Derivatives

Basic Derivatives

• Forwards

• Futures

• Options

• Swaps

Underlying Assets

• Interest rate based

• Equity based

• Foreign exchange

• Commodities

A derivative is a financial security with value based on or derived from an underlying financial asset

The usage is often risk management

Forwards

• Derivative contract to receive or deliver an underlying asset at a particular price, quality, quantity, and place at a future date– Long: Obligation to buy and take delivery of an asset for $K at time T– Short: Obligation to sell and deliver an asset for $K at time T

– ISDA Definition

• Often a risk management contract– The counterparty may be hedging risk also, may be speculating, may be an

arbitrageur or may be a dealer that ‘lays off’ the risk in its net position

Forwards

Forward Price

• Over the counter market e.g., banks

• Arbitrage pricing– Only risk free returns without taking risk– Forward prices are not based on ‘forecasted’ prices

• Example: spot price of gold is $1200/oz, interest rate on money is 4%, storage cost of gold is .5%, and gold lease rate is .125%.

Arbitrage Pricing Say a dealer offers a gold forward (bid and offer) price, F, of $1300, to be settled in gold 1 year from now

At time 0Short a forward contract with forward price $1300Borrow $1200 at 4% for a yearBuy gold at $1200 spotStore the gold @ .5%Flat position: You have obligations, but have not used any of your funds

At 1 yearPay loan and interest

$1200(1+.04)Pay storage fee

$1200(1+.005)Deliver the stored gold and

receive $1300Arbitrage profit of $46.00

Arbitrage Pricing Say a dealer offers a gold forward (bid and offer) price, F, of $1150

At time 0‘Go long’ (buy) a forward contract with forward price $1150Borrow (lease) goldSell the gold in spot market for $1200Loan the $1200 at 4%Flat position

At 1 yearTake delivery on gold

and pay $1150

Return gold and pay lease fee

Receive $1200 deposit @ 4%

Arbitrage profit of $46.50 at no risk

Arbitrage Pricing

F = S ( 1 + r T + s T)

F = $1200 ( 1 + .04 + .005 )

= $1252.50

F = S ( 1 + r T – g T)

F = $1200 ( 1 + .04 - .00125 )

= $1246.50

The forward formula indicates that the $1300 contract is too expensive

forward price = spot price + FV(costs) – FV(benefits)

Sell contracts that are expensive and buy contracts that are cheap when characterized by arbitrage pricing.

The forward formula indicates that the $1150 contract is too cheap

Futures

• Standardized, exchange traded ‘forward’ contracts

• Eliminates counterparty risk

• CME

The Five Pillars of Finance

10

Nobel Prize winner and former Univ. of Chicago professor, Merton Miller, published a paper called the “The History of Finance” Miller identified five “pillars on which the field of finance rests” These include

1. Miller-Modigliani Propositions• Merton Miller 1990 • Franco Modigliani 1985

2. Capital Asset Pricing Model• William Sharpe 1990

3. Efficient Market Hypothesis• Eugene Fama, Robert Shiller 2013

4. Modern Portfolio Theory• Harry Markowitz 1990

5. Options • Myron Scholes and Robert Merton 1997

-$15

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ST-$20

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ST

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PT

ST-$4

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$0

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CT

ST

Options – Value at Expiry

11

Long put

PT = max(K – ST , 0)Long call

CT = max(ST-K , 0)

Short call

-CT = min(K-ST , 0)

Short put

-PT = min(ST –K , 0)

Basic Options – Profit at Expiry

12

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Long put

PT = max(K – ST , 0)-P0

Long call

CT = max(ST-K , 0)-C0

Short call

CT = min(K-ST , 0)+C0

Short put

PT = min(ST –K , 0)+P0

Options vs Forwards

Forward• Long

– Obligation to buy and take delivery of an asset for $K at time T

• Short– Obligation to sell and

deliver an asset for $K at time T

Option• Call

– Long• Right to buy an asset at

price $K at time T – Short

• Obligation to sell an asset at price $K at time T

• Put– Long

• Right to sell an asset at price $K at time T

– Short• Obligation to buy an asset

at price $K at time T 13

Price of European Call Option

Price the call to create a portfolio that returns the risk free rate

Option Pricing1 Period Binomial Lattice Method

Cash flows at time T

Solve for h and B

Cash flows at time 0

Galitz uses the following future value factor instead

Option Pricing1 Period Binomial Lattice Method

‘Risk neutral’ probability of move upward

Present value of future expected cash flow discounted at risk free rate

Recommended calculation of a call option on pages 231 to 233 of handout from Financial Engineering by Lawrence Galitz.

