Data driven recovery of local volatility surfaces

Transcription

1 Data driven recovery of local volatility surfaces Vinicius Albani Uri M. Ascer Xu Yang and Jorge P. Zubelli May 8, 2017 Abstract Tis paper examines issues of data completion and location uncertainty, popular in many practical PDE-based inverse problems, in te context of option calibration via recovery of local volatility surfaces. Wile real data is usually more accessible for tis application tan for many oters, te data is often given only at a restricted set of locations. We sow tat attempts to complete missing data by approximation or interpolation, proposed and applied in te literature, may produce results tat are inferior to treating te data as scarce. Furtermore, model uncertainties may arise wic translate to uncertainty in data locations, and we sow ow a model-based adjustment of te asset price may prove advantageous in suc situations. We furter compare a carefully calibrated Tikonov-type regularization approac against a similarly adapted EnKF metod, in an attempt to fine-tune te data assimilation process. Te EnKF metod offers reassurance as a different metod for assessing te solution in a problem were information about te true solution is difficult to come by. However, additional advantage in te latter approac turns out to be limited in our context. 1 Data manipulation and local volatility surfaces Te basic setup of data assimilation and inverse problems for model calibration consists of an assimilation of dynamics, defined for instance as a discretized PDE, and observed data [RC15, NP15]. Te celebrated Kalman filter, for example, does a forward pass on a weigted least squares problem fitting bot model dynamics and data, and it guarantees variance minimization for a linear problem wit Gaussian noise. However, in practice, te statistical sanctity of te data is often violated before te assimilation process commences. Tis can appen for various reasons and in different circumstances: 1. In cases were te data is scarce, in te sense tat it is observed only at a small set of locations compared to te size of a reasonable discretization mes of a pysical domain, tere would be many models (solutions to te inverse problem) tat explain te data (e.g., [HAO04]). It is ten tempting to complete te data by some interpolation Dept. of Matematics, UFSC, Florianopolis, Brazil, v.albani@ufsc.br Dept. of Computer Science, University of Britis Columbia, Canada, ascer@cs.ubc.ca IMPA, Rio de Janeiro, Brazil, xuyang@impa.br IMPA, Rio de Janeiro, Brazil, zubelli@impa.br 1

2 2 or oter approximation, wereupon te role of an ensuing regularization as a prior is less crucial. 2. Tere may be a idden uncertainty in te locations of data, not only in data values (e.g., [HA08, GH15]). For instance, engineers often prefer to see data given at regular mes nodes, so a quiet constant interpolation, moving data items to te nearest cell vertex, is common practice. 3. Data completion may be necessary to obtain a more efficient algoritm [RKvdDA14, KdSA + 15]. 4. A quiet data completion is often assumed by matematicians in order to enable building teory for inverse problems. Tis includes assumptions of available data on continuous boundary segments [EHN96], or of observed (measured) relationsips between unknown functions tat are presumed to old everywere in a pysical domain. 5. Tere are situations were some form of data completion and oter manipulation is necessary because no one knows ow to solve te problem oterwise [KdSA + 15]. Tese observations raise te following questions: (i) wen (and in wat sense) is it practically acceptable to perform suc data manipulations? (ii) in wic circumstances can one gain advantage by treating te observed data more carefully? and (iii) ow sould one assess correctness of a solution tat as been obtained wit suc manipulated data? Our general observation is tat researcers occasionally, but not always, seem to get away wit suc crimes, in te sense of producing agreeable results. For instance, in [RKvdDA14] te autors obtained agreeable reconstructions so long as te percentage of completed data did not exceed about 50%, but not more. Suc empirical evidence is relatively rare in te literature, owever, and it depends on te problem at and. More insigt is terefore required, and suc may be gained by considering applied case studies. In tis article we focus on a model calibration problem in a setting tat features bot scarce data and uncertainty regarding data location. Furter, it allows us to work wit real rater tan syntetic data, often available troug te internet. Tis problem, wic as ad tremendous impact in matematical finance, concerns te determination of te so-called local volatility surface, making use of derivative prices. A good model for te volatility is crucial for many applications ranging from risk management to edging and pricing of financial instruments. Te classical Black-Scoles-Merton model ad subsumed a constant volatility model σ [BS73] in a simplifying stocastic dynamics for te underlying process [KK01]. However, te constant volatility assumption was quickly contradicted by te actual derivative prices observed in te market. Te disagreement between te Black-Scoles model-implied prices for different expiration dates and negotiated strike prices became known as te smile effect. A number of practical parametric as well as nonparametric models ave been proposed in tis context; see [Gat06] and references terein. Te parametric ones try to fulfill different penomenological features of te observed prices. Yet, in a ground breaking paper Dupire [Dup94] proposed te use of a function σ tat depends on time and te price at tat time. For te case of te European call contracts, e replaced te Black-Scoles equation

