GUIDEBOOKS

# Options Pricing

Explore our straightforward guidebooks focused on options pricing. These guides simplify the concepts of how options are valued, making them accessible to everyone. Ideal for those new to options trading or looking to deepen their understanding, they provide clear explanations and insights into the intricacies of options pricing.

## Delta

Delta in options trading refers to the rate of change in an option’s price for each one-point increase in the price of the underlying asset. In other words, it shows how much the price of an option is likely to move for every rupee change in the underlying asset price.

For instance, if the delta of an option is 0.5, this implies that for each ₹1 move in the underlying asset’s price, the option’s price will theoretically change by 50 paise. If the delta is positive, the option price will increase if the underlying asset’s price increases and decrease if the underlying asset’s price decreases.

It’s also worth noting that the delta of call options is always between 0 and 1, while the delta for put options is between 0 and -1. The closer an option’s delta is to 1 or -1, the more the price of the option responds to changes in the price of the underlying asset.

Remember, the delta changes as the price of the underlying asset changes, making it a dynamic number that traders need to monitor regularly. This is particularly essential for traders who practice delta hedging, a strategy that involves adjusting the number of options held to maintain a delta-neutral position.

## Rho

Rho is an essential concept in options pricing which forms part of the five primary Greek letters used to value derivatives. Rho measures the expected change in an option’s price in response to a change in the risk-free interest rates. Essentially, it depicts the sensitivity of an option’s price to changes in interest rates.

For instance, if an option has a Rho of 0.05, the option’s theoretical price will increase by 0.05 units for each percentage point increase in interest rates. It’s important to note that Rho is more significant for longer-term options as they are much more sensitive to changes in the risk-free interest rate.

Rho is particularly useful for investors who hold a portfolio of options, as it helps them to estimate their options price risk related to interest rate fluctuations. And while it might not be as widely recognized as delta, gamma, theta, or vega, having an understanding of rho can add another tool to an investor’s risk management arsenal. As with all Greeks, Rho is a theoretical measure and should be used carefully, understanding that actual results may vary.

## Gamma

Gamma is one of the “Greeks” in options trading that measures the speed of change of the option’s delta, another Greek that refers to the rate of change of the option’s price relative to a one-unit change in the price of the underlying asset.

Put simply, Gamma tells us how much the Delta will change with every one point move in the underlying asset. It helps traders to calculate the risk associated with options positions. If an option has a high Gamma, the price of the option can start to move dramatically, even with a small change in the price of the underlying asset, which can lead to substantial profits or losses.

But it’s also important to note that the Gamma of an option declines as the option gets closer to expiry. This is because the price of the underlying asset is less likely to make a significant move in a short period of time, so the option’s delta becomes less responsive to changes in the price of the underlying asset. Therefore, Gamma is a dynamic figure that changes as factors such as time to expiry and underlying price levels change.

## Historical Volatility

Historical volatility, often referred to as statistical volatility, is a statistical measure of the dispersion of returns for a particular security or market index. Essentially, it gauges how much a stock, commodity, or index has moved in the past, which is used by traders to anticipate what might happen in the future.

Historical volatility is most commonly calculated by determining the average deviation from the average price of a financial instrument in the given time period.

It is important to remember that historical volatility is just that – historical. It uses past market data, giving a hindsight view. Therefore, while it doesn’t perfectly predict future price changes, it provides an idea of how the asset price has changed over time, which can offer insight into its potential future behaviour.

If a stock has high historical volatility, it means its price has experienced wide swings and, conversely, a stock with low historical volatility has experienced lesser price changes. Traders, particularly those working with options, use this information to help assess market risk and to calculate option pricing.

## Theta (Time Decay)

Theta, also known as time decay, is one of the essential “Greek” metrics used in options pricing. It measures the rate at which an option’s price declines over time, assuming that all other factors remain constant. This is because options are time-limited; hence, the value of the option will decrease as it moves closer to its expiration date.

