Introduction of
Derivatives
Lecture 1
1
Basis
 Basis is the difference between spot &
futures prices before the contract’s
maturity
bt=St-Ft
2
Long Hedge for Purchase of an Asset with
basis risk

Situation: A hedger knows the asset will be needed to buy
at time 2 and want to do long hedge with futures now
(at time 1) while the futures contract’s delivery time is time
3 (later than time 2)
F1 : Futures price at time 1 when hedge is set up
F2 : Futures price at time 2 when asset is purchased
S2 : Asset spot price at time 2
b2 : Basis at time 2, S2 − F2
Cost of asset
S2
Gain on Futures
F2 −F1
Net amount paid
S2 − (F2 −F1) =F1 + b2
The uncertainty of b2 is called the hedging risk or basis risk.
3
Short Hedge for Sale of an Asset
with basis risk
Situation: A hedger knows the asset will be sold at time 2
and want to takes a short hedge in futures at time 1
F1 : Futures price at time 1, when hedge is set up
F2 : Futures price at time 2, when asset is sold
S2 : Asset price at time 2
b2 : Basis at 2
Price of asset
S2
Gain on Futures
F1 −F2
Net amount received
S2 + (F1 −F2) =F1 + b2
4
Minimum Variance Hedge Ratio
the size of the futures position t aken
Hedge ratio (h ) 
the size of the exposure
*
(= size of futures taken to hedge one unit of exposure)
sS
h r
sF
*
Is called the minimum
variance hedging ratio
where
sS is the standard deviation of DS (changes in spots)
sF is the standard deviation of DF (changes in futures)
r is the correlation coefficient between DS and DF.
5
Optimal Number of Contracts
Optimal number of contracts
if no tailing adjustment
(forward contract)
*
h
QA
*
N 
QF
Optimal number of contracts
after tailing adjustment for
daily settlement (futures
contract)
*
h
VA
*
N 
VF
QA Size of position being hedged (units of asset required)
QF Size of one futures contract (units of asset in one contract)
VA
Value of position being hedged (= spot price times QA)
VF
Value of one futures contract (= futures price times QF)
6
Cross Hedging Example
Hedging Using Index Futures
(Page 65)
Stock Index Futures
Hedging
A Stock Portfolio
The optimal number of index futures contracts used for
hedging a portfolio A:
b
VA
VF
where
VA is the current value of the portfolio A
VF is the current value of one index futures contract
b is its beta coefficient of the portfolio (index is taken as the
market portfolio)
After Hedging, the beta of the portfolio become zero.
7
Changing Beta
 What if we don't want to change the portfolio's
Beta to zero
*
 If we want to change its b to b
:
When
b b
,*
b b
When
*,
Long
Short
VA
(b  b )
VF
*
numbers of futures
contracts is required
VA
(b  b )
VF
*
numbers of futures
contracts is required 8
Interest Rates:
Term Structure
Chapter 4
9
Compounding Interest Rate
Discrete compounding and continuous compounding

10
Continuous Compounding
(Pages 84-85)

11
Conversion Formulas
(Page 85)
Rc : continuously compounding rate
Rm: annual rate (APR) with compounding m times per year
Rm 

Rc  m ln1 

m 


Rm  m e
Rc / m

1
Rates used in option pricing are nearly always expressed
with continuous compounding
12
Bond Pricing
Maturity
Zero Rate
(years) (% cont. comp.)
0.5
5.0
1.0
5.8
1.5
6.4
2.0
6.8
 Assuming Par = 100, then C/m = 3, and there
are total 4 cash flows from the two-year
semiannual bond, we should discount each cash
flow using the corresponding zero rate:
3e
0.050.5
 98 .39
 3e
0.0581.0
 3e
0.0641.5
 103 e
0.068 2.0
13
Bond Yield
 The
bond yield is the single discount rate that
makes the calculated bond price equal to the
market price of the bond
 For the previous example, if the market price is
$98.39. Then the bond yield can be got by solving
3e
 y  0 .5
 3e
 y 1.0
 3e
 y 1.5
 103 e
 y  2 .0
 98 .39
 Solution: bond yield y = 0.0676 with continuous
compounding (use try and error or financial calculator)
14
Par Yield
 The
par yield (for a certain maturity) is the
coupon rate that causes the bond price to equal
its face value.

