Greeks for Master Finance

8/9/2019 Greeks for Master Finance

1/73

Instructor: Tingwei WANG

Email: [email protected]

Universit Paris-Dauphine

For Master 224

September 2014


2/73

Risk Management Base asset

- Stocks

Volatility: standard deviation of return

VaR (Value at Risk)

- Bonds

Duration: sensitivity to interest rate change

credit risks


3/73

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 aderivative to the relevant parameters

The focus of risk management of derivatives portfoliois on the Greek letters


4/73

Review of Derivatives Basics What is forward/futures?

What is an option?

How to price forward/futures? How to price an option?


5/73

Types of derivatives Futures/Forward Contracts

- An obligation for both parties to exchange the underlyingasset 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


6/73

ForwardA forward contract is an agreement to buy or sell an

asset at a certain time in the future for a certain price (theforward price)

It can be contrasted with a spot contract which is anagreement to buy or sell immediately (outright purchase)

It is traded in the OTC market


7/73

Futures Definition

- A futures contract is a standardized contract between twoparties to buy or sell a specified asset of standardized

quantity and quality for a price agreed upon today (thefutures 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


8/73

Forward Contracts vs Futures Contracts

Private contract between 2 parties Exchange traded

Not standardized Standardized contract

Usually 1 specified delivery date Range of delivery dates

Settled at end of contract Settled daily

Delivery or final cashsettlement usually occurs

Contract usually closed outprior to maturity

FORWARDS FUTURES

Some credit risk Virtually no credit risk

No basis risk Basis risk

Predictible cash-flows Funding liquidity risk


9/73

Pricing Futures/Forward Price of Futures contract

- Without dividend:

- With dividend:

Mark to market value of forward contract

- Long position:

- Short position:

( )r T t

t tF S e

( )( )r q T t

t tF S e

( )

,( )r T t

t T

f e F K

( )

,( )r T t

t Tf e K F

( )[ ( )] r T tt tF S PV D e


10/73

ExampleABC 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

0.1 0.025

0,1

0

100 1.25 $105.32i

i

F e e


11/73

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 buyan asset by a certain date for a certain price

Put option

- A put option gives the holder of the option the right to sellan asset by a certain date for a certain price


12/73

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,


13/73

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

of70 that expires in 6 months

30

20

10

0

-10

40 50 60 70 80 90 100

Payoff ()

Terminalstock price()


14/73

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

of70 that expires in 6 months

30

20

10

0

-10

40 50 60 70 80 90 100

Payoff ()



15/73

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

underlying asset at the strike price while obligatesthe 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


16/73

P&L of Options P&L = Payoff Option Premium

Example: a European call option of Orange with astrike price of70 that expires in 6 months

30

20

10

0

-10

40 50 60 70

80 90 100

P&L ()


Break-even price

How much is the option premium here?


17/73

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


18/73

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

the option, the stock price will be decreased by theamount of dividends

- The strike price should be adjusted by the amountdividends 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 $2dividend. The strike price will be decreased by $2.


19/73

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

counter stock options, the strike price will not beadjusted in the event of dividend payment!

The option premium actually takes into account thefuture dividend payment

- A call would be cheaper

- A put would be more expensive


20/73

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 twopossible 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 muchis the option worth today?

Traditional cash flow discounting

[( ) ] 0.5 (30 20) 0.5*0$4.55

1 1 10%e

E X Kc

r


21/73

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

payoff as the option, then the price of the portfolioshould 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 becausethe value of base assets rely on fundamental variables

What can be used in a replicating portfolio?

- Base assets: stocks, bonds, commodities, etc.


22/73

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 theunderlying 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


23/73

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

stock at $20 one year later. There are only twopossible outcomes of the stock price. It will be $30

with 50% probability and $10 with 50% probability.