3 3 by a PDE of te form C τ = 1 2 σ2 (τ, K)K 2 2 C C bk, τ > 0, K 0, (1) K2 K wit initial and boundary conditions (for calls) given by C(τ = 0, K) = (S 0 K) +, (2) lim C(τ, K) K = 0, lim K 0 = S 0, were τ is time to maturity, K is te strike price, and C = C(τ, K) is te value of te European call option wit expiration date T = τ. Te parameter S 0 is te asset s price at a given date. One ensuing complication in using (1), owever, is in te calibration of tis model, by wic we mean finding a plausible volatility surface σ(τ, K) tat matces, or explains, given market data on call option values. Te task ere is significantly more callenging tan in te case were σ is a constant. Tis paper deals wit te computational callenges tat tis inverse, or inference problem gives rise to, focussing in particular on te treatment of scarce data and uncertainty in data locations. We sow tat, contrary to a popular approac in te literature, avoiding data completion is te way to go ere. Furter, taking data location uncertainty into account, rater tan simply ignoring it, improves reconstruction quality. Te forward problem involves finding te values of C satisfying te differential problem (1)-(2) for given σ(τ, K) and S 0, evaluated at te points (τ, K) were data values are available. A major difficulty ere is tat te data are scarce. To explain wat we mean by tis, suppose we ave discretized te PDE using, say, te Crank-Nicolson metod on a rectangular mes tat is reasonable in te sense tat te essence of te differential solution is retained in te discrete solution. Ten te data is scarce in tat te number M of degrees of freedom in te discrete C typically far exceeds te number of given data values l: l M. Moreover, te available data in some typical situations are given at locations tat are far from te boundaries of te (truncated) domain on wic te approximated PDE problem is solved. See Figures 1 and 5 below for examples of suc (real) data sets. Now, if te local volatility surface is discretized, or injected, on te same mes as tat of te forward problem, ten tere are rougly M degrees of freedom in σ, wic is again potentially far larger tan te number of data constraints. We can of course discretize σ on a coarser sub-mes (wic in te extreme case would ave only one point, tus leading back to a constant volatility), or parameterize te surface in a more involved manner; see [HKPS07, HK05, AP05a, EE05, BT00, AP05b, CZ13, AZ14] and references terein for furter detail. Here, owever, we stick to a straigtforward nodal representation of tis surface on te full C-mes in te ope of retaining flexibility and detail, wile avoiding artifacts tat may arise from restrictive simplifying assumptions. Tis approac as worked well in geopysical exploration problems [HAO04], among oters. Tus, te problem of finding a volatility surface tat explains te data is often significantly under-constrained in practice. Tis does not make it easy to solve, owever, as te ultimate goal is to obtain plausible volatility surfaces tat can be worked wit, and not just

4 4 to matc data. Our task is terefore to assimilate te data information wit te information contained in te PDE model (1), using any plausible a priori information as a prior in te assimilation process. Suc a priori information can vary significantly, addressing concerns of aderence to te financial model, relative smootness of te volatility surface, and numerical stability issues, among oters. One approac tat as been relatively popular in financial circles is to apply to te data special interpolation/extrapolation metods tat take into account te a priori information of te financial model (e.g., [Ka05]). Tis is used to obtain data values at all points of te rectangular mes on wic C is defined, and subsequently te new data is assimilated wit te information tat te solutions of te Dupire equation for different σ s yield to calibrate (1). An advantage wit tis data completion approac is tat te data is no longer scarce wen we get to te redefined inverse problem. Tis allows for developing some existence and uniqueness teory as well; see [CSZ12, AP05a], and references terein. However, a disadvantage is tat suc data completion constitutes a statistical crime, as te errors in te new data may no longer be considered as independent random variables, see [RKvdDA14]. In fact, we get two solutions C tat are in a sense competing rater tan completing one anoter, since te one satisfying (1), even for te best σ, does not necessarily satisfy te data interpolation conditions and vice versa. In Sections 2 and 5 we terefore examine te performance of tis data completion approac against tat of a scarce data approac tat is based on a carefully tuned Tikonovtype regularization. We ave verified te robustness of our regularization operator by applying also variants of EnKF-like algoritms [JM08, ILS13, CES14], furter described below, obtaining similar results. Using bot syntetic and real data sets, we sow tat te scarce data approac can give better and more reliable results; in our reported experiments tis as appened especially for te real data applications. We ten continue wit te scarce data approac. Te maximum a posteriori (MAP) functional considered in Section 2 is based on te statistical assumption tat te data error covariance matrix is a scalar multiple of te identity. In Section 3 we subsequently consider an algoritm, based on an approac recently proposed in [ILS13] and [JM08], were we attempt to learn more about te error covariance matrix as te iterative process progresses, using ensemble Kalman filter (EnKF) tecniques. Altoug our problem is time-dependent, te time variable ere does not really differ from te oter independent variable in te usual sense. In particular, te unknown surface σ depends on bot K and τ, unlike for instance te material functions in [ILS13, CES14, HAO04], wic are independent of time. Tus, te EnKF-like metods considered use an artificial time [AHD07]. In Section 3 we find tat te EnKF algoritm can be improved in our context by adding smooting prior penalties, just like in Section 2. Te probabilistic setup, altoug general, is not fully effective as a substitute for prior knowledge tat is available in no uncertain terms. Te problem setting used in Sections 2 and 3 regards te asset price S 0 as a known parameter. However, in practice tere is uncertainty in tis parameter. In fact, we ave an observed value wic is in te best case an average over a day of trading, so S 0 sould be treated as an unknown wit an observed mean value and a variance tat is relatively easy to estimate. Tis in turn affects te calibration problem and its solution process. Section 4 deals wit tis additional complication, wic translates into uncertainty in te