In our example, annual coupon( c ) can be got by solving:
c  0.050.5 c 0.0581.0 c  0.0641.5 
c  0.0682.0
e
 e
 e
 100  e
2
2
2
2

 100
 Solution: c = 6.87;
So par yield = 6.87% with continuous compounding
15
Data to Determine Zero Curve
(Table 4.3, page 88)
Bond
Principal
Time to
Maturity (yrs)
Annul
Coupon ($)*
Bond price ($)
100
100
100
100
0.25
0.50
1.00
1.50
0
0
0
$8
97.5
94.9
90.0
96.0
100
2.00
$12
101.6
* Coupons are paid semiannually if it is applicable
Find the zero curve given the above bond price table.
16
Bootstrapping Zero Curve

17
The Bootstrap Method continued
 To calculate the 1.5 year rate we solve
4e
0.10469 0.5
Solve:
 4e
0.10536 1.0
 104 e
 R 1.5
 R 1.5
 96
3.796  3.6  104e
104e
 R 1.5
 96
 88.604
R  ln(104 / 88.604) *1 / 1.5
 0.10681
18
Bootstrapping Zero Curve
Bond
Time to
Annul
Bond price
Principal Maturity (yrs) Coupon ($)* ($)
100
2.00
$12
101.6
This bond has annual coupon $12 pays semiannually. What
is the continuous compounding zero rate for 2 year?
Time
Zero
Period Rates
0.25
10.127%
0.5
10.469%
1.0
10.536%
1.5
10.681%
2.0
10.808%
6 e 0.104690.5  6 e 0.105361.0
 6 e 0.106811.5  106 e  R2
 101.6
which is solved as R=10.808%
19
Formula for Forward Rates
 R f (T1 ,T2 ) is the forward rate for the period between
T1 and T2 is
R2T2  R1T1
T1
R f (T1 ,T2 ) 
 R2  ( R2  R1 )
T2  T1
T2  T1
 R1 is the zero rate to
T1 ; R2 is the zero rate to
T2 ; (continuous compounding)
 it is only approximately true when rates are not
continuous compounding
20
Forward Rate Agreement

A forward rate agreement (FRA) is an agreement that a certain
rate (RK ) will be applied to a loan (L) borrowed for a future time
period (from T1 to T2 )

Assume LIBOR is the risk-free rate, define
 RF : the forward LIBOR rate for the future period between T1
and T2 (discrete compounding, compounding interval is T2-T1)
 RM : the actual LIBOR rate for the period between T1 and T2
observed in the market at T1 (also with compounding interval
T2-T1)

Therefore, for a loan of L borrowed at T1 and paid back at T2
 Without FRA, interest RM *L*(T2-T1) should be paid at T2
 With FRA, interest RK *L*(T2-T1) should be paid at T2
21
Forward Rate Agreement
Saved interest for the borrower at
T2 from an FRA =
(interest paid Without FRA – interest paid With FRA)
L(RM -RK)*(T2-T1)
Usually, FRAs are settled at time T1 rather than T2 , then
the saved interest has value at T1 :
L(R M - R K ) * (T2 - T1 ) is the value of the FRA at T1
1  RM (T2  T1 )
(note: RM is discrete compounding rate from T1 to T2, so 1  RM (T2  T1 )
is the discounting factor from T2 to T1)
22
Forward Rate Agreement
We do not know RM at time 0, but we can use the
forward rate RF as the best prediction.
 Then the expected payoff at T2 for the borrowers is
L(RF - RK)*(T2 - T1)
 for the lender is
L( RK - RF)*(T2 - T1)


Then the value of FRA for the borrower at time 0 is
L(R F - R K ) * (T2 - T1 ) * e

