lOMoARcPSD|4591704
Derivative Securities Notes
Derivative Securities (University of Melbourne)
StuDocu is not sponsored or endorsed by any college or university
Downloaded by Jayde Maluga (jaydemaluga@hotmail.com)
lOMoARcPSD|4591704
Derivative Securities notes:
Lecture 1: Introduction
Derivative


A derivative’s value comes from the underlying security
o It is not valuable on its own
Underlying asset can be real or financial
Characteristics:
 No one issues derivatives  it’s an agreement that gets written
 Always a buyer and seller hence zero net supply
o No upper limit on the size of the market
 Derivative are zero sum games
o One agent wins and other loses
 There can be more derivatives than underlying asset
o Because a derivative is an agreement between two people
o Squeeze the short – own all the underlying asset and get people who have forwards
on asset to buy from you at a higher price
A payoff table
 Tracks payoffs over time
t=0
t=1
t=2
PAYOFF
$
$
$
 Payoffs can be positive or negative depending on nature of cash flows
 In payoff tables – cash flows are the opposite for buying and selling
Securities



A security gives ownership rights over assets and cash flow
o A contract stipulating rights and obligations to parties
Bonds: repay face value at maturity and pay interest (coupons)
Equity: specify ownership and voting rights
o No contractual obligation for repayment; rights to control firm policies (substitute for
no cash flows)
o Stockholders paid by dividends and capital gains
Forwards



Forward contracts are an agreement today to make a trade later
o Terms must be very specific in the contract
o Price paid is the forward price and occurs on the delivery date (T)
Forward contracts negotiated between parties  not on the exchange
3 prices:
o Spot price: price of underlying asset today (changes day to day)
Downloaded by Jayde Maluga (jaydemaluga@hotmail.com)
lOMoARcPSD|4591704
Forward price: price agreed to pay at maturity
Value/price of forward contract: depending on movements in the spot price the
value of the forward contract can gain and lose value
By market convention, no money changes hand when contracts are created
o Can only happen if terms of contract are fair i.e. has zero value
o At initiation, the value of forward contract = 0
o
o

Payoff diagram
 Payoff of a forward contract
o If spot price rises above F the getting it at a
discounted price of F
 Positive payoff
o If spot prices falls below F then paying at a
premium for F
 Negative payoff
 Long position  wants the price to rise as buying underlying asset
 Short position (opposite payoff diagram)  wants the price to fall as selling underlying asset
Futures


Futures is a standardised forward
o Lower transaction costs
o Secondary market on an exchange
o Guaranteed by a clearinghouse (ensure contract is respected)
Different to a forward
o Forward only has one final cash flow but futures have daily cash flows
 Resets the delivery price at maturity for outstanding futures contracts to the
current futures price in freshly minted futures
 Also resets value for a futures contract to zero at end of each trading day
o Different risks
 Only future cash flows have reinvestment risk
Volume and open interest


Trading volume: total number of contracts traded
Open interest: total number of all outstanding contracts, total number of long (or short)
contracts
o The number of long or number of short i.e long = short
Example
 Heather buys 20 contracts from Kyle
TRADER
LONG
SHORT
TRADING
VOLUME
Heather
20
20
Kyle
20
 Heather decided reduce exposure and sells ten contracts to Tate
TRADER
LONG
SHORT
TRADING
Downloaded by Jayde Maluga (jaydemaluga@hotmail.com)
OPEN INTEREST
20
OPEN INTEREST
lOMoARcPSD|4591704
VOLUME
Heather
10
30
Kyle
20
Tate
10
 Kyle reduced short portfolio and buys 5 contracts from Heather
TRADER
Heather
Kyle
Tate
LONG
TRADING
VOLUME
SHORT
5
20
OPEN INTEREST
35
15
15
10
Options


Buyer of the option has the right to exercise BUT NOT the obligation
o Buyer pays a premium for this right
Two types of options:
o European: right can be exercised only at maturity
o American: right can be exercised at any time during life of contract
Call option
 A call option gives the buyer:
o Right to buy but not the obligation to buy
o To buy a specified quantity of a financial or real asset from seller
o On or before a fixed expiration (or maturity) date
o By paying an exercise (or strike) price agreed on today
 Holder exercises the call when she exercises her right and buys the underlying asset by
paying the strike price
 Payoff diagram:
o When spot increase above exercise price, profit for long position
 Negative payoff for short position
o If spot price decreases under exercise price will not exercise call option
o If spot price is the same
as exercise price – indifferent to the option
 At the money
Payoff
Long call
Put option
 Put option gives the owner:
S(T)
o The right to sell the asset to the
writer
o At the strike price
o Until the expiration date
Short call
 A buyer exercises the put when he sells
the underlying asset
and
receives the strike price from the put
writer
 Payoff diagram
o When spot increase above exercise price will not exercise the put option
o If spot price decreases under exercise price:
Payoff
Long put
Downloaded by Jayde Maluga (jaydemaluga@hotmail.com)
S(T)
lOMoARcPSD|4591704
o
Short put
 Profitable position for long put and negative for short put
If spot price is the same as exercise price – indifferent to the option
 At the money
Intrinsic (or Exercise) Value of a Call Option at Expiration
 Call holder’s payoff at expiration (or if exercised early):
{
Cal l ' s intrinsic value= S−K for S > K
0 otherwise
Cal l ' s intrinsic value=max ⁡( S−K , 0)


Call option’s boundary condition i.e. intrinsic or exercise value
option value=intrinsic value+ time value
o
o
Intrinsic value is value of option closed today
Time value declines to zero as get closer to maturity date
Risk






Derivatives are risky themselves
When matched with other risky investments can decrease portfolio risk
o Exchange rate risk, interest rate risk, commodity price risk
Speculators want more risk and hedgers want to reduce risk
o Speculators: don’t want the risk itself BUT want exposure to the underlying asset
o Hedging: to protect yourself against loss by supporting more than one possible result
Someone who needs to buy an assets sets up a long hedge
Someone who needs to sell an asset sets up a short hedge
Managing market or price risk is the focus of this subject
Downloaded by Jayde Maluga (jaydemaluga@hotmail.com)
lOMoARcPSD|4591704
Lecture 2: Trading and Arbitrage
Arbitrage





Arbitrage is a chance to make riskless profit with no investment
o Today  can never lose and later  can gain BUT not lose
o Today  cash in
o Later  no cash out
Arbitrage holds markets together and is invaluable for pricing derivative securities
Well-functioning markets NOT provide arbitrage opportunities
o Prices should be correct
Price derivatives by finding out what price would NOT permit arbitrage
Law of one price: one item = one price
o If not there is an arbitrage opportunity
o If arbitrage opportunity is present due to price discrepancies – as people take
advantage the price will rise and fall so that it equals each other
A (stupid) example
The setup
 NAB offers 6% interest rate on savings account and credit cards
 ANZ offers 7% interest on savings account and credit cards
 Simple arbitrage:
o Borrow $100 from NAB and put in ANZ
Solving the problem: payoff table
Position
Now
One year from now
+ $ 100
−$ 106
Borrow from NAB at 6%
−$ 100
+$ 107
Save at ANZ at 7%
$0
$1
Total portfolio
 Want to eliminate cash flows i.e at NOW total portfolio = $0
o All gains made in one period i.e one year from now (but could be now)
 E.g. match payouts at the end so make arbitrage profit at time = now
o Calculate amount to invest in ANZ so that in one year payoff of investment is the
same as what need to bay back NAB i.e $106
Position
Borrow from NAB at 6%
Save at ANZ at 7%
Now
+ $ 100
106
=−$ 99.07
1 07
+ $ 0.93
Total portfolio
 Goal: get all cash flows at one point in time
o Do this by getting rid of the extra cash flow
One year from now
−$ 106
99.07 ×1.07=+ $ 106
$0
Downloaded by Jayde Maluga (jaydemaluga@hotmail.com)
lOMoARcPSD|4591704
A less (stupid) example
The setup
 Bond A: $100 with 3% interest payments annually, $103 in year 2
 Bond B: $99.50 with 2% interest payments annually, $102 in year 2
 Can’t sell/buy both  need to buy one and sell the other to eliminate cash flows
Arbitrage 1: eliminate cash flows at t=0
 Let’s sell Bond B
o Buy $100 of bond A
o
Sell $100 of bond B i.e sell
Today
1 Year
−$ 100
+ $ 100
Bond A
Bond B
100
=1.005 of bond B
99.5
+$3
100
×2=−$ 2.01
99.5
+ $ 0.99
2 Years
+ $ 103
100
×102=−$ 102.51
99.5
+$ 0.49
0
Position
 Arbitrage opportunity: pay nothing today and receive $0.99c in one year and 0.49c in 2 years
Arbitrage 2: eliminate cash flows at t=2
 Let’s sell Bond B
o Buy $100 of bond A and receive $103 in year 2
o Need to sell an amount of B that will equal $103 in year 2

