Markus Leippold Departement of Banking and Finance

University of Zurich

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Master'and'Bachelor'Thesis'Topic'Proposals!

Markus Leippold Hans Vontobel Professor of Financial Engineering!

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Supervising Team

Felix Stang: Research Interest: Knightian uncertainty (including nonlinear expectations and g-expectations, Epsteins ambiguous volatility model, Robust valuation under model uncertainty), Ross Recovery Theorem, Equilibrium with non-equivalent beliefs, Superreplication and optimal/efficient hedging

Meriton Ibraimi: Research Interest: Fundamental Theorem of Asset Pricing, Knightian Uncertainty, Uncertain Volatility, Equilibrium under heterogenous beliefs, Conic Finance, Arbitrage Pricing Theory, High Frequency Trading

Nikola Vasiljevic: Financial Engineering, Option Pricing, Asset Allocation, Financial Econometrics, Econophysics Pricing and Hedging of Path-dependent and Exotic Options, Calibration Methods, Robust Statistics, Volatility and Jump Risk, Levy Processes, Time Changes

Pascal Marco Caversaccio: Research Interest: Asset Pricing, Copulas and Dependence Modeling, Financial Econometrics, Financial Engineering, Finite Difference Methods, Finite Element Methods, Le vy Processes, Model Calibration, Multilevel Monte Carlo Methods.

Thesis Suggestions

Important notes

The Information given in this document may serve as an entry point for your thesis, i.e., the listed literature is by no means complete or the most recent one. The student is asked to do an appropriate literature search on her/his own.

The idea of the project is to produce a competent piece of work (not a Nobel prize), when faced with an almost infinite amount of information and limited time. It mimics the real world (i.e. a job!) and is diametrically opposite in approach to a taught course. It is groping under uncertainty and mimics a random walk with (we hope) positive drift. A random walk, you will remember, has infinite unconditional variance (i.e. t). That is why there is a fixed date (t

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You should try to write your thesis using TeX or LaTeX. Word is usually a nightmare! I hope this list may give you a good starting point for a successful thesis. Good luck!

Thesis Topics !

1. A chaos expansion approach for the pricing of contingent claims The thesis should consider the chaos expansion approach to option pricing, which is an approximation method for pricing of contingent claims based on the Wiener-Ito chaos expansion. First, the literature should be reviewed and the detailed theoretical derivations should be provided. Second, numerical tests should be conducted in order to examine the performance of the approximation. Ref: Funahashi, H., & Kijima, M. (2013). A chaos expansion approach for the pricing of contingent claims. Journal of Computational Finance, Forthcoming. Funahashi, H., & Kijima, M. (2014). An Extension of the Chaos Expansion Approximation for the Pricing of Exotic Basket Options. Applied Mathematical Finance, 21(2), 109-139.

2. Asian options under model ambiguity The student should explain how model ambiguity arises and why it is important for finance. The model by Epstein and Ji is then described and their result for the superhedging and subhedging price is derived in a rigorous way. Finally, it is shown in this thesis that the arbitrage-free price of Asian options does depend on the ambiguity about volatility, but not about drift. Ref: L. Epstein and S. Ji. Ambiguous volatility and asset pricing in continuous time. Rev. Finan. Stud, 26(7):1740-1786, 2013. L. Denis and C. Martini. A theoretical framework for the pricing of contingent claims in the presence of model uncertainty. The Annals of Applied Probability, 16(2):827-852.

3. Asset allocation with higher moments, regime switching and macroeconomic risk This project should address the problem of international asset allocation in the presence of covariance, co-skewness, and co-kurtosis risk in the setting which distinguishes between bull and bear markets, i.e., in a regime switching model. Furthermore, it is important to consider macroeconomic risk for long-term asset allocation problems. Therefore, the thesis should build on the relevant literature, and combine the framework that considers higher moments under regime switching with macroeconomic factors. The theory should be explained in detail and an empirical study should be conducted to demonstrate the performance of the introduced approach to the asset allocation problem. Ref: Ang, A., & Bekaert, G. (2002). International asset allocation with regime shifts. Review of Financial Studies, 15(4), 1137-1187. Brennan, M. J., & Xia, Y. (2002). Dynamic asset allocation under inflation. The Journal of Finance, 57(3), 1201-1238. Guidolin, M., & Timmermann, A. (2008). International asset allocation under regime switching, skew, and kurtosis preferences. Review of Financial Studies, 21(2), 889-935. Siu, T. K. (2011). Long-term strategic asset allocation with inflation risk and regime switching. Quantitative Finance, 11(10), 1565-1580.

