Optimal Hedging in Linear Risk

LECTURE FIVE

a. Hedging Linear Risk
b. Optimal hedging in linear risk

1

Part 1

HEDGING LINEAR RISK

a. Overview
b. Basis Risk

2

1. Overview
• Risk that has been measured can be managed
•Taking positions that lower the risk profile of the portfolio
Our main goal will be:
find the optimal position that minimize variance of the
portfolio or limit the VaR
Part 1. Hedging Linear Risk - Intro

Then our portfolio consists of two positions:
asset to be hedge & hedging instrument
Initial consideration
Short Hedge: Long Hedge
A company that knows that it is due to •: A company that knows that it is due
sell an asset at a particular time in the to buy an asset at a particular time in
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future the future
Hedge by taking a short futures Hedge by taking a long futures
position position

1. Overview
How hedge can be?
Static hedging

• Consists of setting and leaving a position until maturity of asset
or contract.

• Appropriate if the hedge instrument is linearly related to
the underlying asset price

Dynamic hedging
• Consists of continuously rebalancing the portfolio.
• Associated with options which have non linear payoffs in the
underlying
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Hedging limits the losses, but also the potential profits.
Only makes the outcome more certain – Risk management
focus

1. Overview
Example
US exporter who has been promised a payment of ¥125 millions in 7
months
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2. Basis Risk
• Definition: Basis = Spot – Future

• Occurs
• when the hedge horizon does not match the time to futures
expiration

• when the characteristics of the futures contract differ from
those of the underlying.

Some details
• For investments assets ( stock indices, gold and silver, etc) the basis risk
tends to be small., because there is a well-defined relationship between
the future price and the spot price
F0T=ertS0

• For commodities supply and demands effects can lead to large
variation in the basis
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• Cross hedging, using a futures contract on a totally different asset or
commodity than the cash position. Basis can be large

2. Basis Risk
• Some details (cont)

Additionally of the assets, it is important to consider:

• The choice of the delivery month

• No the same date

• Not the same volatility:

Assets Volatility Date
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Part 2

OPTIMAL HEDGING IN
LINEAR RISK

a. Hedge Ratio: overview
b. The model
c. Regression analysis approach
d. Applications of linear hedging

8

1. Hedge Ratio - Overview
Definition
The “hedge ratio” is the ratio of the size of the position taken in
futures contracts to the size of the exposure (up to now we have
Part 2. Hedging Linear Risk – Hedge Ratio

assumed a hedge ratio = 1).
Or, how many future contracts to hedge a position

A model of a single portfolio is considered with a known variance
and size.

To hedge the risk, (e.g. reduce the variance, of the portfolio only one assets
is available). This assets is called the hedge instrument.

The variance and correlation with the portfolio of hedge
instrument is known.
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Historic data could be used to compute the relevant variances and
correlation or one could opt to use current market consensus

1. Hedge Ratio - Overview

Two scenarios

Values of the portfolios:
• Owns the product and sells the future
• portfolio value is (S - hF) { h because hedges the position
• change in value of the portfolio is ΔS - hΔF

• Buys the future and is short the product
• portfolio value is s hF – S
• change in value of the portfolio is hΔF – ΔS
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Part 2. Hedging Linear Risk – Hedge Ratio 2. Hedge Ratio – The model

2. Hedge Ratio – The model
The model
• Unique asset
• To be hedged (reduce the variance) with one hedging instrument

• Variance and correlation of both instruments are known using
historic data
• Size of portfolio: w1
• Size of hedging instrument: w2
• Standard deviation of w1 is σ21
• Standard deviation of w2 is σ22
The variance of the un-hedged portfolio will be :

The total variance (including asset and hedging instrument) will be:

The hedge instrument is
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added to reduce variance
or eliminate it al together

To find the optimum for the hedge instrument we just have to find the
first derivative with respect to w2 (associated with the hedging

instrument)
FOC to find the
Vh minimum W2
 2w2 2 2  2 w1 1 2  0
w2 =0

To find the optimal position in the hedge instrument, set equation
equal to zero and solve for w2
1
w2   w1 Solve for W2
2
We can find that this is a minimum because
 2Vh Minimum !!
 2 2 2  0
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w2 2
Second derivative is greater than zero

1
w2   w1
2

• The closer ρ is to one, and the larger is the variance of
the product you are hedging,
• the more you hedge

• The larger is the variance of the product used to hedge
the lower the hedge ratio.

