Project Appraisal Options In Investment Appraisal

Options in Investment Appraisal What is an Option? An option contract entitles the holder to buy or sell a designated security or other assets in a certain pre-specified time at a particular price. While the option holder is entitled to buy or sell, he is not obliged to. That is, options carry a right without obligation. Types of Options There are two types of options: • European Options – are those which can be exercised only at a specified time. • American Options – are those that can be exercised at any time during a specified period (called the time to expiration) Some Basic Definitions and Concepts • The option to buy is a call option and the option to sell is a put option.

• The option holder is the buyer of the option and the option writer is the seller. • Option Premium It is the premium paid by the buyer of the option to the seller and is paid at the time of entering into option contract. • Strike Price The holder of the option can buy or sell the asset on which the option was written. It can be exercised only at strike price irrespective of market rate. • Expiration or maturity date The date when the option expires or matures is referred to as the expiration or maturity date. • The act of buying or selling the underlying asset as per the option contract is called exercising the option. • Options traded on an exchange are called exchange-traded options and the options not traded on an exchange are called over-the-counter options.

• Option Series All options of a type are referred to as belonging to the same class of options. • In the Money (ITM) When an option is in ‘ In the Money’, the option, if exercised, will provide with a profit. ( call option) • Out of the Money (OTM) An out of money option is worthless. A call option goes ‘out of the money’ if the market price is less than the exercise price. • At the Money (ATM) An option, the holder of which is indifferent between using the option and not using is said to be ‘ At the Money’ Call Option ATM Exercise price = Market Price ITM Exercise price < Market price OTM Exercise price > Market price

Put option Exercise price = Market price Exercise price > Market price Exercise price < Market price

• Exchange – traded options are standardized in terms of quantity, trading cycle, expiration date, strike prices, types of option, and the mode of settlement. Options and their payoffs just before expiration Call Option It gives the option holder the right to buy an asset at a fixed price during a certain period. While there is no restriction on the kind of asset, the most popular type of call option is the option on stocks. To provide protection to the option holder, the option contract generally specifies that the exercise price and the number of shares will be adjusted for stock splits and stock dividends. Payoff a Call Option ( European call option ) The payoffs of the call option (C) just before expiration depends on the relationship between the stock price(S1) and the exercise price (E). Formally C = S - E if S1 > E “in the money” C=0 if S1 < E “out of money” ( worthless)

Put option The opposite of a call option is called put option. This option gives the holder the right to sell a stock at a fixed price. Payoff of a put option The payoff of a put option (C) just before expiration depends on the relationship between the exercise price (E) and the price of the underlying stock ( S1 ). The payoff of a put option is: C = E – S1 if S1 < E “in the money” C=0 if S1 > E “out of money” Seller’s Position A writer of a call option collects the option premium from the buyer (holder) of the option. In return, he is obliged to deliver the shares, if the option buyer exercises the option. If the stock price ( S1) is less than the exercise price (E) on the expiration date, the option holder will not exercise his option.

In this case, the option writer’s liability is nil. (sell a call ) On the other hand, if the stock price (S1) is more than the exercise price (E), the option holder will exercise the option. Hence, the option writer loses S1 – E. ( selling a put) Combinations Puts and calls represent basic options. They serve as building blocks for developing more complex options. As for example, the payoff for a combination of (i) buying a stock and (ii) buying a put option on the stock. The algebra corresponding to this combination as as follows Payoffs just before Expiration Date If S1 < E If S1 > E (1) Put option E–S 0 + (2) Equity stock S1 S1 =Combination E S1 Thus if you buy a stock along with a put option on that stock at a price E, the payoff will be E if the price S1 < E ; otherwise the payoff will be S1.