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4. Calibration of the Implied Volatility Surface Using High-Frequency Data The implied volatility surface (IVS) is a fundamental building block in computational finance. There are several practical reasons to have a smooth and well-behaved IVS: Market makers quote options for strike-expiry pairs which are illiquid or not listed; Pricing engines, which are used to price exotic options and which are based on far more realistic assumptions than the Black-Scholes model, are calibrated against an observed IVS; The IVS given by a listed market serves as the market of primary hedging instruments against volatility and gamma risk (second-order sensitivity with respect to the spot); Risk managers use stress scenarios defined on the IVS to visualize and quantify the risk inherent to option portfolios. Since the information contained in an IVS is of high importance on an intra-day basis, calibrating and updating the IVS using high-frequency data represents a helpful tool for any financial institution engaged in option trading. Ref: Y. At-Sahalia and J. Jacod. High-Frequency Financial Econometrics. Princeton University Press, 2014. M. R. Fengler. Arbitrage-free smoothing of the implied volatility surface. Quantitative Finance, 9(4):417-428, 2009. C. Homescu. Implied volatility surface: Construction methodologies and characteristics. Working Paper, Wells Fargo Financial, 2011.

5. Early exercise premium decomposition of American-type options Every American-style contingent claim can be decomposed into its European-type counterparty and an early exercise premium. This decomposition technique is well known in literature, and it has been studies for vanilla and barrier American-style options under different dynamics of the underlying, e.g., geometric Brownian motion, stochastic volatility and jump-diffusion models. The goal of this thesis is to review the existing literature on early exercise premium decomposition technique, and derive the main theoretical results. Furthermore, the pricing performance should be tested and benchmarked against another numerical method, e.g., Monte Carlo or finite difference methods. Ref: Carr, P., Jarrow, R., & Myneni, R. (1992). Alternative characterizations of American put options. Mathematical Finance, 2(2), 87-106. Ju, N. (1998). Pricing by American option by approximating its early exercise boundary as a multipiece exponential function. Review of Financial Studies, 11(3), 627-646. Gao, B., Huang, J. Z., & Subrahmanyam, M. (2000). The valuation of American barrier options using the decomposition technique. Journal of Economic Dynamics and Control, 24(11), 1783-1827.

Gukhal, C. R. (2001). Analytical Valuation of American Options on Jump-Diffusion Processes. Mathematical Finance, 11(1), 97-115. Chesney, M., & Gauthier, L. (2006). American Parisian options. Finance and Stochastics, 10(4), 475-506. Cheang, G. H., Chiarella, C., & Ziogas, A. (2013). The representation of American options prices under stochastic volatility and jump-diffusion dynamics. Quantitative Finance, 13(2), 241-253.

6. Calibration Techniques in Option Pricing For the calibration of model parameters people use the so-called market implied approach which relies on the existence of semi-closed- or closed-form expressions for the prices of benchmark derivatives. This implies an inverse problem formulation which means that we need to find a parameter set such that almost all options are matched. Unfortunately, the solution of such a problem entails the optimization of a high-dimensional non-convex problem.

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Ref: D. Bates. Post-87 crash fears in the S&P 500 futures option market. Journal of Econometrics, 94(1): 181-238, 2000. U. K. Chakraborty, editor. Advances in differential evolution, volume 143 of Studies in Computational Intelligence. Springer Verlag, Berlin Heidelberg, 2008. P. Christoffersen and K. Jacobs. The importance of the loss function in option valuation. Journal of Financial Economics, 72(2):291-318, 2004. R. Cont and P. Tankov. Non-parametric calibration of jump-di_usion option pricing models. Journal of Computational Finance, 7(3):1-50, 2004. S. L. Heston. A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies, 6(2):327-343, 1993. K. V. Price, R. M. Storn, and J. A. Lampinen. Differential evolution - A practical approach to global optimization. Natural Computing Series. Springer Verlag, Berlin Heidelberg, 2005.