5 5 data locations of a transformed formulation for (1). In Section 5 we collect our numerical tests, addressing and assessing te various aspects of te metods desribed earlier. We use syntetic data in Section 5.1 to sow te advantage in applying te metod of Section 4 for problems wit uncertainty in te price S 0. In Section 5.2 we use market equity data to fine-tune our regularization functional, as well as to compare Tikonov-type regularization vs te modified artificial time EnKF. In Section 5.3 we use oil and gas commodity market data to furter investigate data completion approaces, sowing tat te scarce data approac is superior. Conclusions are offered in Section 6. 2 Two approaces for andling scarce data Below we assume tat te parameter S 0 is given. 1 Tis assumption will be modified in Section 4. We ten apply a standard transformation canging te independent variable K to te so-called log moneyness variable y = log(k/s 0 ). Tis is followed by canging te dependent variables of te forward and inverse problems to u(τ, y) = C(τ, S 0 exp(y)) and a(τ, y) = 1 2 σ(τ, K(y))2, respectively. We obtain te dimension-less parabolic PDE wit no unbounded coefficients u ( 2 τ + a u y 2 u ) + b u = 0, τ > 0, y R, (3) y y subject to te side conditions u(τ = 0, y) = S 0 (1 exp(y)) +, (4a) lim u(τ, y) y = 0, (4b) lim y = S 0. (4c) We can write (3) (4) as L(a)u = q, wit te linear differential operator L depending on a and operating on u and wit te rigt and side q = q(s 0 ) given. Tus, te forward problem involves finding u satisfying tis parabolic linear differential problem for a given local variance surface a and price S 0. To find a numerical solution for L(a)u = q we first approximate te domain in y by a finite interval, restricting l y y r y for two real values satisfying l y < 0 < r y. Te boundary conditions (4b) and (4c) are ten required to old at r y and l y, respectively. Next, we discretize te PDE problem on a mes wit a fixed step τ in τ and a fixed step y in y. Denote by u i,j te approximation of u(i τ, l y + j y) and by a i,j te injection of te surface a at (i τ, l y + j y), i = 1,..., M τ, j = 0, 1,..., M y + 1, were (M y + 1) y = r y l y, M τ τ = T. Ten, using te Crank-Nicolson metod [AP05a], we ave te difference 1 In te related online calibration setting, data are given for several values of S 0. For eac suc value of S 0 we ten make a variable transformation and find a volatility surface.

6 6 relations u i+1,j u i,j τ + a i,j + a i+1,j 4 y 2 (u i+1,j+1 2u i+1,j + u i+1,j 1 + u i,j+1 2u i,j + u i,j 1 ) a i,j + a i+1,j 2b (u i+1,j+1 u i+1,j 1 + u i,j+1 u i,j 1 ) 4 y = 0, i = 1,..., M τ, j = 1,..., M y. (5) An obvious treatment of te initial and (Diriclet) boundary conditions closes tis system of M = M τ M y equations tat are linear for te variables u i,j. Te mes function u for te approximation u can be conveniently resaped (say, ordered by column) into a vector u R M, retaining te same notation witout confusion. Similarly, we obtain te mes function a as an injection of a(τ, y), resaped into a vector if need be. Ten we can write (5) as L (a )u = q, or u = L (a ) 1 q, (6) were L is a sparse, nonsingular M M matrix and q is te mes injection of q. Te inverse problem is to find a volatility surface σ, approximated troug a, tat explains given observed data d R l. Tese data values approximate u at l locations in te rectangular domain on wic te problem (3)-(4) is defined. Typically, tese locations are far from te boundaries and l M; see Figures 1 and 5. Tus, te data set is sparse, or scarce, and te l M matrix P wic maps grid locations for u to tose of d, using bilinear interpolation as necessary, as many more columns tan rows. Te forward operator, wic predicts te data for a given a, is te matrix-vector product (or projection) P u. Te inverse problem is to find a plausible a for wic te predicted and observed data are sufficiently close, as described below. Note tat te approximate solution of te inverse problem must be positive at all mes points. Tis positivity constraint turns out to old automatically in all our reported calculations (i.e., it is not an active constraint in te encountered optimization problems). 2.1 Regularizing te inverse problem If te data d as Gaussian noise N (0, Γ), were Γ is a symmetric positive definite (SPD) error covariance matrix, ten te maximum likeliood (ML) data misfit function is ϕ(a ) = P u (a ) d 2 Γ 1, (7) were for an SPD matrix C we define te vector energy norm x C = x T Cx. For instance, if Γ = α0 1 I, α 0 > 0, ten te discrepancy principle (see, e.g., [Vog02, EHN96]) yields te stopping criterion (i.e., we sould find a to reduce ϕ until) ϕ(a ) ρ, were ρ = α0 1 l, (8) were l is te number of data points. However, tere are in general many surface meses a tat would satisfy te conditions (8) (i.e., explain te data). We terefore introduce a regularization operator R(a )

7 7 Figure 1: Data locations for a PBR (Petrobras, an oil company) set in te (τ, y) domain wit our coarsest mes in te background. wic is a prior, in te sense tat it represents prior knowledge or belief about our sougt surface. We ten take steps to minimize te MAP merit function Here we introduce te Tikonov-type penalty function R(a ) = α 1 (a i,j a 0 ) 2 i j + α 2 τ 2 j M τ i=1 ϕ R (a ) = ϕ(a ) + R(a ). (9) (a i,j a i 1,j ) 2 + α 3 y 2 i M y +1 (a i,j a i,j 1 ) 2, (10) were α 1, α 2, α 3 0 are parameters to be determined. Te terms involving α 2 and α 3 correspond to smootness of a(τ, y) in τ and y, respectively. Tus, te values a i,j tat are elements of te 2D mes function a sould not fluctuate randomly, as tey ougt to form a reasonably smoot surface. So, we insist on selecting α 2, α 3 > 0, and will modify a given EnKF algoritm accordingly as well. Tis penalizes lack of smootness (or, rougness) in te reconstructed local variance surface. Actual values for tese parameters, determined by experimentation, are reported in Section Before proceeding wit furter details it is important to empasize tat our focus in tis paper is on te data manipulation aspects, and our Tikonov-type regularization described ere is certainly not new as suc. We give its furter details for te sake of completion, and remind te reader tat te performance of te resulting algoritm is not worse tan tat of te two EnKF-type algoritms tat were tested ere. j=1