-R 2T2
The value of FRA for the lender at time 0 is
L(R K - R F ) * (T2 - T1 ) * e
-R 2T2
where R2 is the zero rate from time 0 to T2 with continuous
compounding
23
Introduction to Futures and
Options Markets
Lecture 4
Ch 5: Determination of Forward and
Future Prices
2. Forward price for an investment asset
with No Income, No Cost
The no-arbitrage forward price for forward maturity T
assuming the underlying asset provides no income and
has no cost before T:
• For continuous compounding r :
• For annual compounding r :
S 0:
F0:
T:
r:
F0 = S0erT
F0 = S0(1+r)T
Spot price
Future or forward price
Time from now until delivery date
Risk-free interest rate from now to T
2. No Arbitrage Forward Price
Assets with no income and no cost
•When F0 > S0erT
(short arbitrage)
Arbitrageurs can short forward contracts on the
asset, borrow S0 dollars at rate of r for T years,
and purchase the asset.
•When F0 < S0erT
(long arbitrage)
Arbitrageurs can long forward contracts on the
asset, short selling the asset for S0 dollars,
and invest the proceeds at rate r for T years.
3. Forward Price for Assets with known income
4. Forward Price for Assets with known yield
• Underlying asset provide income: e.g. stock with dividend,
coupon-bearing bonds, ……
– F0 is the current no-arbitrage forward price and S0 is the
current spot price
– Given I is the PV of the known income earn by the
underlying asset during the life of the forward contract
F0 = (S0 – I )erT
– Given q is the average continuous compounding yield earn
by the underlying asset during the life of the forward contract
F0 = S0 e(r–q )T
5. Valuing forward contract
When you first entered a forwards, a delivery price K was
negotiated to make the value of the forward contract f to be
zero, and the forward price F on that day was equal to K.
Now, after some time, your contract delivery price K remains
the same, while the forward price changes to F0, then the
value of the forward contract changes to
f = (F0 - K )e-rT
f = ( K - F0)e-rT
for long position
for short position
The value of the contract is equal to the profit from closing out the contract
5. The value of forward contract
What if the stock pays dividend?
If the stock’s dividend payout from 0 to T has PV
of income I), Then
F0 = (S0 – I )erT
f = (F0 – K )e-rT
= S0 – I – K e-rT
If the stock pays dividend yield income q, Then
F0 = S0 e(r–q )T
f = (F0 – K)e-rT
= S0e-qT – K e-rT
7. Futures prices of stock indices
• A stock index is usually regarded as the price of an
investment asset that pays dividends.
• It is usually assumed that the provided dividends yield is a
known yield
• Given q is the dividend yield of a stock index, then future
price of the stock index is
F0 = S0 e(r–q )T
(interest rate r is continuous compounding)
8. Futures and Forwards on Currencies (Page
117-121)
 A
foreign currency can also be regarded as
security providing a known yield
 The yield of the foreign currency is the foreign
risk-free interest rate
q = rf
 If r is the domestic interest rate, then the forward
price of the foreign currency forward contract
is
( r rf ) T
F0  S0e
31
9. Futures on commodities with storage cost
• Storage costs can be treated as negative income
• If U is the PV of all the storage costs during the
life of a forward contract, then forward price is
F0= (S0 + U)erT
• If storage costs incurred are proportional to the
price of the commodity, which can be treated as
negative yield u (cont. comp.), then forward price
is
F0 = S0e(r+u)T
Futures on commodities
Convenience Yield
Future price for underlying commodity with convenience
yield ( y ) is
F0 = S0e(r + u - y)T
Note: u denotes the cost of storing the commodity
Summary of Forward Prices
The relationship between spot price and futures price for
investment asset is:
F0 = S0ecT
while c is known as the “cost of carry”
•c = r
if underlying has no storage cost and no income
•c = r – q
if underlying has no cost but has income yield q
•c = r – rf
if underlying is foreign currency with IR rf
•c = r – q + u if underlying has income yield q and storage cost u
The relationship between spot price and futures price for
consumption assets is:
F0 = S0e(c - y)T
where c is cost of carry and y is convenience yield
Expected Future Spot Prices
k is the required return of the asset; (higher risk, higher k)
r is the risk-free rate;
No Systematic Risk
(beta = 0)
k=r
F0 = E(ST)
Positive Systematic Risk
(beta > 0)
k>r
F0 < E(ST)
Negative Systematic Risk
(beta < 0)
k<r
F0 > E(ST)
• Positive systematic risk asset example: stock indices
• Negative systematic risk asset example: gold (at least
for some periods)
35
Introduction to Futures and
Options Markets
Chapter 6
Interest Rate Futures
Contents adopted from: Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C.
Hull 2008
1. Day Count Convention
Three Day Count Conventions:
• Actual/Actual: (e.g. US Treasury bonds)
• Actual number of days in every month;
• Actual number of days in every year.
• 30/360: (e.g. US corporate & municipal bonds)
• 30 days for every month;
• 360 days for every year.
• Actual/360: (e.g. US Treasury bills & other money market
security)
• Actual number of days in every month;
• 360 days for every year.
2. Price Quotation of U.S. Treasury Bill
Prices of money market instruments are usually quoted using
discount rate of discounted : (e.g. Treasury bill)
360
P
(100  Y )
n
•Y – the cash price used to buy the Treasury bill now
•n – remaining life of the Treasury bill measured in actual days
•P – the annualized interest provided by the $100 par value treasury
bill using actual/360 day count convention; It is the quoted price.
38
1. Day Count Convention
• C – the annual coupon payment
• M – the number of coupon payment per year
• How to count the days since last coupon payment and
from last to the next coupon payment?
• According to the day count convention of different
instruments in different countries
2. Price Quotation of U.S. Treasury Bond
 Cash price for a bond is the price paid by purchaser
of the bond, is referred to as the dirty price
 Quoted price for a bond is the price quoted in the
market, is referred to as the clean price
Cash price = Quoted price +
Accrued Interest
 Furthermore, quoted price are quoted in dollars and
thirty-seconds of a dollar (1/32)