7. GARCH option pricing models The main goal of this thesis is to review and to analyze different approaches to GARCH option pricing. In particular, the main results for the local risk-neutral valuation relationship approach as well as for the models based on affine and quadratic stochastic discount factors should be derived and discussed in detail. Moreover, An empirical implementation should compare the performance of the introduced discrete-time no-arbitrage pricing models. Ref: Duan, J. C. (1995). The GARCH option pricing model. Mathematical Finance, 5(1), 13-32. Heston, S. L., & Nandi, S. (2000). A closed-form GARCH option valuation model. Review of Financial Studies, 13(3), 585-625. Christoffersen, P., & Jacobs, K. (2004). Which GARCH model for option valuation? Management Science, 50(9), 1204-1221. Hsieh, K. C., & Ritchken, P. (2005). An empirical comparison of GARCH option pricing models. Review of Derivatives Research, 8(3), 129-150. Christoffersen, P., Elkamhi, R., Feunou, B., & Jacobs, K. (2009). Option valuation with conditional heteroskedasticity and nonnormality. Review of Financial Studies, hhp078. Monfort, A., & Pegoraro, F. (2012). Asset pricing with second-order Esscher transforms. Journal of Banking & Finance, 36(6), 1678-1687. Ornthanalai, C. (2014). Levy jump risk: Evidence from options and returns. Journal of Financial Economics, 112(1), 69-90.

8. International capital asset pricing and asset allocation with time-changed Levy models The main goal of this thesis is to examine the performance of time-changed Levy models for international asset pricing and asset allocation problems. In the first step, the literature should be reviewed and the model should be analyzed in detail. In the second step, an empirical study, which includes both calibration exercise and optimal portfolio problem should be conducted and discussed. Ref: Mo, H., & Wu, L. (2007). International capital asset pricing: Evidence from options. Journal of Empirical Finance, 14(4), 465-498.

9. Hedging of Variance Derivatives Variance derivatives are a class of derivative securities where the payoff explicitly depends on some measure of the variance of an underlying asset. Prominent examples of these derivatives include variance swaps and VIX futures and options. Given the direct exposure to the variance, we have seen an extended use of variance derivatives in the risk management because they

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provide a direct exposure to variance without the need to delta-hedge the underlying stock exposure. This fact implies that for market makers of these derivatives the hedging of such product is a crucial issue. Ref: M. Broadie and A. Jain. Pricing and Hedging Volatility Derivatives. Working Paper, 2008. Available here: http://www.columbia.edu/~mnb2/broadie/Assets/VolDerivatives_20080114.pdf. P. Carr and R. Lee. Volatility derivatives. Annual Review of Financial Economics, 1(1):319-339, 2009. J. Crosby. Optimal hedging of variance derivatives. Working Paper, 2012. Available here: http://www.john-crosby.co.uk/pdfs/Crosby_Var_Derivatives_Hedging.pdf. D. Galai. The components of the return from hedging options against stocks. Journal of Business, 56(1):45-54, 1983.

10. Local Stochastic Volatility Models for VIX Options Local stochastic volatility models combine the features of local volatility- and of stochastic volatility models, introducing more flexibility in fitting the smile. In the literature these models are elaborated in detail for equity options (e.g. Standard & Poors 500 (SPX) options). However, since every stochastic volatility model for SPX options embeds an implicit dynamics for the Chicago Board of Options Exchange (CBOE) volatility index (VIX), it is also possible to apply these techniques to the VIX market. Ref: C. Alexander and L. M. Nogueira. Stochastic local volatility. Proceedings of the second international IASTED conference on financial engineering and applications, MIT. 2004. Z. A. Haddou. Stochastic local volatility & high performance computing. Masters thesis. https://www.escholar.manchester.ac.uk/api/datastream?publicationPid=uk-ac-man-scw:213102&datastreamId=FULL-TEXT.PDF, 2013. P. Jckel. Stochastic volatility models: Past, present and future. Slides. http://bfi.cl/papers/Jackel%20-%20Stochastic%20Volatility%20Models%20Past%20Present%20And%20Future.pdf, 2003. Y. Ren, D. Madan and M.Q. Qian. Calibrating and pricing with embedded local volatility models. Risk, 2007. A. W. Van der Stoep, L. A. Grzelak, and C. W. Oosterlee. The Heston stochastic-local volatility model: Efficient Monte Carlo simulation. International Journal of Theoretical and Applied Finance forthcoming, 2013.

11. Market efficiency under non-equivalent beliefs The thesis deals with the question under which conditions a market can be identified as being efficient. It was commonly believed for a long time that one must first specify an equilibrium model before one can give such an identification. But then Jarrow and Larson provided a model independent and rigorous definition of an efficient market and proved two characterization theorems. The goal of this thesis is to follow their approach and discuss their results if one no longer assumes that the probability beliefs of the investors are equivalent, but are only dominated by some probability measure. The thesis finally should give a relationship between an efficient market and asset price bubbles. Ref: R.A. Jarrow and M. Larsson. The meaning of market efficiency. Mathematical Finance, 2012. M. Larsson. Non-Equivalent Beliefs and Subjective Equlibrium Bubbles. Preprint arXiv: 1306.5082v1, 2013.

