Unformatted text preview: Derivatives  Overview Fi8000 Valuation of Financial Assets
Spring Semester 2010 Dr. Isabel Tkatch Assistant Professor of Finance Derivative securities are financial contracts that derive their value from other securities. They are also called contingent claims contingent because their payoffs are contingent on the prices of other securities. Derivatives  Overview
☺Examples
☺ Derivatives  Overview
☺Trading
☺ of underlying assets: venues: Common stock and stock index ☺ Foreign exchange rate and interest rate ☺ Agricultural commodities and precious metals ☺ Futures
☺Examples
☺ Exchanges – standardized contracts ☺ Over the Counter (OTC) – customtailored contracts custom ☺Serve
☺ as investment vehicles for both: of derivative securities: Options (Call, Put) ☺ Forward and Futures ☺ Fixed income and foreign exchange instruments such as swaps Hedgers (decrease the risk level of the portfolio) ☺ Speculators (increase the risk) A Call Option
A European* call option gives the buyer of the option a right to purchase the right underlying asset, at the contracted price (the exercise or strike price) on (th exerc a contracted future date (expiration)
*An American call option gives the buyer of the option (long call) a American right to buy the underlying asset, at the exercise price, on or before on the expiration date Call Option  an Example
A March (European) call option on Microsoft stock with a strike price of $20, entitles the owner with a right to purchase the stock for $20 on the expiration date*. What is the owner’s payoff on the expiration date? What is his profit if the call price is $7? Under what circumstances does he benefit from the position?
* Note that exchange traded options expire on the third Friday of the expiration month. 1 The Payoff of a Call Option
☺ On
☺ Notation
S = the price of the underlying asset (Stock) (we will refer to S0, St or ST) C = the price of a Call option (premium) (we will refer to C0, Ct or CT) X or K = the eXercise or striKe price T = the expiration date t = a time index the expiration date: ☺ If Microsoft stock had fallen below $20, the call would have been left to expire worthless. If Microsoft was selling above $20, the call owner would have found it optimal to exercise. ☺ Exercise
☺ of the call is optimal if the stock price exceeds the exercise price:
Payoff at expiration is the maximum of two:
Max {Stock price – Exercise price, 0} = Max {ST – X, 0}
☺ Profit at expiration = Payoff at expiration  Premium Buying a Call – Payoff Diagram
Payoff = Stock price Max{STX, 0} = ST 0 0 5 0 10 0 15 0 20 0 25 5 30 10
30 25 20 15 10 5 0 0 5 10 15 20 25 30 5 10 15 20 25 30 35 40 Buying a Call – Profit Diagram
Stock price = ST 0 10 20 25 30 35 40
Profit =
Max{STX,0}C X,0}30 25 20 7 7 7 2 3 8 13 15 10 5 0 0 5 10 15 20 25 30 5 10 15 20 25 30 35 40 Buying a Call Payoff and Profit Diagrams
30 25 20 15 10 5 0 0 5 10 15 20 25 30 5 10 15 20 25 30 35 40 5 10 15 20 25 30 30 25 20 15 10 5 0 0 5 10 15 20 25 30 35 40 Writing a Call Option
