
Fi8000 Basics of Options: Calls, Puts Milind Shrikhande

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Fi8000 Basics of Options: Calls, Puts
Derivative securities are financial contracts that derive their value from other securities. They are also called contingent claims because their payoffs are contingent of the prices of other securities.
Derivatives  Overview Examples of underlying assets:  Common stock and stock indexes
 Foreign exchange rate and interest rate
 Futures contracts
 Agricultural commodities and precious metals
Examples of derivative securities:  Options (Call, Put)
 Forward and Futures contracts
 Fixed income and foreign exchange instruments such as swaps
Derivatives  Overview Trading venues:  Exchanges – standardized contracts
 Over the Counter (OTC) – customtailored contracts
They serve as investment vehicles for both:  Hedgers (decrease the risk level of the portfolio)
 Speculators (increase the risk)
The payoff increases as the value (price) of the stock increases The increase is oneforone: for each dollar increase in the price of the stock, the value of our position increases by one dollar
Short Position in a Stock The payoff decreases as the value (price) of the stock increases The decrease is oneforone: for each dollar increase in the price of the stock, the value of our position increases by one dollar Note that the short position is a liability with a value equal to the price of the stock (mirror image of the long position)
Short Stock – a Payoff Diagram
Long vs. Short Position in a Stock – Payoff Diagrams
Long and Short Positions in the Riskfree Asset (Bond) The payoff is constant regardless of the changes in the stock price The payoff is positive for a lender (long bond) and negative for the borrower (short bond)
Lending – a Payoff Diagram
Borrowing – a Payoff Diagram
Lending vs. Borrowing Payoff Diagrams
A Call Option A European* call option gives the buyer of the option a right to purchase the underlying asset, at the contracted price (the exercise or strike price) on a contracted future date (the expiration date) *An American call option gives the buyer of the option (long call) a right to buy the underlying asset, at the exercise price, on or before the expiration date
Call Option  an Example A March (European) call option on Microsoft stock with a strike price $20, entitles the owner with a right to purchase the Microsoft stock for $20 on the expiration date*. What is the owner’s payoff on the expiration date? Under what circumstances does he benefit from the position? * Note that exchange traded options expire on the third Friday of the expiration month.
The Payoff of a Call Option On the expiration date of the option:  If Microsoft stock had fallen below $20, the call would have been left to expire worthless.
 If Microsoft was selling above $20, the call holder would have found it optimal to exercise.
Exercise of the call is optimal at maturity if the stock price exceeds the exercise price:  Value at expiration (payoff) is the maximum of two:
 Max {Stock price – Exercise price, 0} = Max {ST – X, 0}
 Profit at expiration = Payoff at expiration  Premium
Notation S = the price of the underlying asset (stock) (we will refer to S0=S, St or ST) C = the price of a call option (premium) (we will refer to C0=C, Ct or CT) X or K = the exercise or strike price T = the expiration date t = a time index
Buying a Call – a Payoff Diagram
Buying a Call – a Profit Diagram
Buying a Call Payoff and Profit Diagrams
Writing a Call Option The seller of a call option is said to write a call, and he receives the options price called a premium. He has an obligation to deliver the underlying asset on the expiration date (European), for the exercise price which may be lower than the market value of the asset. The payoff of a short call position (writing a call) is the negative of long call (buying a call): Max {Stock price – Exercise price, 0} = Max {ST – X, 0}
Writing a Call – a Payoff Diagram
Moneyness We say that an option is inthemoney when the payoff from exercising is positive  A call options is intomoney when (St–X) > 0
 (i.e. when stock price > strike price)
We say that an option is outofthemoney when the payoff from exercising is zero  A call options is outofthemoney when
 (St–X) < 0
 (i.e. when the stock price < the strike price)
Moneyness We say that an option is atthemoney when the price of the stock is equal to the strike price (St=X) (i.e. the payoff is just about to turn positive) We say that an option is Deepinthemoney when the payoff to exercise is extremely large  A call options is deepinthemoney when
 (St–X) > > 0
 (i.e. when the stock price > > the strike price)
A Put Option A European* put option gives the buyer of the option a right to sell the underlying asset, at the contracted price (the exercise or strike price) on a contracted future date (the expiration date) *An American put option gives the buyer of the option (long put) a right to sell the underlying asset, at the exercise price, on or before the expiration date
Put Option  an Example A March (European) put option on Microsoft stock with a strike price $20, entitles the owner with a right to sell the Microsoft stock for $20 on the expiration date. What is the owner’s payoff on the expiration date? Under what circumstances does he benefit from the position?
The Payoff of a Put Option On the expiration date of the option:  If Microsoft stock was selling above $20, the put would have been left to expire worthless.
