1. Option Pricing.

2. Stochastic Process A variable whose value changes over time in an uncertain way is said to follow a stochastic process. Stochastic processes can be “discrete time” or “continuous time” and also “discrete variable” or “continuous variable”

3. Markov Process It is a particular type of stochastic process where only the present value of a variable is relevant for predicting the future. The past history of the variable and the way in which the present value has emerged from the past are irrelevant. It is consistence with the weak form of market efficiency and means that while statistical properties of the stock prices may be useful in determining the characteristics of the stochastic process followed by the stock price but the particular path followed in the past is irrelevant.

4. Wiener Process It is a particular type of Markov Stochastic Process and has been used in physics to describe the motion of a particular subjected to a large number of small molecular shocks and is sometimes referred to as Brownian Motion. It has two properties; 1.Small change is equal to root of change in time multiplied by a random variable following a standardized normal distribution. 2.For two different short intervals, the small changes are independent.

5. Lognormal Distribution It is the variable whose logarithm values are normally distributed. We need to convert lognormal distributions (stochastic stock price changes) to normal distributions so that one could undertake analysis using confidence limits, hypothesis testing etc.

6. Option Pricing Models DCF criterion cannot be used since risk of an option is virtually indeterminate and hence the discount rate is impossible to be estimated. The two popular models are:  The Binomial Model  Black – Scholes Model

7. The Binomial Model The model assumes, The price of asset can only go up or go down in fixed amounts in discrete time. The price of asset can only go up or go down in fixed amounts in discrete time. There is no arbitrage between the option and the replicating portfolio composed of underlying asset and risk-less asset. There is no arbitrage between the option and the replicating portfolio composed of underlying asset and risk-less asset.

8. The Binomial Model Current stock price = S Current stock price = S Next Year values = uS or dS Next Year values = uS or dS B amount can be borrowed at ‘r’. Interest factor is (1+r) = R B amount can be borrowed at ‘r’. Interest factor is (1+r) = R d < R < u (no risk free arbitrage possible) d < R < u (no risk free arbitrage possible) E is the exercise price E is the exercise price

9. The Binomial Model Depending on the change in stock value, option value will be Cu = Max (uS – E, 0) Cd = Max (dS – E, 0)

10. The Binomial Model. S Su Sd Su 2 Sud Sd 2

11. The Binomial Model We now set a portfolio of ∆ shares and B amount of debt such that its payoff is equal to that of call option after 1 year. Then, Cu = ∆uS + RB……………(1) Cd = ∆dS + RB……………. (2) Solving these equations, (Cu – Cd) (Cu – Cd) ∆ = ; and ∆ = ; and S(u-d) S(u-d) (uCd – dCu) (uCd – dCu) B = B = (u – d)R (u – d)R Hence C = ∆S + B, since portfolio has same payoff as call option.

12. Illustration A stock is currently selling for Rs.40. The call option on the stock exercisable a year from now at a strike price of Rs.45 is currently selling at Rs.8. The risk-free rate is 10%. The stock can either rise or fall after a year. It can fall by 20%. By what percentage can it rise?

13. Black-Scholes Model as the Limit of the Binomial Model The Binomial Model converges to the Black- Scholes model as the number of time periods increases.

14. Black-Scholes Model: The origin 1820s – Scottish scientist Robert Brown observed motion of suspended particles in water. 1820s – Scottish scientist Robert Brown observed motion of suspended particles in water. Early 19 th century – Albert Einstein used Brownian motion to explain movements of molecules, many research papers. Early 19 th century – Albert Einstein used Brownian motion to explain movements of molecules, many research papers. 1900 – French scholar, Louis Bachelier wrote dissertation on option pricing and developed a model strikingly similar to BSM. 1900 – French scholar, Louis Bachelier wrote dissertation on option pricing and developed a model strikingly similar to BSM. 1951 – Japanese mathematician Kiyoshi Ito developed Ito’s Lemma that was used in option pricing. 1951 – Japanese mathematician Kiyoshi Ito developed Ito’s Lemma that was used in option pricing.