The value of the replicating portfolio today is

1

1

30, 10, 10, 0

0.530 20

50

f

ff

u d u d

r

u

rr

d

S S c c

xxS ye

y exS ye

0 0 00.5 20 5 10 5f fr rV xS y e e c


24/73

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 beworth cd

Replicating portfolio

f

ff f

u dr T

u u u d

r Tr T r T d u u d d ud d

u d

c cx

xS ye c S S

c S c S uc dcxS ye c y e eS S u d

S0

Up

Down

Su

Sd


25/73

Value of the call option By law of one price, todays value of the call option

should be equal to todays value of the replicating

portfolio

0 0 0 0

( )

,

f

f f

f f

f

f f

r Tu d d u

u d

r T r T r T r T

u d

r T

u u d d

r T r T

u d

c c uc dcc V xS y S eS S u d

e d u ee c e c

u d u d

e q c q c

e d u e d where q q

u d u d


26/73

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

The expected payoff of the call option is actuallycalculated using q-probability rather than p-probability

- Q-probability is called risk-neutral probability

- Under risk-neutral probability, the expected payoff isdiscounted by risk-free rate and the expected return for allassets is risk-free rate

0

( ) [ ]f fr T r T Q

u u d d

c e q c q c e E c

1f fr T r T

u d

e d u eq q

u d u d


27/73

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

The forward value today is the expected forwardpayoff under risk-neutral probability measurediscounted by risk-free rate

TX S F

0

0

0

[ ][ ]

rT Q

T

rT Q rT

T

rT rT rT

rT

f e E S Fe E S e F

e e S e F

S e F


28/73

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

interval shrinks and the stock price movementsbecome 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


29/73

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

stock price moves continuously and terminal pricesspan the whole set of positive numbers

If we let n go to infinity in the binominal pricingformula, we get the continuous-time version ofoption pricing formula

- First proposed by Black & Scholes (1973) using partialdifferential equation


30/73

Black-Scholes Formula European option on stock without dividend

0 0 1 2

0 2 0 1

2

01

2

02 1

( ) ( )

( ) ( )

ln( ) ( 2) ,

ln( ) ( 2)

( ) is the cumulative normal distribution function

rT

rT

c S N d Ke N d

p Ke N d S N d

S K r T where d

T

S K r T d d TT

N x


31/73

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

will expire in 6 months. The risk-free interest rate is10% per annum and the volatility is 20% per annum.

What is the call price? If it is a put?

Solution

0

20

1 2 1

0 1 2

2 0 1

42, 40, 0.1, 0.2, 0.5

ln( ) ( 2)0.7693, 0.6278

( ) ( ) 4.76

( ) ( ) 0.81

rT

rT

S K r T

S K r T d d d T

T

c S N d Ke N d

p Ke N d S N d


32/73

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

single stock with no dividend payments during thelife of option

We can generalize Black-Scholes formula to price anEuropean option on assets with intermediate cashflows or other derivatives, e.g. stock with dividends,

currencies or futures contract


33/73

Generalized Black-Scholes Formula Change S0 into prepaid forward price

0 0 1 2

0 2 0 1

2

01

2

02 1

( ) ( )

( ) ( )

ln( ) ( 2) ,

ln( ) ( 2)

P rT

rT P

P

P

c F N d Ke N d

p Ke N d F N d

F K r Twhere d

T

F K r Td d T

T


34/73

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

0 0

0 0 1 2

0 2 0 1

2

01

2

02 1

( ) ( )( ) ( )

ln ( ) ( 2) ,

ln ( ) ( 2)

P qT

qT rT

rT qT

qT

qT

F S e

c S e N d Ke N d p Ke N d S e N d

S e K r T where d

TS e K r T

d d TT


35/73

Option on stocks w/ dividends Discrete dividends

0 0

0 0 1 2

0 2 0 1

2

01

2

02 1

( )

[ ( )] ( ) ( )( ) ( )

ln[( ( ) ] ( 2) ,

ln[( ( ) ] ( 2)

P

rT

rT

F S PV D

c S PV D N d Ke N d p Ke N d S N d

S PV D K r T where d

TS PV D K r T

d d TT


36/73

Options on Currencies Continuous compounding interest rates

0 0

0 0 1 2

0 2 0 1

2

01

2

02 1

( ) ( )

( ) ( )

ln ( ) ( 2) ,

ln ( ) ( 2)

f

f

f

f

f

r TP

r T rT

r TrT

r T

r T

F x e

c x e N d Ke N d

p Ke N d x e N d

x e K r Twhere d

T

x e K r Td d T

T


37/73

Options on Futures Contract Blacks Formula

0 0

0 0 1 2 0 1 2

0 2 0 1

2 2

0 01

2

02 1

( ) ( ) [ ( ) ( )][ ( ) ( )]

ln ( ) ( 2) ln ( ) ( 2) ,

ln ( ) ( 2)