8 8 Next, te term governed by te weigt α 1 penalizes te distance from our sougt surface to an a priori function a 0 tat we take to be a constant (not knowing better). A reasonable value for tis constant can often be estimated based on te risk category of te asset under consideration. Te importance of aving tis term is tat witout it te prior may tend to favour surface flatness, weter or not tis is realistic. But for te case of scarce data tere could be too muc freedom in seeking merely a relatively flat surface wic still explains te data. Setting α 1 = 0 migt terefore cause te solution process to beave unstably, as demonstrated furter in Section 5. Having determined values α i, i = 1, 2, 3 for te ensuing experiments te next question is coosing te relative weigt of te data misfit function and te prior, wic amounts to determining a value for α 0 in te expression Γ = α0 1 I. For te sake of completion we mention tat for tis purpose we ave used te well-known L-curve metod [Vog02, EHN96]. Specifically, we applied tis metod in order to find te optimal α 0 wit te values of α 1, α 2 and α 3 fixed. 2.2 Te data completion approac Anoter approac to deal wit te scarce data is to first interpolate/extrapolate te observed data to all M locations of te u-mes. Tis can be done using te Kaale algoritm [Ka05], wic represents a prior tat reflects financial considerations (namely, maintaining te smile, and more). Te inverse problem is subsequently defined on te tus-enriced data set, te matrix P in (7) becoming te identity, and te Tikonov-type regularization previously described is applied for its solution. Tis data completion subsequently allows for te development of a more solid teory for te solution of te corresponding regularized inverse problem. A potential difficulty wit tis approac, owever, is tat te ensuing matcing of te interpolated data values to a field u tat approximately satisfies te PDE problem (3) (4) is applied to data tat can no longer be considered to ave independent, random noise. Te interpolation/extrapolation operator and te PDE discrete Green s function operator could be in conflict. In Sections 5.2 and 5.3 we describe examples using real data sets from different markets wic compare data completion using two different tecniques to te scarce data approac. Te obtained plots in Figures 8, 9, 11 and 12 clearly demonstrate tat te data completion approac can give inferior results. In particular, te reconstructions using data completion often agree neiter wit oter curves nor wit intuition. 3 Artificial time EnKF-type metods Te omotopy, or continuation, approac of embedding a given problem in a larger one wit artificial time, subsequently defining an iterative metod by advancing in te artificial time, is old; see references in [AHD07]. In our context tere ave been recent efforts tat use suc an iterative process to also adaptively learn and improve knowledge of te error covariance [ILS13, CES14]. A Kalman filter setting is obtained as in [ILS13] by defining for te ML data misfit function ϕ(a ) of (7) te augmented state vector ( ) a â = Ψ(a ) =, (11) P u (a )

9 9 togeter wit te artificial dynamics (or prediction) â (n+1) d in (7) are ten approximately matced by = Ψ(a (n) ). Te observations Hâ (n+1), wit H = ( 0 I ). Furter, te ensemble Kalman filter (EnKF) approac applies Monte Carlo approximations in order to obtain ceap estimates for te error covariance matrices tat appear in te Kalman filter metod. It is well-known tat Kalman smooting is equivalent to te solution of te corresponding weigted least squares problem, wile te Kalman filter end result agrees wit tat of te Kalman smooter. However, wat is not taken directly into account in [ILS13, CES14] are te additional regularization terms in te MAP merit function (9). In fact, we generalize (10) by considering min a ϕ R (a ) = ( d P u (a ) ) T Γ 1 ( d P u (a ) ) + (a a 0 ) T D 1 (a a 0 ) + (12) (L τ a 0 L τ a ) T Dτ 1 (L τ a 0 L τ a ) + (L y a 0 L y a ) T Dy 1 (L y a 0 L y a ), were L τ and L y are te scaled discrete sums tat multiply α 2 and α 3 tere, respectively. Te matrices D, D τ and D y are corresponding error covariance matrices, wit D as yet unknown and to be determined in te EnKF process. Next, we modify te data misfit part in (12) using te state augmentation approac (11). For tis we redefine te matrices L τ and L y by L τ ( L τ 0 ), L y ( L y 0 ). Te matrix D is also modified properly, and we set ( ) a0 â 0 =. P u (a 0 ) We can ten rewrite (12) for te augmented variable â of (11) as min â ϕ R (â ) = ( d Hâ ) T Γ 1 ( d Hâ ) + (â â 0 ) T D 1 (â â 0 ) + (13) (L τ â 0 L τ â ) T Dτ 1 (L τ â 0 L τ â ) + (L y â 0 L y â ) T Dy 1 (L y â 0 L y â ). Since (13) is just an enlarged weigted least squares problem, it also corresponds to a Kalman smooter/filter process, and we subsequently build an approximation for it as in [JM08] by defining and solving a tree-stage EnKF. Te details are somewat tedious but straigtforward, so we ave gatered tem in Appendix A. We are led to te following algoritm, were we let â (n,j) denote te augmented state vector of te j-t sample in te n-t iteration. 1. Initialization (a) Generate J samples of a (0), denoted {a(0,j) } J j=1, were a(0,j) D 0 is an initial covariance matrix. (b) Compute P u (a (0,j) ), j = 1,..., J, tus defining {â (1,j) } J j=1. 2. For n = 0, 1, 2,..., until convergence criterion is satisfied, do: N (a 0, D 0 ) and

10 10 (a) Prediction step i. For j = 1,..., J, set â (n+1,j) = Ψ(a (n,j) ) = ii. Define sample mean and covariance matrix as ā (n+1) = 1 J D n+1 = 1 J J j=1 J j=1 â (n+1,j), â (n+1,j) (â (n+1,j) ( ) a (n,j) P u (a (n,j). ) ) T ā (n+1) (ā (n+1) ) T. (b) Analysis step (tree-stage Ensemble Kalman filter) i. Calculate A (1) n+1 = D n+1h T (HD n+1 H T + Γ) 1 B (1) n+1 = (I A(1) n+1 H)D n+1 A (2) n+1 = B(1) n+1 LT τ (L τ B (1) n+1 LT τ + D τ ) 1 B (2) n+1 = (I A(2) n+1 L τ )B (1) n+1 A (3) n+1 = B(2) n+1 LT y (L y B (2) n+1 LT y + D y ) 1 B (3) n+1 = (I A(3) n+1 L y)b (2) n+1 (Here I is an identity matrix of appropriate size.) ii. For j = 1,..., J, update ( ã (n+1,j) = â (n+1,j) + B (3) n+1 H T Γ 1 (d (j) n+1 Hâ(n+1,j) ) + L T τ Dτ 1 (r τ (n+1,j) L τ â (n+1,j) ) + L T y Dy 1 (r y (n+1,j) L y â (n+1,j) ) ), (14) were d (j) n+1 = d + η(j) n+1, η(j) n+1 from N (L τ ā (n+1) iii. For j = 1,..., J, set (c) Convergence test Compute and ceck for convergence., D τ ) and N (L y ā (n+1) N (0, Γ); r(n+1,j) τ and r (n+1,j) y, D y ), respectively. a (n+1,j) = ( I 0 ) ã (n+1,j). a (n+1) = 1 J J j=1 a (n+1,j) are sampled Assuming tat tis algoritm stops after N iterations, it requires (N + 1)J solutions of te forward problem.