Need to sell
103
=1.01 bond B i.e 1.01× 99.5=$ 100.48 of bond B
102
Today
−$ 100
+ $ 100.48
Bond A
Bond B
1 Year
+$3
103
×2=−$ 2.02
102
+ $ 0.98
2 Years
+ $ 103
103
×102=−$ 103
102
0
+ $ 0.48
Position
 Arbitrage opportunity: receive $0.48c today and $0.98c in one year and nothing in two years
Arbitrage 3: adding a bank account
 In addition to the bonds you can borrow or save from ANZ at 2.5% per year
o Easy arb: borrow from Bank and buy bond A
o Easy arb: sell bond B and invest in ANZ
 BUT ANZ can remove one more cash flow
o Borrow enough in t=0 so that owe $0.98c in t=1
o
0.98
=$ 0.96 c
1.025
Today
Bond A and B
ANZ
+ $ 0.48
+ $ 0.96
1 Year
+ $ 0.98
−$ 0.98
+ $ 1.44
0
Position
 arbitrage opportunity: receive $1.44 today and no other cash flows
Downloaded by Jayde Maluga (jaydemaluga@hotmail.com)
2 Years
+ $ 103
103
×102=−$ 103
102
0
lOMoARcPSD|4591704
Lessons from arbitrage example
 We need one asset to remove one cash flow
o Need one asset to remove one source of uncertainty
 Getting all cash flows to one point in time is the goal
 Correct price for Bond A and B are prices that remove any arbitrage opportunities
The payoff table approach
1. Understand the problem, using numbers if necessary
o What do different variables mean? Which security generates what CFs and when?
Draw a timeline
o Plug dates and variable values into formula to get a feel for the result to prove or
how to make arbitrage profits
o Use intuition to sense which variables are overvalued/undervalued
 Sell overvalued and buy undervalued
2. Interpret cash flows
o Have made arbitrage profits if:
 If CFs are positive today and zero on maturity date
 If CFs are zero today and positive on the maturity date
3. Gather variables to one side of the equality and set the net cash flows at one of the dates
to zero
o It is inconvenient to work with nonzero CFs at both the starting and ending dates
o Gather variables to one side of an equality so that you can set today’s or the delivery
date’s net CFs to zero
o To prevent arbitrage, the other net CFs must also be zero
Interest rates


Simple interest is always given as i N ∙ N
o A payment is made N times per year
 E.g. if monthly rate is 1% - the simple interest rate is given as 12%
 E.g. if weekly rate is 0.25% - the simple interest rate is given as 13%
Compound interest provides a true interest rate for a period of time
NT
1+r n ¿ −1
r ( T )=¿
o
r ( T ) =e r

∞
Continuously compounded interest
T
Zero-coupon bond, $1 face value bonds:
NT
1+r n ¿
¿
¿
1
1
B ( T )=
=
1+r (T ) ¿
Downloaded by Jayde Maluga (jaydemaluga@hotmail.com)
lOMoARcPSD|4591704
Hard examples
Question 1
The setup
 Bond 1: two year, 5% annual coupon, $1000 par (price $1000)
 Bond 2: two year, 6% annual coupon, $1000 par (price = $1010)
 ANZ lets you borrow or lend at 2% per year
Solving the problem: payoff table
 Buy bond 2 and sell bond 1
o Want to make profit today and all other cash flows equal
o Need payoff from buying bond 1 to equal payoff bond 2 in year 2
 Payoff bond 1 in t=2 is $ 1,050 and payoff bond 1 in t=2 is $ 1,060
o

1,050
=0.991 of Bond B
1,060
In $ terms need to spend 0.991× 1,010=$ 1,000.47
To equalise these need to buy

To eliminate cash flows in t=1 need to borrow an amount in t=0 which means requires us to
return $9.43 in t=1
o
9.43
=9.25  borrow $9.25 in t=0
1.02
t=0
t=1
t=2
+ $ 1,000
−$ 50
−$ 1,050
1
−$ 1,000.47
+ $ 59.43
+$ 1,050
2
+ $ 9.25
−$ 9.43
ANZ
$ 8.78
0
0
Net position
 Arbitrage opportunity: receive $8.78 in t=0 and no further payments in the future

Look in exercise book for harder examples
Downloaded by Jayde Maluga (jaydemaluga@hotmail.com)
lOMoARcPSD|4591704
Lecture 3: Option trading and arbitrage relations
In words



A long call option will give you 0 if you don’t exercise or S− X if you do
o Exercise only when S− X >0
o Payoff is either S− X or 0 depending on what is bigger
o Written as max ⁡( S− X ,0)
A long put option will give you 0 if you don’t exercise or X −S if you do
o Exercise only when X −S
o Payoff is either X −S or 0 depending on what is bigger
o Written as max ⁡( X −S ,0)
Short options get the negative
Placing specific bets
Directional strategies
 Bullish strategies: pay off when underlying increases in value
o i.e. buying the stock, buying a call
 Bearish strategies: pay off when underlying decreases in value
o i.e. shorting a stock, buying a put
E.g. Vertical Bull Spread
 Profit diagram does not account for time
value of money
 Small range of profit and loss (limit by
taking contrasting positions
 When analysing payoffs for options, at t=T
there may be multiple outcomes
depending on spot price
Option
t=0
t=T
Long in call (X=20)
Short a call (X=22.5)
−Pcall 1
+Pcall 2
S (T )<20
0
0
Total position
Pcall 2−Pcall 1
0
20< S ( T )< 22.5
S ( T ) −20
−( S (T )−22.5)
22.5<S(T )
S ( T ) −20
0
+2.5
S ( T ) −20
Non-directional strategies
E.g. Butterfly spread
 Data for example
Strike price
K 0=$ 17.50
K 1=$ 20.00
K 2=$ 22.50
Call price
Put price
$ 5.50
$ 3.50
$ 2.00
$ 0.10
$ 0.5
$ 1.5
Downloaded by Jayde Maluga (jaydemaluga@hotmail.com)
lOMoARcPSD|4591704
K 3=$ 25.00



$ 1.00
$ 3.00
Combine options to form this shape
when think underlying asset is NOT
volatile
o Profit between 17.5 and 22.5
Need contingencies for each range of
strike prices i.e from 17.5 to 22.5
Generic profit table:
Option
Buy call 1
t=0
t=T
S (T )< X 1
0
−Pcall 1
2× Pcall 2
Sell 2x call 2
−Pcall 3
Buy call 3
Total position
 With numbers:
Option
t=0
Buy call 1
−Pcall 1
Sell 2x call 2
Buy call 3
Total position
0
0
0
X 1 < S ( T )< X 2
S ( T ) −X 1
0
0
S ( T ) −X 1
X 2 < S ( T )< X 3
S ( T ) −X 1
−2(S ( T )−X 2 )
−2 X 2−S ( T )−X 1
X 3 < S(T )
S ( T ) −X 1
−2(S ( T )−X 2 )
S ( T ) −X 3
2 X 2−X 1 −X 3
t=T
S ( T ) <17.5
0
2× Pcall 2
−Pcall 3
0
0
0
17.5< S ( T ) <20
S ( T ) −17.5
0
0
S ( T ) −17.5
20< S ( T )< 22.5
S ( T ) −17.5
22.5< S(T )
S ( T ) −17.5
−2(S ( T )−20)
−2(S ( T )−20)
S ( T ) −22.5
0
22.5−S ( T )
Strategies and straddles
 Buying a call and a put i.e long call and long put with the same strike price creates a straddle
 Buying a call and a put i.e long call and long put with different strike prices creates a strangle
 Straddles and strangles are volatility plays
o Bet on events that can have positive or negative outcomes
Downloaded by Jayde Maluga (jaydemaluga@hotmail.com)
lOMoARcPSD|4591704
Option relations
Put-call parity
 Portfolio of a long call and short put with strike K=F=S(1+ R)
o Long call and sport put look like forward
Option
t=0
t=T
Long forward
0
Sell stock
Lend money
Total position
+ S0
−rT
−F × e
−rT
S 0−F × e
ST < F
S T −F
ST > F
S T −F
−S T
+F
0
−S T
+F
0
S
−(¿ ¿T − X )
¿
Strategy A
Strategy B
Sell (short) call
+ Pcall
0
Long put
Total position
−P put
Pcall −P put
(X −S T )
0


If
0
−rT
is anything but zero – there is an arbitrage opportunity
S 0−F × e
Pcall −P put does not equal S 0−F × e−rT there is an arbitrage opportunity
Stock and dividends



Stocks are made of two parts dividends and capital gains
o When divided paid, stock price falls
 Good for put
 Bad for call
Underlying asset is the stock price minus any dividends over the derivative’s life
With American option  need to verify if exercising immediately is better than waiting
o Calls make you pay out – waiting is good – only exercise if miss out on big divided
o
 Exercise if S CUM >option value
Put send out money – waiting is bad – exercise if there is not much chance of lower
prices
Downloaded by Jayde Maluga (jaydemaluga@hotmail.com)
lOMoARcPSD|4591704
Lecture 4 & 5: Binomial Option Pricing

