Greeks for Master Finance

Instructor: Tingwei WANG
Email: [email protected]
Université Paris-Dauphine
For Master 224
September 2014

Risk Management
 Base asset
- Stocks



Volatility: standard deviation of return
VaR (Value at Risk)

- Bonds



Duration: sensitivity to interest rate change
credit risks

Risk Management of Derivatives
 The value of a derivative depends on the value of

underlying asset and other relevant parameters
 Greek letters describe the sensitivity of the value of a

derivative to the relevant parameters
 The focus of risk management of derivatives portfolio

is on the Greek letters

Review of Derivatives Basics
 What is forward/futures?
 What is an option?
 How to price forward/futures?
 How to price an option?

Types of derivatives
 Futures/Forward Contracts
- An obligation for both parties to exchange the underlying
asset for a pre-determined price
 Swaps
- Exchange of cash flows of different characteristics
 Options
- A right to buy/sell the underlying asset at the strike price

Forward
 A forward contract is an agreement to buy or sell an

asset at a certain time in the future for a certain price (the
forward price)
 It can be contrasted with a spot contract which is an

agreement to buy or sell immediately (outright purchase)
 It is traded in the OTC market

Futures
 Definition
- A futures contract is a standardized contract between two
parties to buy or sell a specified asset of standardized
quantity and quality for a price agreed upon today (the
futures price)
 Specifications need to be defined
- What can be delivered
- Where it can be delivered
- When it can be delivered
 Traded in exchange and settled daily

Forward Contracts vs Futures Contracts
FORWARDS
Private contract between 2 parties
Not standardized
Usually 1 specified delivery date
Settled at end of contract
Delivery or final cash
settlement usually occurs
Some credit risk
No basis risk
Predictible cash-flows

FUTURES
Exchange traded
Standardized contract
Range of delivery dates
Settled daily
Contract usually closed out
prior to maturity
Virtually no credit risk
Basis risk
Funding liquidity risk

Pricing Futures/Forward
 Price of Futures contract
r ( T t )
- Without dividend: Ft  St e
- With dividend:

Ft  St e( r q )(T t ) Ft  [St  PV ( D)]er (T t )

 Mark to market value of forward contract
 r (T t )
( Ft ,T  K )
- Long position: f  e
- Short position: f

 e r (T t ) ( K  Ft ,T )

Example
 ABC stock costs $100 today and is expected to pay a

quarterly dividend of $1.25. If the risk-free rate is 10%
compounded continuously, how much
is the 1-year forward price of ABC stock?
3

F0,1  100e  1.25e
0.1

i 0

0.025i

 $105.32

Options: Call and Put
 There are two basic types of options
- Call option
- Put option
 Call option
- A call option gives the holder of the option the right to buy
an asset by a certain date for a certain price
 Put option
- A put option gives the holder of the option the right to sell
an asset by a certain date for a certain price

Underlying Assets
 Stocks
- Single stock, basket of stocks, stock indices, …
 Bonds
- Treasury bonds, interest rate (caplets and floorlets), …
 Currencies
- Exchange rate option
 Commodities
- Agricultural products, metals, energy, …
 Futures contracts
- On stock indices, bonds, commodities, …

Example: payoff of a call option
 A European call option of Orange with a strike price

of €70 that expires in 6 months
Payoff (€)

30
20

10

Terminal
stock price(€)

0
40
-10

50

60

70

80

90

100

Example: payoff of a put option
 A European put option of Orange with a strike price

of €70 that expires in 6 months
Payoff (€)

30
20

10

Terminal
stock price(€)

0
40
-10

50

60

70

80

90

100

Option Premium (Price)
 Option gives one party the right to buy/sell the

underlying asset at the strike price while obligates
the other party to sell/buy the underlying asset upon
request
- Seller of the option is called the writer
- The writer should be compensated with a premium

 Contrast with futures/forwards
- Futures/forwards bring obligations to both parties
- The initial value of futures/forwards can be set to zero

P&L of Options
 P&L = Payoff – Option Premium
 Example: a European call option of Orange with a

strike price of €70 that expires in 6 months
P&L (€)

30

How much is the option premium here?

20
Break-even price

10
0

40 50 60 70

Terminal
stock price(€)

80 90 100
-10

Moneyness
 At-the-money option
- Spot stock price S is equal to the strike price K
 In-the-money option
- Spot stock price S is larger than the strike price K
- Deep-in-the-money: S>>K
 Out-of-the-money option
- Spot stock price S is smaller than the strike price K
- Deep-out-of-the-money: S<<K

Practical Issues: Dividends
 If a company distributes dividends during the life of

the option, the stock price will be decreased by the
amount of dividends
- The strike price should be adjusted by the amount

dividends on the ex-dividend date

 Example
- Consider a put option to sell 100 shares of a company for
$15 per share. Suppose that the company declares a $2
dividend. The strike price will be decreased by $2.

