### Greeks / Option Sensitivities

Option sensitivities explain how value of option responds to the price of underlying (S), time, risk-free rate or volatility. These are the key risk measures of option position and known as Greeks. They are DELTA, GAMMA, RHO, THETA and VEGA.

Usually Greeks are the key measures used by Market Makers and Option Writers to measure the risk of Listed (exchange traded) options as well as OTC options

Lets us look at each of them.

### Delta

DELTA of an option is the RATE of change of the option price with respect to the price of the underlying asset.

It is a slope of the CURVE that relates the option price to the underlying asset price. (Mathematically slope of the curve represents the rate of change)

Let’s look at this example –

Assume if DELTA of the certain option is 0.7, that means if the underlying asset prices changes by the X amount, the option price changes by 70% of X amount.

(What happened to strike price, /current asset price is used. Hedging is done at the time when the delta is measured. Delta hedging is done as the asset price varies.)

In reality, the relationship between the Stock Price and the Option Price is a Curvature (It is DELTA).

Here’s the question.

**How can DELTA be used to manage the RISK?**

The key reason of using various risk measures is to HEDGE the positions and the portfolios.

Now let’s see how can this measure be useful.

Let’s assume broker has written 10 CALL options (OR SOLD the CALL option contracts). What does it mean? Buyer can buy (opposite party) 1000 shares from this broker.

Also assume the price of the stock is $100, the option price is $10, and the delta is 0.7.

Now the broker can hedge this position by buying 0.7 * 1000 = 700 shares.

The gain on the option position would offset the loss on the stock position. And vice-versa.

Delta of the position is = 0.7 * (-1000) = -700

That means –

Investor loses 700**dS *on the short option position when the stock price increases by *dS*.

**What is the DELTA of the STOCK itself?**

It is 1.0.

So long position of 700 shares has a delta of +700 (= 1.0 * 700)

Now let’s look at the DELTA of the overall position including option position and stock position

That is ZERO (= -700 + 700)

Delta of the Stock position is offsetting option position.

Hence the option position is fully hedged with stock position.

This strategy or the position is called as DELTA NEUTRAL.

**How can broker manage this DELTA NEUTRALITIES over the period? **

The stock price changes so the position needs to be rebalanced.

This strategy of rebalancing the position to make it DELTA NEUTRAL is known as DYNAMIC HEDGING.

Hedging the DELTA of the position is called as DELTA HEDGING.

### Gamma

Gamma is the rate of change of the portfolio’s delta with respect to the price of the underlying asset. In other words, it is a second derivative with respect the asset price.

**Why do we need another measure when there is DELTA?**

Let’s go back to the relationship between the option price and underlying asset price. That is a curvature.

GAMMA is the slope of the curvature.

But DELTA is first degree derivative and it is not the curvature. To correct this difference, GAMMA measure is important. Especially GAMMA measure is important in case of larger changes in underlying asset price.

**So how do we make the portfolio GAMMA NEUTRAL?**

Once portfolio is DELTA NEUTRALised, it is necessary to check its GAMMA.

### Vega

So far none of the GREEKS above considered the VOLATILITY. We assumed that the volatility of underlying asset is constant. But it in reality volatility changes over time. So volatility may have the impact on the value of the position.

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.

Higher absolute Vega means, the portfolio value is very sensitive to small changes in asset volatility.

Lower absolute VEGA means, the portfolio values has little impact from underlying asset volatility.

**What is VEGA and how do you NEUTRALIZE?**

Portfolio can be Vega neutralized by adding a position in traded option (trading offsetting option).

**What is the impact of VEGA NEUTRALIZation?**

To make the portfolio GAMM and VEGA Neutral – it needs two different traded options required.

In reality, traders re-balance at least once a day to maintain DELTA neutrality. It is usually not practical to maintain the Gamma and Vega neutrality on a regular basis. Smaller gamma and vega numbers are accepted. But when they get large, traders try to correct these.

The two other different GREEKS are THETA and RHO.

### Theta

THETA measures the rate of change of the value of the position with respect to the passage of the time with all else remaining constant.

### Rho

RHO measures the rate of change of the value of the position with respect to the interest rate, with all else remaining constant.

Hope this brief introduction help to build deeper insight into option sensitivities.