11 11 4 Uncertainty in te asset price In a typical image processing application of deblurring, denoising or inpainting, data values are prescribed at pixels, so te measurement locations are known. However, wen denoising a point cloud or a surface mes, for instance, tere is no suc distinction between a datum value and location: te unknowns are nodal mes points in 3D, and as suc tey live in a iger dimensional space (namely, 3 rater tan 1). Tis difference affects portability of image processing algoritms to similar problems on surfaces [HA08]. A more subtle instance of location uncertainty arises in our volatility surface context. Te observed value for S 0 is actually an average of a day s trading (say), and as suc contains uncertainty wose variance can fortunately be directly estimated. In tis section we tus make S 0 an unknown tat can be adjusted but sould not stray too muc from a measured (and tus observed) average value Ŝ0. Tis affects te regularization prior, wic now depends also on S 0, so R = R(a, S 0 ), and as a term added to (10): R(a, S 0 ) = α 1 (a i,j a 0 ) 2 + α 5 (S 0 Ŝ0) 2 (15a) i j + α 2 τ 2 j M τ i=1 (a i,j a i 1,j ) 2 + α 3 y 2 i M y +1 (a i,j a i,j 1 ) 2. Te additional parameter α 5 is determined by te daily price variance. Furtermore, in te dimension-less form (3) of te Dupire PDE problem, wic to recall allows bounded PDE coefficients and uniform discretization step sizes in (τ, y), te independent log moneyness variable y = log(k/s 0 ) now contains uncertainty as well! We terefore update also te misfit function (7) to read ϕ(a, S 0 ) = P (S 0 )u (a ) d 2 Γ 1 (15b) j=1 j=1 M y ( + α 4 (1 exp(y j (S 0 ))) + (1 exp(y j (Ŝ0))) +) 2. Te parameter α 4, like α 5, is determined from te variance in S 0. In fact, in our experiments we ave found tat it is safe to set α 5 = 0 and control te uncertainty penalty troug α 4 alone. Note tat in te data projection matrix, P = P (S 0 ). Our optimization problem replacing (9) is now min ϕ R (a, S 0 ) = ϕ(a, S 0 ) + R(a, S 0 ). (16) a,s 0 To solve te extended optimization problem (16) we apply a splitting metod, alternately minimizing (16) for a and for S 0. Wen S 0 is eld fixed, te problem returns to tat considered in Sections 2 and 3. Wen a is frozen in turn, te remaining minimization problem for S 0 is scalar and causes no difficulty. Furtermore, fortunately te coupling between tese variables is weak, so fast convergence of tis splitting metod is observed in all our experiments.

12 12 5 Testing our metods on real and syntetic data In tis section we present a number of tests tat were performed in order to illustrate te points made in te previous sections, as well as to compare te different metodologies. Specifically, we compare results related to te introduction of uncertainty in te value of te reference price S 0 (see Section 4), te effect of data completion as summarized in Section 2.2, te effect of coosing between te EnKF and te Tikonov-type regularization approac, and te introduction of a penalty associated to te mean value of te local variance surface a. Financial markets provide a large quantity of istorical data from equity and commodity prices togeter wit teir corresponding derivative prices. Tis yields a tremendous laboratory for investigating te local volatility surface reconstruction assuming te model of (1)-(2). In everyday practice one is given, for a certain date and current underlying value S 0, various quoted option prices wit future strikes and maturity times. Our task is ten to infer te local volatility surface from suc data. Yet, te quality of te data, as far as our models are concerned, depends igly on te number of contracts being traded. In tis context, we sall often employ te terminology of liquid contracts to tose tat correspond to a substantial volume of negotiations, wereby our models are expected to provide good results. Te computations for all te examples described in tis paper were performed using garden-variety personal computers, wit typical runtimes clocking witin a few minutes. Tus, being concerned in tis work wit addressing several fundamental questions, no special effort was made to optimize te runtime performance of our codes. Te basic setting of our numerical experiments consists of solving te inverse problem described in Section 1, first using syntetic data (tainted by multiplicative Gaussian noise) and ten using real data. In te syntetic data examples, we first assume a ground trut local variance surface a true on a fine grid, from wic we solve te discretized PDE for u as described in Section 2. From tat we sample data on a coarser mes (so as to avoid te so-called inverse crime) and ten subject te data to noise. In te real data examples we selected publicly available option data from te NY stock excange. Te reconstructions are performed by using te tecniques put fort in Sections 2 and Results for uncertain S 0 using syntetic data In order to test our tecniques in situations were we ave full control of te unknown volatility surface, we postulate a local volatility surface wit known form. We concentrate on te issue of underlying price uncertainty, and as we sall see, tis can be well addressed by te metod discussed in Section 4. Te uncertainty in S 0 appears in practice due to possible mismatces among option and underlying price recordings as well as to bid and ask spreads on suc observations.

