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Montes, Juan Miguel (2014) Aspects of Affine Models in the Pricing of Exotic Options and in Credit Risk. [Tesi di dottorato]

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Abstract (inglese)

Affine jump-diffusion term structure models (AJTSMs) are recently receiving much attention in mathematical finance, because they often lead to a tractable analysis of the price distribution functions. This thesis concerns three aspects of mathematical finance, when applied to certain classes of AJTSMs.
The first aspect concerns the pricing problem in the special case when the underlying process Xt is a Continuous-Time Markov Chain. For exotic options, where the claims are time or path dependent, prices can only be estimated by Monte-Carlo simulation, in most cases. We show that this computation is simplified by conditioning first on the number Nt,T of the jumps of the chain. A recursion is proposed to compute the expected discounted payoff given Nt,T=k; Monte Carlo is then used to average out the result over the distribution of Nt,T=k. This leads to a variance reduction by conditioning. We present results of numerical tests which indicate that the method often outperforms plain vanilla Monte Carlo for different kinds of claims.
The second aspect concerns the calibration of a financial model by parameter estimation, when the underlying, a finite state Markov chain, is only partially observed through noisy asset prices. Here, we assume that the jumps of the asset price occurring at the jump-times of the Markov chain are observable as well. Such a model is a special case of the class of models treated in [FR10b]. Their parameter estimation can be addressed via the EM algorithm, following the approach by [EAM08] which, in the case of discrete-time chains, involves the Kalman filter. We extend this approach to to the case of CTMCs via the use of the Wonham filter. Our main contribution is the numerical approximation of the filters and smoothers in the EM algorithm. We compare the classical Euler and Milstein schemes to a new scheme, inspired by [PR10a], that we call a quasi-exact solution and is related to the splitting-up method of [BGR90] and [Gla92]. We prove that such a scheme is of strong convergence order at least 0.5, hence it performs at least as well as the Euler scheme. We present numerical evidence indicating that in fact, in certain cases the method outperforms both the Euler and the Milstein scheme.
The third aspect concerns a unified framework for equity and credit risk modeling, with applications to risk management. Here we treat an affine jump-diffusion model with a single jump-to-default, where the default time is a doubly stochastic random time with intensity driven by an underlying affine factor process. This approach allows for flexible interactions between the defaultable underlying asset price, its stochastic volatility and the default intensity, while maintaining full analytical tractability. We characterize all risk-neutral measures which preserve the affine structure of the model and show that risk management as well as pricing problems can be dealt with efficiently by shifting to suitable survival measures. As an example, we consider a jump-to-default extension of the Heston stochastic volatility model.

[FR10b] R. Frey and W. J. Runggaldier, Pricing credit derivatives under incomplete information: a nonlinear filtering approach., Finance and Stochastics 14 (2010), no. 4, 495–526.

[PR10a] E. Platen and R. Rendek, Quasi-exact approximation of hidden markov chain filters., Communications on Stochastic Analysis 4 (2010), 129–142.

[BGR90] A. Bensoussan, R. Glowinski, and A. Rascanu, Approximation of the zakai equation by the splitting up method, SIAM Journal of Control and Optimization 28 (1990), no. 6, 1420–1431.

[Gla92] F. Le Gland, Splitting-up approximation for spde’s and sde’s with application to non-linear filtering, in: Stochastic Partial Differential Equations and Their Applications, Charlotte 1991, B. L. Rozovskii and R. B. Sowers, editors, Lecture Notes in Control and Information Sciences 176 (1992), 177–187.

