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8/9/2019 Greeks for Master Finance
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Instructor: Tingwei WANG
Email: [email protected]
Universit Paris-Dauphine
For Master 224
September 2014
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Risk Management Base asset
- Stocks
Volatility: standard deviation of return
VaR (Value at Risk)
- Bonds
Duration: sensitivity to interest rate change
credit risks
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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
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Review of Derivatives Basics What is forward/futures?
What is an option?
How to price forward/futures? How to price an option?
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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
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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
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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
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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
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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
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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
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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
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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,
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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()
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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 ()
Terminalstock price()
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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
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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 ()
Terminalstock price()
Break-even price
How much is the option premium here?
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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
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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.
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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
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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
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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.
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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How Factors Change Option Value
Factors (+) Call option Put option
Spot price of underlying asset +
Strike price +
Maturity + ?
Volatility + +
Interest rate +
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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
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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
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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
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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
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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
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Delta
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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
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Gamma & Vega
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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
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Constant volatility In the Black-Scholes model, the volatility of the
underlying asset is constant, which is not true in thereal market
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Volatility Smile
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Volatility Surface
Moneyness K/S0
Maturity T
Impliedvolatility
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Theta
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European Call PremiumOptionpremium
Stock price St
Intrinsic valueMax(St-K,0)
Time value
K
Out-of-the-money
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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
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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
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Rho
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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