13 First experiment wit underlying price uncertainty We start by experimenting wit te uncertainty about te value of S 0 for a ground trut volatility given by ( ) 4πy 25 e τ/2 cos, if 2/5 y 2/5 5 σ(τ, y) = (17) 2/5, oterwise. From (17) we produced call option prices by solving te PDE system (3) (4) discretized over a mes wit step sizes τ = and y = Te maximum time to maturity is τ max = 0.5 years, and te minimum and maximum log-moneyness strikes are y min = 5 and y max = 5, respectively. We also cose, for simplicity, te true underlying asset price as S 0 = 1.0 and te risk-free interest rate as r = 0. Te computed call prices u i,j were ten polluted by a relative noise, u noisy i,j = u i,j ( η i,j ), (18) wit η i,j drawn from te standard normal distribution N (0, 1). Te data is ten composed on a coarse mes by te noisy prices u noisy i,j wit τ i = iδτ, i = 1,..., 5, δτ = 0.1, and y j = jδy, δy = 0.05, j = 0, 1,..., 30. Te observed underlying price is Ŝ0 = T = 37 days 0.5 T = 73 days 0.5 T = 110 days 0.5 T = 146 days 0.5 T = 183 days Figure 2: Reconstructed (continuous line) and true (line wit circles) local volatility surfaces at te five different maturities. Te reconstructed local volatility surface corresponds to te one obtained wit te adjustment algoritm of te underlying asset S 0. As mentioned in Section 4, te minimization of te Tikonov-type functional (16) is acieved by alternating minimizations, namely, te minimization w.r.t. a, wic is performed by a gradient descent metod as in [AAZ17], and w.r.t. S 0, wic is performed wit te Matlab function lsqnonlin. In bot stages, te mes widt used are τ = 0.01 and y = 0.05, te minimum and maximum log-moneyness values are taken as 5 and 5, respectively, and te maximum time to maturity is 0.5. We started te algoritm wit a 0 52 /2 and S0 0 = Ŝ0 = During te minimization w.r.t. a, we took te parameters in te penalty function as α 0 = 10 5, α 1 = 10 2, α 2 = 10 4, α 3 = 1, and α 4 = α 5 = 0, wereas in te minimization w.r.t. S 0, we used α 1 = α 2 = α 3 = α 5 = 0, and α 4 = Te a priori surface was taken as a /2. After 8 iterations, te resulting underlying asset price was 0.999, and te normalized l 2 -distance between te true and te reconstructed local volatility surfaces was At te beginning, wit S 0 = 0.95, te normalized distance was Figure 2 compares between

14 14 te reconstructed local volatility surface and te ground trut at eac maturity. Wit te same scarce data set, as te underlying asset price was adjusted, we found a local volatility surface muc closer to te original one. Table 1 presents te evolution of te normalized l 2 -distance between te true and te reconstructed local volatility surfaces, illustrating te latter observation. Table 1: Normalized l 2 -distance between te true and te reconstructed local volatility surfaces and te value of S 0 at eac step of te algoritm for adjusting S 0. Iteration Normalized Distance S Anoter experiment wit underlying price uncertainty We ave carried out anoter experiment tat deals wit te underlying price uncertainty. Te same general form as given in (17) is used for te volatility surface, except for a little bit of smooting: see te bottom rigt subfigure in Figure 3. However, we make sure to employ parameters tat are close to te ones generally encountered wit real data, and we use data tat follows a practical grid. Te parameters for tis example are given in Table 2. Table 2: Parameters for te example of Figure 3. Ŝ 0 initial spot price 2500 S true optimal spot price 2200 r interest rate 5% te maximum maturity 1.8 Minimum y -3.5 Maximum y 3.5 τ 0.1 y 0.1 a priori surface a 0 2 /2 Following te same algoritm as before, we obtain tat S 0 approximates S true well, and furtermore, te local variance a approximates a true on a coarse grid. Tis is illustrated in Figures 3 and 4. Discussion of te syntetic data results Te experiment wit syntetic data indicates tat including te underlying stock price as one of te unknowns (wic corresponds to andling data location uncertainty in y) leads to better results wen tere is uncertainty about suc value. Indeed, Table 1 and Figures 2 and 3 sow tat te normalized distance between te reconstructions and te true surface decreases considerably wen we combine local volatility calibration wit te adjustment of te underlying asset price, upon using te algoritm presented in Section 4. In te first experiment, te distance between te initial guess price Ŝ0 and te (true) price S true was relatively small, wereas in te second one, te

15 15 Figure 3: Calibration of te local volatility in 5 iterations. Sown, starting from te upper left, are te 1st, 3rd, and 5t iterations, as well as te ground trut (bottom rigt). Figure 4: Te estimated spot price converges to te true price.

16 16 two prices were significantly apart. In bot cases, we can see tat after only a few iterations te ground-trut price was well-approximated. 5.2 Results for real data from equity markets In tis subsection and te next one we consider real data from financial markets. In our experiments we cose options on te Standard and Poors (SPX) index. 3 Figure 5 depicts te locations at wic our data set is given. Figure 5: Locations of te SPX data in te (τ, y) domain wit our coarsest mes in te background. Suc options are fairly liquid ones and tus amenable to te models introduced in Section 1. Te data were collected on 19-Jun-2015 and contain prices for 9 different maturities ranging all te way from one day to two years. Te parameters for all te models are given in Table 3. Note tat te optimal spot price in te table refers to te optimized spot price using te metod described in Section 4. Te parameters α 0, α 1, α 2 and α 3 in te penalty functional (10) or (12) used in tis experiment can be found in Table 4. Figure 6 displays te reconstructed SPX local volatility surfaces at different maturities obtained wit tree metod variants using te original scarce data. Note tat te results generated by te Tikonov-type metod wit a 0 penalty and EnKF are closer, wereas te results witout te a 0 penalty differ inexplicably. 3 Te Standard and Poors is a weigted index of actively traded large capitalization common stocks in te United States.

