### What Is Volatility Surface?

Financial institutions consume market data on continuous basis to value their portfolios. As human body cannot function without blood, financial institutions cannot value their portfolios without market data. Therefore, understanding market data is the key to calculate risk and return of financial transactions and it can also help us understand finance in general.

This article aims to explain what volatility surface is, its usage and how it is constructed.

### What Is Volatility Surface?

Volatility surface contains volatilities that are used to price a number of financial trades e.g. options, swaptions etc. Volatility surface can be of many types, for example FX Volatility Surface or Interest Rate Swaption Volatility Surface.

*I will explain all of the basic components of volatility surface individually.*

#### Volatility Surface Structure

A volatility surface has usually three dimensions: Expiry, Tenor, and Volatility Value. These volatility values are implied volatilities which are produced from the market prices of traded options.

It is therefore important to understand how options work.

#### Options:

Option is a derivative contract that gives the buyer/seller right, but not the obligation, to buy/sell the underlying security. To learn more about options, have a look at my article:

Option contracts can be on a number of underlying securities such as interest rate swap, CDS, Exchange Rates, FX swaps etc. Swaption is a type of option where the buyer has the right to enter into buying an underlying swap such as interest rate swap or CDS.

Price of a swaption is dependent on all of the factors that can influence price of an option or price of the underlying swap, such as term of the swap contract and maturity of the option which is also known as option expiry.

An option can be priced using Black Scholes formula.

#### Black-Scholes Formula

Black-Scholes is a mathematical model that can give us theoretical price of options. It takes in volatility of an option as input to price the option. Volatility of an option measures how stock will move in the future. It is expressed as an annual basis in percentage.

Higher the volatility, more the stock price moves.

Black scholes model assumes that the volatility is constant whereas in practice, volatility is constantly changing.

As a consequence, the market price of options with the same expiry and strike price tend to diverge from their theoretic price which is calculated by using the Black Scholes formula.

This forms the basic concept of Volatility Surface.

There are several models implemented to estimate volatilities. One common and simple methodology is to feed the actual option’s price along with all other inputs back into Black Scholes model to compute theoretical volatility. This volatility is known as **implied volatility.**

It is important for an investor to ensure that their portfolio’s theoretical value is as close as possible to the market value.

Subsequently, volatility smile is constructed from the implied volatilities.

#### What is a Volatility Smile?

If we plot implied volatility for options with different strike prices, we will notice a line graph that resembles shape of a smile.

This chart is known as **Volatility Smile**.

Volatility smile shows that volatilities are higher for at-the-money options.

An at-the-money option is when the strike price equals the price of the underlying asset.

Let’s now understand how implied volatilities are used to construct volatility surface.

#### Constructing Volatility Surface From Volatility Smile

As outlined above, volatility smiles are constructed from implied volatilities. We then calculate volatilities of options with a range of different strike price and expiries. Once we compute volatilities for at the money options with different expiries, we can start constructing a volatility surface out of the volatility smiles.

- To elaborate, at the money swaption trades are selected with a range of option expiries and/or strike prices that are based on swaps over a range of tenors (or maturities). These options are used to compute implied volatility smiles.
- We then extend volatility smiles by interpolating the points using smile interpolation. Piece-wise linear interpolation is usually used due to its simplicity.
- Once a range of volatility smiles are produced for different tenors and expiry terms, we join all of the smiles on terms and tenors and plot the smiles together. A three dimensional surface is then produced. It is known as volatility surface.

As a result, a three dimensional surface is constructed from volatility smiles.

This surface can then be used to price options. With interpolation, we can establish volatilities for a larger range of expiries and tenors.

### How Does Volatility Surface Look?

If we plot volatility surface, where y axis is the volatility point, x axis is the term and z axis is the tenor then we can see a volatility surface:

Volatility surfaces can also be created on different combinations such as from options with different strikes and maturities.

The table below illustrates three dimensional swaption volatility points:

### FX Volatility Example:

If we have been given FX volatility for two currencies and we are required to calculate volatility for the third currency then we can use volatility formula along with exchange rates triangulation to compute the required volatility.

**For example, if we have:**

- Currency 1 per Base Currency e.g. GBP per USD where USD is the base currency
- Currency 2 per Base Currency e.g. EUR per USD

We are required to calculate FX volatility for EUR per GBP then we can use the volatility formula with exchange rate triangulation:

Volatility Currency 2 per Currency 1 Squared = (Volatility Currency 2 per Base Currency Squared) + (Volatility Currency 1 per Base Currency Squared ) — (2 x Volatility Currency 2 per Base Currency x Volatility Currency 2 per Base Currency) x Correlation of (Currency 1, Currency 2)

Correlation can be derived by calculating mean and standard deviation of the volatilities of each currency, potentially by using Person correlation formula.

### Summary

This article demonstrated what volatility surface is, how it is constructed along with its usages.

Hope it helps.

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