Derivatives
A derivative is a financial security with value based on or derived from an
underlying financial asset
The usage is often risk management
Basic Derivatives
Underlying Assets
• Forwards
• Interest rate based
• Futures
• Equity based
• Options
• Foreign exchange
• Swaps
• Commodities
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 0
Short a forward contract
with forward price $1300
Borrow $1200 at 4% for a
year
Buy gold at $1200 spot
Store the gold @ .5%
Flat position: You have
obligations, but have not
used any of your funds
At 1 year
Pay loan and interest
$1200(1+.04)
Pay storage fee
$1200(1+.005)
Deliver the stored gold and
receive $1300
Arbitrage 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 $1150
Borrow (lease) gold
Sell the gold in spot
market for $1200
Loan the $1200 at 4%
Flat position
At 1 year
Take 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
forward price = spot price + FV(costs) – FV(benefits)
Sell contracts that are expensive and buy contracts that are cheap when
characterized by arbitrage pricing.
F
= S ( 1 + r T + s T)
F
= S ( 1 + r T – g T)
F
= $1200 ( 1 + .04 + .005 )
F
= $1200 ( 1 + .04 - .00125 )
= $1252.50
The forward formula indicates
that the $1300 contract is too
expensive
= $1246.50
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
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
10
Options – Value at Expiry
$10
$10
Long put
$8
PT = max(K – ST , 0)
$6
$4
$8
Long call
$6
CT = max(ST-K , 0)
$4
CT
PT $2
$0
-$2 $30
$35
$40
$45
$50
$55
$60
$2
$0
-$2
-$4
-$6
$40
$45
$50
$55
$60
ST
$10
Short put
Short call
-PT = min(ST –K , 0)
$5
$35
-$4
ST
$10
$30
$5
-CT = min(K-ST , 0)
$0
$0
$30
PT
$35
$40
$45
-$5
$50
$55
$60
PT -$5
$30
$35
$40
$45
$50
$55
$60
-$10
-$10
-$15
-$15
-$20
ST
11
ST
Basic Options – Profit at Expiry
$10
$10
$8
Long put
$8
Long call
$6
PT = max(K – ST , 0)-P0
$6
CT = max(ST-K , 0)-C0
$4
$4
$2
$2
$0
-$2 $30
$35
$40
$45
$50
$55
$60
$0
-$4
-$2
-$6
-$4
Short put
$10
$30
$35
$40
$45
$60
CT = min(K-ST , 0)+C0
$5
$5
$55
Short call
$10
PT = min(ST –K , 0)+P0
$50
$0
$0
$30
$30
$35
$40
$45
$50
$55
$60
$35
$40
$45
$50
$55
$60
-$5
-$5
-$10
-$10
-$15
-$15
12
-$20
Options vs Forwards
Forward
• Long
Option
• Call
– Obligation to buy and take
delivery of an asset for $K at
time T
– Long
• Right to buy an asset at
price $K at time T
– Short
• Short
• Obligation to sell an asset
at price $K at time T
– Obligation to sell and
deliver an asset for $K at
time T
• Put
– Long
• Right to sell an asset at
price $K at time T
– Short
13
• Obligation to buy an asset
at price $K at time T
Price of European Call Option
Price the call to create a portfolio that returns the risk free rate
Option Pricing
1 Period Binomial Lattice Method
R=1+r
S×(1+u)×h-(1+r)×B=CUP
D=1+d
S×(1+d)×h-(1+r)×B=CDOWN
Cash flows at time T
U=1+u
Galitz uses the
following future value
factor instead
CUP -CDOWN
h=
S× éë(1+u)-(1+d)ùû
R=er
Solve for h and B
(1+d)×CUP -(1+u)×CDOWN
B=
(1+r)× éë(1+u)-(1+d)ùû
-S×h+B=C
Cash flows at time 0
Option Pricing
1 Period Binomial Lattice Method
Recommended calculation of a call option on pages
231 to 233 of handout from Financial Engineering by
Lawrence Galitz.