Abstract (italiano)

Le strutture a termine affine con diffusione a salti (AJTSMs) stanno recentemente ricevendo molta attenzione in finanza matematica, perché spesso è semplice analizzare le funzioni di distribuzione ad esse associate. Questa tesi riguarda tre diversi aspetti della finanza matematica, applicati su certe classi di AJTSMs.
Il primo aspetto riguarda il problema del prezzaggio, nel caso particolare in cui il processo sottostante Xt sia una Catena Markoviana a Tempo Continuo (CTMC). Per opzioni esotiche, dove il “claim”, cioè il “payoff” del derivato è dipende dal tempo oppure dalle traiettorie, solitamente i prezzi devono essere stimati attraverso simulazioni di tipo Monte Carlo. Mostriamo che, quando si condiziona prima sul numero Nt,T=k dei salti della catena, il calcolo di questa stima si semplifica. Viene proposta una ricorsione per calcolare il valore atteso del “payoff” scontato, dato Nt,T=k; in seguito si calcola il valore atteso del “payoff” rispetto alla distribuzione di Nt,T=k attraverso un metodo Monte Carlo. Questo condizionamento comporta una riduzione della varianza. Presentiamo i risultati di vari test numerici, che indicano che, per diversi tipi di “claims”, il metodo proposto supera spesso un semplice “vanilla” Monte Carlo.
Il secondo aspetto riguarda la calibrazione, cioè la stima dei parametri di un modello finanziario, dove il processo sottostante (una Catena Markoviana finita) è solo parzialmente osservabile tramite i prezzi corrotti del titolo. In questo lavoro, assumiamo che anche i salti del prezzo del titolo corrispondenti ai tempi dei salti della catena Markoviana siano osservabili. Questo è un caso particolare della classe di modelli trattati in [FR10b]. I loro parametri possono essere stimati mediante l’algoritmo “expectation-maximization” (EM), seguendo l’approccio di [EAM08], che, nel caso delle catene a tempo discreto, coinvolge il filtro di Kalman. Estendiamo questo approccio al caso CTMC, usando invece il filtro di Wonham. Il contributo principale di questa parte della tesi è l’approssimazione numerica dei filtri e degli “smoothers” dell’algoritmo EM. Confrontiamo i classici metodi di Eulero e di Milstein con una nuova strategia, simile a [PR10a], che chiamiamo “soluzione quasi-esatta” e che è anche collegata al metodo di “splitting-up” di [BGR90] e [Gla92]. Dimostriamo che tale schema ha un ordine di convergenza forte di almeno 0.5 e che pertanto è almeno tanto efficace quanto lo schema di Eulero. Presentiamo alcuni risultati numerici che indicano che, di fatto, in certi casi il nuovo metodo converge più velocemente di entrambi i metodi di Eulero e di Milstein.
Il terzo aspetto riguarda un quadro unificato per la modellazione del rischio di “equity” e “credit”, con applicazioni alla gestione del rischio. Trattiamo un AJTSM di un’azione con un’unica discontinuità (“jump-to-default”), dove il tempo di fallimento dell’azione è un tempo aleatorio doppiamente stocastico con intensità determinata da un sottostante processo affine. Questo approccio permette una piena trattabilità analitica pur lasciando flessibilità nel definire le interazioni tra il prezzo dell’azione fallibile, la volatilità stocastica e l’intensità del fallimento. Infine caratterizziamo tutte le misure di rischio neutrale che conservano la struttura affine del modello e mostriamo che sia la gestione del rischio che i problemi del prezzaggio possono essere trattati in modo efficiente passando a misure di sopravivenza appropriate. Come esempio, estendiamo il modello di volatilità stocastica di Heston considerando la possibilità di un “jump-to-default”.

[FR10b] R. Frey and W. J. Runggaldier, Pricing credit derivatives under incomplete information: a nonlinear filtering approach., Finance and Stochastics 14 (2010), no. 4, 495–526.

[PR10a] E. Platen and R. Rendek, Quasi-exact approximation of hidden markov chain filters., Communications on Stochastic Analysis 4 (2010), 129–142.

[BGR90] A. Bensoussan, R. Glowinski, and A. Rascanu, Approximation of the zakai equation by the splitting up method, SIAM Journal of Control and Optimization 28 (1990), no. 6, 1420–1431.

[Gla92] F. Le Gland, Splitting-up approximation for spde’s and sde’s with application to non-linear filtering, in: Stochastic Partial Differential Equations and Their Applications, Charlotte 1991, B. L. Rozovskii and R. B. Sowers, editors, Lecture Notes in Control and Information Sciences 176 (1992), 177–187.