• It is even possible that h would be greater than 1.
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1
w2   w1
2

Hull (2005) and Kocken (1997) proved this finding

How?
The variance using the hedge should be less than the variance without hedge

Substitute the w2* in our initial equation

Where w2 is the hedging instrument

Obtain:
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So, we have
1. Variance of portfolio with no hedge
2. Variance of portfolio with hedge Is this hedge efficient?


To check that our finding is accurate, compare the Variance including the
hedging instrument with the variance without hedging instrument

The mathematical reduction leads to:

H  2

So, this hedging is indeed effective in reducing variance !!!
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The proposed hedging strategy using the optimal hedge ratio can be compared
to a less optimal, to illustrate the case:

Take the opposite
position!
Not very optimal
Solve for w2 (that is the hedging instrument)
w1 1
w2  
2
And as done previously, use w2 in the formula for variance

The variance of the portfolio including hedge is given by:
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Again, I can compare both variances

Rearranging the equation leads to:

We have to models to compare:

1 w1 1
w2   w1 w2  
2 2
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Kay point is
H  2 correlation!!


The reduction in variance is given by

H  2
•Both are equivalent when p =1 (when correlation between the portfolio
to be hedged and the hedge instrument is prefect.)

•BUT once correlation drops to 0.5 the linear strategy does not yield any
variance reduction, while the optimal strategy still produce some
reduction is variance.

Variance goes to zero
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Variance is far from zero

Part 2. Hedging Linear Risk – Hedge Ratio 2. Hedge Ratio – The model

Conclusion
• Using a simple model it was shown that various
hedging strategies can influence the total
variance reduction

• In our case an optimal hedging ratio was found for a
simple model. There is no reason why this same
technique wouldn't work form more complicated
models.
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Example Airline company knows that it will buy 1million gallons of fuel in
3 months.

S • St. dev. of the change in price of jet fuel is 0.032.
h   • Hedger could be futures contracts on heating oil (St. dev is
F 0.04
• ρ =0.8
S 0.032
h  0.08  0.64 This is the ratio!
F 0.04

• One heating oil futures contract is on 42,000 gallons.
1000000
0.64  15.2
42000
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3.The Regression Analysis approach
It is also possible to estimate the optimal hedge using regression analysis.
The basic equation is

S    hF Remember the
role of r2
Using OLS theory, it is known that beta is

 xy 
 xy  2   x
 y  y
So beta (the hedge instrument) will be What is this expression?

This is the solution to the minimizing the original objective
function
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The regression analysis approach

It is useful to note that the regression analysis also provides us with some
information as to how good a hedge we are creating.

The r-square of the regression tells how much of the variance in the
change in spot price is explained by the variance in the change of the
futures price.
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An additional consideration

Futures hedging can be successful in reducing market risk

BUT

They can create other risks

• Costs and daily balance: Futures contracts are marked to
market daily , they can involve large cash inflows or outflows
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• Liquidity problems, especially when they are not offset by cash
inflows from the underlying position

Part 2. Hedging Linear Risk – Hedge Ratio 4. Application of this linear hedging

Duration Hedging

Beta Hedging
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4. Application of this linear hedging
Duration Hedging
The modified duration is given by:
P  ( D * P)y

Dollar duration

Duration for the cash and future positions
S  ( DS * S )y •Duration for each asset
•Where S, F are quantities
F  ( DF * F )y of S and F

Variances and covariances are:

 2 S  ( DS * S ) 2  2 (y )
 2 F  ( DF * F ) 2  2 (y )
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 SF  ( DF F )( DS S ) 2 (y )
* *

Duration Hedging
And using the expression that we already found:
1  1, 2

w2   w1  2 Why?
2 2

 SF * *
( DF F )( DS S )
*
DS S
h*   2    *
 F *
( DF F ) 2
DF F
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Duration Hedging
 SF * *
( DF F )( DS S )
*
DS S
h*   2    *
 F *

2
( DF F ) DF F
Example:
Portfolio 10M
Duration 6.8 years
Time to be hedged: 3 months

Future price: 93-02
Notional: $100.000
Duration: 9.2

a. Notional of the future contract b. Number of contracts
This is just convert 93-02

 2
93  
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6.8 * $10,000,000
 h*    79.4
 32 
*100.000  93,062.5 9,2 * $93,062.05
100

Beta Hedging
Beta, or systematic risk, can be viewed as a measure of the exposure
of the rate of return on a portfolio i to movements in the “market”:

Where
• β represents the systematic risk
• α - the intercept (not a source of risk)
• ε - residual.

It is easy to interpret the β as: The change of the
spot is a function of
the sensitivity to the
market movement
(beta and the
And solving for ΔS and ΔF in the ΔV formula change of market)
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Beta Hedging

When N* = Δ(S / F), ΔV=0

So: The optimal hedge with a stock
index futures is given
by beta of the cash position
times its value divided by
the notional of the futures
contract.
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Beta Hedging

Example

Portfolio : $10,000,000
Beta : 1,5 (SPX V Stock)
Current future prices 1400
Multiplier : 250

a. Notional of futures contract

$250 x 1400 = $350.000

b. Number optimal of contracts
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1.5 * $10,000,000
  42.9
1* $350,000


Concluding

REASONS FOR HEDGING AN EQUITY PORTFOLIO

• Desire to be out of the market for a short period of time.
(Hedging may be cheaper than selling the portfolio and buying it back.)

• Desire to hedge systematic risk (Appropriate when you feel that
you have picked stocks that will outpeform the market.)
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Part 3

HEDGING NON LINEAR
RISK

a. Initial considerations (pricing)
b. From Black-Scholes to the Greeks
c. Delta
d. Theta
e. Gamma
f. Vega
g. Rho
33

1. Initial considerations (Pricing) Price of a call is a function of
1. Stock price
Price of a stock is a function of 2. Interest rate
3. Volatility
f t  f ( St , rt ,  t , K , ) 4. Strike price
5. Time

The idea of pricing is finding f when the parameters change

Example One month
Part 3. Hedging NON-Linear Risk

S1=$95 K1=$30 (95-65)

S0=$75 K=65
S1=$63 K1=$0 (63-65)

Riskless Hedge Approach
Option price? We find the optimal number of
stocks to make both portfolios
• Riskless portfolio: find the number of stocks “ϕ” equivalent

$95ϕ-30
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$95ϕ-30=63 ϕ – 0
S0=$75
Number of stocks
$63ϕ-0

1. Initial considerations (Pricing)
Value of portfolio in three months
$95ϕ-30 = 59.06 = 63 ϕ – 0
They are equivalent

One leg Other leg
Payoff will be the same for both scenarios
$30  0
  0.9375 Number of stocks
$95  $63

Payoff regardless the price
Payoff of the
of stock at t+1 0.9375 * 75 – 30 = 59.06 portfolio t+1

Then, the price of a call considering the present value of the call assuming
r=6% and 1 month
Transform the payoff
to t
0.9375 * $75 - C = $59.06 * e –Rf*T
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0.9375 * $75 - C = $59.06 * e 0.06 x 0.833

C = $11.54

Risk neutral approach
• Each path has their own probability
• We try to estimate these probabilities for a risk neutral individual and then
use these risk neutral probabilities to price a call option.