15. Black-Scholes Model: The origin Fischer Black and Myron Scholes worked in Finance Faculty at MIT Published paper in 1973. They were later joined by Robert Merton. Fischer Black and Myron Scholes worked in Finance Faculty at MIT Published paper in 1973. They were later joined by Robert Merton. Fischer left academia in 1983, died in 1995 at 57. Fischer left academia in 1983, died in 1995 at 57. 1997 – Scholes and Merton got Nobel Prize 1997 – Scholes and Merton got Nobel Prize

16. Black-Scholes Model Fischer Black and Myron Scholes, The Journal of Political Economy, 1973 Assumptions: The underlying stock pays no dividends. The underlying stock pays no dividends. It is a European option. It is a European option. The stock price is continuous and is distributed lognormally. The stock price is continuous and is distributed lognormally. There are no transaction costs and taxes. There are no transaction costs and taxes. No restrictions or penalty on short selling No restrictions or penalty on short selling The risk free rate is known and is constant over the life of the option. The risk free rate is known and is constant over the life of the option.

17. Black-Scholes Model C 0 = S 0 N (d 1 ) – E/e rt N (d 2 ) where, C 0 = Present equilibrium value of call option S 0 = Current stock price E = Exercise price e = Base of natural logarithm r = Continuously compounded risk free interest rate t = length of time in years to expiration N (*) = Cumulative probability distribution function of a standardized normal distribution standardized normal distribution

18. Black-Scholes Model C = S N (d 1 ) – Ke -rt N (d 2 ) where, C = Present equilibrium value of call option S = Current stock price K = Exercise price e = Base of natural logarithm r = Continuously compounded risk free interest rate t = length of time in years to expiration N (*) = Cumulative probability distribution function of a standardized normal distribution standardized normal distribution

19. Black-Scholes Model l n (S 0 /E) + (r + ½ σ 2 )t d 1 = σ √t l n (S 0 /E) + (r - ½ σ 2 )t d 2 = σ √t where l n is the natural logarithm

20. Black-Scholes Model l n (S/K e -rt ) d 1 = + 0.5 σ √t σ √t d 2 = d 1 - σ √t where l n is the natural logarithm

21. Illustration The standard deviation of the continuously compounded stock price change for a company is estimated to be 20% per year. The stock currently sells for Rs.80 and the effective annual interest rate is Rs.15.03%. What is the value of a one year call option on the stock of the company if the exercise price is Rs.82?

22. The Linkage between Calls, Puts, Stock, and Risk-Free Bonds. Call Stock Put Risk-Free Bond Black-Scholes Call Option Pricing Model Put-Call Parity Black-Scholes Put Option Pricing Model

23. Put-Call Parity Theorem Payoffs just before expiration Payoffs just before expiration If S 1 E If S 1 E 1. Buy the equity stock S 1 S 1 2. Buy a put optionE-S 1 0 3. Borrow amount equal to exercise price- E- E 1+2+3=Buy a call option 0 S 1 - E

24. Using Black-Scholes Model 1. Find the Standard Deviation of the continuously compounded asset value change and the square root of the time left to expiration 2. Calculate ratio of the current asset value to the present value of the exercise price 3. Consult the table giving %age relationship between the value of the Call Option and the stock price corresponding to the value in steps 1 and 2 4. Value of Put Option = Value of Call Option + PV of exercise price – Stock Price

25. Illustration Find the value of a one year call option as well as a put option, if the current stock price is Rs.120, exercise price is Rs.125 and the S.D. of continuously compounded price change of the stock is 30%. The effective interest rate is 15.03% so that the interest factor is 1.1503.