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Tipo di EPrint:Tesi di dottorato
Relatore:Runggaldier, Wolfgang
Dottorato (corsi e scuole):Ciclo 25 > Scuole 25 > SCIENZE MATEMATICHE > MATEMATICA COMPUTAZIONALE
Data di deposito della tesi:25 Gennaio 2014
Anno di Pubblicazione:24 Gennaio 2014
Parole chiave (italiano / inglese):Derivative Pricing, Exotic Options, Credit Risk, Affine Models, Markov Processes, Stochastic Filtering, Monte-Carlo
Settori scientifico-disciplinari MIUR:Area 01 - Scienze matematiche e informatiche > MAT/06 Probabilità e statistica matematica
Struttura di riferimento:Dipartimenti > Dipartimento di Matematica
Codice ID:6359
Depositato il:19 Mag 2015 16:02
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Le url contenute in alcuni riferimenti sono raggiungibili cliccando sul link alla fine della citazione (Vai!) e tramite Google (Ricerca con Google). Il risultato dipende dalla formattazione della citazione.

Bibliography Cerca con Google

[AR09] R. Ahlip and M. Rutkowski, Forward start options under stochastic volatility and stochastic interest rate, International Journal of Theoretical and Applied Finance 12 (2009), 209–225. Cerca con Google

[Bay08] E. Bayraktar, Pricing options on defaultable stocks., Appl. Math. Financ. 15 (2008), 277–304. Cerca con Google

[Ben02] Michele Benzi, Preconditioning techniques for large linear systems: A survey, Journal of Computational Physics 182 (2002), 418–477. Cerca con Google

[BGR90] A. Bensoussan, R. Glowinski, and A. Rascanu, Approximation of the zakai equation by the splitting up method, SIAM Journal of Control and Optimization 28 (1990), no. 6, 1420–1431. Cerca con Google

[Bif05] E. Biffis, Affine processes for dynamic mortality and actuarial valuations,, Insur. Math. Econ. 37 (2005), 443–468. Cerca con Google

[BKR97] T. Björk, Y. Kabanov, and W. J. Runggaldier, Bond market structure in the presence of marked point processes, Mathematical Finance 7 (1997), no. 2, 211–223. Cerca con Google

[BPP01] V. Bally, G. Pages, and J. Printems, A stochastic quantization method for nonlinear problems., Monte Carlo Methods and Applications 7 (2001), 21–34. Cerca con Google

[BR02] T. R. Bielecki and M. Rutkowski, Credit risk: Modeling, valuation and hedging, Springer, 2002. Cerca con Google

[CC12] C. Ceci and K. Colaneri, Nonlinear filtering for jump diffusion observations, Adv. in Appl. Probab. 44 (2012), no. 3, 678–701. Cerca con Google

[CDMW08] M. Cremers, J. Driessen, P. Maenhout, and D. Weinbaum, Individual stock-option prices and credit spreads, J. Bank. Financ. 32 (2008), 2706–2715. Cerca con Google

[CFK07] P. Cheridito, D. Filipović, and R.L. Kimmel, Market price of risk specifications for affine models: Theory and evidence, J. Financ. Econ. 83 (2007), 123–170. Cerca con Google

[CFK10] P. Cheridito, D. Filipović, and R.L. Kimmel, A note on the dai-singleton canonical representation of affine term structure models, Math. Financ. 20 (2010), 509 – 519. Cerca con Google

[CFMT11] C. Cuchiero, D. Filipović, E. Mayerhofer, and J. Teichmann, Affine processes on positive semidefinite matrices., Ann. Appl. Probab. 21 (2011), 397 – 463. Cerca con Google

Cerca con Google

[CFY05] P. Cheridito, D. Filipović, and M. Yor, Equivalent and absolutely continuous measure changes for jump-diffusion processes, Ann. Appl. Probab. 15 (2005), 1713–1732. Cerca con Google