The seller of a call option is said to write a call, write and he receives the option price called a premium premium. He is obligated to deliver the is underlying asset on expiration date (European), for the the exercise price. 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} 2 Writing a Call – Payoff Diagram
Payoff = Stock price Max{STX,0} = ST 0 0 10 0 15 0 20 0 25 5 30 10 40 20
30 25 20 15 10 5 0 0 5 10 15 20 25 30 5 10 15 20 25 30 35 40 Buying a Call vs. Writing a Call Payoff Diagrams
30 25 20 15 10 5 0 0 5 10 15 20 25 30 5 10 15 20 25 30 35 40 5 10 15 20 25 30 30 25 20 15 10 5 0 0 5 10 15 20 25 30 35 40 Moneyness
☺ We
☺ Moneyness
☺ We say that an option is inthemoney if intheif the payoff from exercising is positive
A call options is intomoney if (St–X) > 0 into(i.e. if stock price > strike price) say that an option is atthemoney if attheif the price of the stock is equal to the strike price (St=X) say that an option is DeepintheDeepinthemoney if the payoff to exercise is extremely if large
☺ (i.e. the payoff is just about to turn positive)
☺ We ☺ We
☺ say that an option is outofthemoney outoftheif if the payoff from exercising is zero
A call options is outofthemoney if outofthe(St–X) < 0 (i.e. if the stock price < the strike price) (i.e. (i.e. if the stock price > > the strike price) A call options is deepinthemoney if deepinthe(St–X) > > 0 A Put Option
A European* put option gives the buyer of the option a right to sell the right underlying asset, at the contracted price (the exercise or strike price) on (th exerc a contracted future date (expiration)
*An American put option gives the buyer of the option (long put) a right American to sell the underlying asset, at the exercise price, on or before the on 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 stock for $20 on expiration date expiration date. What is the owner’s payoff on expiration date? Under what circumstances does he benefit from the position? 3 The Payoff of a Put Option
☺ On
☺ Buying a Put –Payoff Diagram
Payoff = Stock price Max{XMax{XST, 0} = ST 0 20 5 15 10 10 15 5 20 0 25 0 30 0
30 25 20 15 10 5 0 0 5 10 15 20 25 30 5 10 15 20 25 30 35 40 the expiration date: ☺ 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 if the stock price is below the exercise price:
Payoff at expiration is the maximum of two:
Max {Exercise price  Stock price, 0} = Max {X  ST , 0} 0}
☺ Profit at expiration = Payoff at expiration  Premium Writing a Put Option
The seller of a put option is said to write a put, write and he receives the option price called a premium premium. He is obligated to buy the underlying obligated asset on expiration date (European), for the exercise exercise price. 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} 0} Writing a Put – Payoff Diagram
Payoff = Stock price Max{XST,0} Max{X= ST 0 20 5 15 10 10 15 5 20 0 25 0 30 0
30 25 20 15 10 5 0 0 5 10 15 20 25 30 5 10 15 20 25 30 35 40 Buying a Put vs. Writing a Put Payoff Diagrams
30 25 20 15 10 5 0 0 5 10 15 20 25 30 5 10 15 20 25 30 35 40 5 10 15 20 25 30 30 25 20 15 10 5 0 0 5 10 15 20 25 30 35 40 5 10 15 20 25 30 30 25 20 15 10 5 0 Buying a Call vs. Buying a Put Payoff Diagrams – Symmetry?
30 25 20 15 10 5 0 0 5 10 15 20 25 30 35 40 5 10 15 20 25 30 0 5 10 15 20 25 30 35 40 4 Writing a Call vs. Writing a Put Payoff Diagrams – Symmetry?