Practical Issues: Dividends
 For exchange-traded options and many over-the-

counter stock options, the strike price will not be
adjusted in the event of dividend payment!
 The option premium actually takes into account the

future dividend payment
- A call would be cheaper
- A put would be more expensive

Option Pricing (Single Stock/Index)
 Suppose you have an option that allows you to buy a

stock at $20 one year later. There are only two
possible outcomes of the stock price. It will be $30
with 50% probability and $10 with 50% probability.
The expected return of the stock is 10%. How much
is the option worth today?
 Traditional cash flow discounting

E[( X  K ) ] 0.5  (30  20)  0.5*0
c

 $4.55
1  re
1  10%

Law of One Price
 If there exists one portfolio with exactly the same

payoff as the option, then the price of the portfolio
should be the same as the option price
- This indicates that option is replicable if such portfolio exists
- The portfolio is called replicating portfolio
- For base assets, replicating payoff is impossible because

the value of base assets rely on fundamental variables

 What can be used in a replicating portfolio?
- Base assets: stocks, bonds, commodities, etc.

Replicating Option Payoff
 The value of an option on stock can be decomposed

into exercise value and time value
- Exercise value: stock

- Time value: bond

 Does there exist a portfolio composed of the

underlying stock and risk-free bonds that perfectly
replicates the payoff of the option?
- Assume we buy x units of stock and y units of risk-free

bonds and solve for x and y

Replicating Portfolio
 Suppose you have an option that allows you to buy a

stock at $20 one year later. There are only two
possible outcomes of the stock price. It will be $30
with 50% probability and $10 with 50% probability.
Su  30, Sd  10, cu  10, cd  0
 xSu  yerf 1  30  20  x  0.5


 rf
r f 1
y


5
e

 xSd  ye  0
 The value of the replicating portfolio today is

V0  xS0  y  0.5  20  5e

 rf

 10  5e

 rf

 c0

Generalized case
 At t=0, the stock price is S0 and risk-free rate is rf
 At t=T, the stock price either goes up or down. If up,

the stock price is Su=uS0 and the call will be worth cu;
if down, the stock price is Sd=dS0 and the call will be
worth cd
Up

Su

S0
 Replicating portfolio
cu  cd
Down

Sd
x
rf T

 xSu  ye  cu
Su  S d



rf T
 xSd  ye  cd
 y  e  rf T cd Su  cu Sd  e  rf T ucd  dcu

Su  S d
ud

Value of the call option
 By law of one price, today’s value of the call option

should be equal to today’s value of the replicating
portfolio
cu  cd
 r f T ucd  dcu
c0  V0  xS0  y 
S0  e
Su  S d
ud
e

 rf T

e

 rf T

e d
r T u  e
cu  e f
cd
ud
ud
rf T

rf T

(qu cu  qd cd )

e f d
u e f d
where qu 
, qd 
ud
ud
r T

r T

Risk-neutral Probability
 Interestingly, qu plus qd is equal to 1

e f d u e f
qu  qd 

1
ud
ud
 The expected payoff of the call option is actually
calculated using q-probability rather than pprobability
rT

c0  e

 rf T

rT

(qu cu  qd cd )  e

 rf T

E Q [c]

- Q-probability is called risk-neutral probability
- Under risk-neutral probability, the expected payoff is

discounted by risk-free rate and the expected return for all
assets is risk-free rate

Forward Pricing in Q-measure
 The payoff of a long forward contract is

X  ST  F
 The forward value today is the expected forward

payoff under risk-neutral probability measure
discounted by risk-free rate
f 0  e  rT E Q [ ST  F ]
 e  rT E Q [ ST ]  e  rT F
 e  rT  e rT S0  e  rT F
 S0  e  rT F

Multi-period Model
 When the number of periods increases, the time

interval shrinks and the stock price movements
become smaller
 Path distribution
- For n-period model, n+1 possible outcomes
- To reach the highest node, there is only one path: up, up,
up,…all the way up
- To reach the lowest node, also only one path: down, down,
…, all the way down