P rT

rT rT rT

rT

rT

rT

F F e

c F e N d Ke N d e F N d KN d p e KN d F N d

F e K r T F K Twhere d

T T

F e K r Td d T

T


38/73

Factors that affect option price From Black-Scholes formula,

we can see there are five factors that affect optionprice

- Spot price of underlying asset (S0)

- Strike price (K)

- Maturity (T)

- Volatility ( )

- Risk-free interest rate (r)

0 0 1 2 0 2 0 1

2

01 2 1

( ) ( ), ( ) ( )

ln( ) ( 2) ,

rT rT c S N d Ke N d p Ke N d S N d

S K r T where d d d T

T


39/73

How Factors Change Option Value

Factors (+) Call option Put option

Spot price of underlying asset +

Strike price +

Maturity + ?

Volatility + +

Interest rate +


40/73

Greek Letters Greek letters describe the sensitivity of option price

to one of its determinants, ceter is paribus

- Measure sensitivity: partial derivatives

Greek letters are of great importance in riskmanagement

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

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


41/73

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

respect to the underlying asset price

Option price

St

ctSlope = D

Stock price


42/73

Delta Calculate delta

- Call option

- Put option

- Continuous proportional dividend

1 2

1

( ) ( )

( ) 0

rT

tt

tt t

S N d Ke N d c

N dS S

D

2 1

1

( ) ( )( ) 1 0

rT

ttt

t t

Ke N d S N dcN d

S S

D

( ) ( )

1 1( ), [ ( ) 1]t tcall q T t put q T t e N d e N d D D


43/73

Discrete Delta Continuous delta results in losses when asset price

either goes up or down

- Solution: discrete delta

Option price

St

ct

Stock priceSuSd

cu

cd

u d

u d

c c

S S

D


44/73

Compare with the replicating portfolio in binominaltree model

0 0 0

f

ff f

f

u dr T

u u u d r T

r T r T d u u d d ud d

u d

r Tu d d u

u d

c cx

xS ye c S S

c S c S uc dcxS ye c y e eS S u d

c c uc dcc xS y S e

S S u d

0 0 1 2 0 0 2( ) ( ) [ ( )]rT rT c S N d Ke N d S e KN d D

stocks bonds


45/73

Delta


46/73

Gamma Gamma () is the rate of change of delta () with

respect to the price of the underlying asset

- Gamma addresses delta hedging errors caused by

curvature

S

C

Stock priceS

Call price

C

C


47/73

Gamma & Vega


48/73

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 ofhistorical returns

- Implied volatility: observable, backed out from vanilla optionprices using Black-Scholes formula


49/73

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

underlying asset is constant, which is not true in thereal market


50/73

Volatility Smile


51/73

Volatility Surface

Moneyness K/S0

Maturity T

Impliedvolatility


52/73

Theta


53/73

European Call PremiumOptionpremium

Stock price St

Intrinsic valueMax(St-K,0)

Time value

K

Out-of-the-money


54/73

European Put premium

Optionpremium

Stock price St

Intrinsic valueMax(K-St,0)

Time value

K

In-the-money

Negtive time value( )( ) (1 ) 0r T t

t t tp K S c K e


55/73

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

- European put on non-dividend paying stock

2( ) ( ) 0rT

rho call KTe N d

2( ) ( ) 0rT

rho put KTe N d


56/73

Rho


57/73

Example: delta hedging Principle

- Construct a self-financing portfolio with stocks and risk-freebonds to replicate the value of an option

- Self-financing: no additional capital added to the portfolioduring the hedging process

Initiation

- At t=0, the bank sells an option and earns the optionpremium C0

- Then the banks buys units of stocks and unitsof risk-free bonds

0D 0 0 0C S D


58/73

Delta hedging (Contd) On day t, the portfolio value is

On day t+1, before rebalancing the portfolio, the

portfolio value is

At the end of day, the trade rebalances the portfoliowith the updated delta

t t t t S B D

1

2521 1

r

t t t t S B e

D

1 1 1 1

1 1 1 1

t t t t

t t t t

S B

where B S

D

D


59/73

Cumulative P/L of hedging (hedging error) is

The final P/L (total hedging error) is

t t te C

1

2521 1 max( ,0)

T T T

r

T T T T

e C

S B e S K

D


60/73

Decomposition of option value change Daily change of option value

1 1 1 1 1 1( ) ( ) ( ) ( ) ( ) ( )t t t t t t t t t t t t C S C S C S C S C S C S

