Derivatives. Basic Derivatives Forwards Futures Options Swaps Underlying Assets Interest rate based...
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Transcript of Derivatives. Basic Derivatives Forwards Futures Options Swaps Underlying Assets Interest rate based...
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
-$10
-$5
$0
$5
$10
$30 $35 $40 $45 $50 $55 $60PT
ST-$20
-$15
-$10
-$5
$0
$5
$10
$30 $35 $40 $45 $50 $55 $60PT
ST
-$6
-$4
-$2
$0
$2
$4
$6
$8
$10
$30 $35 $40 $45 $50 $55 $60
PT
ST-$4
-$2
$0
$2
$4
$6
$8
$10
$30 $35 $40 $45 $50 $55 $60
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
-$6
-$4
-$2
$0
$2
$4
$6
$8
$10
$30 $35 $40 $45 $50 $55 $60
-$4
-$2
$0
$2
$4
$6
$8
$10
$30 $35 $40 $45 $50 $55 $60
-$20
-$15
-$10
-$5
$0
$5
$10
$30 $35 $40 $45 $50 $55 $60
-$15
-$10
-$5
$0
$5
$10
$30 $35 $40 $45 $50 $55 $60
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
-$15
-$10
-$5
$0
$5
$10
$15
$75 $80 $85 $90 $95 $100 $105 $110
Profi
t
ST
Opt 1
Fwd
Strike
-$15
-$10
-$5
$0
$5
$10
$15
$75 $80 $85 $90 $95 $100 $105 $110
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
$4
$6
$8
$10
$12
$14
$16
$18
$20
$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
-$15
-$10
-$5
$0
$5
$10
$15
$30 $35 $40 $45 $50 $55 $60
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
-$15
-$10
-$5
$0
$5
$10
$15
$30 $35 $40 $45 $50 $55 $60
Call
Forward
Put
TTT fCP TTT fPC
Put – Call Parity and Forwards at Expiry
28
-$15
-$10
-$5
$0
$5
$10
$15
$30 $35 $40 $45 $50 $55 $60
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
-$15
-$10
-$5
$0
$5
$10
$15
$30 $35 $40 $45 $50 $55 $60
Call
Forward
Put
TTT fCP TTT fPC
Protective Put
29
-$15
-$10
-$5
$0
$5
$10
$15
$85 $90 $95 $100 $105 $110 $115 $120
Profi
t
ST
Opt 1
Fwd
Total
Strike
Asset Info Option 1 Option 2 Forward r* 4.00% Call or Put P Call or Put Strike, K 107.00$ 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
-$15
-$10
-$5
$0
$5
$10
$15
$85 $90 $95 $100 $105 $110 $115 $120
Profi
t
ST
Opt 1
Fwd
Total
Strike
Asset Info Option 1 Option 2 Forward r* 4.00% Call or Put C Call or Put Strike, K 107.00$ 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
-$10
-$5
$0
$5
$10
$15
$30 $35 $40 $45 $50 $55 $60
Put
Forward
Call
-$15
-$10
-$5
$0
$5
$10
$15
$30 $35 $40 $45 $50 $55 $60
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