# Is price process of a risky asset a martingale

the risk-neutral measure and its connection to the price process as a martingale. in other words, is price process of a risky asset a martingale?
(Last Updated On: July 16, 2023)

I wanted to discuss a fundamental concept in financial mathematics that interests you: the risk-neutral measure and its connection to the price process as a martingale. in other words, is price process of a risky asset a martingale?

The risk-neutral measure is a critical concept that allows us to price financial derivatives using a simplified assumption about the market. It is widely used in quantitative finance and is crucial in option pricing theory. To truly grasp its significance, it is essential to understand the relationship between the risk-neutral measure and the price process as a martingale.

#### books2 – QUANTLABS.NET

Firstly, let’s briefly define what we mean by the risk-neutral measure. In a financial market context, it is an equivalent probability measure that makes the discounted price process of a risky asset a martingale. Essentially, it is a measure that eliminates the risk associated with the underlying asset’s price movement.

Now, let’s delve into the connection between the risk-neutral measure and the price process being a martingale. A martingale is a mathematical concept that describes a stochastic process in which the expected value of the future value, given the information up to the current time, is equal to the current value. A martingale represents a fair game, where the expected value of future gains or losses is the same as the current value.

When the price process of a risky asset follows a martingale, it implies that the market is efficient and there are no opportunities for arbitrage. In other words, the asset’s price accurately reflects all available information, making it impossible to make riskless profits consistently.

The risk-neutral measure comes into play by allowing us to transform the price process of a risky asset into a martingale. Using this measure, we can determine the fair price of financial derivatives, such as options, based on the assumption that the market is risk-neutral. This assumption simplifies the pricing process by removing the need to consider the complex dynamics of the underlying asset’s price movement.

To summarize, the risk-neutral measure is a powerful tool in quantitative finance that enables us to price derivatives by assuming a risk-neutral market. Making the price process of a risky asset a martingale ensures that the fair value of these derivatives is accurately reflected in the market. Understanding this connection is crucial for pricing options and other financial instruments effectively.

#### books2 – QUANTLABS.NET

I hope this email has clearly explained the risk-neutral measure and its relationship to the price process as a martingale. Please feel free to reach out if you have any further questions or want to discuss this topic in more detail. I’m always eager to engage in fruitful discussions and explore new concepts.

Thank you for your time, and I look forward to your thoughts.

#### books2 – QUANTLABS.NET

maybe you can answer this questions for me:

quant.stackexchange.com/questions/76134/rough-volatility-and-change-of-measure

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