LECTURE FIVE

1

a. Hedging Linear Risk

b. Optimal hedging in linear risk

HEDGING LINEAR RISK

Part 1

2

a. Overview

b. Basis Risk

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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

Then our portfolio consists of two positions:

asset to be hedge & hedging instrument

Short Hedge:

A company that knows that it is due to

sell an asset at a particular time in the

future

Hedge by taking a short futures

position

Long Hedge

•: A company that knows that it is due

to buy an asset at a particular time in

the future

Hedge by taking a long futures

position

Initial consideration

1. Overview

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

Hedging limits the losses, but also the potential profits.

Only makes the outcome more certain – Risk management

focus

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How hedge can be?

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

• Cross hedging, using a futures contract on a totally different asset or

commodity than the cash position. Basis can be large

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Additionally of the assets, it is important to consider:

• The choice of the delivery month

• No the same date

• Not the same volatility:

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2. Basis Risk

• Some details (cont)

Assets Volatility Date

OPTIMAL HEDGING IN LINEAR RISK

Part 2

8

a. Hedge Ratio: overview

b. The model

c. Regression analysis approach

d. Applications of linear hedging

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

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.

Historic data could be used to compute the relevant variances and

correlation or one could opt to use current market consensus

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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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2. Hedge Ratio – The model L

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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 σ2

1

• Standard deviation of w2 is σ2

2

The variance of the un-hedged portfolio will be :

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

The hedge instrument is

added to reduce variance

or eliminate it al together

2. Hedge Ratio – The model L

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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)

To find the optimal position in the hedge instrument, set equation

equal to zero and solve for w2

022 21122

2

2

ww

w

Vh

2

112

ww

We can find that this is a minimum because

Second derivative is greater than zero

02 22

2

2

2

w

Vh

2. Hedge Ratio – The model

FOC to find the

minimum W2

= 0

Solve for W2

Minimum !!

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2

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• 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.

2. Hedge Ratio – The model L

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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:

So, we have

1. Variance of portfolio with no hedge

2. Variance of portfolio with hedge

2

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2. Hedge Ratio – The model

Is this hedge efficient? Le

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

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:

So, this hedging is indeed effective in reducing variance !!!

2H

2. Hedge Ratio – The model L

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

The proposed hedging strategy using the optimal hedge ratio can be compared

to a less optimal, to illustrate the case:

Solve for w2 (that is the hedging instrument)

And as done previously, use w2 in the formula for variance

The variance of the portfolio including hedge is given by:

2

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2. Hedge Ratio – The model

Take the opposite

position!

Not very optimal

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

Again, I can compare both variances

Rearranging the equation leads to:

We have to models to compare:

2

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2H

2

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2. Hedge Ratio – The model

Kay point is

correlation!!

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

The reduction in variance is given by

•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.

2H

2. Hedge Ratio – The model

Variance goes to zero

Variance is far from zero

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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

F

Sh

Airline company knows that it will buy 1million gallons of fuel in

3 months.

• St. dev. of the change in price of jet fuel is 0.032.

• Hedger could be futures contracts on heating oil (St. dev is

0.04

• ρ =0.8

• One heating oil futures contract is on 42,000 gallons.

64.004.0

032.008.0

F

Sh

This is the ratio!

2.1542000

100000064.0

2. Hedge Ratio – The model L

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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

Using OLS theory, it is known that beta is

So beta (the hedge instrument) will be

This is the solution to the minimizing the original objective

function

FhS

y

x

y

xy

xy

2

Remember the

role of r2

What is this expression?

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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.

3. The Regression Analysis approach

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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

• Liquidity problems, especially when they are not offset by cash

inflows from the underlying position

3. The Regression Analysis approach

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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:

Duration for the cash and future positions

Variances and covariances are:

yPDP )*(

Dollar duration

ySDS S )*(

yFDF F )*(

)())((

)()(

)()(

2**

22*2

22*2

ySDFD

yFD

ySD

SFSF

FF

SS

•Duration for each asset

•Where S, F are quantities

of S and F

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Duration Hedging

And using the expression that we already found:

FD

SD

FD

SDFDh

F

S

F

SF

F

SF

*

*

2*

**

2)(

))((*

2

2

2,1

2

112

ww Why?