30 25 20 15 10 5 0 0 5 10 15 20 25 30 5 10 15 20 25 30 35 40 5 10 15 20 25 30 30 25 20 15 10 5 0 0 5 10 15 20 25 30 35 40 Long Position in a Stock
☺ The payoff increases as the value (price) of the stock increases increase is oneforone: for each one orone: dollar dollar increase in the price of the stock, the value of the long position increases by one dollar ☺ The Long Stock – a Payoff Diagram
Stock price = ST 0 5 10 15 20 25 30 Payoff = +ST 0 5 10 15 20 25 30
30 25 20 15 10 5 0 0 5 10 15 20 25 30 5 10 15 20 25 30 35 40 Short Position in a Stock
☺ The ☺ The Payoff payoff decreases as the value (price) of the stock increases ST decrease is oneforone: for each onefordollar increase in the price of the stock dollar increase in the price of the stock, the value of the short position decreases by one dollar that the short position is a liability with a value equal to the price of the stock (mirror image of the long position) ☺ Note Short Stock – a Payoff Diagram
Stock price = ST 0 5 10 15 20 25 30 Payoff = ST 0 5 10 15 20 25 30
30 25 20 15 10 5 0 0 5 10 15 20 25 30 5 10 15 20 25 30 35 40 Long vs. Short Position in a Stock – Payoff Diagrams
30 25 20 15 10 5 0 0 5 10 15 20 25 30 5 10 15 20 25 30 35 40
5 10 15 20 25 30 30 25 20 15 10 5 0 0 5 10 15 20 25 30 35 40 5 Long and Short Positions in the RiskRiskfree Asset (Bond)
☺ The Lending – a Payoff Diagram
Stock price = ST 0 5 10 15 20 25 30 Payoff = +X 20 20 20 20 20 20 20
30 25 20 15 10 5 0 0 5 10 15 20 25 30 5 10 15 20 25 30 35 40 payoff is constant regardless of the changes in the stock price payoff is positive for a lender (long bond) and negative for the borrower (short bond) ☺ The Borrowing – a Payoff Diagram
Stock price = ST 0 5 10 15 20 25 30 Payoff = X 20 20 20 20 20 20 20
30 25 20 15 10 5 0 0 5 10 15 20 25 30 5 10 15 20 25 30 35 40 Lending vs. Borrowing Payoff Diagrams
30 25 20 15 10 5 0 0 5 10 15 20 25 30 5 10 15 20 25 30 35 40
5 10 15 20 25 30 30 25 20 15 10 5 0 0 5 10 15 20 25 30 35 40 Investment Strategies A Portfolio of Investment Vehicles
☺ We Investment Strategies: Protective Put
☺ Long ☺ Buy can use more than one investment vehicle to from a portfolio with the desired payoff. one stock. The payoff at time T is: ST ☺ We We can use any combination of the can use any combination of the instruments instruments (stock, bond, put or call) in any quantity or position (long or short) as our investment strategy. payoff of the portfolio will be the sum of the payoffs of the instruments 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: at The pa at time is Max $20Max { XST , 0 } = Max { $20ST , 0 } Xpayoff of the portfolio at time T will be the sum of the payoffs of the two instruments possible loses of the long stock position are bounded by the long put position ☺ The ☺ The ☺ Intuition: 6 Protective Put – Individual Payoffs
Stock Long Buy price Stock Put 0 0 20 5 5 15 10 10 10 15 15 5 20 20 0 25 25 0 30 30 0
30 25 20 15 10 5 0 0 5 10 15 20 25 30 5 10 15 20 25 30 35 40 Protective Put – Portfolio Payoff
Stock Long Buy price Stock Put 0 0 20 5 5 15 10 10 10 15 15 5 20 20 0 25 25 0 30 30 0 All
(Portfolio)
30 25 20 20 20 20 20 20 25 30 15 10 5 0 0 5 10 15 20 25 30 5 10 15 20 25 30 35 40 Investment Strategies Covered Call
☺ Long ☺ Write Covered Call – Individual Payoffs
Stock Long Write price Stock Call 0 0 0 5 5 0 10 10 0 15 15 0 20 20 0 25 25 5 30 30 10
30 25 20 15 10 5 0 0 5 10 15 20 25 30 5 10 15 20 25 30 35 40 one stock. The payoff at time T is: ST 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: at The pa at time is Max { ST  X , 0 } = Max { ST  $20 , 0 } payoff of the portfolio at time T will be the sum of the payoffs of the two instruments the call is “covered” since, in case of delivery, the investor already owns the stock. ☺ The ☺ Intuition: Covered Call – Portfolio Payoff