Consider a more complex combination that consists of (i) buying a stock (ii) buying a put option on that stock, and (iii) borrowing an amount equal to the exercise price. The payoff from this combination is identical to the payoff from buying a call option. The algebra of this equivalence is as follows: Payoffs just before Expiration Date If S1 < E If S1 > E (9) Buy the equity stock S1 S1 + (2) Buy a put option E - S1 0 (12)Borrow an amount equal to the exercise price -E -E (1) + (2) + (3) = Buy a call option 0 S1 - E

If C1 is the terminal value of the call option ( remember that C1 =Max (S1 – E ), 0), P1 the terminal value of the put option (remember that P1 =Max ( E -S1 ), 0 ), S1 the price of the stock, and E the amount borrowed, C1 = S1 + P1 – E This is referred to as the put-call parity theorem. Option Pricing The value of a call option prior to the time of expiration, depends on some more factors other than stock price and exercise price. The factors are: 1. Stock price(S) 2. Exercise Price (E) 3. Volatility if Stock Price (σ) 4. Time to Expiration (t) 5. Interest Rate (r) The effect of the changes in the above variables can be summarized as follows:

If there is an increase Stock Price Exercise Price Interest Rate Time to Expiration Volatility of the Stock Price

Change in the price of a call option Increase Decrease Increase Increase Increase

Change in the price of a pull option Decrease Increase Decrease Increase Decrease

Apart from the effects of these variables, there are inherent upper and lower bounds on the values of options. Upper Bound: The value of the option is always lesser than the price of the stock. If it is more than the price of the stock, investors can buy the stock rather than the option on the stock. Arbitrage profit can be made by selling the option and buying the stock. Lower Bound: The price of an option never falls below the payoff possible from immediate exercise. For an European call, the price can never be less than the stock price minus the present value of the exercise price.Generally, the value

is a bit higher than the minimum value because the stock price may rise during the time left to expiration giving scope for profits. The additional value is called the time value of option. Option Pricing Models A. The Binomial Model This is the earliest option-pricing model and is also the simplest. The model was formulated for calculating the value of a European call option from which the value of put option can be found. The underlying principles of this model are:  The current price of the stock, the value of the call option and the interest rates are so aligned that there is no possibility of making a risk-less profit by using any combination of calls, puts and borrowing and lending.  For a portfolio to be perfectly hedged, the combination of calls, and lending must be made in such a way that the payoff from the portfolio at the end of the holding

period is independent of the stock price. Investors, being risk-averse, hold only hedged portfolios.  The maximum and minimum values that can be reached by the stock price by the end of the maturity period are known. That is, the expected values are used. Based on the above principles, the value of a call option can be worked from the following formula: Cu x r -d + Cd x u - r u–d u –d C= r

Where, So: Current stock price E: Exercise price u: ( 1+ a) where a is the percentage upward change in the stock price during the maturity period of the option, expressed in decimals d: (1+ b) where b is the percentage downward change in the stock price during the maturity period of the option, expressed in decimals C: The call price α: The number of shares to be purchased per call Cu: Value of the call if the stock price increases. i.e. Max(uso –E, 0) Cd: Value of the call if the stock price decreases. i.e. Max(dso –E, 0) r: (1+rf ) where rf is the risk-free rate of interest in percentage expressed in decimals.

Black and Scholes Model We may set up a portfolio of a stock and a loan in such a way that their payoff is identical to that of a call option and equate their value to the value of a call option. But if the stock prices change continuously, the proportions of the stock, call and loans will also have to be changed continuously. That makes the pricing process tedious. But, the same can be achieved using the Black and Scholes Model for option valuation. The basis of the model is Value of the call option = [delta x share price] – loan. Delta is the amount to be invested in the underlying stock to build a fully hedged portfolio. Delta = N(d1) and the loan is N(d2) x PV(E). Formulas: So In PV(E) d1 = + σ √t and d2 = d1 - σ √t σ √t 2

Where PV(E) = Present value of the exercise price calculated by discounting at the continuously compounded risk-free rate t = Number of periods in years So = Price of the stock now σ = Standard deviation of the continuously compounded rate of return on the stock per period Consider the following data for a certain stock Price of stock now( So) = Rs. 60 Exercise price (E) = Rs. 56 Standard deviation (σ) = 0.3 Years of maturity (t) = 0.5 Interest rate per year = 0.14 Solution:1. Product of the standard deviation and the time 0.3 x √0.5 = 0.2121 2. Ratio of stock price to PV of exercise price 60 / (56/1.14 ) = 1.22

3. Refer table (Value of call option as % of share price) i.e. Stock price divided by PV (E) rounding of 0.2121 to 0.20 and 1.22 to 1.20, we get a value of Rs. 18.50 4. Find the value of the put option from Put-Call parity theorem: Value of the put option = Value of the call option +Exercise price –Stock price That is, P = 18.50 + 56.00 – 60.00 = 14.50 The value of the put option is Rs. 14.50