4. Application of this linear hedging L

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Duration Hedging

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

This is just convert 93-02

FD

SD

FD

SDFDh

F

S

F

SF

F

SF

*

*

2*

**

2)(

))((*

5.062,93000.100*100

32

293

b. Number of contracts

4.7905.062,93$*2,9

000,000,10$*8.6* h

4. Application of this linear hedging L

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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:

And solving for ΔS and ΔF in the ΔV formula

4. Application of this linear hedging

The change of the

spot is a function of

the sensitivity to the

market movement

(beta and the

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.

4. Application of this linear hedging L

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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

9.42000,350$*1

000,000,10$*5.1

4. Application of this linear hedging L

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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.)

4. Application of this linear hedging L

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HEDGING NON LINEAR RISK

Part 3

33

a. Initial considerations (pricing)

b. From Black-Scholes to the Greeks

c. Delta

d. Theta

e. Gamma

f. Vega

g. Rho

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Price of a stock is a function of

The idea of pricing is finding f when the parameters change

Example

),,,,( KrSff tttt

S0=$75

S1=$95

S1=$63

K=65

K1=$30 (95-65)

K1=$0 (63-65)

Option price?

• Riskless portfolio: find the number of stocks “ϕ”

S0=$75

$95ϕ-30

$63ϕ-0

$95ϕ-30=63 ϕ – 0

One month

Price of a call is a function of

1. Stock price

2. Interest rate

3. Volatility

4. Strike price

5. Time

We find the optimal number of

stocks to make both portfolios

equivalent

Number of stocks

Riskless Hedge Approach

1. Initial considerations (Pricing)

Value of portfolio in three months

Payoff regardless the price

of stock at t+1 0.9375 * 75 – 30 = 59.06

Then, the price of a call considering the present value of the call assuming

r=6% and 1 month

One leg Other leg

0.9375 * $75 - C = $59.06 * e –Rf*T

0.9375 * $75 - C = $59.06 * e 0.06 x 0.833

C = $11.54

They are equivalent

Number of stocks

$95ϕ-30 = 59.06 = 63 ϕ – 0

9375.063$95$

030$

Payoff of the

portfolio t+1

Transform the payoff

to t

Payoff will be the same for both scenarios

1. Initial considerations (Pricing) L

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S0=75

S0=95

S0=63

p

1-p

TR

uufePP*

*)]63)1(95[75

387.06395

6375 *

TRf

u

eP

rt

dduu eSPSPS ))1((0Today’s stock price is the

result of both legs

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.

•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)

•It is possible solve Pu

This is the risk neutral probability of

the stock price increasing to $95 at

the end of the month

Generalization

1. Initial considerations (Pricing) L

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A risk neutral individual would assess a 0.387 probability of receiving $30 and a

0.613 probability of receiving $ 0

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

The price of a call is the same!

C = $11.54

S0=75

C =65

Su=95

Cu=30

Sd=63

Cd=0

P=0.387

1-P = 0.613

rt

dduu eCPCPC ))1((0

83.0*6.0

0 ]0*)613.0(30*384.0[ eC

1. Initial considerations (Pricing) L

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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

amounts δS, δT, δr and δσ, with K fixed.

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

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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

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

2. From Black-Scholes to the Greeks L

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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

where is the option price and S

is underlying asset price.

• However, the relationship between option price and stock price is not

linear.

WHY?????

• Intuition the option costs much less than the stock!!

S

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1N(d )

1N(d ) 1

Close to 1 when goes deep in the money

Close to 0 when goes deep out the money

Close to 0 when goes deep out the money

Close to -1 when goes deep in the money

Delta is close to -0.5

Delta is close to 0.5

Calls have positive delta (0 < C < 1)

If the stock price goes up, the price for the call 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,

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.