Time value Price risk

22

1 1 12

2

1 1

( ) ( )1( ) ( ) ( ) ( )

2

1( ) ( )

2

t t t t t t t t t t t t

t t

t t t t t t

C S C S C S C S S S S S

S S

S S S S

D G

Delta exposure Gamma exposure


61/73

Hedging error Option value change

Hedging portfolio value change

Daily Hedging error

2

1 1 1 1

1 1( ) ( ) ( ) ( )

252 2t t t t t t t t t t C S C S S S S S D G

1

2521 1( ) ( 1)

r

t t t t t t S S B e

D

1 1 1

1

22521

( ) ( )

1 1( 1) ( )

252 2

t t t t t

r

t t t t

C C

B e S S

G


62/73

Delta of Futures/Forwards Delta of futures contract

- Without dividend:

- With dividend:

Delta of forward contract

- Long position:

- Short position:

( )r T t

t tF S e

( )( )r q T t t tF S e

( )r T te D

( )( )r q T t e D

( )r T t

tf S Ke

( )r T t

tf Ke S 1D

1D


63/73

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 withDi, the delta of the portfolio is given by

1

n

p i ii w

D D


64/73

Example Suppose a bank has a portfolio of following assets:

- 1. A long position in 1000 call options with strike price 30and 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 putoptions is -0.3

- 3. A long position in 100 shares of underlying stocks

- 4. A short position in a forward contract on 200 shares ofunderlying stocks

1

1000 0.55 500 ( 0.3) 100 1 200 1 600n

p i i

i

w

D D


65/73

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

because the underlying asset or forward/futurescontract on the underlying asset both have zero

gamma

Solution: use a traded option with gamma

- Suppose a portfolio has a gamma equal to

- Adding the traded option, the portfolio gamma becomes

TG

G

0T

T

w

w

G G

G

G


66/73

ExampleA bank writes exotic options to its clients. It accumulates

a negative gamma of -6.000 but is delta-neutral. Toneutralize the negative gamma exposure, the bank

decides to buy call option with a delta of 0.6 and agamma of 1.50. Should the bank buy or sell this calloption? And how many?

Solution

- The bank should buy 4000 call options

60004000

1.5T

w G

G


67/73

After adding the call option to the portfolio, the deltaof the portfolio is not zero anymore!

To make the portfolio delta-neutral again, the bankshould sell 2400 units of underlying asset or sellcertain amounts of forward/futures contract on this

asset

0 4000 0.6 2400p T

wD D


68/73

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 . Toneutralize a portfolio with vega , the number ofoption needed is

Tn

n

0T

T

w

w

n n

n

n


69/73

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 bothgamma neutral and vega neutral

- With one option, it is only possible to neutralize one greekletter in addition to delta

- With two options, two greek letters can be neutralized at thesame time by solving 2 simultaneous equations


70/73

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

vega of -8,000. A traded option has a gamma of 0.5, avega 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

1 2

1 2

1

2

5000 0.5 0.8 0

8000 2.0 1.2 0

400

6000

w w

w w

w

w

400 0.6 6000 0.5 3240pD


71/73

Delta, Theta and Gamma Taylors expansion on the value of a portfolio

Take expectation under Q on both sides

Under Q, the expected return of any asset is r

22

2

2

1( )

2

1 ( )2

d dt dS dS t S S

dt dS dS

D G

2 21[ ] [ ] [( ) ]2

Q Q Q

t t

d dS dS E dt S E S E

S S

D G

[ ] , [ ]Q QdS d

E r dt E r dtS


72/73

Contd

The value of a portfolio composed of derivativeson a non-dividend-paying stock satisfies thedifferential equation

2 21

2rS S r D G

2 2 2

2 2 2 2 2 2

( ) [( ) ] [ ]

[( ) ] [ ] ( )

Q Q

Q Q

dS dS dS Var E E dt

S S S

dS dS E dt E dt rdt dtS S

2 21 [( ) ]2

Q

t

dSr dt dt rSdt S E

S D G

2 21

2r dt dt rSdt S dt D G


73/73

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

with all the properties of options and has a neatsolution to the option price

But, it is based on many strong assumptions that arenot realistic in the real market

- Log-normal distribution of stock price

- Continuous trading w/o transaction cost- Constant volatility and interest rate

-

Documents

Greeks for Master Finance