Greeks and volatiltiy

By A V Vedpuriswar

September 21, 2009

Volatility

Basics of volatility

Volatility is a huge issue in risk management.

The science of volatility measurement has advanced a lot in recent years.

Here we look at some basic concepts and tools.

4

Estimating Volatility

Calculate daily return u1 = ln Si / Si-1

Variance rate per day

We can simplify this formula by making the following simplifications.

ui = (Si – Si-1) / Si-1

ū = 0 m-1 = m

If we want to weight

21

1

)(1

1

uum n

m

i

12

1

2 1

n

m

i

um

n

122

ni un )1( i

5

Estimating Volatility

Exponentially weighted moving average model means weights decrease exponentially as we go back in time.

n 2 = 2n-1+ (1 - ) u2

n-1

= [n-22 + (1- )un-2

2] + (1- )un-12

= (1- )[un-12 + un-2

2] + 2n-22

= (1-) [un-12 + u2

n-2 + 2un-32 ] + 3 2

n-3

If we apply GARCH model,

n2 = Y VL + un-1

2 + 2n-1

VL = Long run average variance rate

Y + + = 1. If Y = 0, = 1-, = , it becomes exponentially weighted model.

GARCH incorporates the property of mean reversion.

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Problem

The current estimate of daily volatility is 1.5%. The closing price of an asset yesterday was $30. The closing price of the asset today is $30.50. Using the EWMA model, with λ = 0.94, calculate the updated estimate of volatility .

7

Solution

ht = λ σ2t-1 + ( 1 – λ) rt-1

2

λ = .94

rt-1 = ln[(30.50 )/ 30]

= .0165

ht = (.94) (.015)2 + (1-.94) (.0165)2

Volatility = .01509 = 1.509 %

Greeks

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Delta

Delta is the rate of change in option price with respect to the price of the underlying asset.

It is the slope of the curve that relates the option price to the underlying asset price.

A position with delta of zero is called delta neutral.

Because delta keeps changing, the investor’s position may remain delta neutral for only a relatively short period of time.

The hedge has to be adjusted periodically.

This is known as rebalancing.

10

Theta

Theta of a portfolio is the rate of change of value of the portfolio with respect to change of time, with all else remaining same.

Theta is also called the time decay of the portfolio.

Theta is usually negative for an option.

As time to maturity decreases with all else remaining the same, the option tends to become less valuable.

11

Gamma

The gamma is the rate of change of the portfolio’s delta with respect to the price of the underlying asset.

It is the second partial derivative of the portfolio price with respect to the asset price.

If gamma is small, it means delta is changing slowly.

In such a case, adjustments to keep a portfolio delta neutral can be made only relatively infrequently.

However, if gamma is large, it means the delta is highly sensitive to the price of the underlying asst.

It is then quite risky to leave a delta neutral portfolio unchanged for any length of time.

12

Vega

The Vega of a portfolio of derivatives is the rate of change of the value of the portfolio with respect to the volatility of the underlying asset.

High Vega means high sensitivity to small changes in volatility.

A position in the underlying asset has zero Vega.

The Vega can be changed by adding options.

If V is Vega of the portfolio and VT is the Vega of the traded option, a position of –V/ VT in the traded option makes the portfolio Vega neutral.

If a hedger requires the portfolio to be both gamma and Vega neutral, at least two traded derivatives dependent on the underlying asset must usually be used.

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Rho

Rho of a portfolio of options is the rate of change of value of the portfolio with respect to the interest rate.

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Problem

Suppose an existing short option position is delta neutral and has a gamma of － 6000. Here, gamma is negative because we have sold options. Assume there exists a traded option with a delta of 0.6 and gamma of 1.25. Create a gamma neutral position.

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Solution

To gamma hedge, we must buy 6000/1.25 = 4800 options.

Then we must sell (4800) (.6) = 2880 shares to maintain a gamma neutral and original delta neutral position.

16

Problem

A delta neutral position has a gamma of － 3200. There is an option trading with a delta of 0.5

and gamma of 1.5. How can we generate a gamma neutral position for existing portfolio while maintaining delta neutral hedge?

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Solution

Buy 3200/1.5 = 2133 options

Sell (2133) (.5) = 1067 shares

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Suppose a portfolio is delta neutral, with gamma= - 5000 and vega = - 8000. A traded option has gamma = .5, vega = 2.0 and delta = 0.6. How do we achieve vega neutrality?

Problem

To achieve Vega neutrality add 4000 options. Delta increases by (.6) (4000) = 2400

So sell 2400 units of asset to maintain delta neutrality.

As the same time, Gamma changes from – 5000 to ((.5) (4000) – 5000 = - 3000.

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Suppose there is a second traded option with gamma = 0.8, vega = 1.2 and delta = 0.5.

if w1 and w2 are the weights in the portfolio,

- 5000 + .5w1 + .8w2 = 0 - 8000 + 2.0w1 + 1.2w2 = 0

w1 = 400 w2 = 6000.

This makes the portfolio gamma and vega neutral.

Now let us examine delta neutrality.

Delta = (400) (.6) + (6000) (.5) = 3240

3240 units of the underlying asset will have to be sold to maintain delta neutrality.