local volatility in option pricing, resources _ fincad

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Pushing the Limits of Local Volatility in OptionPricingDominic Brecher, a Quantitative Analyst at FINCAD, has a PhD in high-energy theoretical physicsfrom the University of Cambridge. In addition, he has conducted postdoctoral research in StringTheory at the Universities of Durham and British Columbia. With several papers published in theJournal of High Energy Physics, Nuclear Physics and Physical Review, he has recently brought hisextensive research experience in physics to the world of financial engineering. Combiningtheoretical requirements and industry practice, Dominic designs derivative pricing models andexplores their consequences in the financial markets. In this article, he explains what he haslearned about the theory of local volatility in option pricing, and discusses its forthcomingpractical implementation in Version 10 of FINCAD XL and FINCAD Developer.

» To download the latest trial version of FINCAD Analytics, contact a FINCAD Representative.

This will include functions to price European and American options, option strategies, portfolios ofoptions, and options with arbitrary piecewise linear payoffs, from the smile as described in thisarticle. All numerical pricing is done through the no-arbitrage PDE. Other functions start insteadwith a given local volatility process defined either by standard parameterizations (such as the CEV,normal and shifted lognormal processes), or a user-defined data table. These functions also priceEuropean and American options, option strategies, portfolios of options, and options with arbitrarypiecewise linear payoffs, and can be used to back out the corresponding implied volatility smile.Using a combination of the two sets of functions, further issues such as the corresponding impliedvolatility smile. Using a combination of the two sets of functions, further issues such as the smiledynamics can be analyzed. Starting with either a given local volatility function, or with a givenimplied volatility smile, the interplay between these functions is shown below.

What's Right and Wrong with Black-Scholes?

Despite its many deficiencies, the Black-Scholes model of option pricing [1] remains widelyused, some 30 years after its inception. It is still the first model that both quants and tradersreach for when given a deal to price. Quants like it because it provides a theoretically consistentframework to price options in almost any asset class, given some (possibly dubious) assumptions.Traders like it because of the tractability of the Black-Scholes formula for option prices, andbecause it is easily tweaked to account for their intuition about the market.

The Black-Scholes formula allows the fair price of a European option to be determined, givenvarious observables - current asset price, strike price, time till maturity, risk-free rate anddividend yield - and a single unobservable constant - the volatility, σ, of the underlying assetprice. Historical volatility is easily determined, though this is not necessarily a good guide as tohow the future asset price will change. One is ultimately left with making a (more or less)educated guess as to the magnitude of σ, the likely average future volatility.

Volatility as a Unit of Currency

Traders have turned this drawback of the Black-Scholes model into a useful feature, by quotingEuropean option prices not in terms of their dollar value, but in terms of the equivalent impliedvolatility. This is the value of σ which must be used in the Black-Scholes formula to give back themarket price of that specific option. If the market price of some call option is C, then the impliedvolatility (a function of asset price S, strike K and maturity T) is defined through the relationship

C(S,K,T) = CBS(S,K, σ(S,K, T), T), (1)

where CBS is the Black-Scholes price of the option.

It is thus always possible to determine the market's view of the future volatility of some assetfrom quotes for options written on that asset: it is just the implied volatility at which the optionstrade.

Using volatility as a unit of currency in this way is only possible due to some underlying propertiesof the Black- Scholes framework. Most importantly, the price of an option is a monotonicallyincreasing function of volatility. If option prices were multi-valued functions of volatility then there

Local Volatility in Option Pricing, Resources | FINCAD http://www.fincad.com/derivatives-resources/articles/loca...

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would be no useful price ↔ volatility correspondence.

The ability to quote option prices in terms of a constant volatility is partly why the Black-Scholesmodel is still so widely used in the market. However, it makes the most important deficiency ofthe Black-Scholes framework transparent: the market-observed volatility is simply not a constant.