Example: 95-16 = $95.50; 112-4 = $112.125;
40
2. T-Bond & T-Notes Futures
Pages 136-141

Price Quote format for T-bond futures and T-bond are
similar: 93-08 = $93.25

Most Recent Settlement Price (of the bond futures) = the
quoted price of the futures at the last settlement time (e.g. 2
p.m for CME group’s bonds)

Cash price for the futures (received by short position at
delivery)
= Most Recent Settlement Price (of the bond future) ×
Conversion factor + Accrued interest (of the bond at delivery
time)
Note: Different bond has different conversion factor and accrued
interest at the time of delivery
41
2. Cheapest – to - deliver
•
For the short position party, the cost of purchasing a bond
for delivery = Cash price of the bond = Quoted bond price
+ accrued interest
•
Then the delivery cost is
= Cash price of the bond at delivery(cash outflow of short position)
– Cash price of the futures at delivery(cash inflow of short position)
= (Quoted bond price + accrued interest)
– (futures settlement price x conversion factor + accrued interest)
= Quoted bond price – (Most recent settlement price x
Conversion factor)
•
Therefore, the short position party will choose the bond with
the least delivery cost for delivery. And it is called the
cheapest-to-deliver bond
5. Quotation of Eurodollar Futures
(Page 142)
Table 6.2 Quoted Prices for December 2012 Eurodollar futures
(Q)
Data
Quoted Futures prices IR=100 - Q
Gain /contract (long party)
July 13, 2012
99.570
(0.43%)
July 16, 2012
99.530
(0.47%)
- $100
July 17, 2012
99.580
(0.42%)
+$125
The Eurodollar futures quote ( Q ) = 100 - the futures
interest rate
• The contract price is defined = 10,000*[100 - 0.25*(100 - Q)]



In July 13, contract price is $998,925 =10,000*[100-0.25(100-99.57)]
In July 16, contract price is $998,825 =10,000*[100-0.25(100-99.53)]
Gain or Loss = 998,925 – 998,825 or = $25 *(99.57-99.53)*100=100