 If Microsoft had fallen below $20, the put holder would have found it optimal to exercise.
Exercise of the put is optimal at maturity if the stock price is below the exercise price:  Value (payoff) at expiration is the maximum of two:
 Max {Exercise price  Stock price, 0} = Max {X  ST , 0}
 Profit at expiration = Payoff at expiration  Premium
Buying a Put – a Payoff Diagram
Writing a Put Option The seller of a put option is said to write a put, and he receives the options price called a premium. He is obligated to buy the underlying asset on the expiration date (European), for the exercise price which may be higher than the market value of the asset. The payoff of a short put position (writing a put) is the negative of long put (buying a put): Max {Exercise price  Stock price, 0} = Max {X  ST , 0}
Writing a Put – a Payoff Diagram
Buying a Put vs. Writing a Put Payoff Diagrams
Buying a Call vs. Buying a Put Payoff Diagrams – Symmetry?
Writing a Call vs. Writing a Put Payoff Diagrams – Symmetry?
Investment Strategies A Portfolio of Investment Vehicles We can use more than one investment vehicle to from a portfolio with the desired payoff. We may use any of the instrument (stock, bond, put or call) at any quantity or position (long or short) as our investment strategy. The payoff of the portfolio will be the sum of the payoffs of the it’s instruments
Investment Strategies Protective Put Long one stock. The payoff at time T is: ST Buy one (European) put option on the same stock, with a strike price of X = $20 and expiration at T. The payoff at time T is: Max { XST , 0 } = Max { $20ST , 0 } The payoff of the portfolio at time T will be the sum of the payoffs of the two instruments Intuition: possible loses of the long stock position are bounded by the long put position
Protective Put – Individual Payoffs
Protective Put – Portfolio Payoff
Investment Strategies Covered Call Long one stock. The payoff at time T is: ST Write one (European) call option on the same stock, with a strike price of X = $20 and expiration at T. The payoff at time T is: Max { ST  X , 0 } = Max { ST  $20 , 0 } The payoff of the portfolio at time T will be the sum of the payoffs of the two instruments Intuition: the call is “covered” since, in case of delivery, the investor already owns the stock.
Covered Call – Individual Payoffs
Covered Call – Portfolio Payoff
Other Investment Strategies Long straddle  Buy a call option (strike= X, expiration= T)
 Buy a put option (strike= X, expiration= T)
Write a straddle (short straddle)  Write a call option (strike= X, expiration= T)
 Write a put option (strike= X, expiration= T)
Bullish spread  Buy a call option (strike= X1, expiration= T)
 Write a Call option (strike= X2>X1, expiration= T)
The Put Call Parity Compare the payoffs of the following strategies: Strategy I:  Buy one call option (strike= X, expiration= T)
 Buy one riskfree bond
 (face value= X, maturity= T, return= rf)
Strategy II  Buy one share of stock
 Buy one put option (strike= X, expiration= T)
Strategy I – Portfolio Payoff
Strategy II – Portfolio Payoff
The Put Call Parity
Arbitrage – the Law of One Price If two assets have the same payoffs in every possible state in the future and their prices are not equal, there is an opportunity to make an arbitrage profit. We say that there exists an arbitrage opportunity if we identify that: There is no initial investment There is no risk of loss There is a positive probability of profit
Arbitrage – a Technical Definition Let CFtj be the cash flow of an investment strategy at time t and state j. If the following conditions are met this strategy generates an arbitrage profit. all the possible cash flows in every possible state and time are positive or zero  CFtj ≥ 0 for every t and j. at least one cash flow is strictly positive  there exists a pair ( t , j ) for which CFtj > 0.
Arbitrage – an Example Is there an arbitrage opportunity if the following are the market prices of the assets: The price of one share of stock is $39; The price of a call option on that stock, which expires in one year and has an exercise price of $40, is $7.25; The price of a put option on that stock, which expires in one year and has an exercise price of $40, is $6.50; The annual risk free rate is 6%.
Arbitrage – an Example In this case we must check whether the put call parity holds. Since we can see that this parity relation is violated, we will show that there is an arbitrage opportunity.
The Construction of an Arbitrage Transaction Constructing the arbitrage strategy: For each asset, use the sign as an indicator of the appropriate investment in the asset. If the sign is negative then the cash flow at time t=0 is negative (which means that you buy the stock, bond or option). If the sign is positive reverse the position.
Arbitrage – an Example In this case we move all terms to the LHS:
Arbitrage – an Example In this case we should: Sell (short) one share of stock Write one put option Buy one call option Buy a zero coupon riskfree bond (lend )
Arbitrage – an Example
Arbitrage – an Example
Arbitrage – an Example
Practice Problems BKM Ch. 20: 112, 1423 Practice Set: 116
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