the underlying asset. Vega may also be known as lambda, kappa, or sigma. vega = change in the value of the portfolio change in the volatility of the underlying asset • The Rho of a portfolio of derivatives is the rate of change of the value of the porfolio to the interest rate. It is a measure of the sensitivity of the portfolio's value to interest rates. rho = change in the value of the porfolio change in interest rates

48. Risk Sensitivities (Greeks) Question 1: Delta is a measure of: A. time decay of the portfolio. B. the sensitivity of the portfolio's value to interest rates. C. the sensitivity of an option's value to a change in the price of the underlying asset. D. none of the above..

49. Risk Sensitivities (Greeks) Question 1: Answer C Delta is a measure of: A. time decay of the portfolio. B. the sensitivity of the portfolio's value to interest rates. C. the sensitivity of an option's value to a change in the price of the underlying asset. D. none of the above..

50. Risk Sensitivities (Greeks) Question 2: Rho is a measure of: A. time decay of the portfolio. B. the sensitivity of the portfolio's value to interest rates. C. the sensitivity of an option's value to a change in the price of the underlying asset. D. the rate of change of the value of the portfolio with respect to the volatility.