What is the Black-Scholes-Merton (BSM) Model?

On the Gate platform, all option Greeks, such as Delta, Gamma, and Theta are derived using the Black-Scholes-Merton option pricing model, commonly referred to as the BSM model. This is one of the most influential models in the field of option pricing. Other well-known pricing models, such as the Heston and SABR models, which are also used in enterprise-level applications across different types of options. These models offer further insights into how Greeks are calculated and how various factors influence option pricing.

While most people don’t need to understand all the mathematical details of the model, having a basic understanding of how it works is still valuable. Whether or not you use it to perform calculations yourself, if you’re trading options, you’ll inevitably come across figures derived from the Black-Scholes model.

Black-Scholes-Merton (BSM) Model

Key Input Parameters:

• Current Price of the Underlying Asset – The present market price of the asset the option is based on
• Strike Price – The agreed-upon price at which the asset can be bought or sold under the contracts
• Time to Expiry – The remaining time until the option expires, typically expressed in years with decimal precision
• Risk-Free Interest Rate – A benchmark rate representing the time value of money
• Implied Volatility (IV) – The market’s expectation of the asset’s contract price volatility

Model Outputs:

• Theoretical Fair Value of the Option (primary output)
• Greek Risk Parameters
• Price Sensitivity Metrics

The BSM model processes these market inputs through a rigorous mathematical framework to generate fair and rational option pricing, offering a quantitative foundation for trading decisions. While the model is based on theoretical assumptions that may not always hold in practice, its core logic remains an essential and widely used benchmark in the field of option pricing.

Impact of BSM Model Parameters on Option Pricing

Within the Black-Scholes-Merton (BSM) option pricing framework, fluctuations in each input parameter directly affect the theoretical value of an option. When all other factors are held constant, the following relationships describe how each variable influences option prices:

Impact of Changes in the Underlying Asset Price:
When the price of the underlying asset increases, the value of call options rises, while the value of put options decreases. This is because asset appreciation makes the right to buy at a fixed strike price more valuable, while reducing the value of the right to sell at a fixed price.

Impact of Changes in Strike Price:
An increase in the strike price leads to a decrease in call option value and an increase in put option value. This effect is opposite to that of rising asset prices. For call options, a higher strike price means you must pay more to acquire the asset, reducing its value. Conversely, for put options, a higher strike price allows the holder to sell at a better price, increasing its value.

Impact of Time to Expiration:
As the expiration date approaches, the value of both calls and puts generally decreases. This is due to the declining time value of options — the less time left, the fewer opportunities for the underlying price to move in a favorable direction.

Impact of Risk-Free Interest Rate:
An increase in the risk-free interest rate typically raises call option prices and lowers put option prices. This is because higher interest rates affect the cost of carry and the present value of future payoffs, altering option valuations accordingly.

Impact of Implied Volatility (IV):
An increase in implied volatility raises the value of both call and put options. Higher volatility signals a greater likelihood that the underlying asset’s price will move significantly in either direction, increasing the option’s potential value.

The Black-Scholes-Merton model captures these dynamics through a structured mathematical framework, serving as a quantitative foundation for option pricing in the market. By gaining a deeper understanding of how each parameter influences option values, traders can better anticipate price movements and make more informed trading decisions.

Delta on Gate’s Options Platform

Where to Find the Greeks on Gate

On Gate’s options trading page, users can select and view relevant Greek values in the top column of each T-shaped options chain.

Greeks are key metrics used to measure the sensitivity of an option’s price to various market variables.

• First-order Greeks: These represent the rate of change of the option’s price with respect to a single underlying factor (e.g., underlying price, volatility, time).
• Second-order Greeks：These measure the sensitivity of the first-order Greeks themselves to changes in market parameters.

In this module, we’ll briefly introduce common Greeks and then dive deeper into each one. Let’s begin with the most fundamental first-order Greek — Delta.

1.What is Delta?
Delta represents “the sensitivity of an option’s price to changes in the price of the underlying asset”, mathematically, it is the partial derivative of the option price with respect to the price of the underlying:

• Call options: 0 ≤ Delta ≤ 1
• Put options: –1 ≤ Delta ≤ 0

What Delta Means on Gate
On Gate, when the price of the underlying asset changes by 1 USDT, the expected change in the theoretical value of the option is equal to Delta:

2.Intuitive Explanation

• Call Options
  When the underlying price increases, the value of the right to “buy at the strike price” rises — hence, Delta is positive.
  Example: If you have the right to buy an asset at 10 USDT, and the market price rises from 10 USDT to 11 USDT, your option becomes more valuable.

• Put Options
  When the underlying price increases, the value of the right to “sell at the strike price” falls — hence, Delta is negative.
  Example: If you have the right to sell an asset at 10 USDT, and the market price rises from 9 USDT to 10 USDT, your option loses value.

3.Example

4.Summary

• Delta is one of the most closely monitored Greeks by traders, as it directly reflects how sensitive an option is to price movements in the underlying asset.
• It helps investors quickly estimate position risk, enabling smarter hedging or position adjustments.
• In upcoming modules, we’ll cover Gamma, Theta, Vega, and other Greeks to help you build a more comprehensive options risk management strategy.