The Volatility Smile

At least since the crash of 1987, the market-observed implied volatilities of European equityoptions have exhibited a distinct "skew" structure. Deep out-of-the-money puts generally trade athigher implied volatilities than out-of-themoney calls. An example - for options on the S&P 500index - is shown in Figure 1. One reason is that this reflects the market's appetite for insuranceagainst stock prices falling. Options on interest rates - caps, floors, swaptions - also generallyexhibit such a volatility skew. Other options exhibit a volatility "smile", in which both deep out-of-the-money calls and puts 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 thefact that a skew, frown, smirk or other such structures are often seen in the market. The impliedvolatility of traded options also varies with the option maturity - the volatility term structure - andso one often talks of an implied volatility surface: σ = σ(K, T) is a function of two variables. Pointson the implied volatility surface for options on the S&P 500 index are shown in Figure 2.

It is easy to modify the Black-Scholes framework to deal with a volatility term structure, but themodel simply does not allow for volatilities to vary with strike price. Once one recognizes this itbecomes clear that, in the words of Rebonato, implied volatility is really only the "... wrongnumber to put in the wrong formula to obtain the right price of plain vanilla options" [2].

Beyond Black-Scholes

Arguably the most important problem in the theory of option pricing is how to take account of thevolatility smile. This is far from a purely academic exercise. Given the skew shown in Figure 1, forexample, what would be the correct volatility to use to price a European option struck at 1295?Whilst fairly easy to answer (one would use some form of interpolation on the volatility smile), it isless clear what volatility to use to price the corresponding American option, and much, much lessclear 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 preciselythis: to calibrate one's pricing of exotic, or even just American, options to the market-observedprices of European plain vanillas. Various models have been developed over the years to gobeyond Black-Scholes in this way. They include:

Jump-diffusion models, introduced by Merton [3];Stochastic volatility models, introduced by Hull and White [4]. The Heston [5] and SABR[6] models are two of the most popular;Local volatility models, introduced by Dupire [7], Derman & Kani [8] and Rubenstein [9];Models in which the underlying stochastic process is a mixture of lognormal distributions,introduced by Brigo and Mercurio [10].


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We will be reviewing local volatility models1 in this article.

Volatility and Stochastic Processes

The constant volatility of the Black-Scholes framework corresponds to the assumption that theunderlying asset follows a lognormal stochastic process in the risk-neutral measure

dS = rS dt + σS dW,

where dW is a Brownian motion, and we will throughout assume a constant risk-free rate r and nodividends. The return on the asset over some time period T is thus normally distributed with astandard deviation of σ √T. This is the basic property which makes the model analyticallytractable.

As the name suggests, local volatility can be thought of as taking the constant volatility, σ, andreplacing it with a functionwhich depends locally both on time and on the value of the underlyingasset: ∑=∑(S, t). The lognormal property is thus no longer true within the local volatilityframework. Rather the goal is to determine the correct underlying process from the market-observed smile, by determining the correct local volatility function. Note that the implied volatilityσ(K, T) is a function of strike and maturity whereas the local volatility ∑(S, t) is a function of assetprice and time.

With this notation, the underlying stochastic process in a local volatility model is2

dS = rS dt + ∑(S, t)S dW (2)

Using standard arguments [11], the corresponding no-arbitrage partial differential equation (PDE)for the price, V , of an option written on the underlying asset is

This is just the Black-Scholes equation, but with the constant volatility replaced by a generalfunction ∑(S, t). Our approach will be to determine this function from the implied volatility smile,then to use standard numerical techniques to solve the PDE(3).

Dupire's Equation

The transition probability function3 associated with the stochastic process (2) satisfies both thebackward (Kolmogorov) and forward (Fokker-Planck) equations. The noarbitrage equation (3) isessentially the backward equation. As shown by Dupire [7] (see also [8] and [12]), the localvolatility function can in principle be determined from the forward equation. From the latter, onecan derive

Both equations (3) and (4) apply in complete generality, to any option written on the asset S. Thelocal volatility function can now be derived by specializing to European plain vanillas, say calloptions with prices C. Applying equation (4) to a complete set of such options (for all strikes andmaturities) and upon re-arranging, we get Dupire's equation

Given the market-observed prices of plain vanillas, this equation in principle determines the localvolatility at time t = T when the stock price is equal to S = K. But only in principle, since oneneeds a complete set of option prices to fully determine the local volatility function; and one onlyhas access to a finite set of such prices. One might wonder whether the right-hand side ofequation (5) is always positive. Indeed it is: no-arbitrage arguments guarantee it [12], although

in practice interpolation and/or extrapolation issues will often lead to negative values of ∑2.

