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# VOLATILITY SMILES & SURFACES

In this article, we will briefly discuss, in an ELI5 manner, the Black-Scholes Formula, volatility surfaces, and volatility smiles. Focusing on what these terms mean, and what they mean for options trading on Dopex. Before progressing make sure to read our previous article “What Are Options” as there are terms used in this article that you may not know but are explained in the previous article.

Without further ado let’s dive in.

**THE BLACK-SCHOLES MODEL**

**What Is The Black-Scholes Model?**

The Black-Scholes formula is a mathematical model designed to give traders a theoretical price of options. The formula uses the volatility of an option as an input to price the option. Option volatility measures how the price of the asset will move in the future. The greater the volatility, the more the asset moves.

**Black-Scholes Assumptions:**

In order to work, the Black Scholes model requires six assumptions to be true:

- The underlying stock does not pay a dividend
- The options are European Style
- Financial markets are always efficient
- All trades are not commissioned
- Interest rates are constant
- Underlying stock returns are log-normally distributed

The Black Scholes model assumes that the volatility of an option is constant but in reality, it is always changing. As a result of this, the market price of options with the same strike price and expiry are prone to diverge from their theoretic price. This in turn forms the basic concept of volatility surfaces.

**VOLATILITY SURFACES**

**What is a Volatility Surface?**

A volatility surface is a three-dimensional plot of the implied volatility of a stock option.

Assuming that the Black-Scholes model is completely correct, the implied volatility surface should be flat. However, in reality, this is not the case. The volatility surface is not flat and instead tends to vary over time due to the assumptions of the Black Scholes model being false at times.

As the time to maturity increases, volatilities across strike prices gravitate towards a constant level. It is key to note that options with a shorter time to maturity have significantly more volatility than options with longer maturities.

For any given strike price, implied volatility can be increasing or decreasing with time to maturity, giving rise to a phenomenon known as a volatility smile.

**VOLATILITY SMILES**

There are several models used to estimate volatilities. The method we will discuss here is the same method that Dopex uses. This method works by feeding the option price along with all other inputs from market data into the Black Scholes model in order to compute theoretical volatility. This volatility is what is known as implied volatility. Implied volatility rises when the underlying asset of an option is further out of the money (OTM)/in the money(ITM) than at the money (ATM).

**What is a Volatility Smile?**

A volatility smile is the shape of the graph that forms as a result of plotting the strike price and implied volatility of a group of options with the same underlying asset and expiration date but different strike prices. When the implied volatility is plotted for each of the different strike prices, a U-shape resembling a smiling face forms, hence the term volatility smile.

The Black-Scholes model does not predict the volatility smile. As mentioned before, the BSM predicts that when plotted against varying strike prices, the implied volatility curve is flat. The Black Scholes Model assumes that the implied volatility would be the same for options with the same expiry and underlying asset, even if their strike prices differ.

This, however, is merely theoretical. In practice, extreme events can occur causing significant price shifts in options. This potential/possibility for significant price shifts is factored into implied volatility. Which explains why implied volatility varies as options move more ITM or OTM.

**DOPEX AMM PRICING FORMULA**

The Dopex protocol consists of a number of moving parts to provide a liquid, well incentivized platform replicating successful option platforms such as Deribit and Robinhood with it’s own AMM.

The Dopex AMM uses assets from the asset pools along with Black-Scholes pricing, accounting for volatility smiles to allow anyone to purchase options based on strikes of their choice for future expiries.

The cost of options purchased is calculated on-chain based on the Black-Scholes formula — using implied volatility and asset prices retrieved via Chainlink adapters — and passed through a function to determine volatility smiles based on the realized volatility of the asset and past data.

**THE PROBLEM AT HAND**

Due to the fact that implied volatility is subjective without active/liquid order books. At the moment, it is very difficult to price options for assets without liquid option markets like the $BTC and $ETH markets.

**How Dopex Solves It**

The Dopex pricing model solves this by essentially creating a decentralized consensus model between delegates to quote on multipliers to create volatility surfaces without giving away their internal models on how they are getting these multipliers.

**About Dopex**

*Dopex is a decentralized options protocol that aims to maximize liquidity, minimize losses for option writers and maximize gains for option buyers — all in a passive manner.
Dopex uses option pools to allow anyone to earn a yield passively. Offering value to both option sellers and buyers by ensuring fair and optimized option prices across all strike prices and expiries. This is thanks to our own innovative and state-of-the-art option pricing model that replicates volatility smiles.*

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