= 0.4290 The next question that you should ask is whether the call is overpriced or under-priced 45 18 SOLUTION TO EXAMPLE. The call is overpriced, so we should sell 10,000 call options and buy 4,290 units of the underlying. Sell 10,000 calls at 9 and you get 90,000

Buy 4,290 units of the underlying at 65 =278,850 Net cash flow = 90,000-278,850 = - 188,850 So we invest 188,850. The value of this combination at expiration will be If ST = 84.50, we will get 4,290(84.50) - 10,000(14.50) = 217,505 If ST = 50.70, we will get 4,290(50.70) - 10,000(0) = 217,503 Thus, we receive a risk-free return almost twice the risk-free rate. We could borrow the initial outlay of $188,850 at the risk-free rate and capture a riskfree profit without any net investment of money. 45 19

BINOMIAL PUT OPTION PRICING Consider a one-period binomial model in which the underlying is at 65 and can go up 30 percent or down 22 percent. The risk-free rate is 8 percent. Determine the price of a European put option with exercise price of 70. We have u = 1.30 and d = 1 - 0.22 = 0.78. S+ = S*u = 65*1.30 = 84.5 S- = S*d = 65*0.78 = 50.70 Then find the option values at expiration: p+ = Max(0,70 - 84.50) = 0 p- = Max(0,70 - 50.70) = 19.30 45

20 The risk-neutral probability is = (1.08 -.078)/(1.30-0.78) = 0.5769 and 1 - = 0.4231. The put price today is p= (0.5769*0+0.423*19.30)/1.08 = 7.56 45 21 EXERCISE FOR CLASS

Suppose a stock currently trades at a price of $150. The stock price can go up 33 percent or down 15 percent. The risk-free rate is 4.5 percent. A. Use a one-period binomial model to calculate the price of a put option with exercise price of $150. B. Suppose the put price is currently $14. Show how to execute an arbitrage transaction that will earn more than the risk-free rate. Use 10,000 put options. C. Suppose the put price is currently $1 1. Show how to execute an

arbitrage transactions that will earn more than the risk-free rate. Use 10,000 put options 45 22 BLACK SCHOLES PRICING FORMULAS 45 23 The underlying pays no dividends during

the option's life ASSUMPTI ONS OF THE BLACK AND SCHOLES MODEL Option model is a European one. Markets are efficient No commissions are charged. Interest rates remain constant and known Returns are log normally distributed

45 24 BLACK-SCHOLES OPTION PRICING FORMULA 45 25 CALL OPTION PRICE AND DELTA

45 26 PUT OPTION PRICE AND DELTA 45 27 EXERCISE ON BLACK SHOLES MERTON OPTION PRICING

Consider the following example. The underlying price is 52.75 and has a volatility of 0.35. The continuously compounded risk-free rate is 4.88 percent. The option expires in nine months; therefore, T = 9/12 = 0.75. The exercise price is 50. what is the call price of this option? First calculate d1 and d2 using the formula 45 28 CALCULATING OPTION PRICE d1 = 0.4489 d2 = 0.4489 0.35*() = 0.1458

For our use, we will round off both d1 and d2 to two decimals and look up the table. So, we have d1 = 0.45 and d2 = 0.15 Hence, N(d1) = 0.6736 And N(d2) = 0. 5596 Hence, c = 52.75*(0.6736) 50* e-0.0488*3/4 * (0.5596) = 8.5580, and the value of the put option on the same would be p = 50* e-0.0488*3/4 (1-0.5596) 52.75*(1-0.6736) = 4.0110 45 29 EXAMPLE ON OPTION

PRICING Use the Black-Scholes-Merton model to calculate the prices of European call and put options on an asset priced at 68.5. The exercise price is 65, the continuously compounded risk-free rate is 4 %, the options expire in 110 days, and the volatility is 0.38. There are no cash flows on the underlying The time to expiration will be T = 110/365 = 0.3014. Then d1 and d2 are d1 = 0.4135 d2 = 0.4135 0.38* = 0.2049, Now calculate N(d1) and N(d2) N(0.41) = 0.6591 N(0.20) = 0.5793, hence c= 7.95 and p = 3.67 45

30 EFFECT OF CASH FLOW ON OPTION PRICING we subtract the present value of the dividends from the underlying price and use this adjusted price to obtain the boundary conditions or to price the options using put-call parity. We do the same using the Black-Scholes-Merton model For stocks, we used a continuously compounded dividend yield; for currencies, we used a continuously compounded interest rate. In the case of stocks, we let 6 represent the continuously compounded dividend rate.