Stock Long Write price Stock Call 0 0 0 5 5 0 10 10 0 15 15 0 20 20 0 25 25 5 30 30 10 All
(Portfolio)
30 25 20 Other Investment Strategies
☺ Long
☺ ☺ straddle 0 5 10 15 20 20 20 15 10 5 0 0 5 10 15 20 25 5 10 15 20 25 30 35 40 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 30 Buy a call option (strike= X1, expiration= T) Write a Call option (strike= X2>X1, expiration= T) 7 The Put Call Parity
Compare the payoffs of the following strategies:
☺ Strategy
☺ ☺ Strategy I – Portfolio Payoff
Stock Buy Buy price Call Bond 0 0 20 5 0 20 10 0 20 15 0 20 20 0 20 25 5 20 30 10 20 All
(Portfolio)
30 25 20 15 10 5 0 0 5 10 15 20 25 30 5 10 15 20 25 30 35 40 I: Buy one call option (strike= X, expiration= T) Buy one riskfree bond risk(face value= X, maturity= T, return= rf) rf
☺ Strategy
☺ ☺ II Buy one share of stock Buy one put option (strike= X, expiration= T) 20 20 20 20 20 25 30 Strategy II – Portfolio Payoff
Stock Buy Buy price Stock Put 0 0 20 5 5 15 10 10 10 15 15 5 20 20 0 25 25 0 30 30 0 All
(Portfolio)
30 25 20 15 10 5 0 0 5 10 15 20 25 30 5 10 15 20 25 30 35 40 The Put Call Parity
If two portfolios have the same payoffs in every possible state and time in the future, their prices must be equal: 20 20 20 20 20 25 30 C+ X =S+P T (1 + rf ) Arbitrage – the Law of One Price
If two assets have the same payoffs in every possible state in the future but their prices are not equal, there is an opportunity to make an arbitrage profit. We say that there exists an arbitrage profit 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. (ii) at least one cash flow is strictly positive there exists a pair ( t , j ) for which CFtj > 0.
(i) 8 Example
Is there an arbitrage profit 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, The 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%. Example
In this case we should 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 profit opportunity. C+ X $40 = $7.25 + = $44.986 T (1 + rf ) (1 + 0.06)1 S + P = $39 + $6.50 = $45.5 The Construction of an Arbitrage Transaction
Constructing the arbitrage strategy:
1. Example
In this case we move all terms to the LHS: Move all the terms to one side of the equation so their sum will be positive; 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. ( S + P) −⎜C +
⎝ i.e. S + P −C − ⎛ 2. X⎞ ⎟ = $45.5 − $44.986 = $0.514 > 0 (1+ rf )T ⎠ X >0 (1+ rf )T Example
In this case we should:
1. 2. 3. 4.
Time: → Strategy: ↓ State: → Short stock Write put Buy call Buy bond Total CF Example
t=0 t=T 40 ST < X = 40 ST > X = 40 40 Sell (short) one share of stock Write one put option one put option Buy one call option Buy a zero coupon riskfree bond (lend) risk CF0 CFT1 CFT2 9 Example
Time: → t=0 t=T 40 ST < X = 40 ST > X = 40 40 Time: → Strategy: ↓ State: → Short stock Write put Buy call Buy bond Total CF +S=$39 +P=$6.5 C=($7.25) C=(X/(1+rf)=($37.736) X/(1+rf)=( Example
t=0 t=T 40 ST < X = 40 ST (XST) (X0 X
ST (XST)+X (X Strategy: ↓ State: → Short stock Write put Buy call Buy bond Total CF +S=$39 +P=$6.5 C=($7.25) C=(X/(1+rf)=($37.736) X/(1+rf)=( ST > X = 40 40 ST 0 (STX) X
ST (XST)+X (X S+PS+PCX/(1+rf) = 0.514 0.514 S+PS+PCX/(1+rf) = 0.514 > 0 0.514 =0 =0 Practice Problems
BKM 7th Ed. Ch. 20: 112, 1423 114BKM 8th Ed Ch 20: BKM 8th Ed. Ch. 20: 514, 1622, 26, CFA: 12 161Practice Set: 116 1 10 ...
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