Pricing options little harder than pricing other financial instruments
o Approach pricing from no arbitrage argument
Price of option will depend on what happens to the underlying asset

The (One-Period) Binomial Model

Assume stock price either moves up or down by same amount over one period
o E.g. stock price rise 10% or fall by 10%
Gives approx. answers

Applications and uses of the Binomial Model
 Binomial because assume ONLY two outcomes
 Make assumption about stock price movements
o Derivative depends on the price of the underlying
 NOT reasonable to assume that ONLY two values exist in the future
 However, very useful in pricing American options
Basic example
 Option with strike price K=$ 53 ,
o What is price of call option?
 Set up payoff table
t=0
−c
−m∗S0
+b
−c−m∗S 0 +b
Long call
Buy shares
Borrow
Total position

m and
S up =$ 65, S down=$ 40, i=10 %
t=1
S down=$ 40
0
+m∗40
−b∗1.1
40 m−1.1 b
t=1
S up=$ 65
65−53=$ 12
+m∗65
−b∗1.1
12+65 m−1.1 b
b are parameters  choose so that total position in t=1 equals 0
o Given m and b substitute into total position in t=0 and solve for c
when t otal position=0

Formula shortcut (for any derivative security):
m=
b=
−X U − X D
SU −S D
m S D + X D m SU +X U
=
1+r
1+r
o
m  makes a stock’s spread across states offset the derivatives


If m is negative from the calculation means selling
o Positive CF when sell
o Offset call that pay off when derivative is doing well
If m is positive from the calculation means buying
o Negative CF when buy
Downloaded by Jayde Maluga (jaydemaluga@hotmail.com)
lOMoARcPSD|4591704
o
b
o Offset put that payoff when derivative is doing poorly
 gets size of payoffs right
Ups and downs of derivative pricing

up = $1 if spot increases (otherwise 0) and down = $1 if spot decreases (otherwise 0)
t=0
Long up
Buy shares
Borrow
Total position
t=1
−up
−m∗S0
+b
−up−m∗S0 +b
t=0
S down=$ 40
0
+m∗S D
−b∗(1+r )
m S D −b(1+r )
t=1
S down=$ 40
+1
+m∗S D
−b∗(1+r )
1+m S D −b(1+r )
t=1
S up =$ 65
+1
+m∗SU
−b∗(1+r )
1+m SU −b(1+r )
t=1
−down
Long down
−m∗S0
Buy shares
+b
Borrow
−down−m∗S
Total position
0 +b
 Up and down are just derivatives (have to do with underlying)
o If underlying asset goes up (in the up state) NOT the derivative
o If underlying asset goes down (in the down) NOT the derivative
o E.g. derivative pays off $100 in up and $27.54 in down
 Derivative is 100 “ups” and 27.54 “downs”
S up =$ 65
0
+m∗SU
−b∗(1+r )
m SU −b(1+r )
State prices – up
m=
b=

−X U − X D −1−0
=
SU −S D
SU −S D
m S D +X D m S D +0 m S D
=
=
1+r
1+r
1+r
Therefore, price of an “up” is:
S
SD
S0 − rTD
1+r
e
X0=
∨( if continuosuly compounded ) X 0=
S U −S D
S U −S D
S0 −



Give it symbol θU = price of up
“Up” is what happens to underlying asset price
o The price of derivative and strike price is irrelevant
Once have up  applicable to any derivative security that has the same tree
State prices – down

Add up to risk-free asset i.e.
1
1+r f
Downloaded by Jayde Maluga (jaydemaluga@hotmail.com)
lOMoARcPSD|4591704

θ D=
If buy “up” and “down” – guaranteed $1
o Price “up” and “down” must be equal to the price of a $1 risk free bond
1
−θU
1+r f
E.g. state prices

S 0=50. S U =65. S D=40 .r =10 %
40
1+0.1
θU =
=0.5454
65−40
50−
θ D=

1
−0.5454=0.3636
1+0.1
If call with strike price of $53
( 0 ) ( 0.3636 ) + ( 12 ) ( 0.5454 ) =$ 6.545

If put with strike of $48
( 8 ) ( 0.3636 ) + ( 0 )( 0.5454 )=$ 2.909

If forward with $55 delivery price
( 15 ) ( 0.3636 ) + (−10 ) ( 0.5454 )=$ 0
Risk-neutral probabilities
 Multiply state prices by (1+r) so up and down prices add to 1
π U =( 1+r ) θU π D =(1+r ) θ D
Works like probability but discounted at risk-free rate
Called risk-neutral probability
 Not actually probabilities but can use them as such to calculate expected
values etc…
At the end, need to divide by 1+r because multiplied by 1+r at the start
o
o

Examples

Qantas is trading for $ 2.00 might go up to $ 2.50
o Call option with K=$ 1.90 and r f =2 %
Approach 1: hedging approach
t=0
Long call
Buy shares
Borrow
Total position
−c
−m∗2.00
b
or down to
t=1
S down=$ 1.75
0
+m∗1.75
−b∗1.02
1.75 m−1.02b
Downloaded by Jayde Maluga (jaydemaluga@hotmail.com)
$ 1.75
t=1
S up =$ 2.50
2.5−1.9=$ 0.6
+m∗2.50
−b∗1.02
0.6+2.5 m−1.02 b
lOMoARcPSD|4591704
m=
−0.6−0
−0.8∗1.75
=−0.8 b=
=−1.37
2.5−1.75
1.02



m=−0.8  sell 0.8 shares (therefore, −(−0.8 ) =0.8 )
b=−1.37  lend 1.37
Therefore, position at t=0
−c±(−0.8∗2.00 )+ (−1.37 )=0
c=$ 0.23
Approach 2: state prices
1.76
1.02
θU =
=0.38
2.5−1.75
2.00−
θ D=

1
−0.38=0.6
1+0.02
Price of call is
c= payoff down ( θ D ) + payoff up (θU )
c=( 0 )( 0.6 )+ ( 0.6 ) ( 0.38 )=0.23
Approach 3: risk-neutral probabilities
π U =( 1+0.02 ) ( 0.38 ) π D =( 1+ 0.02 ) ( 0.60 )
π U =0.39 π D =0.61

X0=
Calculate price from expected cash flows i.e
X0=
E [ CF ]
1+r
( 0 )( 0.61 ) + ( 0.6 )( 0.39 )
=0.23
1+0.02
Summary
 Want a hedge  use m and b approach
 Want price ONLY  use state prices
The Binomial Model
Background
 Assumptions:
o No frictions (transaction costs)
o No credit risk (protection against default)
o Well-functioning, competitive markets
o No intermediate cash flows
o No arbitrage
Downloaded by Jayde Maluga (jaydemaluga@hotmail.com)
lOMoARcPSD|4591704



o
o
( 1+r )∗S ( 0 )> S U  best case bank better than stock
OR suppose ( 1+r )∗S ( 0 )< S D  worst case stock better than stock
Nobody want one of the assets: must have S D < ( 1+r )∗S ( 0 )< S U
Suppose
No interest rate uncertainty  constant
Binomial process  stock trades in discrete time and price evolves according to a
binomial prices
Multi-period binomial tree option pricing
Example
 Identify state prices for each two options (r=9.531%
continuously compounded)
o 50  65 OR 40


θU =0.5454∧¿
o
25  35 or 20

θU =0.4545∧θ D=0.4545
o
35  50 or 25
θU =0.4909∧θ D=0.4182

Using state prices calculate price backwards of call option
o E.g. call option with X=$35 and T=2
50  65 or 40
 Calculate payoffs for each state (in
regard to X=$35):
o
UP=30
o
DOWN =5
 Using state prices calculate value of
option at this point
c=( 0.5454 ) (30 )+ ( 5 ) ( 0.3636 )
c=$ 18.18

θ D=0.3636
25  35 or 20
 Calculate payoffs for each state (jn
regard to X=$35):
o
UP=0
o
DOWN =0
 Using state prices calculate value of
option at this point
c=( 0.4545 )( 0 )+ ( 0 ) ( 0.4545 )
c=$ 0
Using these prices for call at T=1 now calculate price of call at T=0
o Price of call now representative of payoff in up and down state
c=( 0.4909 )( 18.18 ) + ( 0.4182 ) ( 0 )
c=$ 8.92
Desirable properties of binomial trees
 Can do as many times as would like
o Using smaller steps will get a better approximation
o Usually want at least 50
 The process:
o Find two adjacent outcomes where we know the value
o Find state prices (up and down)
o Value the latest option
o Repeat until at start of trees
Downloaded by Jayde Maluga (jaydemaluga@hotmail.com)
lOMoARcPSD|4591704