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12. Modelling the interbank lending market Banks act as lenders and borrowers to each other and the recent financial crisis has shown that it is important to model these interbank linkages and their balance sheets in order to better understand how shocks evolve throughout this financial system. We want to model a network consisting nodes and edges, where each node corresponds to a bank and each link describes the amount of lending between these banks. The linkages are drawn randomly from a given distribution and we want to model the evolvement of a shock to the balance sheets of banks in the system. A key point is the modelling of the balance sheet of each bank, since we have to capture several effects such as haircuts on repurchase agreements, interest on short-term debt and possible fire sales. A thorough discussion of the literature will be necessary, since the topic seems to be very controversially discussed. Ref: Markus K. Brunnermeier, Deciphering the liquidity and credit crunch, 2007-2008. Prasanna Gai, Andrew Haldane, Sujit Kapadia, Complexity, concentration and contagion. Nicole Boyson, Jean Helwege, Jan Jindra, Crises, Liquidity shocks, and fire sales at commercial banks. Did Christian Laux, Christian Leuz, Fair-Value accounting contribute to the financial crisis?

13. Multifactor Stochastic Volatility Models Single-factor stochastic volatility models can capture the slope of the smile, however, these models cannot explain large independent fluctuations in the corresponding level and slope over time. Multifactor stochastic volatility models try to resolve these issues by introducing additional volatility factors. Ref: D. Bates. Post-87 crash fears in the S&P 500 futures option market. Journal of Econometrics, 94(1): 181-238, 2000. P. Christoffersen, S. L. Heston, and K. Jacobs. The shape and term structure of the index option smirk: Why multifactor stochastic volatility models work so well. Management Science, 55(12): 1914-1932, 2009. C. Gouriroux, J. Jasiak, and R. Sufana. The Wishart Autoregressive process of multivariate stochastic volatility. Journal of Econometrics, 150(2):167-181, 2009. S. L. Heston. A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies, 6(2):327-343, 1993. M. Leippold and F. Trojani. Asset pricing with matrix jump diffusions. Working Paper, University of Zurich, University of Lugano, 2010.

14. Multilevel Monte Carlo Methods for Option Pricing Monte Carlo simulation has become an essential tool in the pricing of derivatives security and in risk management. Multilevel Monte Carlo methods are an extension of the ordinary Monte Carlo methods and represent a new tool for variance reduction in the estimates. Ref: M. B. Giles. Multilevel Monte Carlo path simulation. Operations Research, 56(3): 607-617, 2008. M. B. Giles and L. Szpruch. Multilevel Monte Carlo methods for applications in finance. In Recent Developments in Computational Finance, World Scientific / Imperial College Press, 2013. S. Heinrich. Multilevel Monte Carlo methods. In Large-Scale Scientific Computing, volume 2719 of Lecture Notes Computational Science, pages 58-67. Springer Berlin Heidelberg, 2001.

15. No Arbitrage under model uncertainty and transaction costs Bouchard and Nutz consider a robust notion of no-arbitrage under a quite general setting: They allow for model uncertainty and proportional transaction costs. After explaining the model

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assumptions and their approach, the student should consider transaction costs which can be described by a convex function. Does the robust fundamental theorem of asset pricing still hold in this generalized version? Ref: B. Bouchard and M. Nutz. Consistent Price Systems under Model Uncertainty. Preprint arXiv: 1408.5510v1, 2014. E. Bayraktar and Y. Zhang. Fundamental theorem of asset pricing under transaction costs and model uncertainty. Preprint arXiv: 1309.1420v2, 2013.

16. Numerical methods for the pricing of American options under stochastic volatility and jump-diffusion Numerical methods are often used for pricing of path-dependent options. This thesis should provide a survey of numerical techniques for pricing of American options for stochastic volatility and jump-diffusion models. It should comprise a detailed description and ample numerical experiments, which compare the performance of different numerical methods. The implications for the computation of the option prices, the greeks, and the early exercise boundary should be analyzed. Ref: Glasserman, P. (2004). Monte Carlo methods in financial engineering (Vol. 53). Springer. Duffy, D. J. (2006). Finite Difference methods in financial engineering: a Partial Differential Equation approach. John Wiley & Sons. Hilber, N., Reichmann, O., Schwab, C., & Winter, C. (2013). Computational methods for quantitative finance. Springer Finance.

