Barrier option volatility smile
FINCAD offers the most transparent solutions in the industry, providing extensive documentation with every product. This is complemented by an extensive library of white papers, articles and case studies. In addition, he has conducted postdoctoral research in String Theory at the Universities of Durham and British Columbia. With several papers published in the Journal of High Energy PhysicsNuclear Physics and Physical Reviewhe has recently brought his extensive research experience in physics to the world of financial engineering.
Combining theoretical requirements and industry practice, Dominic designs derivative pricing models and explores their consequences in the financial markets. This will include functions to price European and American options, option strategies, portfolios of options, and options with arbitrary piecewise linear payoffs, from the smile as described in this article.
All numerical pricing is done through the no-arbitrage PDE. Other functions start instead with a given local volatility process defined either by standard parameterizations such as the CEV, normal and shifted barrier option volatility smile processesor a user-defined data table. These functions also price European and American options, option strategies, portfolios of options, and options with arbitrary piecewise linear payoffs, and can be used to back out the corresponding implied volatility smile.
Barrier option volatility smile a combination of the two sets of barrier option volatility smile, further issues such as the corresponding implied volatility smile.
Using a combination of the two sets of functions, further issues such as the smile dynamics can be analyzed. Starting with either a given local volatility function, or with a given implied volatility smile, the interplay between these functions is shown below.
Despite its many deficiencies, the Black-Scholes model of option pricing [ 1 ] remains widely used, some 30 years after its inception. It is still the first model that both quants and traders reach for when given a deal to price. Quants like it because it provides a theoretically consistent framework to price options in almost any asset class, given some possibly dubious assumptions. Traders like it barrier option volatility smile of the tractability of the Black-Scholes formula for option prices, and because it is easily tweaked to account for their intuition about the market.
Historical volatility is easily determined, though this is not necessarily a good guide as to how the future asset price will change. Traders have turned this drawback of the Black-Scholes model into a useful feature, by quoting European option prices not in terms of their dollar value, but in terms of the equivalent implied volatility. If the market price of some call option is C, then barrier option volatility smile implied volatility a function of asset price S, strike K and maturity T is defined through the relationship.
It is thus always possible to determine the market's view of the barrier option volatility smile volatility of some asset from quotes for options written on that asset: Using volatility as a unit of currency in this way is only possible due to some underlying properties of the Black-Scholes framework.
Most importantly, the barrier option volatility smile of an option is a monotonically increasing function of volatility. The ability to quote option prices in terms of a constant volatility is partly why the Black-Scholes model is still so widely used in the market. However, it makes the most important deficiency of the Black-Scholes framework transparent: At least since the crash ofthe market-observed implied volatilities of European equity options have exhibited a distinct "skew" structure.
Deep out-of-the-money puts generally trade at higher implied volatilities than out-of-themoney calls. One reason is that this reflects the market's appetite for insurance against stock prices falling. Options on interest rates - caps, floors, swaptions - also generally exhibit such a volatility skew. Other options exhibit a volatility "smile", in which both deep out-of-the-money calls and barrier option volatility smile trade at higher implied volatilities than their at-the-money counterparts.
We refer to the variation of implied volatility with strike price as the "volatility smile" despite the fact that a skew, frown, smirk or other such structures are often seen in the market. The implied volatility of traded options also varies with the option maturity - the volatility term structure - and so one often barrier option volatility smile of an implied volatility surface: Points on a volatility surface: The underlying price is It is easy to modify the Black-Scholes framework to deal with a volatility term structure, but the model simply does not allow for volatilities to vary with strike price.
Once one recognizes this it becomes clear that, in the words of Rebonato, implied volatility is really only the " Arguably the most important problem in the theory of option pricing is how to take account of the volatility smile.
This is far from a purely academic exercise. Given the skew shown in Figure 1, for example, what would be the correct volatility to use to price a European option barrier option volatility smile at ?
Whilst fairly easy to answer one would use some form of interpolation on the volatility smileit is less clear what volatility to barrier option volatility smile to price the corresponding American option, and much, much less clear what value to use to price an exotic contract, say a knock-in barrier option.
The real motivation for considering the effects of the volatility smile in option pricing is precisely this: Various models have been developed over the years to go beyond Black-Scholes barrier option volatility smile this way.
The constant volatility of the Black-Scholes framework corresponds to the assumption that the underlying asset follows a lognormal stochastic process in the risk-neutral measure. Barrier option volatility smile is the basic property which makes the model analytically tractable. The lognormal property is thus no barrier option volatility smile true within the local volatility framework. Rather the goal is to determine the correct underlying process from the market-observed smile, by determining the correct local volatility function.
Using standard arguments [ 11 ], the corresponding no-arbitrage partial differential equation PDE for the price, Vof an option written on the underlying asset is. Our approach will be to determine this function from the implied volatility smile, then barrier option volatility smile use standard numerical techniques to solve the PDE 3.