[CHJ09] P. Christoffersen, S. Heston, and K. Jacobs, The shape and the term structure of the index option smirk: why multifactor stochastic volatility models work so well., Manage. Sci. 55 (2009), 1914–1932. Cerca con Google

[CJN12] D. Coculescu, M. Jeanblanc, and A. Nikeghbali, Default ttime, no-arbitrage conditions and changes of probability measures,, Financ. Stoch. 16 (2012), 513–535. Cerca con Google

[CK12] T.K. Chung and Y.K. Kwok, Handbook on computational economics and finance, ch. Equity-credit modeling under affine jump-diffusion models with jump-to-default, p. forthcoming, Oxford University Press, 2012. Cerca con Google

[CL06] P. Carr and V. Linetsky, A jump to default extended cev model: An application of bessel processes., Financ. Stoch. 10 (2006), 303–330. Cerca con Google

[CM99] P. Carr and D.B. Madan, Option valuation using the fast fourier transform., J. Comput. Financ. 2 (1999), 61–73. Cerca con Google

[CM10] P. Carr and D.B. Madan, Local volatility enhanced by a jump to default., SIAM J. Financial Math. 1 (2010), 2–15. Cerca con Google

[CPS09] L. Campi, S. Polbennikov, and A. Sbuelz, Systematic equity-based credit risk: A cev model with jump to default., J. Econ. Dyn. Control 33 (2009), 93–108. Cerca con Google

[Cre99] A. Di Crescenzo, A probabilistic analogue of the mean value theorem and its applications to reliability theory., J. Appl. Prob. 36 (1999), 706–719. Cerca con Google

[CS08] P. Carr and W. Schoutens, Hedging under the heston model with jump-to-default, Int. J. Theor. Appl. Financ. 11 (2008), 403–414. Cerca con Google

[CT03] J. Campbell and G. Taksler, Equity volatility and corporate bond yields., J. Financ. LVIII (2003), 2321–2349. Cerca con Google

[CW10] P. Carr and L. Wu, Stock options and credit default swaps: A joint framework for valuation and estimation., J. Financ. Economet. 8 (2010), 409–449. Cerca con Google

[CW12] P. Cheridito and A. Wugalter, Pricing and hedging in affine models with possibility of default., SIAM J. Financial Math. 3 (2012), 328–350. Cerca con Google

[DF09] E. Mayerhofer D. Filipovic, Affine diffusion processes: Theory and applications, Radon Series in Computational and Applied Mathematics 8 (2009), 1–40. Cerca con Google

[DFS03] D. Duffie, D. Filipovic, and W. Schachermayer, Affine processes and applications in finance, Annals of Applied Probability 13 (2003), 984–1053. Cerca con Google

[DK96] D. Duffie and R. Kan, A yield-factor model of interest rates, Mathematical Finance 6 (1996), 379–406. Cerca con Google

Cerca con Google

[DPS00] D. Duffie, J. Pan, and K.J. Singleton, Transform analysis and asset pricing for affine jump-diffusions, Econometrica 68 (2000), no. 6, 1343–1376. Cerca con Google

[Dri05] J. Driessen, Is default event risk priced in corporate bonds?, Rev. Financ. Stud. 18 (2005), 165–195. Cerca con Google

[DS94] F. Delbaen and W. Schachermayer, A general version of the fundamental theorem of asset pricing, Math. Ann. 300 (1994), 463–520. Cerca con Google

[DS00] Q. Dai and K. Singleton, Specification analysis of affine term structure models, Journal of Finance 55 (2000), 1943–1978. Cerca con Google

[Duf99] G. R. Duffee, Estimating the price of default risk, The Review of Financial Studies 12 (1999), no. 1, 197–226. Cerca con Google

[Duf02] G.R. Duffee, Term premia and inerest rate forecasts in affine models, Journal of Finance LVII (2002), no. 1, 405–443. Cerca con Google

[Duf05] D. Duffie, Credit risk modeling with affine processes., J. Bank. Financ. 29 (2005), 2751–2802. Cerca con Google