So one-basis-point (= 0.01) decrease in the futures quote corresponds to
a loss of $25 per contract for the long party.
43
5. Convexity Adjustment
For continuous compounding forward rate and futures
rate from T1 to T2 , we have
" convexity adjustment " :
1 2
Forward rate = Futures rate  s T1T2
2
s - - is the standard deviation of the changes in
short - term interest rate.
6. Duration
Duration of a bond that provides cash flow ci at time
ti is
 ci e
ti 

i 1
 B
n
 yt i



where B is its price and y is its yield (cont. com.)
It leads to:
DB
  DDy
B
6. Duration
Duration Matching:
•This involves hedging against interest rate risk by matching
the durations of assets and liabilities
•It provides protection against small parallel shifts in the zero
curve
•Duration-based hedge ratio:
VF
PD P
N 
VF DF
*
Contract Price for one Interest Rate Futures
DF Duration of Asset Underlying Futures at futures Maturity
P
Value of portfolio being Hedged
DP Duration of Portfolio at Hedge Maturity
Swaps
Chapter 7
Options, Futures, and Other Derivatives, 8th Ed, Ch 7, Copyright © John C. Hull 2013
47
Typical Uses of an
Interest Rate Swap
 Converting a liability from
 Fixed (floating) rate to floating (fixed) rate
 Converting an investment from
 Fixed (floating) rate to floating (fixed) rate
48
Interest-rate Swap with Financial
Intermediary
Without Financial Intermediary:
5.2%
5%
Intel
Microsoft
LIBOR+0.1%
LIBOR
With Financial Intermediary:
Swap 1 (bid rate)
5.2%
Swap 2 (offer rate)
4.985%
Intel
5.015%
Bank
LIBOR
Microsoft
LIBOR
LIBOR+0.1%
Swap 1 and swap 2 are two offsetting swaps. The net payments for both
Microsoft and Intel are increased because of the spread earned by the bank.
Interest-rate Swap with Financial
Intermediary
Without Financial Intermediary:
4.7%
5%
Intel
Microsoft
LIBOR
LIBOR - 0.2%
With Financial Intermediary:
Swap 1 (bid rate)
Swap 2 (offer rate)
5.015%
4.985%
Intel
LIBOR-0.2%
Bank
LIBOR
4.7%
Microsoft
LIBOR
The net income for both Microsoft and Intel are decreased because of
the spread earned by the bank
Introduction to Futures and
Options Markets
Chapter 9 -- Mechanics of Options Market
Payoff of European Call Option
The Payoff for long an European Call option is
if ST > K, payoff = ST - K;
if ST < K, payoff = $0.
or
max( ST  K , 0)
Then the Payoff for short an European Call option is
if ST > K, payoff = K - ST ;
if ST < K, payoff = $0.
or
Try to draw the payoff graph!
 max( ST  K , 0)
Payoff of European Put Option
The Payoff for long an European Put option is
if ST > K, payoff = $0;
if ST < K, payoff = K - ST .
or
(no exercise)
(exercise)
max( K  ST , 0)
Then the Payoff for short an European Put option is
if ST > K, payoff = $0;
if ST < K, payoff = ST - K.
or
 max( K  ST , 0)
Try to draw the payoff graph!
Stock Dividends & Stock Splits
 Suppose there is a stock option contract with a strike price
K to buy N shares of stock,
 If the underlying stock has an
n-for-m split
 m shares are transferred to n shares
 Then
 the strike price is adjusted to
mK/n
 the no. of shares that can be bought is adjusted to nN/m
 If one share of stock pays 0.5 share stock dividends, then
it is equivalent to 1.5-for-1 split
54
Margin for Selling Call Option(Page 222-224)

When a naked call option written in US, the
sellter need to maitain a margin which is the
greater of:
1.
2.
A total of 100% of the proceeds of the sale
plus 20% of the underlying share price less
the amount (if any) by which the option is out
of the money
or
A total of 100% of the proceeds of the sale
plus 10% of the underlying share price
55
Margin for Selling Put Option