We use the stock price as S e-T 0 45 31 EXAMPLE Use the Black-Scholes-Merton model adjusted for cash flows on the underlying to calculate the price of a call option in which the underlying is priced at 225, the exercise price is 200, the continuously compounded riskfree rate is 5.25%, the time to expiration is 3 years, and the volatility is 0.15. The effect of cash flows on the underlying is indicated below for two

alternative approaches: A. The present value of the cash flows over the life of the option is 19.72. B. The continuously compounded dividend yield is 2.7 percent. 45 32 CALL PRICE IN FIRST CASE Adjust the price of the underlying to So = 225 - 19.72 = 205.28. Then insert into the Black-Scholes-Merton formula as follows: So = 225e-0.027(3) = 207.49, so then we have

d1= 0.8364 d2= 0.5766 N(0.84) = 0.7995 N(0.58) = 0.7190 c= 41.28 and in the second case In second adjust the stock by the dividend paid, i.e. 45 33 CALL PRICE IN SECOND CASE d1=0.8776

d2= 0.8776 0.15* = 0.6175 Hence, N(0.88) = 0.8106 N(0.62) = 0.7324 c= 43.06 45 34 EXERCISE FOR CLASS Consider an asset that trades at $100 today. Call and put options on this asset are available with an exercise price of $100. The options expire in 275 days, and the volatility is 0.45. The continuously compounded risk-free rate is 3 percent.

A. Calculate the value of European call and put options using the Black- Scholes- Merton model. Assume that the present value of cash flows on the underlying asset over the life of the options is $4.25. B. Calculate the value of European call and put options using the Black- Scholes- Merton model. Assume that the continuously compounded dividend yield is 1.5 percent. 45 35

SOLUTION TO EXERCISE 45 36 SOLUTION TO PART B 45 37 OPTION GREEKS MEASUREMENT OF RISK Delta:

Delta is the sensitivity of an options price to a change in the price of underlying variable. It is calculated as percentage change in option price for a 1% change in the underlying asset price. It is also known as hedge ratio. Gamma: The Gamma is the change in the Delta with respect to change in the underlying asset price. 45 38

OPTION GREEKS Theta: Theta is the sensitivity of the value of an option to changes in time, everything else remaining constant (spot, volatility, strike and interest rate and forward) Vega: Vega measures the sensitivity of the option price with respect to changes in volatility. Vega is thus the variation in percentage of the value of the option for a 1% change of implied volatility. Vega is large if the option has a long time to expiry or close to ATM 45

39 OPTION GREEK.. Rho: The rho of a portfolio of options is the rate of change of the value of the portfolio with respect to the interest rate: rho (call) = /r 45 40

OPTIONS GREEK Delta is the sensitivity of the option price to a change in the price of the underlying. Gamma is a measure of how well the delta sensitivity measure will approximate the option price's response to a change in the price of the underlying. Rho is the sensitivity of the option price to the risk-free rate. Theta is the rate at which the time value decays as the option approaches expiration.

Vega is the sensitivity of the option price to volatility. 45 41 EXAMPLE ON OPTION GREEK More formally, the delta is defined as Delta = Change in option price/Change in underlying price Consider the following example. The underlying price is 52.75 and has a volatility of 0.35. The continuously compounded risk-free rate is 4.88 percent. The option expires in nine months; therefore, T = 9/12 = 0.75. The exercise price is 50, what is the value of the call option and the put option on the underlying?

First we calculate the values of d and d : 1 2 45 42 IMPLIED VOLATILITY In a market in which options are traded actively, we can reasonably assume that the market price of the option is an accurate reflection of its true value. Thus, by setting the Black-Scholes-Merton price equal to the market price, we can work backwards to infer the volatility. This procedure

enables us to determine the volatility that option traders are using to price the option. This volatility is called the implied volatility. 45 43 MONTE CARLO SIMULATION METHOD OF OPTION PRICING Monte Carlo Option Price is a method often used in Mathematical finance to calculate the value of an option with multiple sources of uncertainties and random features, such as changing interest rates, stock prices or exchange

rates, etc. 45 44 MONTE CARLO OPTION PRICING In the formula, the random terms on the right-hand side can be considered as shocks or disturbances that model functions in the stock price. After repeatedly simulating stock price trajectories and computing appropriate averages, it is possible to obtain estimates of the price of a European call option.

45 45 QUESTIONS FOR THE CLASS In case you are bearish on market, which of the following options would you use? A) Call option buy, b) Put option buy, c) sell call option, d) sell put option A higher interest rate would lead to higher or lower price of call option and why? A high volatility in the market will lead to higher call price or lower call

price and why? If delta of a call option is close to 1, what does it indicate? A) the option is far out of money, b) the option is deep in the money, c) the option is at the money If you have sold 400 call option of ICICI Bank and its delta is 0.35, how many shares would you need to own to hedge your position? A) 400, b) 45 140, c)1143, d) 200 46

THANK YOU. 45 47