Call and Put. Options. American and European Options. Option Terminology. Payoffs of European Options. Different Types of Options

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1 Call and Put Options A call option gives its holder the right to purchase an asset for a specified price, called the strike price, on or before some specified expiration date. A put option gives its holder the right to sell an asset for a specified price, called the strike price, on or before some specified expiration date. Option Terminology Call Put Long = Buy the right Short = Sell the right = Write Key Elements Exercise or Strike Price X Premium or Price Maturity or Expiration American and European Options An American option can be exercised at any time before expiration or maturity. A European option can only be exercised at the expiration or maturity date. Stock Price S t Different Types of Options Stock Options Index Options Futures Options Foreign Currency Options Interest Rate Options Payoffs of European Options Suppose you long a call option, which gives you the right to buy an ounce of gold for $300 in three months. What is your payoff if gold price is $50, $300, $350 in three months? $350.

2 Payoffs Payoff to long call $300. $50. So the payoff to a long call is always nonnegative. (unlike forward) Hence, it is expensive. payoff Payoff=max(0, S T -K) 0 K S T Payoffs of European Options Suppose you long a put option, which gives you the right to sell an ounce of gold for $300 in three months. What is your payoff if gold price is $50, $300, $350 in three months? $50. Payoffs $300. $350. So the payoff to a long put is always nonnegative. Payoff to long put Market and Exercise Price Relationships payoff 0 Payoff=max(0, K-S T ) K S T In the Money Exercise of the option would be profitable Call: market price > exercise price Put: exercise price > market price Out of the Money Exercise of the option would not be profitable Call: market price < exercise price Put: exercise price < market price At the Money Exercise price and asset price are equal

3 Payoffs and Profits at Expiration: Calls Payoff to Call Holder (S T - X) if S T >X 0 if S T < X Profit to Call Holder Payoff - Premium (Price) Payoff to Call Writer -(S T - X) if S T >X 0 if S T < X Profit to Call Writer Payoff + Premium Payoffs and Profits at Expiration: Puts Payoffs to Put Holder 0 if S T > X (X - S T ) if S T < X Profit to Put Holder Payoff Premium Payoffs to Put Writer 0 if S T > X -(X - S T ) if S T < X Profits to Put Writer Payoff + Premium Put-Call Parity Relationship Suppose you buy a European call and short a European put with the same strike price and maturity. Then your payoffs at expiration would be S T < X S T > X Payoff for Call Owned 0 S T -X Payoff for Put Written - ( X - S T ) 0 Total Payoff S T -X S T -X Arbitrage & Put-Call Parity Since the payoff on a combination of a long call and a short put is always S T -X, which is a levered equity, the cost of entering into either position must be equal: C P = S X / + ( ) T R f If the above relation does not hold, arbitrage will be possible. Time Value of Options: Call Factors Influencing Option Values: Calls Option value before expiration Value of Call Intrinsic Value Factor Stock price Exercise price Volatility of stock price Dividend rate Effect on value increases decreases increases decreases Time value X Stock Price

4 Black-Scholes Option Valuation The price of a European call option is given by d d C ln = ln = rt = SN( d) Xe N( d ) ( S / X ) + ( r + σ / ) T σ T ( S / X ) + ( r σ / ) T = d σ T σ T Black-Scholes Option Valuation where C = Current call option value S = Current stock price N(d) = probability that a random draw from a normal dist. will be less than d. X = Exercise price. e =.788, the base of the natural log. r = Risk-free interest rate T = time to maturity of the option in years. ln = Natural log function σ = Standard deviation of annualized cont. compounded rate of return on the stock Call Option Example Example: S o = 00 X = 95 r =.0 T =.5 (quarter) σ =.50 What is the value of the call option? ( ) + (. +.5 / ) ln 00/95 d =.5.5 d = d.5.5 =.8.5 =.43 Probabilities from Normal Dist. => N (.43) =.6664 Table 7. d N(d) Interpolation Probabilities from Normal Dist. => N (.8) =.574 Table 7. d N(d) Call Option Value Now we use these values to solve for the call option price: C = SN C = $3.70 rt ( d ) Xe N( d ) C = e

5 Put Option Value If the Black-Scholes formula only gives the value of a call option, then how can we find the value of a put option on the same stock? By using put-call parity: P = C + Xe P = e P = $6.35 rt S Implied Volatility Of the five variable inputs used to solve for a call option price, the stock volatility has the most potential to impact the option s value. Using the Black-Scholes formula and an observed market price of an option, we can solve for volatility. This is called an implied volatility because it s value is implied by the other five variables in the Black-Scholes formula. Hedging Foreign Exchange Risk US firm expects to receive million Pound in three months. The firm wants to protect against a decline in profit that would result from a decline in the pound. Short or sell pounds for future delivery to avoid the exposure. That is, the firm enters a futures contract that agrees to sell million pound in three month, at say, $.5 per pound. Hedging Foreign Exchange Risk Sell pound futures at $.5/. If the pound goes up to $.6/ in three months, mil is worth $3. mil You lose (.6-.5) mil = $ 0. mil Your total payoff is $3 mil Hedging Foreign Exchange Risk If the pound drops to $.4/ in three months, mil is worth $.8 mil You win in futures (.5-.4) mil = $ 0. mil Your total payoff is $3 mil This guarantees that that firm will receive $3 million in three month regardless of the future exchange rate. Portfolio Insurance Consider, for example, a fund manager with a $00 million portfolio whose value mirrors the value of the S&P 500. Suppose that the S&P 500 is at a level of 000 and the manager wishes to insure against the value of the portfolio dropping below $90 million in the next six months.

6 Using Put Options One approach is to buy six-month option contracts on the S&P 500 with a strike price of 900 and maturing in 6 months. If the index drops below 900, the put options will become in the money and provide the manager with compensation for the decline in the value of the portfolio. Using Put Options Buy put options with strike of 900 If the S&P 500 index goes up to 00 Your portfolio is worth approximately $0 million You option is out of the money hence worthless Your total value is $0 mil Using Put Options If the S&P 500 index drops to 800 Your portfolio is worth approximately $80 million You option is in the money and it is worth ( ) = $0 mil Your total value is $90 mil Of course, insurance is not free.