17 17 Table 3: Parameters for te equity data examples. S 0 initial spot price S 0 optimal spot price r interest rate 5% te maximum maturity 2.5 Minimum y -4.5 Maximum y 1.5 τ 0.05 y 0.1 initial a /2 Table 4: Parameters of te penalty functional (10) or (12) wit SPX data. Parameter α 0 α 1 α 2 α 3 Value 4.e+8 1.e+6 or 0 1.e+6 1.e+6 Table 5: Residuals of te 6 metod variants. Tikonov-type EnKF Scarce Comp. Scarce (no a 0 ) Comp. (no a 0 ) Scarce Comp. Residual Figure 6: Reconstructed SPX local volatility surfaces at different maturities obtained wit tree metod variants using scarce data.

18 18 In Figure 7 we consider SPX local volatility surfaces at different maturities obtained wit te metods of Sections 2 and 3. Tey were computed using completed data as in [Ka05]. Note tat te tree metod variants produce similar results around y = 0. (Tis is called at-te-money.) For y > 0.5 (te so-called in-te-money and out-of-te-money regions), if we do not add te a 0 penalty, te two wings blow up. Figure 7: Reconstructed SPX local volatility surfaces at different maturities obtained wit Tikonov-type and EnKF metods using completed data. Tese results are inferior to te corresponding ones for scarce data, displayed in Figure 6. Figure 8 presents reconstructed SPX local volatility surfaces obtained wit all six metod variants. Wen put togeter, we can see tat te different metods lead to results tat get closer as te maturity gets longer around y = 0. Figure 8: Reconstructed SPX local volatility surfaces obtained wit six metod variants. Figure 9 is a zoom-in of Figure 8 to te region around y = 0. In tis region one expects a larger number of liquid contracts. Observe tat te plotted curves are generally divided into two groups: one is obtained from te original (scarce) data and te oter from te completed data. Tis penomenon is clearer in te figures for te earliest and latest dates T = 25 and T = 332. Taken togeter, Figures 8 and 9 again demosntrate te superiority of te scarce data approac, as well as te regularization metods tat employ a 0. One concept tat is prevalent in practical applications in te so-called implied volatility. It is defined as te volatility tat would be observed for a standard call (or put) contract to give te observed price if te classical Black-Scoles formula were used. In oter words, it is te constant volatility tat wen plugged into (1) - all else being equal - would yield te corresponding quoted price. In Figure 10 we see tat te reconstructions of implied volatility

19 19 Figure 9: Reconstructed SPX local volatility surfaces obtained wit six metod variants for different maturities in te at-te-money (y = 0) neigborood. from local volatility do better for intermediate term and long term maturities. However, wen te maturities are sort, te results split into two groups according to weter we use te original scarce data or te completed data. Figure 10: Implied (Black-Scoles) volatility corresponding to te local volatility surfaces obtained wit te six metod variants compared to te market one. It is important to empasize tat te red curves in Figure 10 represent implied volatility, not te prices (i.e., te red curves are not simple observations to be fitted). In financial markets, a misfit suc as is displayed ere for te sort maturities may or may not be acceptable depending on issues suc as bid and ask spread as well as risk premia and absence of arbitrage opportunities. Moreover, te simple fact tat a problem is under-determined does not mean tat we sould be able to fit it well wit a given model suc as (1). In

20 20 fact, even if suc fitting were possible in a different (e.g., interpolation) sense, te prices obtained could be subject to arbitrage opportunities, and ere indeed te practical use would be excluded. (Tere is incidentally a well-known and somewat similar effect wen over-fitting learning examples in macine learning.) In fact, if only sort term contracts are of interest, ten a better agreement among te curves in te first two subplots of Figure 10 can be obtained by simply dropping te data columns related to long term contracts! Tis is wat some practitioners do in suc a case, and we ave verified tat it works to significantly improve agreement among te curves, altoug not necessarily our state of knowledge about te actual financial implications. In order to assess te misfit between our reconstructed results and te practical implied volatility in our metods, we make use of tree different figures of merit. Note tat our inverse problem solution process is not directly trying to minimize suc quantities, and tus tey sould be only taken as an additional quality control quantity. Let Ii,j L (Iba i,j ) denote te implied volatility corresponding to te reconstructed local volatility (from te average of bid and ask option prices) wit strike K i and maturity τ j. In te SPX example, we restrict te strikes to be between 1890 and Furter, V i,j is te volume of te option price wit strike K i and maturity τ j, and N vol is te number of contracts tat ave a nonzero volume. We define RMSE = RW MSE = RR = i,j i,j i,j (I L i,j Iba i,j )2 /N vol, (I L i,j Iba i,j )2 V i,j /N vol, (I L i,j Iba i,j )2 / Te resulting values are presented in Table 6. i,j (root mean square error) (I ba i,j )2, (root weigted mean square error) (relative residual). Table 6: Measures of data misfit of te 6 models. Tikonov-type EnKF Scarce Comp. Scarce (no a 0 ) Comp. (no a 0 ) Scarce Comp. RMSE RWMSE RR Discussion of te equity data results From te above experiment we conclude tat using real or completed data sets we get quite different results. Witin te completed data set, if we discard te a 0 penalty, te two wings of te local volatility surface are not stable, for bot metods of Sections 2 and 3. Tis is apparent in all te figures as well as Table 5. From te results involving reconstruction of te implied volatility as sown in Table 6, te Tikonov-type metod using te original (scarce) data as te best residuals in all tree measures, wit te EnKF metod a close second.

