Understanding the Volatility Smile: Slope, Curvature, and Arbitrage
In the world of options pricing, the volatility smile is a fascinating phenomenon that has significant implications for traders and risk managers. But what exactly is it, and why does it matter? This post breaks down the concepts of slope, curvature, and arbitrage within the context of the volatility smile. We’ll explore how these elements interact, using examples to illustrate key points.
Why Should You Care About the Volatility Smile?
When trading options, understanding the volatility smile can give you insights into market sentiment and potential arbitrage opportunities. Essentially, the smile shows how implied volatility (IV) varies with different strike prices. This variation can tell you a lot about the risk-neutral probability density (RND) and help you price options more accurately.
Let’s go through some key concepts
Non-Arbitrage Conditions
In options pricing (martingale pricing framework), non-arbitrage conditions ensure that there are no opportunities for riskless profit. These conditions affect both the call price and the implied volatility (IV).
- Slope of the Call Price: The first derivative of the call price with respect to the strike price.
- Curvature of the Call Price: The second derivative of the call price with respect to the strike price.
- Slope and Curvature of the Implied Volatility: How the IV changes with the strike price.
The risk-neutral PDF (RND) is directly accessible from the call price. We can deduce the conditions to be fulfilled by the functional “call price” as well as the conditions for the implied volatility (slope and curvature). Finally, an expression of the risk-neutral probability density is obtained by taking the Black-Scholes model as a reference. The RND is expressed as a kind of series expansion à la Edgeworth “around” the normal law.
Equations for Non-Arbitrage Conditions
A summary of the bounds induced by the assumption of no arbitrage on call (put) and implied volatility is given below (derivations are provided in the appendices, and the results are equivalent to those of [1, 2]).
Let’s consider examples to Illustrate Key Points
Example 1: Call and Forward Call in the Heston Model
The Heston model is a popular framework for modeling volatility. It’s designed to be arbitrage-free, making it a great starting point for understanding the volatility smile.
The Heston model ensures no arbitrage exists by preventing the total variance curves from intersecting across different maturities. The subsequent figures depict the recovered risk-neutral distribution (RNDs) extracted from these volatility smiles. However, the right wing of the total variance curve visually appears concave. That’s not a smile but a frown.
I’ve included figures below illustrating the full and BS RNDs recovered from these volatility smiles.
Example 2: One-Period Binomial Model
The binomial model is simple yet powerful. It can illustrate how a volatility frown might occur, which is the opposite of a smile.
Suppose the stock price is S_0 at t=0, and there are only two possible future values (binomial process). Let’s say at t=T=1, S_1 can be either x_u*S_0 with a risk-neutral (RN) probability p, or x_d S_0 with an RN probability 1−p. For convenience, the risk-free rate is set to r=0. Under the RN measure, we have:
The RN probabilities are:
The call price:
This model is risk-neutral by construction and thus fulfills the constraints, but produces a frown.
There will be an arbitrage opportunity if the call is priced with the wrong model. Consider the previous binomial example and assume you used the Black-Scholes model to price the same call, using the true variance p(1−p) rather than the implied volatility. This arbitrage is illustrated by the difference between the true price (using the “true model” i.e., the binomial model) and the BS price with the binomial variance
This model arbitrage occurs due to the misspecification of the RND. The price difference is shown in Figure 5.
Example 3: The Bimodal Model
Next, let’s consider a more realistic model where the true RND is bimodal, achieved through a mixture of two lognormal distributions. In this scenario, I anticipate a volatility smile with a potential hump. While an unimodal model like the Black-Scholes (BS) model typically produces a smile with a single extremum, the bimodal model can exhibit a second extremum. This happens because the BS model is used as a reference, causing the implied volatility to increase to accommodate the second mode of the true RND.
Put simply, wherever the true call price exceeds the BS price, the implied volatility will be higher than the true volatility since the BS price is a function of the total variance. This concept is illustrated by mixing two lognormal distributions. The true and recovered RNDs are shown in Figure-6.
The call price is evaluated using the standard martingale pricing framework:
The true RND is depicted in Figure-6. When dealing with a convex combination of two regular and continuous distributions, the expectation and variance are:
Moreover, the relations between the expectation, the variance, and the parameters µk et σk of the kth lognormal law are
For illustrative purposes, here are calculated the true price and the Black-Scholes price using the lognormal equivalent volatility σ_eq_LN derived by substituting Var_bimod[X] into the equation of σ²_k. The resulting price difference is shown in Figure-7.
The smile induced by a bimodal RND has quite an unusual form but fulfills the no-arbitrage constraints. It is visually no so clear, but there is no crossing in the total variance plot.
As illustrated in Fig-6, the RND is correctly recovered form this peculiar smile.
Conclusion
Understanding the volatility smile and its implications is crucial for options traders. The smile tells you how implied volatility changes with strike price, which in turn helps you understand market expectations and potential arbitrage opportunities.
- Non-arbitrage conditions: Ensure that there’s no riskless profit.
- Risk-neutral density (RND): Provides a probabilistic view of future prices.
- Practical examples: Show how different models behave and how mispricing can lead to arbitrage.
By grasping these concepts, you can better navigate the options market and make more informed trading decisions. Happy trading!
References
1. Carr, Peter. Implied Vol Constraints. Bloomberg Working Paper, 2004.
2. Gatheral, Jim. The Volatility Surface: Statics and Dynamics.
3. Laurini, Márcio. Imposing No-Arbitrage Conditions in Implied Volatility Surfaces Using Constrained Smoothing Splines. Insper Working Paper, 2007.
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Appendix
A — Absence of arbitrage and consequences: bounds derivation
Consequence on the slope of the call price
Consequence on the curvature of the call price
B — Constraints on the smile
Bounds on the smile slope
From (15) and (18) we get a condition on the IV slope :
using the BS model as the reference, it comes
Bound on the smile curvature
Doing the same job with the condition (17) and (19), comes a condition on the curvature of the implied volatility :
C — Recover the RND from the smile
