Volatility risk in financial derivatives

• Impact of volatility in the option value;
• Vega is the derivative of the option price with respect to the market volatility;
• Dynamics of the volatility smile (backbone);
• Particularities of the FX desk;

Hello all!

From a technical point of view, one of the most interesting things in finance (at least for me) is the probabilistic nature of the input variables and output results. It is pretty much like the life itself. Being essentially probabilistic, one should handle an inherent factor: risk. Risk is then associated with the effects that a change in the previous considered probability distribution might have in the current assumed position. In principle, the probability distribution is dependent of different parameters, that may change, generating risk. In this post, the focus will be on the volatility risk, that is, how changes of market volatility affect financial derivatives.

First steps

Let´s consider the risk-neutral equation for pricing of financial derivatives that can be read

$\frac{X(t)}{\beta(t)}=E\left[ \frac{X(T)}{\beta(T)} | F(t) \right]$

where $X(t)$ is the derivative value at $t$ and $\beta(t)$ denotes an unit investment in the money market account. The operator $E[... | F(t)]$ indicates the conditional expectation considering the information (filtration) available at $t$, that is, $F(t)$. The probability density employed in the equation above reflects the risk-neutral one, which we usually denominate as $Q$. One interested in working directly with the associated probability distribution, should consider spend some time solving the related Kolgomorov equations. This probability density encompasses the market variables observed at the filtration time $t$.

Vega: to Greek or not to Greek

Vega

In order to quantify the risk associated with each parameter included in the pricing model, one calculates the derivatives of the instrument value with respect to each of these parameters. These derivatives are called Greeks. The concept behind them is quite intuitive. The idea is to estimate the possible movements of the value considering changes in the market variables. Mathematically, this is expressed by a Taylor expansion, that is

$X(b)=X(b) + \frac{\partial X}{\partial b}(b-b_0)+\frac{1}{2}\frac{\partial^2 X}{\partial b^2}(b-b_0)^2+...$

where $b$ indicates the set of market variables plus time and each derivative of the expansion denotes a different Greek. The most known Greek is Delta, which is the derivative of the option value with respect to the spot value of the underlying asset. Gamma is the second derivative with respect to the value of the underlying asset. Many other Greeks exist and in fact a complete study of each Greek would require several posts(!). The derivative with respect to the market volatility 

$\vartheta = \frac{\partial X}{\partial \sigma}$

is called Vega (which is not a Greek letter, but several other things like a Street Fighter character).
A portfolio free of volatility risk is named Vega neutral portfolio, which can be obtained by adding the Vegas of the different positions that compose the portfolio. For sake of curiosity, I recommend this paper that discusses hedge of volatility risk by using financial derivatives whose underlying variable is the volatility of determined assets.
Higher values of Vega indicate that the present value of the financial instrument are more susceptible to changes in the market volatility. This is related to non linear increase of probability for the option ends up in the money with respect to the variance of the spot value. In fact, as a general rule, higher volatility of the underlying asset means higher prices of the financial instrument. 

Here, an important comment shall take place. Pricing models for financial derivatives are calibrated against implied volatilities of the more liquid instruments with the same underlying asset. This means that the realized volatility is actually indirectly captured by the model, since the implied volatility of the more liquid instrument denotes a expectation of the market and somehow just reflects the realized volatility. It means that from the point of view of the Vega, the source risk is the implied volatility.
The computation of Vega is usually one of the most complicated among the Greeks. This is because one should bump the parameter that directly affects the calibration of the model, which might be computationally expensive, mainly for multi-factor models, such as the Libor Market model. Other interesting point that should be considered is the specification of the exact procedure for the volatility bumping. Since one deals with a volatility surface, it is necessary to determine if the bump is absolute or relative and also if the magnitude will be the same for the whole surface.

Backbone

The must-read paper published by Hagan and his collaborators "Managing smile risk" presents the SABR model that computes the volatility of forward prices of an asset as a stochastic variable, that is, volatility is assumed to be a random variable described by a stochastic differential equation. The great advantage of this model compared to the former well-established local volatility model is that Hagan´s model encompasses the correct dynamics of the volatility smile with respect to changes in the underlying asset forward price. 

Put in other words, the SABR model allows the correct computation of vega and delta! Wait! Why delta as well? The reason is because the volatility smile (surface) might change with the underlying asset forward price. The "line" that guides the movement of the ATM volatility with the underlying forward price is called backbone. Therefore, when bumping the underlying price, the associated backbone should be considered. By the way, two questions to the reader: what other Greeks are affected by the backbone? What can be the intuitive shape of the backbone? 

FX smile parameters

Each desk has its own particularity when dealing with the market variables, including the volatility surface. In the FX desk, the horizontal axis of the smile is given in terms of the absolute (adjusted or not) BS delta and not in terms of the strike price, as usual for some other desks. In this case, for instance, two parameters can be used to describe the smile, named 25 risk reversal and 25 butterfly. 

The number 25 is associated with the points at which delta is 0.25 for puts and calls, which are symmetric with respect to the ATM point. The 25 risk reversal denotes the difference between the volatility of calls at 25 delta and puts at 25 delta. The 25 butterfly denotes the distance of the ATM point (that is, 50 delta) from the average of the 25 delta out of the money calls and puts.

FX smile example

It is quite clear that the values of the risk reversal and butterfly are directly related to the risk associated the volatility smile. For example, large (positive or negative) risk reversals indicate that a Vega neutral portfolio using options with the same underlying assets include different amounts of calls and puts.

I hope you enjoyed the post. Leave your comments and share!

Peace profound for all of you.

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Diogo de Moura Pedroso

LinkedIn: www.linkedin.com/in/diogomourapedroso