To break long T periods into shorter periods 
o
N
o
E.g.
T
N
 period
1
5
breaking up the year into five periods
Keeping risk-neutral probabilities constant
 Risk neutral probabilities is the future value of state price
o State prices is value of security in one state as of today
o Risk-neutral probabilities = value of state prices tomorrow
 Calculate risk-neutral probabilities with continuous compounding at T broken into periods
( TN ) ∙θ
π U =exp r ∙
o
U
Want to factor out the
S 0 of
S0 −
θU =
SD
e rT / N
S U −S D
r∗T
e N −D
π U=
U −D
π D =1−π U

U
D are returns  called up and down factors
o Each step increases stock price to S∗U or decreases to S∗D
and
Have just removed constant from each step
o To keep theta and pi constant – need ratios to stay constant
U , D and r stay the same in your tree so do the risk-neutral probabilities
o *not required to be constant
o

If
Makes the tree recombine
 In order for the tree to come back together, we need:
S ∙U ∙ D=S ∙ D ∙ U
o

This occurs when 
D=
1
U
*model does not require
Makes the tree match our uncertainty
 Ideal for spread of outcomes (up and downs) to match uncertainty
o So that U−D=risk
 Cox, Ross and Rubenstein suggest that U should be one positive standard deviation and D
should be one negative standard deviation
o Standard deviation in finance is interchangeable with volatility
Downloaded by Jayde Maluga (jaydemaluga@hotmail.com)
lOMoARcPSD|4591704
∴U =e σ √∆ t∧D=e−σ √∆ t
o

∆ t=
T
N
- time of whole tree divided by number of steps
*model does not require
Call options on dividend paying stocks

Call option value increase when stock price increase
o If holding option when dividend paid – don’t get dividend
o Exercise option before big dividend otherwise call option will lose value after
dividend as price decreases
Example
Call option:T =3 K =$ 60 r =10 % U =1.5 D=0.5÷¿ $ 10 at t=$ 2

U
and
D and r constant can calculate state prices for whole tree:
40
1+0.1
θU =
=0.5454
120−40
80−
θ D=
1
−0.5454=0.3636
1+0.1
European option call prices
 For t=2 calculate 3 call prices:
o TOP  ( 0.5454 )( 255−60 ) +( 0.3636) ( 85−60 )=$ 115.45
( 0.5454 )( 75−60 ) + ( 0.3636 ) ( 0 ) =$ 115.45=$ 8.18
o BOTTOM  ( 0.5454 )( 0 )+ ( 0.3636 ) ( 0 )=$ 0
o

MIDDLE 
For t=1 calculate 2 call prices using prices for the call options in t=3 as new payoffs
Downloaded by Jayde Maluga (jaydemaluga@hotmail.com)
lOMoARcPSD|4591704
( 0.5454 )( 115.45 )+ ( 0.3636 ) ( 8.18 )=$ 65.95
o BOTTOM  ( 0.5454 )( 8.18 )+ ( 0.3636 )( 0 )=$ 4.46
o

TOP 
Using call prices for t=1 can calculate call option at t=0
o
c=( 0.5454 ) ( 65.95 )+ ( 0.3636 ) ( 4.456 )
c=$ 37.6
o
American option call prices
 These prices mean something for American
because can exercise option at any point
o Need to determine if the option is worth
more alive of exercised
 Evaluate stock before price drop
(due to dividend)
 Options:
o If S CUM −K > c alive : exercise option

o

If
Gain from exercise is greater
than value of option
S CUM −K < c alive : keep option
 Option has more value than exercising at the present time
Early exercise ONLY occurs immediately prior to a dividend payment (for call option)
o As $ 120> $ 1 55.45 it is optimal to exercise the option
Downloaded by Jayde Maluga (jaydemaluga@hotmail.com)
lOMoARcPSD|4591704
Lecture 6: Monte Carlo Valuation:
The basics


Relies on expectation calculations
If think risk-neutral probabilities look like a normal distribution
o Then the mean return is the risk-free rate
o Calculate derivate payoff, X (S) , for every possible outcome of S


Use normal distribution to get
E [ X ( S) ]
NOT feasible to write out all the possible outcomes
o Need to generate bunch random variables and calculate average payoff to derivative
 Discounted at risk-free rate
Example
 Euro call option,
o
X =100,T =1, S ( 0 )=$ 100, r f =5 % , σ =25 %
Generate random numbers corresponding to returns
Random number/returns
0.467663
0.149375
0.052878
-0.03458
-0.16357
S(T)
e0.467663*100 = 159.6259
e0.149375*100 = 116.1108
e0.052878*100 = 105.4301
e-0.03458*100 = 96.60136
e-0.16357*100 = 84.91096
Avg X
PV (Avg X)
X(S)
59.62589
16.11078
5.430129
0
0
16.23336
15.44165
Understanding the Monte Carlo


Monte Carlo computes an estimate of the price by using
o Risk-neutral distribution of the unvarying asset
o Generate set of possible future values of the underlying asset
 For which the derivative payoff is computed
 Then, averaged and discounted using e−rT to obtain an estimate for price
Monte Carlo is ONLY an approximation
o Good for complex derivatives that depend on how a stock price evolves through time
o Bad for American options because uncertainty over length of option
The lookback option

Lookback allows you to secure the maximum price of the stock

A (floating) lookback put pays off:
max ( Smax −ST , 0 ) =Smax −ST
S max ≥ S T because by definition S max cannot be smaller than any S T
o Will only know S max a year from now
o

Hard to price
o Payoff will depend on path stock has taken
Downloaded by Jayde Maluga (jaydemaluga@hotmail.com)
lOMoARcPSD|4591704


Use Monte Carlo to compute value of lookback option from the paths
o Simulate a lognormal distribution to generate stock price paths
Generic calculation with 100 paths
S i ;max −S i ;T
1
Ṕ0=
∑
100 i=1
e−rT
100
Monte Carlo properties
Approximating stock price distribution
 Law of large numbers
Si
f (¿)→ E [ f (S) ]
n
1
∑¿
n i=1

o
o
Averages move to expectations as sample goes to infinities
For an infinite amount of draws – the Monte Carlo estimate equals the true price
o
Need many stock price paths to approximate the true distribution of
S T well
Returns are lognormally distributed so that can have tail
o Returns bounded by zero, hence have long tail to the right
o Because of tail it pulls the average away from zero
 To adjust for this, need to make adjustment to average calculations:
(
)
1
μ=ln ( S 0 ) + r− σ 2 T
2
Sampling error
 Central Limit Theorem (CLT)
Measure of variability of the Monte Carlo estimate  confidence intervals

o
o
Z −score∗st error
( √σn ) ± average=CI
If fix the number of paths and repeat the Monte Carlo simulation, we would
generate a different set of stock prices and obtain a different option price estimate
 Different estimate is due to sampling error
 Range of estimates tells us about variability
The average of n independent and identically distributed random variables will
converge to a normal distribution
 Random variable is discounted option payoff for one path
2
N (μ MC ,σ MC )

 Average is the Monte Carlo option price
Calculating mean and variance of Monte Carlo option
o The mean (is the Monte Carol estimate)
Downloaded by Jayde Maluga (jaydemaluga@hotmail.com)
lOMoARcPSD|4591704
100
S i ;max −S i ;T
1
μMC = Ṕ0=
=$ 7.9872
∑
100 i=1
e−rT
 Take the average of all the estimates of the simulation
Variance of discounted option payoffs (random variable)
o
f ( Si ) −7.9872¿ =45.0628
¿
¿
100
1
2
f ( S i )− Ṕ0 ¿ = ∑ ¿
99 i=1
¿
n
1
2
σ f (S ) =
∑¿
n ¿−1 i=1
2
¿
i
Variance of the Monte Carlo option price estimate (the average of the random
variable)
o
1
σ 2MC = ∗variance discounted options payoffs
n
σ 2MC =σ 2Ṕ =
0
1 2
1
σ =
45.0628=0.4506
n¿ f (S ) 100
i
σ MC =σ Ṕ =√ 0.4506=0.6713
0
Confidence interval with 95% probability
o
[ Ṕ −1.96 σ
0
Ṕ 0
, Ṕ0 +1.96 σ Ṕ ] =[ 6.67,9 .3 ]
0

Range will contain true option price
Monte Carlo is slow
 Increasing number of paths by a factor of 100 decreases the standard deviation of the Monte
Carlo by a factor of √ 100=10
o Adding simulations doesn’t tighten error bounds
Generating stock price paths



Monte Carlo can handle many different stock price distributions as long as know how to
generate path
Depending on option want to price the path will look different
o Can just be S 0 and S T (if payout ONLY depends of price at maturity i.e.
European)
o Or may need more values in between start and end i.e. for lookback
Each simulation  assume returns over a step are normally distributed (U and D from
binomial tree)
Downloaded by Jayde Maluga (jaydemaluga@hotmail.com)
lOMoARcPSD|4591704
Sn
S n−1
1 2
σ T 2
rT 2
σ T
(¿) N
−
,
N
N
N
Rn =ln ¿
(
)
1 2
σ T
o
σ2 T
rT 2
∧variance=
μ= −
N
N
N
o Generate lots of random numbers R1 , R2 with normal distribution