[DZ86] A. Dembo and O. Zeitouni, Parameter estimation of partially observed continuous time stochastic processes, Stochastic Processes and Their Applications 23 (1986), 91–113. Cerca con Google

[EAM08] R. J. Elliott, L. Aggoun, and J. B. Moore, Hidden markov models: Estimation and control, 2nd ed. ed., Stochastic Modelling and Applied Probability, Springer, 2008. Cerca con Google

[EHJ00] R. Elliott, W. Hunter, and B. Jamieson, Financial signal processing: A self calibrating model, Working Paper, Federal Reserve Bank of Chicago (2000). Cerca con Google

[EK06] E. Eberlein and W. Kluge, Exact pricing formulae for caps and swaptions in a levy term structure model., Journal of Computational Finance 9 (2006), 99–125. Cerca con Google

[Ell93] R. Elliott, New finite-dimensional filters and smoothers for noisily observed markov chains, IEEE Transactions On Information Theory 39 (1993), no. 1. Cerca con Google

[Fil09] D. Filipovic, Term structure models, a graduate couse, Springer, 2009. Cerca con Google

[FKK72] M. Fujisaki, G. Kallianpur, and H. Kunita, Stochastic differential equations for the non-linear filtering problem, Osaka J. Math 9 (1972), 19–40. Cerca con Google

[FM14] C. Fontana and J.M. Montes, A unified approach to pricing and risk management of equity and credit risk, Journal of Computational and Applied Mathematics 259 (2014), 350–361. Cerca con Google

[Fon12a] C. Fontana, Four essays in financial mathematics, Ph.D. thesis, University of Padua, 2012. Cerca con Google

[Fon12b] , Mathematical and statistical methods for actuarial sciences and finance., ch. Credit risk and incomplete information: a filtering framework for pricing and risk management., pp. 193–201, Springer, Milan, 2012. Cerca con Google

[FR10a] C. Fontana and W. J. Runggaldier, Credit risk and incomplete information: Filtering and EM parameter estimation, International Journal of Theoretical and Applied Finance 13 (2010), 683–715. Cerca con Google

[FR10b] R. Frey and W. J. Runggaldier, Pricing credit derivatives under incomplete information: a nonlinear filtering approach., Finance and Stochastics 14 (2010), no. 4, 495–526. Cerca con Google

[Fri07] C. Fries, Mathematical finance: Theory, modelling and implementation, Wiley, 2007. [FZ02] D. Filipovic and J. Zabczyk, Markovian term structure models in discrete time., Annals of Applied Probability 7 (2002), no. 2, 710–729. Cerca con Google

[Gla92] F. Le Gland, Splitting-up approximation for spde’s and sde’s with application to nonlinear filtering, in: Stochastic Partial Differential Equations and Their Applications, Charlotte 1991, B. L. Rozovskii and R. B. Sowers, editors, Lecture Notes in Control and Information Sciences 176 (1992), 177–187. Cerca con Google

[Gla04] P. Glasserman, Monte carlo methods in financial engineering., Springer-Verlag, 2004. Cerca con Google

[GS09] R.M. Gaspar and T. Schmidt, Financial risks: New developments in structured product and credit derivatives, ch. CDOs in the light of the current crisis, pp. 33–48, Economica, Paris, 2009. Cerca con Google

[GT08] M. Grasselli and C. Tebaldi, Solvable affine term structure models,, Math. Financ. 18 (2008), 135 – 153. Cerca con Google

[Han07a] R. Van Handel, Filtering, stability, and robustness., Ph.D. thesis, California Institute of Technology, 2007. Cerca con Google

[Han07b] , Lecture notes on stochastic calculus, filtering and stochastic control., 2007. Cerca con Google

[Hes93] S.L. Heston, A closed-form solution for options with stochastic volatility, with applications to bond and currency options., Rev. Financ. Stud. 6 (1993), 327–343. [HKP00] P. Hunt, J. Kennedy, and A. Pelsser, Markov-functional interest rate models, Finance and Stochastics 4 (2000), no. 4, 391–408. Cerca con Google