In addition to this, numerical effects for deep out-of-the-money options, when the denominator4

of equation (5) tends to zero, lead to badly behaved local volatility functions - the numerator musttend to zero at the same rate in order that the local volatility function remain finite. For thisreason, it is often preferable to rewrite equation (5) directly in terms of implied volatilities ratherthan option prices.

From Implied to Local Volatility Directly

The implied volatility of an option with price C(S,K,T) is defined through the Black-Scholes formulaas in equation (1). The derivatives of the call price in equation (5) can thus be computed throughthe chain rule, and substituting for these leads to the following result [12, 13]:

and t0 and S0 are respectively the "market" date, on which the volatility smile is observed, and theasset price on that date.

Equation (6) is the basic result. Not only is it difficult to analyze analytically, but it is also notparticularly wellbehaved numerically. Small changes in the implied volatility lead to large changesin the local volatility, and in fairly unpredictable ways.

However, equation (6) can in principle be used to determine the local volatility function from themarket-observed smile. This can then be used through the PDE (3) to give option prices which arecalibrated to the European plain vanillas seen in the market.


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Pricing Techniques

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 onlybe as good as one's interpolation scheme, and research continues into the pros and cons of variousmethods (see, e.g., [14]). To price an American option, or anything at all exotic, we need to usethe local volatility function in conjunction with a numerical method to solve equation (3). Theinterpolation from the implied volatility surface to various strikes and/or option maturities is justthe first step. We then have to compute the local volatility through equation (6) at each relevantpoint in (S, t) space.

Implied Trees

The original work of Dupire [7], Derman & Kani [8] and Rubenstein [9] all used binomial ortrinomial trees to encode the local volatility function. To construct the tree, one ensures that, ateach (S, t) node, an option with (K = S, T = t), is priced correctly.

The procedure is notoriously unstable. Even after just a few time-steps, negative branchingprobabilities are encountered which one has to re-set by hand, although the problem is somewhatbetter if one uses an approach based on forward rather than spot prices [15]. Moreover, the localvolatility function which one can back out from such an implied tree does not have particularlyintuitive properties, as discussed at length in [2].

In addition to this, the tree-based approach would have all the usual problems when applied, forexample, to more exotic instruments such as barrier options. For these reasons, a PDE approach tooption pricing is preferred.

PDEs

Given a local volatility function, it is easy to apply standard numerical techniques to solve theno-arbitrage PDE (3) for option prices. We use the Crank-Nicolson scheme for well-documentedreasons [13]. The application of local volatility to such finite difference methods was firstdiscussed in [12].

We will not review the Crank-Nicolson method in any detail here. Suffice it to say that we have setup a grid of points in (S, t) space, and that the derivatives in equation (3) have beenapproximated using finite differences. The grid points are labeled by i and j as Smin ≤ Si ≤ Smax

and t0 ≤ tj ≤ T0 where the option maturity is denoted by TB0B. The next step is to determine thevalue of ∑ at every grid point (i, j). As for the implied trees discussed above, we need to be able toreproduce option prices for all values of K = Si and T = tj in our grid.

The procedure used in [12] to determine the discrete local volatility function is numericallyinvolved and somewhat unstable. For these reasons, we prefer to work with the result (6), whichrelates the local volatility function directly to the implied volatility smile. A simple finite differenceapproximation for the derivatives in equation (6), along the lines discussed in [2], is the simplestway to generate the discrete local volatility function, and seems to give results at least as good asany other procedure.

Whichever approach one takes, the problem is ultimately how to interpolate and/or extrapolate theobserved implied volatility surface. Given a finite set of data points, how do we generate impliedvolatilities for all values of K = Si and T = tj? This is a real problem since the results depend verysensitively on the interpolation method.

