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Numerical methods for the calibration problem in finance and mean field game equations
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
2017 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

This thesis contains five papers and an introduction. The first four of the included papers are related to financial mathematics and the fifth paper studies a case of mean field game equations. The introduction thus provides background in financial mathematics relevant to the first four papers, and an introduction to mean field game equations related to the fifth paper.

In Paper I, we use theory from optimal control to calibrate the so called local volatility process given market data on options. Optimality conditions are in this case given by the solution to a Hamiltonian system of differential equations. Regularization is added by mollifying the Hamiltonian in this system and we solve the resulting equation using a trust region Newton method. We find that our resulting algorithm for the calibration problem is both accurate and robust.

In Paper II, we solve the local volatility calibration problem using a technique that is related to - but also different from - the Hamiltonian framework in Paper I. We formulate the optimization problem by means of a Lagrangian multiplier and add a Tikhonov type regularization directly on the parameter we are trying to estimate. The resulting equations are solved with the same trust region Newton method as in Paper II, and again we obtain an accurate and robust algorithm for the calibration problem.

Paper III formulates the problem of calibrating a local volatility process to option prices in a way that differs entirely from what is done in the first two papers. We exploit the linearity of the Dupire equation governing the prices to write the optimization problem as a quadratic programming problem. We illustrate by a numerical example that method can indeed be used to find a local volatility that gives good match between model prices and observed market prices on options.

Paper IV deals with the hedging problem in finance. We investigate if so called quadratic hedging strategies formulated for a stochastic volatility model can generate smaller hedging errors than obtained when hedging with the standard Black-Scholes framework. We thus apply the quadratic hedging technique as well as the Black-Scholes hedging to observed option prices written on an equity index and calculate the empirical errors in the two cases. Our results indicate that smaller errors can be obtained with quadratic hedging in the models used than with hedging in the Black-Scholes framework.

Paper V describes a model of an electricity market consisting of households that try to minimize their electricity cost by dynamic battery usage. We assume that the price process of electricity depends on the aggregated momentaneous electricity consumption. With this assumption, the cost minimization problem of each household is governed by a system of mean field game equations. We also provide an existence and uniqueness result for these mean field game equations. The equations are regularized and the approximate equations are solved numerically. We illustrate how the battery usage affects the electricity price.

##### Abstract [sv]

Den här avhandlingen innehåller fyra artiklar och en introduktion. De första fyra av de inkluderade artiklarna är relaterade till finansmatematik och den femte artikeln studerar ett fall av medelfältsekvationer. Introduktionen ger bakgrund i finansmatematik som har relevans för de fyra första artiklarna och en introduktion till medelfältsekvationer relaterad till den femte artikeln.

I Artikel I använder vi teori från optimal styrning för att kalibrera den så kallade lokala volatilitetsprocessen givet marknadsdata för optionspriser. Optimalitetsvillkor ges i det här fallet av lösningen till ett Hamiltonskt system av differentialekvationer. Vi regulariserar problemet genom att släta ut systemets Hamiltonian och vi löser den resulterande ekvationen med en trust region Newtonmetod. Den resulterande algoritmen är både noggrann och robust i att lösa kalibreringsproblemet.

I Artikel II löser vi kalibreringsproblemet för lokal volatilitet med en teknik som är besläktad med - men också skiljer sig från - det Hamiltonska ramverket i Artikel I. Vi formulerar optimeringsproblemet med en Lagrangemultiplikator och använder en Tikhonovregularisering direkt på den parameter vi försöker uppskatta. De resulterande ekvationerna löses med samma trust region Newtonmetod som i Artikel II. Även i detta fall erhåller vi en noggrann och robust algoritm för kalibreringsproblemet.

Artikel III formulerar problemet att kalibrera en lokal volatilitet till optionspriser på att sätt som skiljer sig helt från vad som görs i de två första artiklarna. Vi utnyttjar linjäriteten hos Dupires ekvation som ger optionspriserna och kan skriva optimieringsproblemet som ett kvadratiskt programmeringsproblem. Vi illusterar genom ett numeriskt exempel att metoden kan användas för att hitta en lokal volatilitet som ger en bra anpassning av modellpriser till observerade marknadspriser på optioner.