[JLY05] R.A. Jarrow, D. Lando, and F. Yu, Default risk and diversification: Theory and empirical implications, Mathematical Finance 15 (2005), no. 1, 1–26. Cerca con Google

[KD01] H. J. Kushner and P. G. Dupuis, Numerical methods for stochastic control problems in continuous time, Springer-Verlag, 2001. Cerca con Google

[KR09] M. Keller-Ressel, Affine processes theory and applications in finance, Ph.D. thesis, TU Wien, 2009. Cerca con Google

 Cerca con Google

[Lan00] C. Landen, Bond pricing in a hidden Markov Model of the short rate., Finance and Stochastics 4 (2000), 371–389. Cerca con Google

[MFE05] A.J. McNeil, R. Frey, and P. Embrechts, Quantitative risk management concepts, techniques and tools,, Princeton University Press, 2005. Cerca con Google

[MPR13] J. Montes, V. Prezioso, and W. Runggaldier, Monte carlo variance reduction by conditioning for pricing with underlying a continuous-time finite state markov process., submitted preprint, University of Padua (2013). Cerca con Google

[NEK01] L. Martellini N. El Karoui, A theoretical inspection of the market price for default risk., Working Paper, University of Southern California (2001). Cerca con Google

[Nor03] R. Norberg, The markov chain market, ASTIN Bulletin (2003). Cerca con Google

[Nor05] , Anomalous pdes in markov chains: Domains of validity and numerical solu- Cerca con Google

tions., Finance and Stochastics 9 (2005), 519–537. Cerca con Google

[Øks10] B. Øksendal, Stochastic differential equations, Springer-Verlag, 2010. Cerca con Google

[Pao07] M.S. Paolella, Intermediate probability: A computational approach, Wiley, Chichester, 2007. Cerca con Google

[PBL10] E. Platen and N. Bruti-Liberati, Numerical solution of stochastic differential equations with jumps in finance., Springer, 2010. Cerca con Google

[Pla82] E. Platen, An approximation method for a class of ito processes with jump component, Liet. Mat. Rink. 22 (1982), no. 2, 124–136. Cerca con Google

[PP12] K. B. Petersen and M.S. Pedersen, The matrix cookbook, 2012, Version 20121115. Cerca con Google

[PR10a] E. Platen and R. Rendek, Quasi-exact approximation of hidden markov chain filters., Communications on Stochastic Analysis 4 (2010), 129–142. Cerca con Google

[PR10b] V. Prezioso and W. J. Runggaldier, Interest rate derivatives pricing when the short rate is a continuous time finite state markov process, Preprint (2010). Cerca con Google

[Pre10] V. Prezioso, Interest rate derivatives pricing when the short rate is a continuous time finite state markov process, Ph.D. thesis, University of Padua, 2010. Cerca con Google

[Pro05] P. E. Protter, Stochastic integration and differential equations, 2nd, ver 2.1 ed., Stochastic Modelling and Applied Probability, Springer, 2005. Cerca con Google

[Sch03] P.J. Schönbucher, A note on survival measures and the pricing of options on credit default swaps., working paper, Department of Mathematics, ETH Zürich (2003). [SS07] M. Shaked and G. Shanthikhumar, Stochastic orders, Springer Series in Statistics, 2007. Cerca con Google

[WH06] B. Wong and C.C. Heyde, On changes of measure in stochastic volatility models,, J. Appl. Math. Stoch. Anal. (2006), 1–13. Cerca con Google

Cerca con Google

[Won65] W.M. Wonham., Some applications of stochastic differential equations to optimal nonlinear filtering, SIAM J. Control 2 (1965), 347–369. Cerca con Google

[Wu83] C.F.J. Wu, On the convergence properties of the EM algorithm, The Annals of Statistics 11(1) (1983), 95–103. Cerca con Google

[ZD88] O. Zeitouni and A. Dembo., Exact filters for the estimation of the number of transitions of finite-state continuous time markov processes., IEEE Transactions on Information Theory 34 (1988), 890–893. Cerca con Google

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