Despite its flaws and limitations, the Black-Scholes model became the benchmark to interpret option prices. Specifically, option prices are reverse-engineered to calculate implied volatilities, the same way that bond prices are transformed into yields, which are easier to understand. This process, combined with interpolation and extrapolation techniques, gives rise to an entire surface along the strike and maturity dimensions.
The Black-Scholes model assumes a single constant volatility parameter to price options. In practice, however, every listed vanilla option has a different implied volatility (K, T) for each strike K and maturity T. Figure 2.1 shows what an implied volatility surface looks like.
For a fixed maturity T the curve is called the implied volatility smile or skew and exhibits a downward-sloping shape, as shown in Figure 2.2. Note that in other asset classes, such as interest rates or currencies, the smile tends to be symmetric rather than downward-sloping.
A direct consequence of the implied volatility smile is that the Black-Scholes model gives inaccurate call spread and put spread prices. To illustrate this point, Figure 2.3 shows the Black-Scholes price of a one-year call spread with strikes $100 and $110 as a function of the single Black-Scholes volatility parameter σ. We can see that the curve peaks at σ ≈ 30.8% for a maximum price of $3.78. Thus, no single value of σ may reproduce any market price above $3.78. Interestingly, this phenomenon is not symmetric: $90–$100 put spreads can be priced with a single volatility parameter, but the value of σ will be significantly off the level of implied volatility for each put.
The smile does not mean that Black-Scholes is wrong and should be rejected. Practitioners typically price over-the-counter (OTC) vanilla options using Black-Scholes and an appropriate volatility interpolation or extrapolation scheme, for the simple reason that implied volatilities are derived from listed option prices in the first place. In other words, the fact that the Black-Scholes model may be faulty is fairly irrelevant for vanilla option pricing.
However, the vanilla hedge ratios or Greeks are model-dependent and should be adjusted for the smile. To see this, notice first how a change in spot price impacts the moneyness of an option: after a $1 uptick from an initial $100 underlying spot price, an out-of-the-money call struck at $110 is now only $9 out of the money. In the presence of the smile, we may want to use a different implied volatility to reprice the call, and we must make an assumption on the behavior of the smile curve:
Other rules would obviously produce different results.
In Chapter 3 we will see that European exotic payoffs can in theory be replicated by a static portfolio of vanilla options along a continuum of strikes. In the absence of arbitrage, the price of the exotic option must match the price of the portfolio. Thus it would be inaccurate to use the Black-Scholes model to price the exotic option in the presence of the smile.
As a fundamental example consider the digital option that pays off $1 at maturity T if the final spot price ST is above the strike K, and 0 otherwise. The Black-Scholes value for the digital option is simply:
where S is the spot price, r is the continuous interest rate for maturity T, and σ is the constant Black-Scholes volatility parameter.
Digital options are difficult to delta-hedge because their delta becomes very large around the strike as maturity approaches. Equity exotic traders will typically overhedge them with tight call spreads. For example, to overhedge a digital paying off $1,000,000 above a strike of $100 and 0 otherwise, a trader might buy 200,000 call spreads with strikes $95 and $100.
Figure 2.4 shows how in general a quantity 1/ϵ of call spreads with strikes K − ϵ and K will overhedge a digital option struck at K. In the limit as ϵ goes to zero, we obtain an exact hedging portfolio whose price is:
where σ*(K, T) is the implied volatility for strike K and maturity T and VBS is the Black-Scholes vega of a vanilla option. Because the equity smile is mostly downward-sloping we typically have and thus the digital option is worth more than its Black-Scholes value.
Implied volatilities derived from listed option prices are only available for a finite number of listed strikes and maturities. However, on the OTC market, option investors will ask for quotes for any strike or maturity, and it is important to be able to interpolate or extrapolate implied volatilities.