R=1+r
D=1+d
Return rate and future value factor notation
U=1+u
R-D
p=
U-D
C=
(10.27)
p×CUP + (1-p)×CDOWN
1+r
(10.28)
C ×(1+r) = p×CUP + (1-p)×CDOWN
‘Risk neutral’
probability of move
upward
Present value of future
expected cash flow
discounted at risk free rate
Option Pricing
1 Period Binomial Lattice Method
Option Pricing
1 Period Binomial Lattice Method
Option Pricing
1 Period Binomial Lattice Method
Example on pages 231 to 233 of handout from Financial Engineering
by Lawrence Galitz. Same as question 4 on quiz
R =1+r =1+.1 =1.1
D =1+d =1-.1 =0.9
U =1+u =1+.2 =1.2
R-D
p=
U-D
1.1-0.9
=
1.2-0.9
.2
= = 0.6666
.3
C=
p×CUP + (1-p)×CDOWN
1+r
.6666×20+ (.3333)×0
=
1+.1
40
=
3.3333
= $12.12
Black Scholes Eqn & Solution
European Call Options
∂V 1 ∂2V 2 2 ∂V
*
*
+
×
s
×S
+
×S×r
=r
×V
2
∂t 2 ∂S
∂S
A fully hedged portfolio returns the
risk free rate
S: spot price of underlying asset
V: value of derivative
s: std deviation of underlying return
rates
t: continuous time
r* is the expected risk-free rate of
return (continuously compounded)
S 
ln 0   r *  .5  σ 2  T
K
d1   
σ T
S 
ln 0   r*  .5  σ 2  T
K
d2   
σ T
This formula is the solution to the B-S
PDE for the European call option with
its initial and boundary conditions
Options vs Forwards
$15
$10
Opt 1
$0
$75
$80
$85
$90
$95 $100 $105 $110
-$5
Fwd
Strike
$15
-$10
-$15
$10
ST
$5
Profit
Profit
$5
Opt 1
$0
$75
$80
$85
$90
$95 $100 $105 $110
-$5
Fwd
Strike
-$10
-$15
ST
21
Put – Call Parity
Portfolio of one share of stock, S, one long put, P, one short call, C
Same strike, K, and time to expiry T
P0 = S0 + P0 – C0
Long
Put
Long
Stock
K
Short
Call
PT = ST + PT – CT
P0 = K e – r T=S0 + P0 – C0
ST ≤ K
K e – r T = S0 + P0 – C0
PT = ST + ( K – ST ) – 0
=K
C0 – P0 = S0 - K e – r T
ST > K
PT = ST + 0 - ( ST - K )
=K
P0 = K e – r T
22
Option Value Components
At expiry
Prior to expiry
Value of a
forward with
contract price
K
Time Value
Intrinsic Value
Out of the money
In the money
St
K
23
Call Value as Expiry Approaches
$20
$18
$16
$14
$12
(T-t)=1.0
$10
(T-t)=0.5
(T-t)=0.25
$8
(T-t)=0.0
$6
$4
$2
$0
$30
$35
$40
$45
$50
$55
$60
Call & Put Price Example
Not on Quiz
Current stock price, S0 = $40.00
Expected (continuously compounded) rate
of return, m* = 16.00 %
Annual volatility, s = 20%
Strike price, K: $45.00
Risk free (continuously compounded) rate
of return, r*: 6%
Time to expiry, T = 1.0 years
~d   K  erT  N
~d 
C0  S0  N
1
2
 $40 
ln
  .08  1.0
$50

d1  
 .18892
.2  1.0
 $40 
ln
  .04  1.0
$50

d2  
 -.38892
.2  1.0
~ d   K  e rT  N
~ d 