Thankfully such effects, as long as they are localized, do not lead to large differences in optionprices. In that sense, the actual behaviour of the local volatility function is not too relevant tooption pricing.

Interpolation of the Implied Volatility Surface

Given a set of implied volatilities as in Figure 2, there are various interpolation methods one mightuse. Both parametric and non-parametric methods have been discussed in the literature, aselection of these including:

Standard bi-linear or bi-cubic splines;Entropy minimization techniques [16];Smoothing techniques such as Nadaraya-Watson smoothing [17];Dimension reduction techniques such as Principle Component Analysis [14]

Each method has its benefits and drawbacks. The only certain thing is that the local volatilityfunction depends very sensitively on the implied volatility σ and its derivatives, in particular on thederivatives of σ with respect to the strike price K.

This dependence can be made less sensitive by using smoothing techniques. A simple suchtechnique - using smooth cubic splines - was discussed originally in the context of probabilitydensity functions [18], and applied to the implied volatility surface in [19]. The latter work alsoshowed how to apply no-arbitrage constraints to the procedure in a relatively straightforward way.

Thin Plate Splines

The smoothing technique is applied in the strike direction only. In the context of probabilitydensity functions [18], one is after all only interested in the implied volatility as a function ofstrike price, for options of a single maturity. For the grid-based option pricing methods at hand,however, we need the implied volatility as a function of strike price, for all maturities T = tj in ourgrid. We will thus construct a smooth cubic spline in strike space at each timestep in our grid. Togenerate the implied volatilities to be smoothed at each time-step, we first need to construct apre-smoothed surface. This can be done using a bi-linear or bi-cubic spline, but a better method isto use a thinplate spline [19]. One may, for instance, only have data for a few different strikes ateach maturity, and these strikes may differ across maturities. Such a data set would not bewell-suited to bi-linear or bi-cubic interpolation. A thin plate spline is the natural two-dimensionalgeneralization 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 by , the objective function is


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given by

where The coefficients ai, bi can be found through straightforward linear algebra[20]. In our case, the dimensionless coordinates (K/S0, T/T0) should be used.

Our procedure is thus as follows:

Construct a thin-plate spline through the given set of implied volatility data points. This isour presmoothed surface.Starting from the final time-step in the grid, evaluate the thin-plate spline at that value ofT = tj and at some representative number of strikes (say 20), between K = Smin and K =Smax.Construct a smooth cubic spline through these points (see below).Evaluate the resulting cubic function at all values of K = Si for that time-step.Compute the derivatives of the implied volatility in equation (6) using standard finitedifferenceapproximations. (The ∂/∂T term needs the smooth spline for at least two adjacenttime-steps.)Put everything together as in equation (6) to compute the value of ∑(Si, tj) at each gridpoint Si at that time-step.Use these values in the Crank-Nicolson scheme to solve for option prices, and then move tothe next time-step.

Smooth Cubic Splines

The smooth cubic splines are easy to construct. The objective function is piecewise cubic in K:

where i runs over the points at which the thin-plate spline is evaluated by each timestep (i = 1, . .. 19 in the above). The usual continuity conditions hold on g(K) and its first two derivatives. A

smooth cubic spline is then constructed as the minimizer ∧g of the penalized sum of squares5

the penalty function being equal to the integrated (squared) curvature. The constant λ is thesmoothing parameter, controlling the weight of the penalty function. Using the moneynesscoordinate K/SB0B in equation (7) ensures that λ is dimensionless. For λ ≥ 0, there is a uniqueminimum; and there exists a simple algorithm involving only some elementary linear algebra tocompute it [20]. For λ = 0, one recovers the standard natural cubic spline, going through all datapoints. For λ â†’ ∞, one instead generates a least-squares fit. Figure 3 shows an example of asmooth cubic spline through the volatility smile of figure 1, for options on the S&P 500 index.