Delta

S

Relation Delta (∆), at/in/out the money

Values of delta

3. Delta ∆ L

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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 )

• 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

• 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

end up in-the-money at expiration).

Relation Delta (∆), at/in/out the money

Values of delta

After 1

After 4

3. Delta ∆ L

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Relation Delta (∆) time to maturity

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

As expiration approaches, changes in

the stock value will cause more dramatic

changes in delta

Delta reflects the probability that the option will finish in-the-money

Logical

3. Delta ∆ L

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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)

3. Delta ∆

Delta of a portfolio

The delta of a portfolio of options is just the weighted sum of the

individual deltas

The weights wi equal the number of underlying option contracts

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Delta (∆)

The delta of an option depend on the kind of option

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

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

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

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

1N(d )

1N(d ) 1

q

1e N(d )

q

1e N(d ) 1

3. Delta ∆ L

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Delta neutral hedging is defined as keeping a portfolio’s value neutral to

small changes in the underlying stock’s price.

3. Delta hedging

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.

• 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.

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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.

Delta measure can be combined

with other risk measures to yield

better risk measurement.

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

time to expiration.

This relation shows that:

• The price of the option declines as maturity approaches

•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,

It is not necessary to make a theta hedge portfolio

against the effect of the passage of time.

t

( 1)t t

If (time to maturity) this derivative is < 0 T t

Rate of change of the option price respected

to the passage of time

Note that t is different from τ

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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

)( rT

t KeSPC rT

t KeSC 0PBecause

So...

},0max{ rTKeSC Recall the restrictions on the value of a call option

Lower bound

0rAlso because

KSC

European call option notation

Now, the American Option worth more

CA>C KSCC A

Hence it will always be better to sell the

option rather than exercise it early

What about PUT options???

Relation time and American call option

American and European options: longer dated

options give more opportunities for profit

KSKeSCC t

rT

tt

A

t

4. Theta Θ Here we have a relation between time and option’s price.

How this relation changes when American Call is exercised early?

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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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At the money V Out/In the money

Time decay of an at-the-money call option

At-the-money options will

experience more significant dollar

losses over time than in- or out-of-

the-money options with the same

underlying stock and expiration

date.

90 DAYS: lose $.30 of its value in one month

60-DAY option, lose $.40 of its value over

the course of the following month.

30-DAY option will lose the entire remaining

$1

4. Theta Θ

Time is more

important ATM

options

Time decay is

stronger near

expiration

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Theta is the amount the price of calls and puts decrease for a one-day change in the

time to expiration.

For a European call option on a non-dividend stock, theta can be

written as:

For a European put option on a non-dividend stock, theta can be

shown as

rt s1 2

SN (d ) rX e N(d )

2

rt s1 2

SN (d ) rX e N( d )

2

4. Theta Θ

Summarizing

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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

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.

2

2S S

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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.

5. Gamma Γ L

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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.

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

The price of near-term at-the-money options will exhibit the

most explosive response to price changes in the stock.

5. Gamma Γ L

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For a European call option on a non-dividend stock, theta can be

written as:

For a European put option on a non-dividend stock, theta can be

shown as

1

t s

1N d

S

1

t s

1N d

S

5. Gamma Γ L

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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

To make a gamma-neutral portfolio, we should trade

options

o o

*

o o/

Example

Delta and gamma: 0.7 and 1.2.

A delta-neutral portfolio has a gamma of -2,400.

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

Gamma of portfolio

Gamma of portfolio

Gamma of option

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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

This changes delta from 0 to 0.62 * 2000 = 1240

Sell 1240 shares of underlying to regain delta neutrality

*

o o/

5. Gamma Γ L

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options20005.1

3000

Relation gamma, delta and price of portfolio

(Delta-gamma approximation)

21change in option value change in stock price (change in stock price)

2

Given that the option value is not a linear function of underlying stock price

Gamma makes the correction.

St of XYZ = $657

Call option = $120

Delta = 0.47

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 =

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

$148.56

This approximation comes from

the Taylor series expansion near

the initial stock price

5. Gamma Γ L

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LECTURE FIVE

60

End Of The Lecture