Essentially, Theta provides an estimate of the amount that an option’s price will decrease each day, all else remaining equal. If an option has a Theta of -0.05, for example, its value will decrease by 5 paise per day. Theta is typically negative for purchased options, meaning the price drops as time passes.

However, for sellers of options, who benefit as the price decreases, this time decay and associated negative Theta can actually work in their favour.

## Implied Volatility

Implied volatility (IV) is a fundamental concept used in options pricing. Essentially, it is a metric that reflects the market’s view on the likelihood of changes in a security’s price. It is often used to price options contracts, and it’s derived from an option’s premium.

Here’s how it works: options pricing models typically use variables like the price of the underlying asset, the option’s strike price, the time until the option expires, and a risk-free interest rate. However, these models also need an expected volatility input. This volatility is the implied volatility and is backed out from the market price of the option.

This implied volatility isn’t a direct forecast of future price volatility. Instead, it’s a reflection of what the market thinks future volatility could be. So, higher implied volatility generally means that larger price swings are expected, and such options will be pricier due to the increased risk for the option holder. Conversely, lower implied volatility indicates a lesser expected price movement.

Keep in mind, implied volatility does not have directional bias. It doesn’t predict the direction of the price movement, just the degree.

## Option Greeks

Option Greeks are statistical values that indicate the risk factors that could affect the value of an option’s contract. There are five main Greeks: Delta, Gamma, Theta, Vega, and Rho.

**The Delta**measures how the price of an option is expected to change per ₹1 change in the price of the underlying asset. It is essentially a measure of the option’s sensitivity to the underlying asset’s price.**The Gamma**reflects the rate of change for delta with respect to the underlying asset’s price. It represents the risk from the rate of change of the price of the underlying asset.**Theta**represents the rate of change between the option’s price and the time, or the time sensitivity of an option’s price. It measures the time decay of the option’s price.**Vega**represents the sensitivity of a derivative’s price to changes in the volatility of the underlying asset. It measures how much the option’s price is estimated to change for every 1% change in the underlying asset’s volatility.**Rho**, less commonly used, measures the impact of an interest rate change on the price of an option. It gauges how a change in the interest rate can impact the option’s price.

These Greeks allow traders to assess and manage the risk associated with different options positions better.

## Vega

Vega is an important concept in options pricing. In simple terms, it measures the sensitivity of an option’s price to changes in the volatility of the underlying asset. In other words, it tells you how much the price of an option will change for every 1% change in market volatility.

For instance, if an option has a Vega of 0.10, this means its price will increase by ₹0.10 for every 1% increase in volatility, all else being equal. Conversely, if market volatility decreases by 1%, the price of the option will decrease by ₹0.10.

Vega is particularly useful for options traders who lean more towards speculative or hedging strategies, as these traders tend to be more concerned with the price performance of an option, rather than owning the underlying asset itself. By understanding how Vega impacts an option’s price, these traders are better equipped to plan their trades around fluctuations in market volatility.

## Black-Scholes Model

The Black-Scholes Model is a mathematical model used to calculate the theoretical price of options. It was developed in 1973 by economists Fischer Black and Myron Scholes, with contributions from Robert Merton. The principles behind the model are rooted in the idea that markets are efficient and that the movement of asset prices can be predicted through a linear, or ‘normal’, distribution.

The entire calculation is built around five primary variables: the current price of the asset, the option’s strike price (the price at which the asset can be bought or sold using the option), the time until the option expires, the risk-free interest rate, and the volatility of the asset. The model assumes constant volatility and interest rates throughout the option’s life.

While the model may seem complex, its primary purpose is to help investors and traders assess whether an option is overpriced or underpriced. Although it’s widely used and regarded as an integral part of financial theory, the Black-Scholes Model should be used with caution, as its assumptions do not always hold in the real world. Market conditions can change rapidly, and unpredictable events can disrupt the assumed normal distribution of asset prices.