When a naked Put option is written, the sellter
need to maitain a margin which is the greater of:
1.
2.
A total of 100% of the proceeds of the sale
plus 20% of the underlying share price less
the amount (if any) by which the option is out
of the money
or
A total of 100% of the proceeds of the sale
plus 10% of the strike price (or exercise
price)
(Note: the underlined part is different from call option)
56
Properties of Stock Options
Chapter 10
Fundamentals of Futures and Options Markets, 8th Ed, Ch 10, Copyright © John C. Hull 2013
57
Notation

c : European call option

price
price

p : European put option
price

S0 : Stock price today

K : Strike price

T : Life of option

s: Volatility of stock price
C : American Call option

P : American Put option
price

ST :Stock price at option
maturity

D : Present value of
dividends during option’s life

r : Risk-free rate for
maturity T with cont. comp.
58
Lower Bound and Upper Bound for
European Call Option Prices “c” (No
Dividends)
 If c is higher than this upper bound, no one will buy the option
(people can buy the stock directly)
 If c is less than this lower bound, there is an arbitrage
opportunity. How?
59
Lower andUpper Bound for European
Put Option Prices (No Dividends)
max(Ke –rT – S0, 0) ≤ p ≤ Ke –rT
• The European put option price p cannot be more than the
PV of strike price (Ke –rT), otherwise, no one would buy it
(the option allow you to sell stock at most for K in the
future)
• If p is less than this lower bound, there are also arbitrage
opportunities.
60
The Put-Call Parity
c + Ke -rT = p + S0
Assume a Europan call and a European put have the
same underling of no dividend stock, same K, same T
Then
the call option price + PV of strike price K
= put option price + current stock price
61
American Options
Upper Lower Bounds, Put-Call Parity
•
62
The Impact of Dividends
to Lower Bounds of Option Prices
Lower Bounds for European Call and Put with Dividend:
c  max( S0  D  Ke
p  max( D  Ke
 rT
 rT
, 0)
 S 0 , 0)
63
Put-Call Parity
for stock with Dividend
 European options on stock with dividend ( D > 0)
c + D + Ke -rT = p + S0
 American options on stock with dividend ( D > 0 )
S0 - D - K < C - P < S0 - Ke -rT
Note: When D = 0, they are consistent with the formula for non-dividend
stock
64
Trading Strategies
Involving Options
Chapter 11
Fundamentals of Futures and Options Markets, 8th Ed, Ch 11, Copyright © John C. Hull 2013
65
Profit of European Call Option
The Payoff for long an European Call option is
if ST > K, payoff = ST - K;
if ST < K, payoff = $0.
Profit = max( ST  K , 0) -C
Then the Payoff for short an European Call option is
if ST > K, payoff = K - ST ;
if ST < K, payoff = $0.
Profit =  max( ST  K , 0) + C
Payoff of European Put Option
The Payoff for long an European Put option is
if ST > K, payoff = $0;
if ST < K, payoff = K - ST .
(no exercise)
(exercise)
Profit = max( K  ST , 0) -P
Then the Payoff for short an European Put option is
if ST > K, payoff = $0;
if ST < K, payoff = ST - K.
Profit =  max( K  S , 0) + P
T
Profit of Options
Profit($)
Profit($)
Long Call with K=100
and Call price = 5
Long Put with K=100
and Put price = 5
Stock price($)
100
100
Stock price($)
5
5
Profit($)
Short Call with K=100
and call price = 5
Short Put with K=100
and put price =5
Profit($)
5
5
100
Stock price($)
100
Stock price($)
Profit of Long Stock
Long the Stock
Profit
S0
ST
(a.1)
The profit of long the stock = ST-S0
The profit of Short the stock = S0-ST
69
Introduction to Binomial
Trees
Chapter 12
70
Multiple steps
S0u
D
B
E S0ud
S0 A
C
S0d
F
Risk-Neutral Probabilities
Given u and d, the risk-neutral probability p can be
computed:
e rDt  d
p
ud
where Dt is the time of one step in the tree
How is this formula derived?
72
The Probability P
ad
p
ud
a  e rDt for a nondividen d paying stock
a  e ( r  q ) Dt for a stock index wher e q is the dividend
yield on the index
ae
( r  r f ) Dt
for a currency w here r f is the foreign
risk - free rate
a  1 for a futures contract
73
Matching u and d with volatility
One way of matching the volatility to u and d is to
set:
ue
s
Dt
d 1 u  e
s
Dt
where
 s is the volatility of underlying price
 Dt is the length of one time step
74
Delta
 Delta (D) is the sensitivity of option value to the
underlying value