There are various criteria which can be used to determine what is the "best" value of λ [20],although not all of these produce good results in the financial context [21]. At any rate, thesmoothing procedure acts to reduce the (absolute value of the) curvature of the implied volatilitysurface in the strike direction. The net effect is to make the local volatility function dramaticallysmoother. On the other hand the resulting spline(s) will no longer go through all (if any) of theoriginal implied volatility data points. One might object that this takes us away from the originalmarket data, but perhaps this does not matter too much. After all, there is always some room formanoeuvre due to the bid-ask spread. Indeed, for deep out-of-the-money options this spread canbe very large when converted to volatility space.

An Example: Pricing an Option with Local Volatility

To make the above discussion more explicit, consider pricing an at-the-money European calloption on the S&P 500 index, calibrated to the implied volatility data of Figure 2, and expiring inT0= 11 months. For definiteness, we take the continuously compounded risk-free rate anddividend yield to be, respectively, 6% and 2%. To price the option, we set up a grid in (S, t) spacewith 40 steps in each direction. We start with the smoothing parameter λ = 0, an naively applythe procedure given in the previous section. The implied volatility surface at points in our grid isshown in Figure 4. An immediate problem should be apparent - the implied volatility is negativefor large regions of the grid!


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Ignoring this problem for the time being, the local volatility function used in our grid is shown inFigure 5. Although it cannot be seen from the figure, for some parts of our grid ∑squared alsoturns out to be negative! (We have set ∑ = 0 in those regions.) Apart from this (neverthelessimportant) fact, the local volatility function looks fairly well behaved.

The numerical price and first few risk statistics of the European option, for various numbers ofsteps in (S, t) space are given by:

40 x 40 steps 100 x 100 steps 200 x 200 steps

price 94.0553 94.3058 94.3133

Δ 0.4480 0.4455 0.4455

Γ 0.00219 0.00218 0.00204

Θ -0.1190 -0.1181 -0.1185

The expected Black-Scholes price is 94.7719, found by reading off the value of σ = 15.3996%from the (S = 1258.05, t = 11months) grid point. The Greeks are not comparable betweenmodels, whereas the price should be (in the continuous limit).

Negative values ∑2 of are obviously bad but, in the case at hand, they do not have much affect on

the option prices. The regions in which ∑2 < 0 are fairly localized, and the values are not too

negative. In other cases, ∑2 << 0 in large parts of the grid which leads to non-sensical optionprices, since it is a signal of arbitrage opportunities.

Extrapolation of the Implied Volatility Surface

As we found above, extrapolation of the implied volatility surface to the boundaries of our grid canlead to negative values of σ. The thin plate spline we construct through the original impliedvolatility data points asymptotes to a linear function (in (K,T) space), which will fairly easily giverise to such behaviour. To avoid this, we need to extrapolate in a more sensible way.

Given an implied volatility smile, extrapolation to constant values of σ at large and small strikeshas the effect of pasting lognormal tails onto the probability density function of the asset price.Since liquid market data for deep out-of-the-money options is anyway hard to come by, we choosethis extrapolation procedure in our numerical scheme.

A simple extrapolation to constant values outside of the range of implied volatility data will lead todiscontinuities in ∂σ/∂K which would in turn lead to spikes in the local volatility function. To avoidthis, we instead add points to the implied volatility data out to the boundaries of our grid, beforeconstructing the thin-plate spline. At each option maturity in the data shown in figure 2, we addextra points at constant values of σ, out to K = Smin and Smax. If need be, we also extrapolate inthe T direction in a similar manner.

Having added points in this way, we then construct the thin plate spline and proceed as above.Again with the smoothing parameter λ = 0, the resulting implied volatility surface for our grid isshown in figure 6. Note that it is now everywhere positive. The price we pay is that it isconsiderably less smooth. In particular, our extrapolation procedure leads to a certain amount of


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overshoot; but there are no discontinuities in ∂σ/∂K.

Although the extrapolation procedure should stop the implied volatility from becoming negative, it

may still allow regions in the grid for which ∑2 < 0. To avoid this latter behaviour we further

introduce a cut-off at, say, 2%, and demand that ∑2 > = 2% everywhere. (For the case at hand,this cut-off is not necessary, but for other implied volatility data it certainly is.) At any rate, theimproved local volatility function in our grid is shown in figure 7. Both plots in figures 6 and 7 areobtained using the FINCAD function aaOption LV smile tbl, available in the forthcoming Version 10release of FINCAD XL and FINCAD Developer.