S0=95
p

S0=75
1-p S0=63

•For a risk neutral investor, the current stock price is the expected payoff
discounted at the risk-free rate of interest (Rf=6%) and T=0.083 (month)
 R f *T
75  [95Pu  (1  Pu )63)] * e Generalization

Today’s stock price is the S0  ( Pu Su  (1  Pd )Sd )e rt
result of both legs
•It is possible solve Pu
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75e Rf *T  63 This is the risk neutral probability of
Pu   0.387 the stock price increasing to $95 at
95  63 the end of the month

A risk neutral individual would assess a 0.387 probability of receiving $30 and a
0.613 probability of receiving $ 0
Su=95
P=0.387 Cu=30
S0=75
C =65
1-P = 0.613 Sd=63
Cd=0

The price of a call will have the same idea than in the previous slide

C0  ( Pu Cu  (1  Pd )Cd )e rt

C0  [0.384 * 30  (0.613) * 0]e0.6*0.83

The price of a call is the same!

C = $11.54
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2. From Black-Scholes to the Greeks
Using the Black – Scholes we know that the price of a call option depends on:

•Price of the underlying asset (S)
•Strike price (K)
•Time to maturity, (T)
•Interest rate, (r) and

•Volatility,
The first order approximation shows the effect of price when change some
factors

This show the effect of varying each of the parameters, S,T, r and σ by small
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amounts δS, δT, δr and δσ, with K fixed.

So each of the partial effect is given by a Greek letter

2. From Black-Scholes to the Greeks

Each of the partial effects is given a Greek letter

Delta ∆ = δΠ/ δS Option price changes when the price of the
underlying asset changes

Theta Θ=- δΠ/ δT Option price changes as the time to maturity
decreases.

Rho ρ = δΠ/ δr Option price changes as the interest rate changes

Vega ν = δΠ/ δσ Option price changes as the volatility changes
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Gamma Γ = δ2Π/ δS2 Measures the rate of change of the option's as
the price of the underlying changes (Acceleration)
by a Greek letter

3. Delta ∆
Delta (∆): how much will the price of an option move if the stock moves $1

• Delta varies from node to node
• Defined as the first partial derivative
with respect to price



S
where is the option price and S
is underlying asset price.

• However, the relationship between option price and stock price is not
linear.
WHY?????
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• Intuition the option costs much less than the stock!!

3. Delta ∆
Relation Delta (∆), at/in/out the money
Values of delta
Delta   N(d1 )
Close to 1 when goes deep in the money
Delta is close to 0.5
Close to 0 when goes deep out the money

S

Close to 0 when goes deep out the money
Delta is close to -0.5

  N(d1 )  1
 Calls have positive delta (0 < C < 1)
If the stock price goes up, the price for the callto -1 when goes deep in the money
Close will go up.
 Puts have a negative delta (-1< P < 0)
If the stock goes up the price of the option will go down.
 So...as expiration nears,
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Delta for in-the-money calls will approach 1, reflecting a one-to-
one reaction to price changes in the stock.
Delta for out-of the-money calls will approach 0 and won’t react
to price changes in the stock.

3. Delta ∆
Relation Delta (∆), at/in/out the money
Values of delta

I have a call option
• K= $50
60 days prior to expiration S=$50. (at-the-money option)  Δ should be 0.5
C=$2.

• Case 1: St to $51, C goes up from to $2.50 ( S:C = 1:0.5 )
After 1 • Case 2: St+1, to $52? (Higher probability that the option will end up
in-the-money at expiration)
• What will happen to delta? … increases to 0.6
• C to $3.10 ($.60 move for a $1 movement in the stock)
• Case 4: St to $49?
• C to $1.50, reflecting the .50 delta
After 4 • Case 5: St to $48, the option might go down to $1.10.
• Delta would have gone down to .40 (lower probability the option will
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end up in-the-money at expiration).