The transition probability function3 associated with the stochastic process 2 satisfies both the backward Kolmogorov and forward Fokker-Planck equations. The noarbitrage equation 3 barrier option volatility smile essentially the backward equation.
As shown by Dupire [ 7 ] see also [ 8 ] and [ 12 ]the local volatility function can in principle be determined from the forward equation. From the latter, one can derive. Both equations 3 and 4 apply in complete generality, to any option written on the asset S. The local volatility function can now be derived by specializing to European plain vanillas, say call options with prices C. Applying equation 4 to a complete set of such barrier option volatility smile for all strikes and maturities and upon re-arranging, we barrier option volatility smile Dupire's equation.
But only in principle, since one needs a complete set of option prices to fully determine the local volatility function; and one only has access to a finite set of such prices. One might wonder whether the right-hand side of equation 5 is always positive. In addition to this, numerical effects for deep out-of-the-money options, when the denominator 4 of equation 5 tends to zero, lead to badly behaved local volatility functions - the numerator must tend to zero at the same rate in order that the local volatility function remain finite.
For this reason, it is often preferable to rewrite equation 5 directly in terms of implied volatilities rather than option prices. The derivatives of the call price in equation 5 can thus be computed through the chain rule, and substituting for these leads to the following result [ 1213 ]:. Equation 6 is the basic result. Not only is it difficult to analyze analytically, but it is also not particularly wellbehaved numerically.
Small changes in the implied volatility lead to large changes in the local volatility, and in fairly unpredictable ways. However, equation 6 can in principle be used to determine the local volatility function from the market-observed smile. This can then be used through the PDE 3 to give option prices which are calibrated to the European plain vanillas seen in the market. To price a European option from the smile, one only needs to interpolate from the market-observed implied volatilities to the strike and maturity of the option in question.
The price will only be as good as one's interpolation scheme, and research continues into the pros and cons of various methods see, e. To price an American option, or anything at all exotic, we need barrier option volatility smile use the local volatility function in conjunction with a numerical method to solve equation 3.
We then barrier option volatility smile to compute the local volatility through equation 6 at each relevant point in S, t space. The procedure is notoriously unstable. Even after just a few time-steps, negative branching probabilities are encountered which one has to re-set by hand, although the problem is somewhat better if one uses an approach based on forward rather than spot prices [ 15 ].
Moreover, the local volatility function which one can back out from such an implied tree does not have particularly intuitive properties, as discussed at length in . In addition to this, the tree-based approach would have all the usual problems when applied, for example, to more exotic instruments such as barrier options. For these reasons, a PDE approach to option barrier option volatility smile is preferred.
Given barrier option volatility smile local volatility function, it is easy to apply standard numerical techniques to solve the no-arbitrage PDE 3 for option prices. We use the Crank-Nicolson scheme for well-documented reasons [ 13 ].
The application of local volatility to such finite difference methods was first discussed in [ 12 ]. We will not review the Crank-Nicolson method in any detail here. Suffice it to say that we have set up a grid of points in S, t space, and that the derivatives in equation 3 have been approximated using finite differences.
The procedure barrier option volatility smile in [ 12 ] to determine the discrete barrier option volatility smile volatility function is numerically involved and somewhat unstable. For these reasons, we prefer to work with the result 6which relates the local volatility function directly to the implied volatility smile. A simple finite difference approximation for the derivatives in equation 6along the lines discussed in [ 2 ], is the simplest way to generate the discrete local volatility function, and seems to give results at least as good as any other procedure.
This is a real problem since the results depend very sensitively on the interpolation method. Thankfully such effects, as long as they are localized, do not lead to large differences in option prices. In that sense, the actual behaviour of the local volatility function is not too relevant to option pricing.
Given a set of implied volatilities as in Figure 2, there are various interpolation methods one might use. Both parametric and non-parametric methods have been discussed in the literature, a selection of these including:.
Each method has its benefits and drawbacks. This dependence can be made less sensitive by using smoothing techniques. Barrier option volatility smile simple such technique - using smooth cubic splines - was discussed originally in the context of probability density functions [ 18 ], and applied to the implied volatility surface in [ 19 ]. The latter work also showed how to apply no-arbitrage constraints to the procedure in a relatively straightforward binary options for beginners to open a demo account frontstocks and also 60 sec what is the best bin. The smoothing technique is applied in the strike direction only.
In the context of probability density functions , one is after all only interested in the implied volatility as a function of strike price, for options of a single maturity. We will thus construct a smooth cubic spline in strike space at each timestep in our grid. To generate the implied volatilities to be smoothed at each time-step, we first need to construct a pre-smoothed surface.
This can be done using a bi-linear or bi-cubic spline, but a better method is to use a thinplate spline [ 19 ]. One may, for instance, only have data for a few different strikes at each maturity, and these strikes may differ across maturities.
Such a data set would not be well-suited to bi-linear or bi-cubic interpolation. A thin plate spline is the natural two-dimensional generalization of the cubic spline, in that it is the surface of minimal curvature through a given set of two-dimensional data points. Denoting these points bythe objective function is given by.