These random numbers will represent random returns

Start at



S 0 , use R1 to get S 1=e R S0
R
Start at S 1 , use R2 to get S 2=e S1
Do until get S N  go to a large number M
1
2
Summary:
o Derivatives depend on
o
Each simulation


S1 … SN
M : calculate X m ( S1 … S N )
i.e. need to calculate the derivative payoff (in relation to some X)
Each simulation will have different derivative payoff
X m( S1… SN )
M
X m ( S1 … S N ) −rT
×e
price=
M
o
Calculate the average
o
Discount at r 
Lecture 7 – Black-Scholes-Merton:
Introduction



BSM relies on a lot of assumptions:
o No market frictions
o No credit risk
o Competitive and well-functioning markets
o No intermediate cash flows
o No arbitrage opportunities
o No interest rate uncertainty
o Trading takes place continuously in time
o The stock price follows a lognormal probability distribution
Suppose:
o Price path of underlying asset is continuous
o Return on the underlying asset is independently distributed through time and
variance is non-stochastic (non-random)
o Risk free rate is non-stochastic
Depending on length of investment can have many, many paths:
Downloaded by Jayde Maluga (jaydemaluga@hotmail.com)
lOMoARcPSD|4591704
1
2 =2 paths
o Stock moves up or down monthly: 212 =4,096 paths
o Stock moves up or down each trading day: 2250 paths
o
Stock moves up or down 1 time:
Building the BSM:

Need to answer two questions:
1. What is the expected value of the asset?
2. What is the expected cost of exercise?
What is the expected cost of exercise?
K ∙ P ( exercise ) ∙ e−rT + 0 ∙ P(no exercise)∙ e−rT
Sometimes exercise option and pay strike price, sometimes exercise the option and
pay zero
Exercise will only occur when S ( T ) >K
o
P ( exercise )=P ( S ( T )>K )
o

o
K
ln S (T )>ln ¿
P¿


Assume that returns and normally distributed which means stock prices are
going to be lognormally distributed
Assumptions:
o
(
)
1
ln S ( T ) =ln S ( 0 ) + r− σ 2 ∙T +σ √ T ∙ Z
2
(1):

o


Ending price is starting price + adjustment for mean + randomness
(2): Z N (0,1)  probability that exercise based on standard normal
distribution
Use this assumption to calculate the unexpected part of the return
Solve (1) for Z and then look up probability from normal table
Probability of exercising
1 2
ln S ( 0 )−ln K +(r − σ )∙ T
2
d 2=
σ √T



This is the z-score associated with exercising
o This point that need to cross to exercise
o Just the average divided by the standard deviation
ln S ( 0 )  starting stock
ln K
o


 number that underlying asset needs to get over
ln S ( 0 )−ln K → the return that have to get
1 2
r− σ  mean
2
This is just the z-score associated with the exercise decision
o Can find probability on a z-table = probability of exercise
Downloaded by Jayde Maluga (jaydemaluga@hotmail.com)
lOMoARcPSD|4591704

This is a “pseudo” of exercising the option
o The true probability of exercising is:
μ−
1
2
S
+( ¿ σ )∙ T
(
K)
ln
2
σ √T
d 2=¿
Expected value of asset
1
ln S ( 0 )−ln K +(r − σ 2 )∙ T
2
d 1=
+σ √ T
σ √T


When we exercise the option – always occur when stock price is in-the-money
Just the z-score associated with the exercise decision with an “UP” steep added
o Up-step accounts from fact that when exercise call – the stock price is higher
o Adjust today stock price so that it incorporates the idea that only exercise when
good stuff has happened
 Getting a valuable asset when exercise
Putting it together
 What is the expected value of the asset?
S (0)∙ N (d 2+σ √T )
o
Starting stock price multiplied by probability exercise, with adjustment for only
exercise if stock price is high
N ( d 1 )=N (d 2+σ √T )
o

What is the expected cost of exercise?
−rT
K ∙e
∙ N (d 2)
o
Probability that exercise multiplied by price have to pay
c=S ( 0 ) ∙ N ( d 2+σ √T ) −K ∙ e−rT ∙ N (d 2)

Digital options – either on or off
o Cash or nothing – if stock prices passes a barrier I give $1 or nothing
 Like a call option – bet on stock prices being high or low
o Asset or nothing – if stock prices passes a barrier I give asset or nothing
 Receive shares conditional on share prices
E.g. BSM valuation

S=$ 65. K=$ 70. σ=35 % p . a . t=0.25 years .r =0.0953 . er ×1=1.1 p . a .
Downloaded by Jayde Maluga (jaydemaluga@hotmail.com)
lOMoARcPSD|4591704
1
0.0953− (0.35 ¿2 )∙ 0.25
2
¿
65
+¿
70
ln¿
d 2=¿
( )
d 2=−0.1998164
o
Convert
d 2 into Z
score
N ( d 2 )=0.420812
d 1=d 2 ++σ √ T
d 1=−0.1998164 +0.35 √ 0.25=−0.3748164
N ( d 1 )=0.353898

Because of symmetrical distribution of standard normal
o When d 1 <0 use 1−N (−d 1)
BSM put options
 ONLY for European – resolved at end of maturity
o With put options, the benefit is receiving the strike price
−rT
p=c+ e
K −S
p=S ∙ N ( d 2+ σ √ T ) −e−rT K ∙ N ( d 2 ) +e−rT ∙ K −S
d
¿
(¿ 2¿)−1
N¿
p=S ( N ( d 2+ σ √T ) −1 )−e−rT K ¿
p=−S ∙ N ( −d 2−σ √ T ) +e
−rT

K ∙ N (−d 2)
Can only be written as:
−d
N (¿ ¿1)
p=K ∙ e−rT ∙ N (−d 2 ) −S 0 ∙ ¿
o
o
N (−d 2 ) is the probability that S ( T ) < K
−d
N (¿¿ 1) is the replication ratio
¿
Downloaded by Jayde Maluga (jaydemaluga@hotmail.com)
lOMoARcPSD|4591704
BSM call options with dividends
 Need to alter present value of stock price to include dividends:
o
PV ( S ( t ) )=S−PV (¿)

Call value for an option on an underlying asset with dividends is:
c=( S−PV ( ¿ ) ) ∙ N ( d 2+ σ √ T ) −e
−rT
KN (d 2)
1 2
ln (S−PV ( ¿ ) )−ln K +(r − σ )∙ T
2
+σ √ T
d 1=
σ √T
1 2
ln (S−PV ( ¿ ) )−ln K +(r − σ )∙ T
2
d 2=
σ √T
Understanding the formula


Call values depend on:
o Stock price
o Strike price
o Time to maturity
o Riskless interest rate
o Stock return’s volatility
It does NOT depend on stock’s expected return
o People can think returns on stock will be different but will be priced the same
Downloaded by Jayde Maluga (jaydemaluga@hotmail.com)
lOMoARcPSD|4591704
Lecture 8 – The Greeks and Implied Volatility:
The Greeks

d
(¿¿
2)
Black-Scholes:
c=S ∙ N ( d 1) −K e−rT ∙ N ¿

N ( d 1 ) and N ( d 2 ) try and tell you what line the option is on
Binomial: −X 0 −m∙ S 0 +b=0
o
m is equivalent to having N ( d 1 ) amount of shares
o
Binomial just choose two points – up and down
 Want it to be as close to BSM as possible
 At the edges the BSM changes more than binomial
Depending on position on line – determines how much stock you have
o If deep in-the-money had almost one stock
o If deep out-of-the-money have almost 0 stock
o

o

N ( d 1 ) hence the larger m
Greeks are useful for understanding sensitivity of options values to changes in inputs
o Can use delta and gamma to hedge option price risks caused by changes in these
parameters when BSM assumptions are satisfied
o Hedge against uncertainty of underlying asset
Delta
 Delta,
m=
The more in-the-money the larger
Δ , gives the number of shares needed to replicated a long call
−X u− X d dc
=Δ=N (d1 )
Su−S d dS
o
N (d 1 ) indicates how far in-the-money the option is


When close to 0, never exercise, if 1, may as well exercise
Therefore, delta changes depending how deep in-the-money the option is
Downloaded by Jayde Maluga (jaydemaluga@hotmail.com)
lOMoARcPSD|4591704

If
N ( d 1 )=x then need x shares to hedge
o Same as m which is essentially the derivative of the binomial model

m creates risk-free portfolio by buying m amount of shares


Remove risk by comparing difference in option (up and down) and spot price
(up and down)
 If option move half as far as stock  need ½ of stock to hedge
Depending on how in/out of-the-money, the delta will change
o According to BSM to hedge, need to continuously be adjusting position
o Delate changes quicker when there is more curviness in BSM
Example

o
ln
9
, σ=15 % , S=$ 100,r =5 %( cc)
12
How do hedge against movements in stock price?