3. Delta ∆
Relation Delta (∆) time to maturity

As expiration approaches, changes in
the stock value will cause more dramatic
changes in delta
Logical

St = $50
K = $50
Two days from expiration
Delta =.50
• Case 1: St+1= $51. Delta should be high (0.9) in just ONE day
• Case 2: St+1= $49. Delta might change from .50 to .10 in ONE day
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Delta reflects the probability that the option will finish in-the-money

3. Delta ∆
In-the-money options will move more than out-of-the-money
options (Remember the graph)

 Short-term options will react more than longer-term options
to the same price change in the stock.

(From previous slide)

Delta of a portfolio

The delta of a portfolio of options is just the weighted sum of the
individual deltas
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The weights wi equal the number of underlying option contracts

3. Delta ∆
Delta (∆)
The delta of an option depend on the kind of option

• For a European call option on a non-dividend stock

  N(d1 )

• For a European put option on a non-dividend stock

  N(d1 )  1
•For a European call option on a dividend-paying stock
  eq N(d1 )
•For a European put option on a dividend-paying stock
  e q  N(d1 )  1
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3. Delta hedging
Delta neutral hedging is defined as keeping a portfolio’s value neutral to
small changes in the underlying stock’s price.

Stock price : $100
Call option : $10
Current delta : 0.4

A financial institution sold 10 call option to its client, so that the client
has right to buy 1,000 shares at time to maturity.

To construct a delta hedge position
• Financial institution should buy 0.4 x 1,000 = 400 shares of stock
• If the stock price goes up to $1, the option price will go up by
$0.4. In this situation, the financial institution has a $400 ($1 x 400
shares) gain in its stock position, and a $400 ($0.4 x 1,000 shares)
loss in its option position.
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• If the stock price goes down by $1, the option price will go down
by $0.4. The total payoff of the financial institution is also zero.

But...

3. Delta hedging
Delta changes over different stock price.

If an investor wants to maintain his portfolio in delta neutral, he
should adjust his hedged ratio periodically. The more frequently
adjustment he does, the better delta-hedging he gets.

Underlying stock price of $20, the
investor will consider that his
portfolio has no risk.

As the underlying stock prices
changes (up or down), the delta
changes and he will have to use
different delta hedging.
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Delta measure can be combined
with other risk measures to yield
better risk measurement.

4.Theta Θ
Theta is the amount the price of calls and puts decrease for a one-day change in the
time to expiration.

 Rate of change of the option price respected
 to the passage of time
t
If   T  t (time to maturity) this derivative is < 0
   

   (1) Note that t is different from τ
t  t 
This relation shows that:

• The price of the option declines as maturity approaches
Same idea

•When time passes, the time value of the option decreases
• Longer dated options are more valuable.

BUT
• The passage of time on an option is NOT uncertain,
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It is not necessary to make a theta hedge portfolio
against the effect of the passage of time.

Here we have a relation between time and option’s price.
4.Theta Θ How this relation changes when American Call is exercised early?
Relation time and American call option
American and European options: longer dated
options give more opportunities for profit

Early exercise of an American Option is NON OPTIMAL

If an American Call is exercised before T, the payoff could be

St – K
Put – Call parity condition European call option notation
C  P  (St  KerT )
Because
P0 So... C  St  Ke rT
Recall the restrictions on the value of a call option
Lower bound
C  max{ 0, S  Ke  rT }
Also because
r 0 C SK
Now, the American Option worth more
C A>C CA  C  S  K
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Ct  Ct  St  KerT  St  K Hence it will always be better to sell the
A
option rather than exercise it early
What about PUT options???

4.Theta Θ

Early exercise of an American Option is NON OPTIMAL

Intuitive reasons

1. Delaying exercise delays the payment of the strike price.
Option holder is able to earn interest on the strike price for a
longer period of time.
2. More movements: Assume that you exercised your option today,
what if tomorrow a big crazy thing will occur and the price of an
underlying asset just shoots?