Artikel IV behandlar hedgingproblemet i finans. Vi undersöker om så kallad kvadratiska hedgingstrategier formulerade för en stokastisk volatilitetsmodell kan generera mindre hedgingfel än vad som erhålls med hedging i den standardmässiga Black-Scholes modellen. Vi tillämpar således teorin för kvadratisk hedging så väl som hedging med Black-Scholes modell på observerade priser för optioner skrivna på ett aktieindex, och beräknar de empiriska felen i båda fallen. Våra resultat indikerar att mindre fel kan erhållas med kvadratisk hedging med de använda modellerna än med hedging genom Black-Scholes modell.

Artikel V beskriver en modell av en elmarknad som består av hushåll som försöker minimera sin elkostnad genom dynamisk batterianvändning. Vi antar att prisprocessen för el beror på den aggregerade momentana elkonsumtionen. Med detta antagande kommer kostnadsminimeringen för varje hushåll att styras av ett system av medelfältsekvationer. Vi ger också ett existens- och entydighetsresultat för dessa medelfältsekvationer. Ekvationerna regulariseras och de approximerade ekvationerna löses numeriskt. Vi illustrerar hur batterianvändningen påverkar elpriset.

##### Place, publisher, year, edition, pages
Kungliga Tekniska högskolan, 2017. , p. 28
##### Series
TRITA-MAT-A ; 2017:02
##### National Category
Computational Mathematics
##### Identifiers
ISBN: 978-91-7729-519-8 (print)OAI: oai:DiVA.org:kth-214082DiVA, id: diva2:1140192
##### Public defence
2017-10-13, F3, Lindstedtsvägen 26, Stockholm, 13:00
##### Note

QC 20170911

Available from: 2017-09-11 Created: 2017-09-11 Last updated: 2017-09-13Bibliographically approved
##### List of papers
1. Local volatility calibration with optimal control in a Hamiltonian framework
Open this publication in new window or tab >>Local volatility calibration with optimal control in a Hamiltonian framework
##### Abstract [en]

We formulate the calibration of a local volatility function that makes a solution to Dupie's equation match market data as an optimal control problem for which optimality conditions are given by a Hamiltonian system. Regularization is added by mollifying the Hamiltonian functional in this system. We have direct access to the Jacobian matrix of the Hamiltonian system, and can therefore employ a Newton based method in the solving phase, whereas other studies tend to use gradient based methods or quasi Newton algorithms. We illustrate our method by calibrating a volatility function to market data on the Euro Stoxx 50 index and find that our algorithm is both accurate and robust.

##### Keywords
Finance, local volatility, calibration, optimal control, inverse problems
##### National Category
Computational Mathematics
Mathematics
##### Identifiers
urn:nbn:se:kth:diva-213954 (URN)
##### Note

QC 20170908

Available from: 2017-09-07 Created: 2017-09-07 Last updated: 2017-09-11Bibliographically approved
2. Local volatility calibration with optimal control in a Lagrangian framework
Open this publication in new window or tab >>Local volatility calibration with optimal control in a Lagrangian framework
##### Abstract [en]

We develop a Lagrangian based method for solving the calibration problem of identifying a local volatility function that makes the solution to Dupire's pricing equation match observed market quotes on options. Our method uses well established techniques from the field of inverse problems, but differs from other published methods in that our formulation makes it possible to use a Newton method with the analytic Jacobian of the system corresponding to first order optimality conditions of our problem. We give a numerical example with market data of options on the Euro Stoxx 50 index and find that our method is efficient and robust in its ability to identify a volatility function that fit observed data.