Changes in stock option value over the changes in
the underlying stock price
change in option val ue
D
change in underlying value

For each node in binomial tree, the “change” means
the variation in the next step. So the D value varies from
node to node.
75
Valuing Stock Options:
The Black-Scholes-Merton Model
Chapter 13
76
The Black-Scholes Formula for
no-dividend stock
c  S 0 N ( d1 )  K e  rT N ( d 2 )
p  K e  rT N (  d 2 )  S 0 N (  d1 )
where d1 
d2 
ln( S 0 / K )  ( r  s 2 / 2)T
s T
2
ln( S / K )  ( r  s / 2)T
0
s T
 d1  s T
• N(x) is the standard normal distribution function
77
Options on Stock Paying Known
Cash Dividend
c  ( S 0 - I) N ( d1 )  K e  rT N ( d 2 )
p  K e  rT N (  d 2 )  ( S 0 - I) N (  d1 )
where d1 
d2 
ln(( S 0 - I) / K )  ( r  s 2 / 2)T
s T
ln(( S 0 - I) / K )  ( r  s 2 / 2)T
s T
 d1  s T
I is the present value of the cash dividend
Options on Stock Paying Known
Dividend Yields
c  S 0 e -qT N ( d1 )  K e  rT N ( d 2 )
p  K e  rT N (  d 2 )  S 0 e -qT N (  d1 )
where
d1 
ln( S 0 e  qT / K )  ( r  s 2 / 2)T
s T

ln( S 0 / K )  ( r  q  s 2 / 2)T
s T
d 2  d1  s T
q is the continuous compounding dividend yield.
Find N(x) in Normal Distribution Table
X
.05
.06
.07
.08
.09
0.0
0.5199
0.5239
0.5279
0.5319
0.5359
N(0.0809) = N(0.08) + 0.09[N(0.09) - N(0.08)]
= 0.5319 + 0.09 x (0.5359 – 0.5319)
= 0.53226
X
.05
.06
.07
.08
.09
-0.1
0.4404
0.4364
0.4325
0.4286
0.4247
N(-0.1666) = N(-0.16) - 0.66[N(-0.16) – N(-0.17)]
= 0.4364 - 0.66 x (0.4364 - 0.4325)
= 0.433826
The Greek Letters
Chapter 17
81
Delta
- If Delta of a call option on a stock is 0.6, then
when the stock price changes by a small
amount ds, the option price changes by about
60% of that amount, i.e. dc = 0.6 x dS.
dc
- It is denoted by D 
dS
Delta of long call option is positive: when stock
price increase, call option value increase
Delta of underlying stock price is 1 (
dS
 1)
dS
Delta Hedging
•Delta hedging means using the combined
position in option and stock, to creat a portfolio
with zero delta.
•
•
•
•
Maintaining a delta neutral portfolio of options and
stocks
Gain/loss on the option position is offset by loss/gain
on stock position
Delta of options will change as stock price changes
and time passes
Hedge position in stocks therefore should be
rebalanced from time to time
Delta of a Portfolio
The delta of a portfolio of options is the quantity
weighted average of the deltas of the individual
option in the portfolio:
n
D   wi D i
i 1

D i is the delta of the ith option

w i is the quantity of the ith option

For example, the portfolio with long 100,000 call option
with delta 0.53 and short 50,000 put option with delta
- 0.5 has portfolio delta 78,000
84
Delta for Stock Options
Delta for Options:
dc
- qT
 e N ( d1 )
(delta for call option on stock)
dS
dp
 e -qT N ( d1 )  1 (delta for put option on stock)
dS
Delta for Futures:
since F0  S 0 e (r -q)T , so
dF
 e (r -q)T
dS
where
S is the spot price of the underlying asset
q is the income yield of the asset