21 Results for real data from commodities and data completion Commodities ave been traded extensively in different markets trougout te world for centuries. In many of tose markets, a number of liquid options on suc assets are also traded. Here again, data from suc markets are freely available, and modelling suc data is very relevant for financial analysts and risk management applications. In te present set of examples, we consider te adaption of Dupire s model to te context of option prices on commodity futures introduced in [AAZ17]. For te present purpose, it consists of essentially te model presented in Sections 1 and 2. We cose data from WTI 4 futures and teir options, as well as Henry Hub 5 contracts. Te end-of-te-day WTI option and future prices were traded on 06-Sep-2013, wit te maturity dates, 18-Oct-2013, 16-Nov-2013, 17-Dec-2013, 16-Jan-2014, 15-Feb-2014 and 20- Mar Te end-of-te-day Henry Hub option and future prices were traded on 06- Sep-2013, wit te maturity dates, 29-Oct-2013, 27-Nov-2013, 27-Dec-2013, 29-Jan-2014, 26-Feb-2014 and 27-Mar Te option prices were divided by teir underlying future prices, so S 0 = 1. In wat follows, by residual we mean te merit function (7) wit Γ = α0 1 I, α 0 > 0. Te penalty function (10) was used wit α 1 = α 2 = α 3. A gradient descent metod was applied in te minimization of te resulting Tikonov-type regularization functional. Te parameter α 0 was cosen by a discrepancy-based metod inspired by te classical Morozov s principle. See [AAZ17] for furter details. To complete te data (wen tis alternative was used in tis subsection) we applied linear interpolation, taking into account te boundary and initial conditions in (4). Te boundary conditions were applied at y = ±5. Wen using completed data, we evaluated te local volatility only in te maturity times given in te original data, and interpolated it linearly in time, at eac step in te minimization. Also, wenever y > 0.5 was encountered, we set a(τ, y) = a(τ, 0.5 sign(y)). Wen using te given scarce data, at eac iteration, for eac maturity time τ in te data set, we interpolated a(τ, ) linearly in te intervals [ 5, y min ), and (y max, 5], assuming tat a(τ, 5) = max{0.08, a(τ, y min )} and a(τ, 5) = max{0.08, a(τ, y max )}, were y min and y max are te minimum and maximum log-moneyness strikes in te data set corresponding to τ. Figures 11 and 12 display reconstructed local volatility surfaces for te different maturities, comparing between using te given data and te completed data. Figures 13 and 14 present a comparison between te implied volatilities of bot metods and te market ones, in order to assess ow accurate te reconstructions are. One of te main advantages of te local volatility model is te capability of fitting te market implied smile, wic as an important relationsip wit market risk. Te implied volatilities were evaluated using te Matlab function blsimpv, and we used te interest rate as te dividend yield. In all of tese experiments, we ave used te mes widts τ = 1/365 and y = 0.05, te annualized risk-free interest rate was taken as 5%, and b = 0 in (3), since futures ave no drift. 4 West Texas Intermediate (WTI) is a grade of crude oil used as a bencmark in oil pricing. 5 Henry Hub (HH) natural gas futures are standardized contracts traded on te New York Mercantile Excange (NYMEX).

22 T = 53 days T = 82 days 0.5 T = 42 days T = 71 days T = 112 days 0 T = 145 days T = 102 days 0 T = 132 days T = 173 days T = 202 days 0 T = 162 days 0 T = 193 days Figure 11: Reconstructed local volatility for different maturity dates for Henry Hub call option prices, comparing between completed data (green line wit pentagram) and scarce data (blue line) results. Figure 12: Reconstructed local volatility for different maturity dates for WTI call option prices, comparing between completed data (green line wit pentagram) and scarce data (blue line) results. Table 7 displays te parameters obtained in te tests of local volatility calibration wit Henry Hub and WTI call prices, wit scarce and completed data. In tis table, by residual, we mean te l 2 -distance between te evaluated quantity and te data, normalized by te l 2 -norm of te data. Table 7: Parameters obtained in te local volatility calibration wit Henry Hub and WTI call prices using sparse data and completed data. WTI Henry Hub Comp. Data Sparse Data Comp. Data Sparse Data α 0 1.0e4 1.0e3 1.0e3 1.0e3 α 1 = α 2 = α Price Residual 2.16e e e e-2 Implied Vol. Residual 1.26e e e e-2 Discussion of te real commodity data results Observing te market implied volatilities and te implied volatilities obtained wit bot metods in Figures 13 and 14, te results wit scarce data present a muc better smile aderence tan wen using completed data. So, completing te data can be seen as an unnecessary introduction of noise or inconsis-

23 Figure 13: Henry Hub prices: completed data (green line wit pentagram), scarce data (blue continuous line), and market (red squares) implied volatilities. Figure 14: WTI prices: completed data (green line wit pentagram), sparse data (blue continuous line), and market (red squares) implied volatilities. tency. It can be better noticed wen observing te implied volatilities at deep in-te-money (y < 0.1) and deep out-of-te-money (y > 0.1) strikes. In tese regions, for almost all cases, te results wit scarce data practically matced te implied volatility, wereas wit data completion, te resulting implied volatilities presented iger values. For financial market practitioners, iger implied volatilities can be translated to iger risk. So, using data completion could lead investors to be more conservative tan necessary. 6 Conclusions Te questions of ow to treat observed data in order to produce agreeable solutions, and relatedly, ow muc to trust te quality of a given data set collected by anoter agent, are prevalent in many nontrivial applied inverse problems. Standard teory appears to ave little to contribute to teir satisfactory resolution, and more carefully assembled experience is required. In tis article we ave igligted some of te issues involved troug an important application in matematical finance, and we ave proposed metods tat improve on tecniques available in te open literature. Te problem of constructing a local volatility surface as similarities wit some applications in areas suc as geopysics and medical imaging, in tat it boils down to te calibration of a diffusive PDE problem, reconstructing a distributed parameter function, i.e., a surface, rater tan a few unrelated parameter values. Te difficulty of dealing wit scarce data, wic igligts te need for a careful practical selection of a prior, is common,