Return rate and future value factor notation

Option Pricing1 Period Binomial Lattice Method

Option Pricing1 Period Binomial Lattice Method

Option Pricing1 Period Binomial Lattice Method

Example on pages 231 to 233 of handout from Financial Engineering by Lawrence Galitz. Same as question 4 on quiz

Black Scholes Eqn & SolutionEuropean Call Options

A fully hedged portfolio returns the risk free rateS: spot price of underlying assetV: value of derivative s: std deviation of underlying return ratest: continuous timer* is the expected risk-free rate of return (continuously compounded)

Tσ

Tσ.5rKS

lnd

Tσ

Tσ.5rKS

lnd

2*0

2

2*0

1

This formula is the solution to the B-S PDE for the European call option with its initial and boundary conditions

Options vs Forwards

21

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$75 $80 $85 $90 $95 $100 $105 $110

Profi

t

ST

Opt 1

Fwd

Strike

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Profi

t

ST

Opt 1

Fwd

Strike

Put – Call Parity

22

Portfolio of one share of stock, S, one long put, P, one short call, CSame strike, K, and time to expiry T

PT = ST + PT – CT

ST ≤ K

PT = ST + ( K – ST ) – 0

= K

ST > K

PT = ST + 0 - ( ST - K )

= K

P0 = K e – r T

P0 = S0 + P0 – C0P0 = K e – r T=S0 + P0 – C0

K e – r T = S0 + P0 – C0

C0 – P0 = S0 - K e – r T

Long Stock

Short Call

Long Put

K

Option Value Components

23

In the moneyOut of the money

Intrinsic Value

Time Value

At expiry

Prior to expiry

K

St

Value of a forward with contract price K

Call Value as Expiry Approaches

$0

$2

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$30 $35 $40 $45 $50 $55 $60

(T-t)=1.0

(T-t)=0.5

(T-t)=0.25

(T-t)=0.0

Call & Put Price ExampleNot on Quiz

25

38892.-1.0.2

1.0.04$50$40

lnd

18892.1.0.2

1.0.08$50$40

lnd

2

1

23.2$ 34867.e45$42508.40$

dN~eKdN~S C0.106.

2Tr

100

2Tr

100 dN~eKdN~S C

61.4$ 57492.40$65133.e45$

dN~SdN~eKP0.106.

102Tr

0

61.4$00.40$38.42$23.2$

SKeCP 0Tr

00

Current stock price, S0 = $40.00Expected (continuously compounded) rate of return, m* = 16.00 %Annual volatility, s = 20%

Strike price, K: $45.00Risk free (continuously compounded) rate of return, r*: 6%Time to expiry, T = 1.0 years

Option Pricing

26

If this variable increases

The call price The put price

Stock price, S Increases Decreases

Exercise price, K Decreases Increases

Volatility of asset, s Increases Increases

Time to expiry, T-t Increases Either

Risk free interest rate, r

Increases Decreases

Dividend payout Decreases Increases

)d(N~Ke)d(N~S C 2Tr

100

*

)d(N~Ke)d(N~S- P 2Tr

100

*

Tσ

Tσ.5rKS

lnd

Tσ

Tσ.5rKS

lnd

2*0

2

2*0

1

Put – Call Parity and Forwards at Expiry

27

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Put

Forward

Call

TTT

TrTT

TrTTT

fPCeKSf

eKSPC*

*

Long call = Long put + long forwardLong forward = Long call + short put

TTT

TTT

PCffPC

TTT

TTT

CPffCP

Long put = Long call + short forwardShort forward = Long put + short call

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Call

Forward

Put

TTT fCP TTT fPC

Put – Call Parity and Forwards at Expiry

28

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Put

Forward

Call

TTT

TrTT

TrTTT

fPCeKSf

eKSPC*

*

Long call = Long put + long forwardLong forward = Long call + short put

TTT

TTT

PCffPC

TTT

TTT

CPffCP

Long put = Long call + short forwardShort forward = Long put + short call

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Call

Forward

Put

TTT fCP TTT fPC

Protective Put

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Profi

t

ST

Opt 1

Fwd

Total

Strike

Asset Info Option 1 Option 2 Forward r* 4.00% Call or Put P Call or Put Strike, K 107.00$ s 15.00% Strike, K 107.00$ Strike, K Long / Sht L

S0 100.00$ Long / Sht L Long / Sht Num 1T 0.50 Number 1 Number

Premium 7.203$ Premium

Covered Call

30

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Profi

t

ST

Opt 1

Fwd

Total

Strike

Asset Info Option 1 Option 2 Forward r* 4.00% Call or Put C Call or Put Strike, K 107.00$ s 15.00% Strike, K 107.00$ Strike, K Long / Sht L

S0 100.00$ Long / Sht S Long / Sht Num 1T 0.50 Number 1 Number

Premium 2.322$ Premium

-$15

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Put

Forward

Call

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Call

Forward

Put

Put – Call Parity and Forwards before Expiry

31

ttt

trtt

trttt

fPCeKSf

eKSPC*

*

Long call = Long put + long forwardLong forward = Long call + short put

ttt

ttt

PCffPC

ttt

ttt

CPffCP

Long put = Long call + short forwardShort forward = Long put + short call

ttt fCP ttt fPC