Instead of exercising your American call option you should
have sold it to someone else
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4.Theta Θ
90 DAYS: lose $.30 of its value in one month Time decay is
60-DAY option, lose $.40 of its value over stronger near
the course of the following month. expiration
30-DAY option will lose the entire remaining
$1

Time decay of an at-the-money call option

Time is more
important ATM
At the money V Out/In the money options

At-the-money options will
experience more significant dollar
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losses over time than in- or out-of-
the-money options with the same
underlying stock and expiration
date.

4.Theta Θ
Theta is the amount the price of calls and puts decrease for a one-day change in the
Summarizing time to expiration.

For a European call option on a non-dividend stock, theta can be
written as:
St s
  N(d1 )  rX  e r N(d 2 )
2 
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For a European put option on a non-dividend stock, theta can be
shown as S
   t s  N(d1 )  rX  e r N(d 2 )
2 

5. Gamma Γ
Gamma is the rate that delta will change based on a $1 change in the stock price.
Or
The rate of change of delta respected to the rate of change of underlying asset price
  2
  2
S S
Delta is the “SPEED” at which option prices change, gamma as the “ACCELARATION”

 Gamma shows how often we should rebalance

 If Γ is large then it will be necessary to change Δ by a large
amount as S changes.

 Options with the highest gamma are the most responsive to
changes in the price of the underlying stock.
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5. Gamma Γ
Delta is a dynamic number that changes as the stock
price changes, doesn’t change at the same rate for
every option based on a given stock.

St = 50
K = 50
Delta = 0.5

 The price of at-the-money options will change more significantly than the
price of in- or out-of-the-money options.
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Part 3. Hedging NON-Linear Risk 5. Gamma Γ

The price of near-term at-the-money options will exhibit the
most explosive response to price changes in the stock.
 As your option moves in-the-money, delta will approach 1 more rapidly. If
you’re an option buyer, high gamma is good as long as your forecast
is correct.
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 If you’re an option seller and your forecast is incorrect, high gamma
is the enemy. That’s because it can cause your position to work against you
at a more accelerated rate

5. Gamma Γ

For a European call option on a non-dividend stock, theta can be
written as:
1
 N  d1 
St s 

For a European put option on a non-dividend stock, theta can be
shown as
1
 N  d1 
St s 
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5. Gamma Γ
Make a position gamma neutral

 Suppose the gamma of a delta-neutral portfolio is Γ
 Suppose the gamma of the option in this portfolio is ΓO,
 The number of options added to the delta-neutral portfolio is w0.

Then, the gamma of this new portfolio is
Gamma of portfolio
o o  
To make a gamma-neutral portfolio, we should trade
Gamma of portfolio

o*   / o options
Example Gamma of option
Delta and gamma: 0.7 and 1.2.
A delta-neutral portfolio has a gamma of -2,400.
Lecture 5

To make a delta-neutral and gamma-neutral portfolio, we should add a long
position of 2,400/1.2=2,000 shares and a short position of 2,000 x 0.7=1,400
shares in the original portfolio.

5. Gamma Γ
One more example

 Suppose a portfolio is delta neutral with a gamma of -3000
 Suppose the delta and gamma of the option is 0.62 and 1.50
 Make a portfolio gamma neutral by buying

3000  2000options o*   / o
1.5

 This changes delta from 0 to 0.62 * 2000 = 1240
 Sell 1240 shares of underlying to regain delta neutrality
Lecture 5

5. Gamma Γ
Relation gamma, delta and price of portfolio
(Delta-gamma approximation)
Given that the option value is not a linear function of underlying stock price
Gamma makes the correction.
1
change in option value    change in stock price    (change in stock price)2
2

St of XYZ = $657 This approximation comes from
Call option = $120 the Taylor series expansion near
Delta = 0.47 the initial stock price
Gamma = 0.01.

Price of the call option if XYZ stock price suddenly begins trading at $699

C(St+h) = C(St) + ∆ (Change St) + (1/2) (Change St)2 * Γ =
Lecture 5

120 + 42 * 0.47 + (1/2) (422) * 0.01 =
$148.56

LECTURE FIVE

End Of The Lecture

60

Optimal Hedging in Linear Risk

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