18 - The Greek Letters

The Greek Letters

• Option Greek Letters: Delta, Theta, Gamma, Vega, and Rho

• Each Greek Letter [or simply called as a Greek] measures

the risk to the option writer associated with an option

position from a specific dimension.

• Particularly, the option writers [the intermediaries] get

exposed to such risks that gets transferred to them from the

hedgers. Of course, they face it on their netted position only.

• The value of a Greek denotes the level of a specific risk

associated with a particular option position. Those values,

being a function of So, K, r, SD, and T, keep on changing.

This calls for the traders to re-balance the Greeks again and

again, so as to keep the risk at its acceptable level.

• Of course, it is a difficult task, but there are no better

alternatives to that.

Textbook Illustration (Page-402)

A bank [i.e. a trader] has sold for $300,000 a European call option on 100,000 shares of a non-dividend paying stock with

S0 = 49, K = 50, r = 5%, s = 20%, T = 20 weeks

The Black-Scholes-Merton value of the option is $240,000 for the lot size of 100,000.

This leaves it with a profit of $60,000, as of now. However, is it a sure profit [for ever]?

In reality, the profit could be more or less than this figure, oreven negative! So, what could be the reaction of the trader to this kind of a situation?

Alternatives Available to Option Traders/Writers

Take no action: Leave it a Necked Position, and face the risk

Convert it into a Covered Position

Stop Loss Strategy

[Buy the underlying when St>K; sell it when St<K]

Hedging using Greeks (Delta, Theta, Gamma, Vega, Rho)

Hedging with Greeks

Delta

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

respect to the underlying. Thus, it is the first derivative of

the option value with respect to the spot price.

An Option Delta of 0.6 says that when the price of the stock

increases by Re.1, the price of option would increase by 0.6.

Option

price

A

BSlope = D

Stock priceThe hedge position must be frequently rebalanced!!!

Delta Hedging

This involves maintaining a delta-neutral portfolio.

The delta of a European call on a non-dividend paying stock is

N (d 1).

The delta of a European put on the stock is N (d 1) – 1.

[By the way, the delta of the underlying asset (e.g. stock) is

always 1.]

For Example: A trader has written 100 call options on a stock.

If the Option Delta is 0.6, he should buy 0.6*100 shares = 60

shares. If the price of the share goes up by Re. 1, he gains Rs.

60 on the long position on stock, and looses Re. 0.6*100 calls

= Rs. 60 on calls shorted. Thus, his portfolio of short calls +

long stock is a delta-neutral portfolio. However, now, at the

new price of the stock, the Delta would change! So, again to

be rebalanced, and re-rebalanced!!!

Gamma

Gamma (G) is the rate of change of delta (D) with respect to

the price of the underlying asset. Thus, it is the second partial

derivative of the option value with respect to the spot price.

Gamma measures the curvature shown previously. So,

Gamma addresses the delta hedging error caused by the

curvature of the delta function.

Gamma Hedging

Since the delta of the stock is 1, its Gamma (G) is always zero.

So the same stock cannot be used to make the portfolio gamma

neutral.

Gamma can be changed by adding such instruments like other

options whose value is not linearly related to the underlying.

Gamma neutrality, just like delta neutrality is only for a short

period. However, delta neutrality provides protection against

relatively small movements in the underlying, whereas gamma

neutrality provides protection against relatively large

movements in the underlying.

High gamma value would call for frequent rebalancing.

Vega

Vega (n) is the rate of change of the value of a derivative

with respect to volatility.

Other aspects of vega are same as that of gamma.

Vega Hedging

The tips are same as for gamma hedging.

Theta (Q) of a derivative is the rate of change of the value

with respect to the passage of time.

The theta of a call or put is usually negative. This means

that, if time passes (with the price of the underlying asset

and its volatility remaining the same), the value of a long

position in the option declines.

Theta

Rho

Rho is the rate of change of the value of a

derivative with respect to the interest rate.

Managing Delta, Gamma, & Vega

D can be changed by taking a position in the underlying.

To adjust G & n it is necessary to take a position in an option or other derivative.

There is hardly any relevance for managing Theta and Rho actively.

Hedging in Practice

Traders usually ensure that their portfolios are delta-neutral

at least once a day.

Whenever the opportunity arises, they improve gamma and

vega.

As portfolio becomes larger hedging becomes less expensive.