Option Price is a function of 5 factors.

Spot price

Strike price

Volatility

Time

Interest rate.

Y = f(x1, x2, x3, ….. xK)

### Three Important Concepts in Risk Management

Sensitivity Analysis: Sensitivity analysis shows the change in the value of Y by the change of 1 factor (x1 or x2 or xK... etc.), cet par.

Scenario Analysis: We create different scenarios, like conservative, moderate and optimistic.

Simulation: We shock all the factors simultaneously (x1, x2, xK....).

Option greeks show the sensitivity analysis of options with respect to each factor.

### Definitions of Option Greeks

Option's Delta: It represents a change in the option's price with respect to a change in the underlying's price.

Option's Gamma: It represents a change in the delta with respect to a change in the underlying's price.

Option's Vega: It represents a change in the option's price with respect to change in the volatility of the underlying.

Option's Theta: It represents a change in the option's price with respect to the passage of time.

Option's Rho: It represents a change in the option's price with respect to change in interest rate (have least impact).

Going back to the option pricing factors, the following greeks influence the respective pricing factors:

Spot price - Delta and Gamma

Strike price - Remains constant.

Volatility - Vega

Time - Theta

Interest rate - Rho

To understand the behavior of option greeks, we need to analyze how all the greeks behaves with the change in 3 key parameters:

Moneyness of the Option

Volatility of the Underlying

Time to Maturity

## Understand all the Option Greeks in Detail

### Option's Delta

a) As the share price increases, the price of the call option increases and vice versa. Hence the delta of a call option is always positive. (0 to 1)

b) As the share price increases, the price of the put option decreases and vice versa. Hence the delta of a put option is always negative. (0 to -1)

Relationship of Delta with Moneyness: Delta of an ATM option is close to 0.5. Delta of an option increases as the option moves ITM. (for call options - moves from 0.5 towards 1 and for put options - moves from -0.5 towards -1 ). Delta of an option decreases as the option moves OTM. (for call options - moves from 0.5 towards 0 and for put options - moves from -0.5 towards 0 )

Relationship of Delta with Volatility: As volatility increases, the delta of an option decreases, and as volatility decreases, the delta of an option increases.

[ Hint: Check the formula of (d1) in BSM ]

Relationship of Delta with Time: As time passes, the delta of ITM options tends to move towards 1, and the delta of the OTM option tends to move towards 0. Further ATM delta shows more jumps as options move closer to expiry.

Rationale: For ITM and OTM options, longer maturity options have more time than shorter maturity options for the movement of underlying prices to impact the moneyness.

| Suggestion: Revise the concept of Delta Hedging |

### Option's Gamma

It is a second-order derivative. Gamma measures the curvature of the relationship between the option price and stock price.

Relationship of Gamma with Moneyness: Gamma is highest for ATM options and it starts decreasing as the option moves ITM or OTM.

Rationale: The curvature in the relationship between an option and stock prices are highest on ATM (in the middle). As the option moves towards deep ITM, the option's delta starts increasing and moves towards 1 (getting close to linear movement). Similarly, as the option moves towards deep OTM, the delta of the option starts decreasing and moves towards 0 ( getting close to linear movement).

Relationship of Gamma with Volatility: As volatility decreases, Gamma increases, and as volatility increases, gamma decreases.

[ Hint: Inline with Delta ]

Relationship of Gamma with Time: With the passage of time, the gamma of ATM options goes up and the Gamma of ITM/OTM options moves towards 0.

Rationale: With the passage of time, the delta of ITM options tends to move towards 1, and delta of OTM options tends to move towards 0 and tries to move in a linear fashion, and hence the curvature decreases.

[ Hint: Passage of time always intensifies the movements except for vega ]

| Suggestion: Revise the concept of Delta-Gamma Hedging |

### Option's Vega:

Option buyers love volatility as it increases the probability of large shifts in the underlying prices.

Therefore vega of long option positions (both call and put) is positive and the vega of short option positions (both call and put) is negative.

Relationship of Vega with Moneyness: Vega is highest for ATM options and it starts decreasing as the option moves ITM or OTM.

Rationale: If the option is deep ITM/OTM, the increase/decrease in volatility (all other factors remaining constant) will not change the worthiness of an option much.

Relationship of Vega with Volatility: As volatility increases, the vega of an option increases and as volatility decreases, the vega of an option decreases (straightforward).

Relationship of Vega with Time: As time passes, the vega of an option decreases. Longer maturity options have a higher vega than shorter maturity options.

Rationale: The price of an option is a combination of intrinsic value and extrinsic value (time value/volatility premium)

The premium of near-term options is majorly made up of the intrinsic value component. However, time value/volatility premium makes up a major proportion of the option price for longer maturity options.

| Suggestion: Revise the concept of Implied Volatility |

### Option's Theta

As time passes, the value of options decreases (cet par). Long options (both call and put) have negative theta and short options have positive theta.

Short positions enjoy time decay.

Relationship of Theta with Moneyness: Theta is highest (negative) for ATM options and it starts approaching towards 0 as the option moves ITM or OTM.

Rationale: Similar to vega, if the option is deep ITM/OTM, the passage of time (all other factors remaining constant) will not change the worthiness of an option much.

Important Concept: Long positions (both call and put) in options suffer time decay. However, a call option suffers more time decay than a put option.

The rationale behind this can be understood from the BSM formula. The BSM formula takes the present value of strike price in the calculation. The formula works in such a way that the present value of strike price decreases the value of call option but increases the value of put options. Further, this phenomenon can sometimes lead the near-term put options more valuable than far-term puts (having positive theta).

Relationship of Theta with Time: As time passes, the rate of time decay increases. Near-term options have higher theta than long-term options.

Rationale: If an option is expiring in 10 days, the option premium will suffer a time decay of 10% in 1 day i.e. 1/10. However, if the option is expiring in 2 days, the option premium will suffer a time decay of 50% in 1 day i.e. 1/2.

### Option's Rho

Options can be used to replicate the exposure of the underlying asset. A call option is a substitute for buying a stock and a similarly put option is a substitute for selling a stock. Buying or selling stock requires an upfront transfer of cash. However, taking an option position can provide leverage. Buying a call option can save financial costs and similarly selling a put option leads to loss of interest income that could have been earned if sold the underlying stock.

Hence rho of a call option is positive and the rho of a put option is negative.

Relationship of Rho with Time: Rho of short equity options is negligible.

### Numerical Example on Option Greeks

Taking a call option with Spot price = 100 and Risk-free interest rate = 10%:

| Hint: Relate these results with the above concepts |