17. Numerical methods for the pricing of barrier and Parisian options under stochastic volatility and jump-diffusion

Numerical methods are often used for pricing of path-dependent options. This thesis should provide a survey of numerical techniques for pricing of standard and Parisian-style barrier options for stochastic volatility and jump-diffusion models. It should comprise a detailed description and ample numerical experiments which compare the performance of different numerical methods. The implications for the computation of the option prices and the greeks. Ref: Glasserman, P. (2004). Monte Carlo methods in financial engineering (Vol. 53). Springer. Duffy, D. J. (2006). Finite Difference methods in financial engineering: a Partial Differential Equation approach. John Wiley & Sons. Hilber, N., Reichmann, O., Schwab, C., & Winter, C. (2013). Computational methods for quantitative finance. Springer Finance.

18. Option Hedging with transaction costs The goal of this thesis is to come up with a hedging strategy (or several hedging strategies) generating high mean-variance risk-return trade-off. The project involves implementation of the hedging strategy (ies) in a binomial (or multinomial) lattice. Ref: H.E. Leland. Option Pricing and Replication with Transaction Costs. Journal of Finance, 47, 271- 294, 1985. S. Grannan. Minimizing Transaction Costs of Option Hedging Strategies.Mathematical Finance 6(4), 341-364, 1996. V. Zakamouline. Hedging of Option Portfolios and Options on Several Assets with Transaction Costs and Nonlinear Partial Differential Equations. Int. J. Contemp. Math. Sciences, 3(4), 159-180, 2008

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19. Optimal hedging with transaction costs and model uncertainty

In a complete market, an investor can perfectly hedge any claim. However, if the market is not complete or he is not willing to pay for a perfect hedge, he could do superreplication, quantile hedging and optimal hedging. After describing these terms, the thesis discusses optimal hedging with transaction costs and model uncertainty in detail. Ref: M. Kirch. Efficient hedging in incomplete markets under model uncertainty. PhD thesis (Humboldt- Universita t zu Berlin, Germany), 2001. K. Kamizono. Partial Hedging under proportional transaction costs. PhD thesis (Columbia University), 2001. Fo llmer, H., Leukert, P. Efficient Hedging: Cost versus Shortfall Risk, Finance and Stochastics, 4, 117-146, 2000.

20. Optimal option portfolios with higher order coherent risk measures The concept of coherent risk measures has become the standard both in academia and practice. This thesis addresses the question of optimal option portfolios with higher order coherent risk measures, which have been recently introduced in the literature. The contribution of the thesis is twofold. First, it should review the literature and explain in detail the concept of (higher order) coherent risk measures, and its applicability to problems of option portfolios. Secondly, an empirical study should address the question of implementation and performance of the proposed approach(es). Ref: Krokhmal, P. A. (2007). Higher moment coherent risk measures. Quantitative Finance, 7, 373387. Cheridito, P. & Li, T. (2009). Risk measures on Orlicz hearts. Mathematical Finance, 19(2), 189-214. Dentcheva, D., Penev, S., & Ruszczyski, A. (2010). Kusuoka representation of higher order dual risk measures. Annals of Operations Research, 181(1), 325-335. Topaloglou, N., Vladimirou, H., & Zenios, S. A. (2011). Optimizing international portfolios with options and forwards. Journal of Banking & Finance, 35(12), 3188-3201. Matmoura, Y., & Penev, S. (2013). Multistage optimization of option portfolio using higher order coherent risk measures. European Journal of Operational Research, 227(1), 190-198.

21. Recovering risk-neutral density and its moments from American options Option-implied risk-neutral distribution and its moments are important for empirical asset pricing, volatility forecasting, and portfolio selection. Although there exists a huge body of research for European-style derivatives, most of the options traded at organized exchanges are American-style. Therefore, it is important to be able to obtain option-implied distribution based on the American option prices, which are often the only options traded in the market. The thesis should provide a detailed review of the literature on American option-implied risk-neutral density, and conduct an extensive empirical study using market data. Ref: Breeden, D. T., & Litzenberger, R. H. (1978). Prices of state-contingent claims implicit in option prices. Journal of business, 621-651. Rubinstein, M. (1994). Implied binomial trees. The Journal of Finance, 49(3), 771-818.