d 1=
X =$ 105, T =
Short a European call with:
Calculate
( )
N (d 1 )
100
0.15 2 9
+(0.05+
)∙
105
2
12
√
9
0.15
12
=−0.022
N ( d 1 )=0.4920

Because we shorted the European call – to hedge movements in stock price, will need to buy
0.4920 of stock
Gamma
 Gamma,
Γ , how does delta change when the underlying price changes
d Δ d2 c
1
= 2=
N ' (d 1 )
dS d S Sσ √ T

d1 ¿2
−1
¿
o Where
2
1 ¿
N ' ( d1 ) =
e
√2 π
Want Γ as close to zero as possible because it indicates that a simple delta hedge may be
sufficient i.e. no need to rebalance constantly
o If Γ=x , then for every unit shift of prices, the delta hedge will shift by
Example

9
, σ=15 % , S=$ 100,r =5 % ( cc )
12
o Same as before  already know that d 1=−0.022
Short a European call with:
X =$ 105, T =
Downloaded by Jayde Maluga (jaydemaluga@hotmail.com)
x
lOMoARcPSD|4591704
−0.022¿2
−1
¿
2
1 ¿
Γ=
e
√2 π
Γ=0.0307

Interpretation: ∆=0.4920 is only appropriate delta hedge for a small amount of stock
o After one unit change  the new delta hedge is:
∆=0.4920+ Γ =0.5227
Delta and gamma are useful for hedging
X =$ 105, T =

E.g. Short a European call with:

Want portfolio hedge to equal zero
9
, σ=15 % , S=$ 100,r =5 %(cc)
12
∆ derivativesecurity + N S ∆stock =0
N S =¿ number of stock
∆ stock=1 because change in stock is always the same
Where
o
o

However, to keep delta hedge relatively consistent (so no need to constantly adjust)  want
gamma hedge of the portfolio to be equal to zero
o Hedging with gamma gets rid of curviness of BSM
Γ derivative security + N o Γ o =0
N o=¿ number of options
o


K=S=$ 100 , r=5 % , σ=15 % ,T =
New option data:
ln
d 1=
( )
2
100
0.15
3
+(0.05+
)∙
100
2
12
√
3
0.15 ∙
12
1
N ( d1 ) =
e
√2 π
'
Γo=
3
12
Γ o i.e. calculate d 1 and put into formula
Calculate
o

Gamma hedge by buying new option i.e. different strike price, time to
maturity etc..
−0.204 2
2
1
√
=0.2042
2
=0.3907
∙ 0.3907=0.05209
3
100 ∙ 0.15∙
12
Substitute
Γ o =0.05209 into Γ derivative security +N o Γ o =0
Downloaded by Jayde Maluga (jaydemaluga@hotmail.com)
lOMoARcPSD|4591704
o
From previously

Γ derivative security =−0.0307
Negative because the call option was shorted
−0.0307+ N o ( 0.05209 ) =0
N o=


0.0307
=0.5893
0.05209
For every option shorted, need to buy 0.5893 of new option to gamma hedge the portfolio
o However, this does not fully ensure delta hedge
New delta hedged portfolio need to consider:
o One shorted European call and long call 0.5893 of new option
∆ new portfolio =∆original option + N 0 ∆ 0
o
From previous calculations:

∆ original option =−0.492


 negative because shorted the option
N 0=0.5893
∆ 0  is N (d 1 ) of new option i.e.
N ( 0.2042 )=0.5809
∆ new portfolio =−0.492+ ( 0.5893 ) ( 0.5809 )=−0.1497

Therefore, to delta hedge need:
−0.1497+N S × 1=0
N S =0.1497


o Need to buy 0.1497 of underlying asset for each short position
Still need to delta hedge - but doing so more effectively now
o NOT needed to buy as many shares
o Better protection against large price movements
Overall portfolio:
o Short original European call
o Long 0.5893 of new option with different strike price and time to maturity
o Buy 0.1487 of stock
Delta and gamma hedging put options
 A long call and a short put equals a forward contract
o Delta of forward contract equals to one  as no rate of change in price movements
– it is a constant one-to-one relationship
∆ call−∆ put=1
∆ put=∆ call−1
∆ put=N ( d 1 )−1
Downloaded by Jayde Maluga (jaydemaluga@hotmail.com)
lOMoARcPSD|4591704
o
Because of symmetrical shape of normal distribution can re-write
∆ put=−N (−d1 )

Gamma for a forward contract is derivative of 1 which equals zero
Γ call−Γ put=0
o
Gamma is the same for put and call options when have same X and T
Other Greeks
 Rho: change in c with change in interest rates
 Vega: change in c with change in volatility
 Theta: change in c with change in time
 Problems with rho and vega hedging:
o BSM assumes constant (or at least known) interest rates and volatility
o “Hedging” is removing uncertainty
o If need to hedge interest rates and volatility – NOT use BSM
 Need to use a different model
 Delta and gamma are valid procedures when BSM assumptions are satisfied
Volatility in the BSM Model
Observable inputs
 Most inputs of BSM are easily obtained
o Strike price and time to maturity are specified in option contract
 For actively traded stocks use last transaction price for spot price
 For less liquid stocks use average of bid and ask process
 Risk-free interest rate comes from Treasury security prices
o Use Treasury bill that matures closes to option’s expiration date
o Compute T-bill price from ask price
o Solve for the continuously compounded annual interest rate
Estimating volatility
 Volatility is hardest input to estimate, two methods:
Historical volatility
 Using statistics and past prices
o Stock return’s sample standard deviation
 Based on taking a sample from actual distribution
( )
St
St −1

E.g. take log returns i.e.

o To turn weekly measure into annual figure multiply by 52
o To turn daily measure into annual figure multiply by 252
How many observatins should use:
o More data the better
o But too long may contain information not relevant to current distribution
ln
for every week/day etc…
Downloaded by Jayde Maluga (jaydemaluga@hotmail.com)
lOMoARcPSD|4591704

How long time interval between observations?
o If too short, then market frictions and microstructure issues can significantly impact
estimation i.e. abnormally large orders
o Daily eliminates these problems
Implied volatility
 Implicit method using option market prices
 Solved by equation:
c market =BSM (σ implied)



Implied
volatility
Use (S , K , r , T ) and c market as inputs and then compute, σ by solving the
equation
o From given price and inputs, calculate the volatility that produced that price
o Calibration: estimates a parameter from a model by using market prices as inputs
 Guess and check work
Volatility not as simple as BSM suggest
o Changes for different stock prices (not major though)
Can compare expected volatility to actual volatility to make investment decisions
o If expected volatility > calculated volatility, then should then should sell as
overestimate the volatility
o If expected volatility < calculated volatility, then
should buy as market underestimate the
volatility of an option
Spot
price
Implied volatility graph
 BSM assume that the volatility stay the same no matter
the strike price
o In reality there is a smile
 The lowest implied volatility is where the option is at
the money
 The changes of volatility at different strike prices are not
major
o Usually insignificant
Strike
price
Reasons for the smile in option prices
 Volatility changes
o Buyer of option will face risk that value will change in response to:
 Stock price – hedged by taking appropriate position on underlying asset
 Volatility – risk ONLY hedged by taking position n security whose value also
changes in response to changes in stock’s volatility
o Options determined by stochastic volatility option pricing model
 Jumps in the stock price process
o BSM assume price touches every value
o In reality there are jumps
 Changes option values not be perfectly replicated by stocks and bonds
Downloaded by Jayde Maluga (jaydemaluga@hotmail.com)
lOMoARcPSD|4591704
Option prices determined by a jump diffusion option pricing model
 Volatilities implied from option prices can exhibit a smile
Option prices are affected by transaction costs
o Smile is most pronounced for deep-out-of-the-money near maturity put options
 Prices in middle depressed and hard/costly to hedge
o Suppose everyone agrees that the put option’s payoff is almost certain to be zero
 The short/the writer/the seller of the option has margin money posted
 The long/the buyer of the option does not have margin money posted
o If the short does not earn sufficient interest to compensate for having capital tied up
as margin money, then it is costing the short to remain short in a valueless position
o If the short wishes to avoid this cost, the short must close out her position she must
go long i.e. buy a put
o

Downloaded by Jayde Maluga (jaydemaluga@hotmail.com)
lOMoARcPSD|4591704
Lecture 9 – Forwards and Futures:
Forwards