Continuous-time option pricing
 When the number of periods approaches infinity, the

stock price moves continuously and terminal prices
span the whole set of positive numbers
 If we let n go to infinity in the binominal pricing

formula, we get the continuous-time version of
option pricing formula
- First proposed by Black & Scholes (1973) using partial

differential equation

Black-Scholes Formula
 European option on stock without dividend

c0  S0 N (d1 )  Ke  rT N (d 2 )
p0  Ke  rT N (d 2 )  S0 N (d1 )
ln( S0 K )  (r   2 2)T
where d1 
,
 T
ln( S0 K )  (r   2 2)T
d2 
 d1   T
 T
N ( x) is the cumulative normal distribution function

Example
 Current stock price is $42. A call with strike price $40

will expire in 6 months. The risk-free interest rate is
10% per annum and the volatility is 20% per annum.
What is the call price? If it is a put?
 Solution
S0  42, K  40, r  0.1,   0.2, T  0.5
d1 

ln( S0 K )  (r   2 2)T

 T

 0.7693, d 2  d1   T  0.6278

c  S0 N (d1 )  Ke  rT N (d 2 )  4.76
p  Ke rT N (d 2 )  S0 N (d1 )  0.81

Generalized Black-Scholes Formula
 Black-Scholes formula can only be applied to a

single stock with no dividend payments during the
life of option
 We can generalize Black-Scholes formula to price an

European option on assets with intermediate cash
flows or other derivatives, e.g. stock with dividends,
currencies or futures contract

Generalized Black-Scholes Formula
 Change S0 into prepaid forward price

c0  F0P N (d1 )  Ke  rT N (d 2 )
p0  Ke  rT N (d 2 )  F0P N (d1 )
ln( F0P K )  (r   2 2)T
where d1 
,
 T
ln( F0P K )  (r   2 2)T
d2 
 d1   T
 T

Option on stocks w/ dividends
 Continuous dividends (stock index)

F0P  S0e  qT
c0  S0 e  qT N (d1 )  Ke  rT N (d 2 )
p0  Ke  rT N (d 2 )  S0e  qT N (d1 )
where d1 
d2 

ln ( S0 e  qT K )  (r   2 2)T

 T
ln ( S0 e  qT K )  (r   2 2)T

 T

,
 d1   T

Option on stocks w/ dividends
 Discrete dividends

F0P  S0  PV ( D)
c0  [ S0  PV ( D)]N (d1 )  Ke  rT N (d 2 )
p0  Ke  rT N (d 2 )  S0 N (d1 )
where d1 
d2 

ln[( S0  PV ( D) K ]  (r   2 2)T

 T
ln[( S0  PV ( D) K ]  (r   2 2)T

 T

,
 d1   T

Options on Currencies
 Continuous compounding interest rates

F0P  x0 e
c0  x0 e

 rf T

 rf T

p0  Ke

 rT

N (d1 )  Ke  rT N (d 2 )
N (d 2 )  x0e

where d1 
d2 

ln ( x0 e

 rf T

 rf T

N (d1 )

K )  (r   2 2)T

 T
ln ( x0 e

 rf T

,

K )  (r   2 2)T
 d1   T
 T

Options on Futures Contract
 Black’s Formula

F0P  F0 e  rT
c0  F0 e  rT N (d1 )  Ke  rT N (d 2 )  e  rT [ F0 N (d1 )  KN (d 2 )]
p0  e  rT [ KN (d 2 )  F0 N (d1 )]
where d1 
d2 

ln ( F0 e  rT K )  (r   2 2)T

 T
ln ( F0 e  rT K )  (r   2 2)T

 T



ln ( F0 K )  ( 2 2)T

 T

 d1   T

,

Factors that affect option price
 From Black-Scholes formula,
c0  S0 N (d1 )  Ke rT N (d 2 ), p0  Ke  rT N (d 2 )  S0 N (d1 )

where d1 

ln( S0 K )  (r   2 2)T

, d 2  d1   T

 T
we can see there are five factors that affect option
price
- Spot price of underlying asset (S0)
- Strike price (K)
- Maturity (T)

- Volatility (  )
- Risk-free interest rate (r)

How Factors Change Option Value
Factors (+)

Call option

Put option

Spot price of underlying asset

+

–

Strike price

–

+

Maturity

+

?