With this local volatility function, the price and risk statistics of our European option are


price 94.2248 94.6228 94.7043

Δ 0.4916 0.4872 0.4866

Γ 0.00190 0.00186 0.00187

Θ -0.1246 -0.1218 -0.1231

where now σ = 15.4061% at the (S = 1258.05, t = 11months) grid point, giving a Black-Scholesprice of 94.8011. These results are generated by the FINCAD function aaOption LV smile p, alsoavailable in the forthcoming Version 10 release of FINCAD XL and FINCAD Developer. Althoughchanging the local volatility function drastically, the extrapolation procedure has not made such abig difference to these numbers.

Adding Some Smoothing

With no smoothing, the local volatility function has some rather large spikes as can be seen fromFigure 7. The function is better behaved once we introduce some smoothing into the interpolation

of the implied volatility surface. Setting λ= 10-3 gives an implied volatility surface as in Figure 8.


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The corresponding local volatility function is shown in Figure 9, both plots obtained from theFINCAD function aaOption_LV_smile_tbl. As expected, the smoothing procedure gives rise to amuch better behaved function. Higher values of λ would smooth out the function even more.

With the local volatility function as in Figure 9, the price and risk statistics of our European option,using the FINCAD function aaOption_LV_smile_p, are:


price 94.1935 94.6149 94.7165

Δ 0.5322 0.5313 0.5308

Γ 0.00217 0.00217 0.00216

Θ -0.1370 -0.1358 -0.1363

where σ = 15.4036% at the (S = 1258.05, t = 11months) grid point, giving the Black-Scholesprice of 94.7898. It is perhaps surprising how little difference the smoothing parameter - and sothe precise form of the local volatility function - makes to these numbers.

Conclusions

Rebonato has discussed general features of the local volatility function for different impliedvolatility surfaces [2]. For those inputs, we find similar behaviour in the "atthe-money region" ofthe grid. Outside of that region, however, the local volatility function can be quite badly behaveddue mainly to extrapolation effects: extrapolation to constant values of σ will ultimately lead tocurvature in the implied volatility surface, which in turn produces spikes in the local volatilityfunction. On the other hand, these tend to occur in regions of the grid which are deeply in- orout-ofthe- money. Although not necessarily true for more exotic structures, the precise form of thelocal volatility function in many cases seems to have little impact on plain vanilla European andAmerican option prices.

The three local volatility functions in figures 5, 7 and 9 presumably give such similar option pricesbecause the implied volatility surfaces are similar in the at-the-money region of the grid (roughly0.8S0 . Si . 1.2S0, covering the original data of Figure 2). The corresponding local volatilityfunctions in this region are shown in figures 10, 11 and 12, the behaviour in the three cases beingqualitatively similar. The relevant region of the surface widens as ti increases (see figure 2). Forexample, at t = 0, the only relevant value of Σ is at S = S0

Mathematically, computing the local volatility function from the implied volatility surface is aninverse problem which is not "well-posed" [2]. Certainly, differences in the extrapolation procedurelead to large differences in the resulting local volatility function. That said, the extrapolation is todeep in- or out-of-the-money regions of the grid and the prices of plain vanilla options seem fairlyinsensitive to these regions. One can experiment with enforcing different extrapolation methodssimply by adding extra points to the original implied volatility data; the local volatility function inthe region shown in figures 10, 11 and 12 does not change much from case to case. All these


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issues can be explored in the forthcoming Version 10 of FINCAD XL and FINCAD Developer.

_________Footnotes1Sometimes called deterministic volatility models.2The factor of S is sometimes absorbed into the definition of the local volatility function. In ourconventions, Σ has the same dimensions as σ, namely 1/√time.3This is the probability _(S"t"t, St) that the asset price equals S' at time t'given its value of St attime t.4The denominator of equation (5) is proportional to the probability density function of theunder-lying asset, so asymptotes to zero for out-of-the-money options.5In [18], mainly to minimize the effect of deep out-of-the-money options, the smoothing wasdone in call option _ space rather than K space; and the data points were weighted by the relevantvalues of the options' vega. This procedure is numerically impractical to perform in a grid-basedpricing method.