At-Sahalia, Y., & Lo, A. W. (1998). Nonparametric estimation of state-price densities implicit in financial asset prices. The Journal of Finance, 53(2), 499-547. Jackwerth, J. C. (1999). Option-implied risk-neutral distributions and implied binomial trees: A literature review. The Journal of Derivatives, 7(2), 66-82.

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At-Sahalia, Y., & Duarte, J. (2003). Nonparametric option pricing under shape restrictions. Journal of Econometrics, 116(1), 9-47. Tian, Y. S. (2011). Extracting risk-neutral density and its moments from American option prices. The Journal of Derivatives, 18(3), 17-34.

22. Ross' Recovery Theorem and its critics This thesis deals with the question under which assumptions one can uniquely recover physical probabilities from Arrow-Debreu state prices. Ross considers in his theorem a finite state space, but unfortunately Hansen et al. show that the setup cannot be easily extended to continuous time, which is then also discussed by Qin and Linetsky in more detail. The goal of this thesis is to discuss these papers and their assumptions. Ref: S.A. Ross. The Recovery Theorem. To appear in The Journal of Finance, 2013. J. Borovic ka, L.P. Hansen and J.A. Scheinkman. Misspecified Recovery. Working paper, 2014. Qin and V. Linetsky. Ross Recovery in Continuous Time. Working paper, 2014.

23. Static hedging of path-dependent options Many types of path-dependent options can be efficiently priced and dynamically hedged in the Black-Scholes model. However, in a more general diffusion setting the problem is more challenging, and several authors have suggested static hedging approach instead. This thesis should review the existing literature on the static hedging of path-dependent options, with an emphasis on barrier and American options. Furthermore, an extensive implementation exercise is necessary to demonstrate how well the proposed methods perform. Ref: Carr, P., & Chou, A. (1997). Breaking barriers. Risk, 10(9), 139-145. Carr, P., Ellis, K., & Gupta, V. (1998). Static hedging of exotic options. The Journal of Finance, 53(3), 1165-1190. Nalholm, M., & Poulsen, R. (2006). Static hedging of barrier options under general asset dynamics: Unification and application. The Journal of Derivatives, 13(4), 46-60. He, C., Kennedy, J. S., Coleman, T. F., Forsyth, P. A., Li, Y., & Vetzal, K. R. (2006). Calibration and hedging under jump diffusion. Review of Derivatives Research, 9(1), 1-35. Maruhn, J. H., & Sachs, E. W. (2009). Robust static hedging of barrier options in stochastic volatility models. Mathematical Methods of Operations Research, 70(3), 405-433. Chung, S. L., & Shih, P. T. (2009). Static hedging and pricing American options. Journal of Banking & Finance, 33(11), 2140-2149. Chung, S. L., Shih, P. T., & Tsai, W. C. (2013). Static hedging and pricing American knock-in put options. Journal of Banking & Finance, 37(1), 191-205.

24. VWAP option pricing models Volume-weighted average price (VWAP) options became popular class of derivative securities in the last couple of years. Due to the inherent volume risk which cannot be hedged the market is not complete, pricing of VWAP options represents a challenging task. The thesis should provide a detailed overview of the literature on VWAP securities and conduct a numerical and/or empirical study of the performance of VWAP option pricing models. Ref: Konishi, H. (2002). Optimal slice of a VWAP trade. Journal of Financial Markets, 5(2), 197-221. Stace, A. W. (2007). A moment matching approach to the valuation of a volume weighted average price option. International Journal of Theoretical and Applied Finance, 10(01), 95-110. Biakowski, J., Darolles, S., & Le Fol, G. (2008). Improving vwap strategies: A dynamic volume approach. Journal of Banking & Finance, 32(9), 1709-1722.

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Buryak, A., & Guo, I. Effective and simple VWAP options pricing model. 2014. arXiv:1407.7315

25. Uncertain Volatility This project is based on two papers: one describing how to model implied volatility directly instead of modelling first the instantaneous volatility as a stochastic process and then obtaining the process of implied volatilities, and the other paper is a purely mathematical finance paper describing the foundations of the theory of uncertain volatilities. The goal is to combine these two papers and see how the former paper can be transformed into the language of the latter. Ref: Mete Soner, Quasi-sure stochastic analysis through aggregation. Liuren Wu, Peter Carr, A New Framework for Analyzing Volatility Risk and Premium Across Option Strikes and Expiries.