Interpolation is relatively easy: for a given maturity, if the at-the-money option has 20% implied volatility and the $90-strike option has 25% implied volatility, it intuitively makes sense to linearly interpolate and say that the $95-strike option should have a 22.5% implied volatility. Similarly, for a given strike, we can linearly interpolate implied volatility through time.
One issue with linear interpolation, however, is that it produces a cracked smile curve. More sophisticated interpolation techniques, such as cubic splines, are often used to obtain a smooth curve. Figure 2.5 compares the two methods.
It must be emphasized that unconstrained interpolation methods may produce arbitrageable volatility surfaces. Several papers listed in Homescu (2011) discuss how to eliminate arbitrage.
On the other hand extrapolation is a difficult endeavor: how to price a five-year option if the longest listed maturity is two years? There is no definite answer to this question, and we must typically resort to a volatility surface model (see Section 2.4).
Note that extrapolating a cubic spline fit tends to produce unpredictable results and should be avoided at all costs.
Not every surface f(K, T) is a candidate for an implied volatility surface . Denote the call and put values induced by , respectively. To preclude arbitrage we must at least require:
These1 inequalities place upper and lower bounds on and its derivatives. For example, by the chain rule applied to , we obtain , and thus is equivalent to the upper bound .
When designing an implied volatility surface model, it is important to check that these constraints are satisfied.
The implied volatility surface must also satisfy certain asymptotic properties. Perhaps the most notable one for fixed maturity is that implied variance, the square of implied volatility, is bounded from above by a function linear in log-strike as kF → 0 and kF → ∞:
where kF = K/F is the forward-moneyness and β ∈ [0, 2] is different for each limit. This result is more rigorously expressed with supremum limits and we refer the interested reader to Lee (2004).
Every large equity option house maintains several proprietary models of the implied volatility surface, which are used by their market-makers to mark positions. By definition these models are not in the public domain, and we must regrettably leave them in the dark. Fortunately some researchers have published their models and we now present a selection.
There are two ways to model the volatility surface :
Here, it is worth distinguishing between two kinds of implied volatility : market-implied volatility , which is computed from market prices; and model-implied volatility , which is induced by a volatility surface model attempting to reproduce . However, for ease of notation we will often keep the notation σ* when there is no ambiguity nor need for such distinction.
A popular parameterization of the smile for fixed maturity is the SVI model by Gatheral (2004). SVI stands for stochastic volatility-inspired and has the simple functional form:
where a, b, ρ, m, and s are parameters depending on T, and kF = K/F is the forward-moneyness. Intuitively a controls the overall level of variance, m corresponds to a moneyness shift, ρ is related to the correlation between stock prices and volatility and controls symmetry, s controls the smoothness near the money (kF = 1), and b controls the angle between small and large strikes.
Figure 2.6 shows an example of the shape of the smile produced by the SVI model, which is plausible.
The SVI model is connected to stochastic volatility models (see Gatheral-Jacquier (2011)). Specifically, the authors show how Equation (2.4) is the limit-case of the implied volatility smile produced by the Heston model as the maturity goes to infinity.
To ensure the no-arbitrage condition Equation (2.1) we must have . In his original 2004 talk, Gatheral claims that this condition is also sufficient to ensure Equation (2.2), but a recent report by Roper (2010) suggests otherwise.
An attractive property of the SVI model is that it is relatively easy to satisfy Equation (2.3) since its parameters are time dependent. This is also a drawback: as a surface, the SVI model has too many parameters. To circumvent this issue, Gurrieri (2011) put forward a class of arbitrage-free SVI models with term structure using 11 time-homogenous parameters.
It should be noted that, being a function of and thus , the SVI model incorporates the sticky-delta rule and thus produces a higher delta than Black-Scholes, as shown in Figure 2.7.
Any alternative to Black-Scholes will generate an implied volatility surface, which may be used to appraise the quality of the model. This is a major source of implied volatility surface models.