References

[1] Black, F. and Scholes, M. (1973) "The Pricing of Options and Corporate Liabilities,"Journal ofPolitical Economy 81 (3), 637-659.

[2] Rebonato, R. (2004) Volatility and Correlation: the Perfect Hedger and the Fox,2nd ed.,Chichester, John Wiley & Sons Ltd.

[3] Merton, R. (1976) "Option Pricing when Underlying Stock Returns are Discontinuous,"Journal ofFinancial Economics 3, 125-144.

[4] Hull, J. C. and White, A. (1988) "An Analysis of the Bias in Option Pricing caused by aStochastic Volatility," Advances in Futures and Options Research 3, 29-61.

[5] Heston, S. L. (1993) "A Closed-form Solution for Options with Stochastic Volatility withApplications to Bond and Currency Options," Review of Financial Studies 6 (2), 327-43.

[6] Hagan, P. S., Kumar, D., Lesniewski, A. S. and Woodward, E. (2002) "Managing Smile Risk,"Wilmott September 2004, 84-108. 17

[7] Dupire, B. (1994) "Pricing with a Smile," RISK 7 (1), 18-20.

[8] Derman, E. and Kani, I. (1994) "Riding on a Smile," RISK 7 (2), 32-39.

[9] [Rubenstein, M. (1994) "Implied Binomial Trees," Journal of Finance 49 (3), 771-818.

[10] Brigo, D. and Mercurio, F. (2000) "A mixed-up Smile," RISK 13 (9), 123-6.

[11] Merton, R. (1973) "Theory of Rational Option Pricing," Bell Journal of Economics andManagement Science 4, 141-83.

[12] Andersen, L. B. G. and Brotherton-Ratcliffe, R. (Winter 1997/1998) "The Equity OptionVolatility Smile: an Implicit Finite-Difference Approach," The Journal of Computational Finance 1(2), 5-37.

[13] Wilmott, P. (2006) Paul Wilmott on Quantitative Finance, 2nd ed., Chichester, John Wiley &Sons Ltd.

[14] Fengler, M. R. (2005) "Semiparametric Modeling of Implied Volatility," Berlin Heidelberg,Springer-Verlag.

[15] Barle, S. and Cakici, N. (1998) "How to Grow a Smiling Tree," Journal of FinancialEngineering 7 (2), 127-46.

[16] Avellaneda, M., Friedman, C., Holmes, R. and Samperi, D. (1997) "Calibrating VolatilitySurfaces via Relative- Entropy Minimization," Applied Mathematical Finance 4 (1), 37-64.

[17] Ait-Sahalia, Y. and Lo, A. W. (1998) "Nonparametric Estimation of State-Price DensitiesImplicit in Financial Asset Prices," The Journal of Finance 53 (2), 499-547.

[18] Clews, R., Panigirtzoglou, N. and Proudman, J. (2000) "Recent Developments in ExtractingInformation from Options Markets," Bank of England Quaterly Bullletin February 2000; Bliss, R. R.and Panigirtzoglou, N. (2000) "Testing the Stability of Implied Probability Density Functions,"Bank of England Working Paper No 114; Bliss, R. R. and Panigirtzoglou, N. (2004) "Option-ImpliedRisk Aversion Estimates," The Journal of Finance 59, 407-46.

[19] Fengler, M. R. (2005) "Arbitrage-Free Smoothing of the Implied Volatility Surface," HumboldtUniversit ⋅⋅at zu Berlin SFB 649 Discussion Paper.

[20] Green, P. J. and Silverman, B. W. (1994) Nonparametric Regression and Generalized LinearModels, Boca Raton Florida, Chapman & Hall.

[21] Weinberg, S. A. (2001) "Interpreting the Volatility Smile: an Examination of the InformationContent of Option Prices," International Finance Discussion Papers, Federal Reserve System,Number 706.

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