Forward contract initially has zero value
o At maturity the long gets (S−F )  losses not bounded by zero like an option
o Negotiated between long and short position
When contract begins:
o Forward price stays constant for length of contract
o Underlying asset value changes
 Want to calculate the worth of forward contraction after created but before
expires
Pricing – cost of carry
 Need an asset at T - have three choices:
o 1. Wait and buy asset at T for S (T )
 Face risks in price movements
o 2. Buy asset at time 0 and hold until T
 Incur interest rate
o 3. Agree to buy asset at time T for F
 Fair price F is related to cost of carrying asset from time 0 to
T
rT
F=S e
o
rT
Se

o
 cost of carry  how much cost to own the asset
e
rT
 future value of spot price
Forward price NOT depend on future spot price or even expected value  purely
determined by opportunity cost/arbitrage
Using no arbitrage to price forwards
 E.g.1. S=$ 100, r=5 % ,T =6 months , F=$ 102
o When short the forward, get forward price in T=6 months
t=0
Long spot
Borrow
−$ 100
$ 100
T=6 months
S (T )
−$ 100∙ e
0.05∗6
12
=−$ 102.53
0
Short forward
$ 102−S (T )
0
−$ 0.53
Total
o To turn positive – just flip trades i.e. short spot, lend and long forward

E.g.2.
S=$ 100, r=5 % ( borrow ) ,r =3 % ( lend ) T =6 months , F=$ 102
t=0
Short spot
Lend
$ 100
−$ 100
Long forward
0
T=6
−S (T )
0.03∗6
$ 100∙ e 12 =+ $ 101.51
S ( T ) −102
Downloaded by Jayde Maluga (jaydemaluga@hotmail.com)
lOMoARcPSD|4591704
o

−$ 0.49
0
Total
If borrowing and lending rates differ cannot just reverse trade
 Need to incorpate transaction costs
Results:
o Cash-and-carry deems upper bound of $102.53 on forward price for e.g.1.
 i.e. if F>102.53 then e.g.1. is profitable
o Reverse cash-and-carry deems lower bound of $101.51 on forward price for e.g.2.
 i.e. F<101.51 then e.g.2. is profitable
o Any price between is possible and cannot be exploited for profit
 Demand/supply or other transaction costs will determine exact price
 Range increase with more transaction costs
Adding transaction costs
 E.g.1. S=$ 100, r=5 % ( borrow ) ,r =3 % ( lend ) T =6 months , F=$ 102
o BUT trading is costly
 A forward contract costs $0.25 per contract when initiated  NO costs on
expiration
 A stock incurs $1 transaction cost i.e. buying the spot today costs $101
selling yields $99
 What is the range of no arbitrage prices?
o Option 1: cash-and-carry
 When long spot need to pay extra dollar when buy and sell
 Initiating a forward contract costs $0.25
t=0
S ( T ) −$ 1
−($ 100+ $ 1)
$ 101.25
Long spot
Borrow
Short forward
Total
o
0.05∗6
−$ 101.25 e 12 =−$ 103 . 81
F−S (T )
F−$ 1−$ 103.81
−$ 0.25
0
Cash-and-carry: F ≤ $ 104.75

Option 2: reverse cash-and-carry
 When short a stock need to pay extra dollar when buy and sell
 Initiating a forward contract costs $0.25
t=0
Short spot
Lend
$ 100−$ 1
−$ 98.75
Long forward
Total
−$ 0.25
0

T=6
T=6
−( S ( T )−$ 1)
0.03∗6
$ 98.75 e 12 =$ 10 0.24
S ( T ) −F
$ 100.24−$ 1−F
 Reverse cash-and-carry: F ≥ $ 99.22
When add transaction costs widen the range of no-arbitrage prices
What is the existing value of a forward contract?
 If want to get out of forward contract – how does one value it?
 E.g.1. Last year gold prices were 1225/ounce with r=2%.
Downloaded by Jayde Maluga (jaydemaluga@hotmail.com)
lOMoARcPSD|4591704
Entered 2-year forward contract (bought) where price = 1275/ounce i.e. 1225e 2r
Today gold price $1325 and r=0.0114
 As gold prices higher – buyer of contract in better position as can buy at a
cheaper price than the market
o Seller wants to get out of the contract – how much will it cost
 Buy an offsetting forward contract (have a different forward price)
If wait till the end with offsetting forward contract:
o
o

t=1
T=2
1274.99−S(T )
0
Existing forward (short i.e.
sold)
Borrow
Buy New forward where
S ( T ) −1340.19
(F=PV ( 1325 ) =1310)
0
Total
1275−1340=−$ 65.20
−$ 65.20
o New Forward price = S erT =1325∗e0.0114 =1340.19
o

Can lock in a loss of
Alternatively, can get out now i.e. in t=1
t=1
Existing forward (short i.e.
sold)
Lend
Buy New forward where
T=2
0
1274.99−S(T )
−PV ( 65.20 )=−64.46
¿ $ 65.20
S ( T ) −1340.19
−64.46
0
( F=PV ( 1325 ) =1310)
Total
Because calculate the loss in t=2, can lend that amount in t=1 so that can offset the
difference in t=2
Value in two different forward contracts is the difference between forward prices
o Then take present value to get in terms of cash today
o

f ( t )=S ( t )−F × e−r T

Value of forward equals the spot price minus the strike price discounted backwards
o Same structures as BSM – however BSM takes into account the probability of certain
actions occurring i.e. exercising – n(d1) and n(d2)
o No probability involved with forwards as have to make transaction
Convenience yield
 Some assets give benefit of ownership i.e. dividends (owning a future does not give
entitlement to dividends)
o Need to add spot in payoff table for dividend  borrow or lend to match it

S−PV (¿)  need to subtract because agent will miss this cash flow

 Dividends are convenience yield for those who own it
 Conveience yield is a negative cost of carry
Other assets have cost of carry associated costs
o E.g. expensive to carry gold – need a vault and insurance
Downloaded by Jayde Maluga (jaydemaluga@hotmail.com)
lOMoARcPSD|4591704
o

If
 There is a cost to owning gold
Think of as a negative dividend – make line in payoff table for incurred costs

S +PV ( costs )  owner of asset gets paid for the underlying and storage
service
 Need to borrow more to cover carry cost
rT
then there must be additional cash flow i.e. cost of carry or convenience yield
F≠S e
Futures market



Are exchange traded contacts
o Traded anonymously on an exchange
o Long and short cannot monitor each other
o Contract between initial parties is immediately replaced with separate contracts with
the clearing house
 Clearinghouse is both long and short
Clearinghouse is between short and long
o They ensure all trades goes through and promise delivery
At expiration the following transaction occurs:
o Long position pays cash to clearing house who then pays the short position
o Short position gives good to clearinghouse who then gives it to long position
Margin requirements
 Clearinghouse manages credit risk by enforcing margin requirements
 Each trader posts margin i.e. ‘good faith deposit’
o Margin can be adjusted based on the creditworthiness of individuals
Extreme case:
 Positions:
o Long deposit treasuries with payoff equal to future price at expiration
o Short deposits underlying asset
 Not feasible:
o For short position – good may not even exist yet (need to grow) or maybe perishable
o Long position loses any leverage implicit in futures and borrowing the full margin
may be costly (transaction costs)
Realistic case
 Risk exposure is only to the change in the futures price between initiation and reversal
 Feasible solution:
o Require relatively small initial margin
o Deduct loses from/add gains to each trader’s margin account at close of each day
o If margin account falls below a maintenance margin (i.e. 2x daily limit move)
 Trader receives margin call i.e. need to top up margin account
 If trader fail to meet the call – the clearinghouse submits a marked order on
their behalf to close their position
o Clearinghouse perfectly hedged throughout
Downloaded by Jayde Maluga (jaydemaluga@hotmail.com)
lOMoARcPSD|4591704
Price used in the determination of the end—of-day mark-to-market cash flows is
termed the settlement price
E.g. New contract requires $ 2,000 initial margin
o When account falls below $ 1,500 get a margin and call and need to top up
account to $ 3000 (maintenance margin level)
o Buy a future contract @ $ 10,000 then next day at $ 11,500 and then @
o

$ 8750




Day 1: margin account is $ 2,000
Day 2: gains  margin account $ 2,000+$ 1,500=$ 3,500
Day 3: loses  margin account $ 3,500−$ 2,750=$ 750< $ 1500
 Receive margin call and need to top up account to $ 3,000 i.e.
deposit $ 2,250
Day 4: margin account is at $3,000
Standardisation of futures contract
 Anonymous trading in futures is facilitated through standardisation:
o Delivery periods
o Delivery locations
o Quantity, and
o Quality
 Allows margin accounts and marking-to-market to guarantee performance
o Buyers and sellers are able to match to multiple other buyers and sellers
o Traders who won’t reach margin calls can be replaced by other traders who can
o No negotiation between either party as have varying investment horizons
Delivery options and the short squeeze
 Short squeeze: someone buys up all the future contracts on an asset
 To reduce likelihood of squeeze there are typically many variants of the underlying asset i.e.
grades of wheat etc…
o Leads to cheapest-to-deliver underlying asset
 Party attempting a squeeze will go long the spot and long the futures
o At expiration suppose he/she holds all the spot
o Short needs either to buy the spot at whatever price is demanded in order to deliver
it, or to close short position by going long in an offsetting future contract position
o ‘squeezer’ will control futures market price
o harder for ‘squeezer’ to control available spot inventory with delivery options
 Possibility of squeeze reduced by having a lengthened delivery period
o More difficult to control a significant portion of the available cash/spot inventory for
a month than it is for a day
Downloaded by Jayde Maluga (jaydemaluga@hotmail.com)
lOMoARcPSD|4591704
Lecture 10 – Risk Management:
Hedging


m u=
One risk with derivatives is uncertainty of price of underlying asset
Binomial model – hedge with m:
−X DU −X DD
−X UU − X UD
md =
SUU −SUD
S DU −S DD
o
Can further hedge this:
mu −md
su −s d