The stochastic alpha, beta, rho (SABR) model of Hagan and colleagues (2002) assumes that the underlying forward price dynamics are described by the coupled diffusion equations:
where W and Z are standard Brownian motions with and α, β, ρ are constant model parameters.
In the special case β = 1 an analytical formula for implied volatility is available for short maturities, which allows to fit the parameters to observed option prices. Near the money the formula has the Taylor expansion expression:
The SABR model is popular for interest rates where the smile is more symmetric than for equities.
The Heston (1993) model is perhaps the most popular approach for stochastic volatility. It assumes the following underlying spot price dynamics coupled to an instantaneous variance process:
where W and Z are standard Brownian motions with and are constant model parameters that must satisfy the Feller condition to ensure strictly positive instantaneous variance vt at all times. The instantaneous rate of return μt on the underlying asset can be anything since it will disappear under the risk-neutral measure.
Rouah (2013) provides a valuable resource detailing the theoretical and practical aspects of the Heston model, including code examples.
Although no analytical formula for implied volatility is available, the popularity of the Heston model is largely due to the existence of quasi-analytical formulas for European options making the computation of implied volatilities very quick.
Figure 2.8 compares a Heston fit to the S&P 500 implied volatility surface. We can see that the Heston model produces a plausible shape but is too flat for short expiries.
One limitation of the Heston model is that instantaneous variance is a somewhat elusive concept, which cannot be measured in practice. However an analytical formula for the total expected variance over a period [0, T] is available:
Hence the expected total annualized variance is approximately constant at v0 regardless of maturity, which is inconsistent with empirical market observations (see Section 5.2).
Recently Carr and Wu (2011) proposed a sophisticated framework for the underlying forward price dynamics, which results in a closed-form formula for implied volatility. The forward price process solves the diffusion equation:
where W is a standard Brownian motion and v is an arbitrary stochastic instantaneous variance process. Furthermore, the entire implied volatility surface is assumed to evolve through time according to the diffusion equation:
where Z is a standard Brownian motion with and μ, ω, ρ may be stochastic.
Carr-Wu then show that in this setup the implied volatility surface at any time t is fully determined by a quadratic equation, which depends on μt, ωt, ρt and the unspecified vt.
This very general framework can produce a wide range of implied volatility surfaces. Choosing and yields the Log-Normal Variance (LNV) model in which is solution to the quadratic equation:
where κ, w, η, θ, v, ρ are constant parameters and k denotes moneyness K/S0.
The LNV model thus combines two attractive features: a functional parametric form for the implied volatility, and known dynamics for the evolution of the underlying spot price as well as the implied volatility surface itself.
Consider an underlying asset S with spot price S and forward price F. Let r denote the continuous interest rate for maturity T, be the upper and lower bounds on the slope of the smile corresponding to the no call or put spread arbitrage condition (2.1). Given , show that that where .
Assume zero interest rates and dividends. Consider the Black-Scholes formula for the European call struck at K with maturity T:
where S is the underlying asset's spot price, σ is the volatility parameter, and N(.) is the cumulative distribution function of a standard normal.
Consider a one-year vanilla call with strike K = 1, and let (S) be its implied volatility at various spot price assumptions S. Assume zero rates and dividends and denote the call price:
Using your favorite optimization software (Matlab, Mathematica, etc.) find the parameters for the SVI model corresponding to a least square fit of the following one-year implied volatility data:
Strike (%forward) | 20% | 50% | 70% | 90% | 100% | 110% | 130% | 150% | 160% |
Implied volatility | 45.5% | 34.6% | 29.4% | 24.0% | 22.3% | 19.9% | 16.4% | 14.9% | 14.3% |
Answer: a = 0.0180, b = 0.0516, ρ = −0.9443, m = 0.2960, s = 0.1350 using initial condition a = 0.04, b = 0.4, ρ = −0.4, m = 0.05, s = 0.1, and bounds a > 0, 0 < b < 2, −1 < ρ < 1, s > 0.