C0  S0  N
1
2
 $40  .42508  $45  e .061.0  .34867
 $2.23
~  d   S  N
~  d 
P0  K  e rT  N
2
0
1
 $45  e .061.0  .65133  $40  .57492
 $4.61
P0  C 0  e rT  K  S0
 $2.23  $42.38  $40.00
 $4.61
25
Option Pricing
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
r * T
~
~(d )
C0  S0  N(d1 )  e
K N
2
~(d )  er* T  K  N
~(d )
P0  - S0  N
1
2
S 
ln 0   r *  .5  σ 2  T
K
d1   
σ T
S 
ln 0   r*  .5  σ 2  T
K
d2   
σ T
26
Put – Call Parity and Forwards at
Expiry
C T  PT  S T  K  e r T
*
Long call = Long put + long forward
Long forward = Long call + short put
Long put = Long call + short forward
Short forward = Long put + short call
C T  PT  fT
PT  C T  fT
fT  S T  K  e r T
*
C T  PT  fT
 fT  PT  C T
fT  C T  PT
C T  PT  fT
$15
PT  C T  fT
$15
$10
$10
$5
$5
Put
$0
Forward
$30
-$5
-$10
-$15
$35
$40
$45
$50
$55
$60
Call
$0
Call
Forward
$30
$35
$40
$45
$50
$55
$60
Put
-$5
-$10
-$15
27
Put – Call Parity and Forwards at
Expiry
C T  PT  S T  K  e r T
*
Long call = Long put + long forward
Long forward = Long call + short put
Long put = Long call + short forward
Short forward = Long put + short call
C T  PT  fT
PT  C T  fT
fT  S T  K  e r T
*
C T  PT  fT
 fT  PT  C T
fT  C T  PT
C T  PT  fT
$15
PT  C T  fT
$15
$10
$10
$5
$5
Put
$0
Forward
$30
-$5
-$10
-$15
$35
$40
$45
$50
$55
$60
Call
$0
Call
Forward
$30
$35
$40
$45
$50
$55
$60
Put
-$5
-$10
-$15
28
Protective Put
Asset Info
r* 4.00%
s 15.00%
S0 $ 100.00
T
0.50
Option 1
Call or Put
Strike, K
Long / Sht
Number
Premium
Option 2
P
Call or Put
$ 107.00 Strike, K
L
Long / Sht
Forward
Strike, K
$ 107.00
Long / Sht
L
Num
1
1
Number
$ 7.203 Premium
$15
$10
$5
Profit
Opt 1
Fwd
$0
$85
$90
$95
$100
$105
$110
$115
$120
Total
Strike
-$5
-$10
-$15
ST
29
Covered Call
Asset Info
r* 4.00%
s 15.00%
S0 $ 100.00
T
0.50
Option 1
Call or Put
Strike, K
Long / Sht
Number
Premium
Option 2
C
Call or Put
$ 107.00 Strike, K
S
Long / Sht
Forward
Strike, K
$ 107.00
Long / Sht
L
Num
1
1
Number
$ 2.322 Premium
$15
$10
$5
Profit
Opt 1
Fwd
$0
$85
$90
$95
$100
$105
$110
$115
$120
Total
Strike
-$5
-$10
-$15
ST
30
Put – Call Parity and Forwards before Expiry
C t  Pt  S t  K  e r t
*
Long call = Long put + long forward
Long forward = Long call + short put
Long put = Long call + short forward
Short forward = Long put + short call
C t  Pt  ft
Pt  C t  ft
ft  S t  K  e r t
*
C t  Pt  ft
 ft  Pt  C t
ft  C t  Pt
Ct  Pt  ft
$15
$15
$10
$10
$5
$5
Pt  Ct  ft
Put
$0
Forward
$30
$35
$40
$45
$50
$55
$60
Call
$0
Call
Forward
$30
-$5
-$5
-$10
-$10
-$15
-$15
$35
$40
$45
$50
$55
$60
Put
31