Volatility

+

+

Interest rate

+

–

Greek Letters
 Greek letters describe the sensitivity of option price

to one of its determinants, ceteris paribus
- Measure sensitivity: partial derivatives

 Greek letters are of great importance in risk

management
- They measure the risk exposure of holding an option to all

the possible factors
- Traders contruct hedging portofolio based on Greek letters

Delta
 Delta (D) is the rate of change of the option price with

respect to the underlying asset price
Option price

Slope = D
ct
St

Stock price

Delta
 Calculate delta
- Call option
 rT


S
N
(
d
)

Ke
N (d 2 ) 
ct
t
1

Dt 

 N (d1 )  0
St
St
- Put option
 rT


Ke
N (d 2 )  St N (d1 ) 
ct

Dt 

 N (d1 )  1  0
St
St
- Continuous proportional dividend
 q (T t )
put
 q (T t )
Dcall

e
N
(
d
),
D

e
[ N (d1 )  1]
1
t
t

Discrete Delta
 Continuous delta results in losses when asset price

either goes up or down
- Solution: discrete delta
Option price

cu

cu  cd
D
Su  S d

ct
cd
Sd

St

Su Stock price

 Compare with the replicating portfolio in binominal

tree model

cu  cd

x  S  S
 xSu  yerf T  cu

u
d



rf T
 xSd  ye  cd
 y  e  rf T cd Su  cu Sd  e  rf T ucd  dcu

Su  S d
ud
c c
 r T ucd  dcu
c0  xS0  y  u d S0  e f
Su  S d
ud

c0  S0 N (d1 )  Ke rT N (d2 )  D0 S0  [e rT KN (d2 )]
stocks

bonds

Delta

Gamma
 Gamma (G) is the rate of change of delta (D) with

respect to the price of the underlying asset
- Gamma addresses delta hedging errors caused by

curvature
Call price
C’’
C’

C

S

S’

Stock price

Gamma & Vega

Vega
 Vega (n) is the rate of change of the option price with

respect to implied volatility
- Vega is always positive for vanilla options but not always for

exotic options

 Real volatility V.S. Implied Volatility
- Real volatility: unobservable, calculated using a period of
historical returns
- Implied volatility: observable, backed out from vanilla option
prices using Black-Scholes formula

Constant volatility
 In the Black-Scholes model, the volatility of the

underlying asset is constant, which is not true in the
real market

Volatility Smile

Volatility Surface

Implied
volatility

Maturity T
Moneyness K/S0

Theta

European Call Premium
Option
premium

Time value
Intrinsic value
Max(St-K,0)

K
Out-of-the-money

Stock price St

European Put premium
Option
premium

Negtive time value

pt  ( K  St )  ct  K (1  e r (T t ) )  0

Time value
Intrinsic value
Max(K-St,0)
K
In-the-money

Stock price St

Rho
 Rho is the rate of change of the value of the option

price with respect to the interest rate
- For currency options there are 2 rhos

 Calculations
- European call on non-dividend paying stock
rho(call )  KTe rT N (d2 )  0
- European put on non-dividend paying stock
rho( put )   KTe rT N (d2 )  0

Rho

Example: delta hedging
 Principle
- Construct a self-financing portfolio with stocks and risk-free
bonds to replicate the value of an option
- Self-financing: no additional capital added to the portfolio
during the hedging process

 Initiation
- At t=0, the bank sells an option and earns the option
premium C0
- Then the banks buys D 0 units of stocks and C0  D0 S0 units
of risk-free bonds

Delta hedging (Cont’d)
 On day t, the portfolio value is
t  Dt St  Bt
 On day t+1, before rebalancing the portfolio, the

portfolio value is1
t 1  Dt St 1  Bt e

r

252

 At the end of day, the trade rebalances the portfolio

with the updated delta
 t 1  Dt 1St 1  Bt 1

where Bt 1   t 1  Dt 1St 1

 Cumulative P/L of hedging (hedging error) is
et  t  Ct

 The final P/L (total hedging error) is
eT  T  CT
 DT 1ST  BT 1e

r

1
252

 max( ST  K , 0)

Decomposition of option value change
 Daily change of option value

Ct 1 (St 1 )  Ct (St )  Ct 1 (St 1 )  Ct (St 1 )  Ct (St 1 )  Ct (St )
Time value

Price risk

Ct ( St )
1  2Ct ( St )
2
Ct ( St 1 )  Ct ( St ) 
( St 1  St ) 
(
S

S
)
t 1
t
St
2 St 2
1
 Dt ( St 1  St )  Gt ( St 1  St ) 2
2
Delta exposure Gamma exposure

Hedging error
 Option value change
1
1
Ct 1 ( St 1 )  Ct (St )  
 Dt ( St 1  St )  Gt ( St 1  St ) 2
252
2
 Hedging portfolio value change
t 1  t  Dt (St 1  St )  Bt (e

r

1
252

 1)

 Daily Hedging error

 t 1  ( t 1   t )  (Ct 1  Ct )
 Bt (e

r

1
252

 1)   

1
1
 Gt ( St 1  St ) 2
252 2

Delta of Futures/Forwards
 Delta of futures contract
r ( T t )
D  er (T t )
- Without dividend: Ft  St e
- With dividend:

Ft  St e( r q )(T t )

D  e( r q )(T t )

 Delta of forward contract
 r (T t )
 St D  1
- Long position: f  Ke
- Short position:

f  St  Ke r (T t ) D  1

Greeks of a portfolio
 Greeks of a portfolio are simply the weighted greeks

of each individual asset in the portfolio
 Example: delta of a portfolio
- Suppose a portfolio consists of a quantity wi of asset i with
Di, the delta of the portfolio is given by
n

D p   wi Di
i 1

Example
 Suppose a bank has a portfolio of following assets:
- 1. A long position in 1000 call options with strike price 30
and an expiration date in 3 months. The delta of each
option is 0.55
- 2. A short position in 500 put options with strike price 20
and an expiration date in 6 months. The delta of each put
options is -0.3
- 3. A long position in 100 shares of underlying stocks
- 4. A short position in a forward contract on 200 shares of
underlying stocks
n

D p   wi Di  1000  0.55  500  (0.3)  100 1  200 1  600
i 1

Gamma Neutral Portfolio
 A delta-neutral portfolio is not gamma-neutral

because the underlying asset or forward/futures
contract on the underlying asset both have zero
gamma
 Solution: use a traded option with gamma GT
- Suppose a portfolio has a gamma equal to G
- Adding the traded option, the portfolio gamma becomes
wGT  G  0
w

G
GT

Example
 A bank writes exotic options to its clients. It accumulates

a negative gamma of -6.000 but is delta-neutral. To
neutralize the negative gamma exposure, the bank
decides to buy call option with a delta of 0.6 and a
gamma of 1.50. Should the bank buy or sell this call
option? And how many?
 Solution
G
6000
w

 4000
GT
1.5
- The bank should buy 4000 call options

 After adding the call option to the portfolio, the delta

of the portfolio is not zero anymore!

D p  0  wDT  4000  0.6  2400
 To make the portfolio delta-neutral again, the bank

should sell 2400 units of underlying asset or sell
certain amounts of forward/futures contract on this
asset

Vega Neutral Portfolio
 The method of constructing a vega neutral portfolio

is the same as gamma neutral portfolio
 Suppose a traded option has a vega of n T . To

neutralize a portfolio with vega n , the number of
option needed is
wn T n  0
w

n
nT

Gamma-vega Neutral
 A gamma neutral portfolio is in general not vega

neutral, and vice versa
 It is possible to make a delta neutral portfolio both

gamma neutral and vega neutral
- With one option, it is only possible to neutralize one greek

letter in addition to delta
- With two options, two greek letters can be neutralized at the
same time by solving 2 simultaneous equations

Example: gamma-vega neutral
 A delta neutral portfolio has a gamma of -5,000 and a

vega of -8,000. A traded option has a gamma of 0.5, a
vega of 2.0 and a delta of 0.6. Another traded option has
a gamma of 0.8, a vega of 1.2 and a delta of 0.5.
 Let w1 and w2 be the quantities of the two options

5000  0.5w1  0.8w2  0

8000  2.0w1  1.2w2  0
 w1  400
D p  400  0.6  6000  0.5  3240

 w2  6000

Delta, Theta and Gamma
 Taylor’s expansion on the value of a portfolio


1  2
2
d 
dt 
dS 
(
dS
)
t
S
2 S 2
1
 dt  DdS  G(dS ) 2
2

 Take expectation under Q on both sides
E Q [

d
dS 1
dS
]  dt  DSt E Q [ ]  GSt2 E Q [( ) 2 ]

S
2
S

 Under Q, the expected return of any asset is r
dS
d
E Q [ ]  r  dt , E Q [
]  r  dt
S


Cont’d
1
dS
rdt  dt  DrSdt  GSt2 E Q [( )2 ]
2
S
dS
dS
dS
)  E Q [( ) 2 ]  E Q [ ]2   2 dt
S
S
S
dS
dS
E Q [( ) 2 ]   2 dt  E Q [ ]2   2 dt  (rdt ) 2   2 dt
S
S

Var (

1 2 2
rdt  dt  DrSdt  GS  dt
2

 The value  of a portfolio composed of derivatives

on a non-dividend-paying stock satisfies the
differential equation
1
  rS D   2 S 2G  r
2

Problems with Black-Scholes
 Black-Scholes is a very nice model that is consistent

with all the properties of options and has a neat
solution to the option price
 But, it is based on many strong assumptions that are

not realistic in the real market
- Log-normal distribution of stock price
- Continuous trading w/o transaction cost
- Constant volatility and interest rate

- ……