##### Keywords
Finance, local volatility, calibration, optimal control, Lagrangian multiplier, inverse problems
##### National Category
Computational Mathematics
##### Identifiers
urn:nbn:se:kth:diva-213961 (URN)
##### Note

QC 20170911

Available from: 2017-09-07 Created: 2017-09-07 Last updated: 2017-09-13Bibliographically approved
3. Local volatility calibration with constrained linear combinations of solutions to Dupire's equation
Open this publication in new window or tab >>Local volatility calibration with constrained linear combinations of solutions to Dupire's equation
##### Abstract [en]

We exploit the linearity of Dupire's partial differential equation to formulate the problem of calibrating a local volatility function to observed market data as a quadratic programming problem under linear constraints. Whereas this inverse problem is traditionally treated as fully non-linear optimization problem in an infinite space, we formulate a problem on the form $\min_{\alpha \in R^n} \norm{A\alpha - b}^2$, where $A$ is a matrix and $b$ is a vector, under linear constraints. We illustrate the method by solving our optimization problem with input data consisting of prices on options written on the Euro Stoxx 50 equity index.

##### Keywords
Finance, local volatility, calibration, Dupire's equation, linearity, quadratic programming
##### National Category
Computational Mathematics
Mathematics
##### Identifiers
urn:nbn:se:kth:diva-213964 (URN)
##### Note

QC 20170908

Available from: 2017-09-07 Created: 2017-09-07 Last updated: 2017-09-18Bibliographically approved
4. Empirical performance of quadratic hedging strategies applied to European call options on an equity index
Open this publication in new window or tab >>Empirical performance of quadratic hedging strategies applied to European call options on an equity index
##### Abstract [en]

Quadratic hedging is a well developed theory for hedging contingent claims in incomplete markets by minimizing the replication error in a suitable $L^2$-norm, but it is not widely used among market practitioners and relatively few papers evaluate how well it works on real market data. Here, we develop a framework for comparing hedging strategies, and use it to empirically test the performance of quadrating hedging of European call options on the Euro Stoxx 50 index modeled with an affine stochastic volatility model with and without jumps. As comparison, we use hedging in the standard Black-Scholes model. We find that the quadratic hedging strategies significantly outperform hedging in the Black-Scholes model for out of the money options and options near the money of short maturity when only spot is used in the hedge. When in addition another option is used for hedging, quadratic hedging outperforms Black-Scholes hedging also for medium dated options near the money.

##### Keywords
Finance, equity, hedging strategies, quadratic hedging, stochastic volatility with jumps, Black-Scholes model
##### National Category
Computational Mathematics
Mathematics
##### Identifiers
urn:nbn:se:kth:diva-213966 (URN)
##### Note

QC 20170911

Available from: 2017-09-07 Created: 2017-09-07 Last updated: 2017-09-11Bibliographically approved
5. A mean field game model of an electricity market with consumers minimizing energy cost through dynamic battery usage
Open this publication in new window or tab >>A mean field game model of an electricity market with consumers minimizing energy cost through dynamic battery usage
##### Abstract [en]

This work contains a model of an electricity market consisting of consumers who own batteries that they charge and discharge in an optimal way. The goal of each individual customer is to minimize their total electricity cost, not by changing how much they consume, but by utilizing an optimal strategy for their battery usage. For each consumer we therefore have a value function. Since all consumers are assumed to be equal, their associated value functions are also equal. The optimization problem to determine the optimal battery usage depends on the electricity price, which in turn depends on the total electricity consumption. The consumption is given as a solution to a Kolmogorov forward equation, which involves the battery usage. Hence the Hamilton-Jacobi-Bellman and Kolmogorov equations need to be solved together as a coupled system of PDEs. We devise a numerical scheme for this system and show some simulations. We also prove a result on the existence and uniqueness of solutions to the system of PDEs.

##### Keywords
Mean field games, mfg, optimal control, electricity market, existence theorem
##### National Category
Computational Mathematics
Mathematics
##### Identifiers
urn:nbn:se:kth:diva-214024 (URN)
##### Note

QC 20170911

Available from: 2017-09-08 Created: 2017-09-08 Last updated: 2017-09-11Bibliographically approved

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Cite
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