 What happens to m
BSM – can delta or gamma hedge
o Delta: −N (d 1 )
as stock prices move
'
o
Gamma:
−N ( d 1)
sσ √ T
Different types of risk




Models used so far have been constructed excluding credit, liquidity and operational risk
o Risk only felt through uncertainty of prices
Credit risk: risk that counterparty fail to execute terms of a contract
o Natural conflict in forward contract that is negotiated independently
Liquidity risk: occurs when quantity impact of trade on price obtained
o Market power  buying large quantity of any security changes the price because
buyer needs to induce seller to seller
o Asymmetric information  belief seller informed and selling because security worth
less than current price – selling price declines
o Simply the inability to trade when need to
Operational (and legal) risk – inherent in running financial institution
Risk management

Corporate financial risk management is management of the risk of a firm’s equity capital
Assets, A t
Current assets
 Cash and cash equivalents
 Accounts receivable
 Inventories
Long-term assets
 Financial assets
 Property, plant and equipment
 Management decides:
Liabilities, Lt
 Accounts payable
 Financial liabilities
 Pension fund obligations
Equity, Et
 Ownership shares
Downloaded by Jayde Maluga (jaydemaluga@hotmail.com)
lOMoARcPSD|4591704
Purchase/sale of firm’s assets
 Includes paying out cash as dividends and raising cash by issuing shares
o Accumulation of liabilities i.e. debt
Equity is a residual claim on the firm
A t −Lt =Et  actions on assets/liabilities directly impact equity
o
o


For risk management purposes, equity holders are concerned with changes in equity value
over predetermined time interval i.e. [ t , t+ ∆ t ]
o Where ∆ t can be a day, week, month or year
∆ Et ≡ Et +∆ t −Et =∆ A t−∆ Lt
The loss distribution
 Assign a certain value that is acceptable to lose and calculate probability of that loss
o The risk of a change in the equity value is completely characterised by gain
distribution i.e.
Prob { ∆ E t ≤ x } =Prob { ∆ A t−∆ Lt ≤ x }

o In a time period cannot lose more than the change in equity
E.g. Firm will become insolvent if change in the value of equity is more negative than the
negative of the existing value of equity
P rob { ∆ Et ≤−E t }=Prob { ∆ At −∆ Lt ≤−Et }

When computing loss distribution need to take into account all risks:
o Market, credit, liquidity, operational
o Can be very difficult – different asset/liabilities have separate probability
distributions
Example
 Set-up:
o
o
o
r=5 % , σ 2=30 % ,T =1 and
A 0=$ 100, A T =$ 105, L0=$ 80.3, LT =K =$ 90
Debt is a single zero-coupon bond
 Major assumption – simplify to one large payoff at maturity
All assets jointly are lognormally distributed
1
ln (100 )+(0.05− (0.3 ¿2 )× 1,0. 32 ×1)
2
1
A t =ln ln ( A 0 ) + r− σ 2 T , σ 2 T =ln ¿
2
(

(
)
)
Hence:
Downloaded by Jayde Maluga (jaydemaluga@hotmail.com)
lOMoARcPSD|4591704
(
)
1 2
( A 0 ) +¿ r− 2 σ T
( x )−¿ ln
σ √T
ln ¿
(
)
1
( A0 ) +¿ r − 2 σ 2 T
≤¿
( AT )−¿ ln
σ √T
ln ¿
¿
P rob { A t ≤ x } =Pr ¿
o
Can find probability by identifying the associated z-score from normal distribution
(
)
1 2
( A 0 ) +¿ r− 2 σ T
( x )−¿ ln
σ √T
ln ¿
¿
N¿
o
x=K
If replace
(
(strike price) then:
)
1 2
( A 0 )+¿ r − 2 σ T
( K )−¿ ln
σ √T
ln ¿
(
1
)
( A 0 )−ln ( K ) +¿ r − 2 σ 2 T
ln
σ √T
−¿
¿
¿=N ¿
N¿
What is the probability that the asset value fall far enough to a point that is not
acceptable
In real-world need true probabilities (not risk-neutral probabilities)
o NO easy way to compute probability distribution of
o

∆ A−∆ L=( A T − A 0 )−(LT −L0) as not lognormal
o
Solution  approximate lognormal distribution with a normal distribution
A 0 ( 1+ rT ) , A20 σ 2 T =N (105,900)
AT ≈ N ¿

Loss distribution can be approximated as:
∆ E=∆ A−∆ L= A T − A 0−LT +L0 =−$ 4.70
Downloaded by Jayde Maluga (jaydemaluga@hotmail.com)
lOMoARcPSD|4591704
N (−4.70,900)
Value-at-Risk (VaR)

VaR is the acceptable loss associated with an assigned probability of
α%
Pr { ∆ Et ≤−Va Rα } =Pr { ∆ A t−∆ Lt ≤−Va Rα }=α
o
o
With probability α , the loss in equity value will be larger than the VaR (i.e. the
change in equity value will be more negative than –VaR)
 Choosing probability and then find z-score and convert to value
OR with a probability 1−α , the loss will be smaller than VaR
Computing the VaR
 1-year horizon where α =10 %=0.1
o Find the level at which there is a 10% chance of being there or below
−VaRα =z−score α + st .dev −mean
P r { ∆ Et ≤−Va R0.1 }=Pr { ∆ A t −∆ Lt ≤−Va R 0.1 }=0.1
Pr
{
} (
)
−Va R 0.1−(−4.7 )
∆ A−∆ L−(−4.7) −Va R 0.1−(−4.7)
≤
=N
=0.1
√ 900
√ 900
√ 900
−Va R 0.1−(−4.7 )
√ 900
o
−1
=N ( 0.1 )=−1.2816 (¿ table usingrow ∧coloum values)
Need to convert back into figures:
−Va R 0.1=−1.2816 × √ 900+ (−4.7 )=−$ 43.14

There is a 10% probability that losses are worse than $ 43.14
o VaR always represented by negative sign i.e. this above is a $43.14 loss
Downloaded by Jayde Maluga (jaydemaluga@hotmail.com)
lOMoARcPSD|4591704
Using VaR
 Given VaR, management/regulators can decide whether firm is properly capitalised for risks
that is taken by comparing VaRa to the firms existing equity capital position, Et
o Does firm have enough equity to absorb losses

If VaRa <Et then firm’s insolvency probability is less than tolerated
properly capitalised
o Can increase risk until get closer to the boundary

If

a % , and firm is
VaRa > Et then firm’s insolvency probability is greater than a % and firm needs to:
o Add more equity capital to balance sheet
o Restructure risk of assets and liabilities to reduce loss probability  hedge
E.g. management/regulators happy with at most 10% probability of going bankrupt
Va R 0.1=−43.14 and Et =19.7
o

Probability of default is 31%  distance of blue box along x-axis
o Doesn’t matter about distribution of the tail  ONLY interested in boundary value
o Blue region is undercapitalisation

Suppose buy put option on assets that pays if A T < $ 90
A T < $ 90 changes probability of losses to zero for certain level
o
o
o

Limit maximum loss in equity to $19.7
 10% chance of going below 14 (<19.7)
10% VaR smaller than current equity value
Problems with VaR
If used as sole measure has many shortcomings
o Not consider magnitude of losses beyond boundary
o VaR can penalise diversification BUT diversification of asset probability is beneficial
as it reduces risk
Downloaded by Jayde Maluga (jaydemaluga@hotmail.com)
lOMoARcPSD|4591704

E.g. Diversification penalises VaR
Loss
$0
$0.5
$1
Prob(loss A)
0.991
o



Prob(loss B)
0.991
0.0009
0.0009
What is least risky portfolio? $1 in A or $0.5 in A and B
If invest $1 in A 
VaR0.001 ( A )=$ 0
because NO loses occur with probability
greater than 0.001
If invest $0.5 in A and B
o Lose 0.000000081+0.0179838>0.001
Because VaR is larger – the diversified portfolio
would requite more capital
o This is the wrong decision
o Using ONLY VaR in this case would
encourage risk taking
Downloaded by Jayde Maluga (jaydemaluga@hotmail.com)
Prob(loss (A+B)/2)
0.99820081
0.0179838
0.000000081