26. Illustration Step 1: Standard Deviation × √Time = 0.30 × √1 = 0.30 = 0.30 Step 2: The ratio of stock price to the PV of exercise price = 120 ÷ 125/1.1503 exercise price = 120 ÷ 125/1.1503 = 120/108.7 = 1.10 = 120/108.7 = 1.10 Step 3: Consulting the table we get 16.5% of the stock price as the value of call option i.e. stock price as the value of call option i.e. 120×0.165 = 19.8 120×0.165 = 19.8 Step 4: Value of Put Option = 19.8 + 108.7 – 120 = 8.5 = 19.8 + 108.7 – 120 = 8.5

29. Variants of Black-Scholes Model To price European options on dividend paying options and American options on non-dividend paying stocks (Robert Merton, 1973 and Clifford Smith, 1976). To price European options on dividend paying options and American options on non-dividend paying stocks (Robert Merton, 1973 and Clifford Smith, 1976). American call options on dividend-paying stocks (Richard Roll, 1977; Robert Whaley, 1981; and Richard Geske and Richard Roll, 1984) American call options on dividend-paying stocks (Richard Roll, 1977; Robert Whaley, 1981; and Richard Geske and Richard Roll, 1984)

30. Variants of Black-Scholes Model When price changes are discontinuous (Cox, Rubinstein and Ross, 1979). This was published in Journal of Financial Economics as “Option Pricing: A Simplified Approach”. This is the Binomial Model for Option Pricing. When price changes are discontinuous (Cox, Rubinstein and Ross, 1979). This was published in Journal of Financial Economics as “Option Pricing: A Simplified Approach”. This is the Binomial Model for Option Pricing.

31. Garman - Kohlhagen Model The foreign currency option pricing model is equivalent to the Black-Scholes mode except that the spot rate is discounted by the foreign interest rate and appears instead of the Stock Price.

32. Sensitivity of Option Premiums An option’s intrinsic value is the amount by which it is in the money and the time value is the difference between actual premium and the intrinsic value i.e. premium = intrinsic value + time value. An option’s intrinsic value is the amount by which it is in the money and the time value is the difference between actual premium and the intrinsic value i.e. premium = intrinsic value + time value. At the money option has highest likelihood of gaining intrinsic value as compared to that of losing. It has no value to lose but 50-50 chance of gaining. At the money option has highest likelihood of gaining intrinsic value as compared to that of losing. It has no value to lose but 50-50 chance of gaining.

33. Sensitivity of Option Premiums Delta (δ): Change in option price relative to the price of underlying asset. Reverse of Delta is used to calculate a hedge ratio. Gamma: The rate of change of Delta. It is the second derivative of option price with respect to price of the asset and is also known as option’s curvature. High gamma makes option less attractive.

34. Sensitivity of Option Premiums Lambda: Change in option price relative to change in volatility. Its value lies between zero and infinity and declines as option approaches maturity Theta: Change in option price relative to Time to Expiration. The value of theta lies between zero and total value of option. Rho: Change in option value in relation to interest rates and varies from type to type of the options.

35. Sensitivity of Option Premiums Implied volatility is obtained by finding the S.D. that when used in the Black-Scholes model makes the model price equal to market price of the option. The pattern of implied volatility across expirations is often called the term structure of volatility, and the pattern of volatility across exercise prices is often called the volatility smile or skew.

36. Illustration If on February 1, one wants to price a March European call option of a company, Where S=Rs.92.00 K= Rs.95.00 t=50 days, or (50/365=0.137 years) r = 7.12% σ =35% and the company does not pay any dividends

37. Return Relatives If P(0) is the beginning wealth and P(T), the ending wealth, the price relative R(0,T) is given by P(T)/P(0). Since P(T) is a random variable, P(T)/P(0) is also a random variable. If P(0) is the beginning wealth and P(T), the ending wealth, the price relative R(0,T) is given by P(T)/P(0). Since P(T) is a random variable, P(T)/P(0) is also a random variable. Holding period return is the effective return r(T) is related to R(0,T). Holding period return is the effective return r(T) is related to R(0,T). Continuous holding period return r c (T) is related to price relative by: r c (T) = ln(R(0,T)). Continuous holding period return r c (T) is related to price relative by: r c (T) = ln(R(0,T)). Thus all these are random variables. Thus all these are random variables.