Stochastic Modelling of New Phenomena in Financial Markets

The Global Financial Crisis (GFC) has revealed a number of new phenomena in financial markets, to which stochastic models have to be adapted. This dissertation presents two new methodologies, one for modeling the “basis spread“, and the other for “rough volatility“. The former gained prominence during the GFC and continues to persist, while the latter has become increasingly evident since 2014. The dissertation commences with a study of the interest rate market. Since 2008, in this market we have observed “basis spreads“ added to one side of the single-currency floating-for-floating swaps. The persistence of these spreads indicates that the market is pricing a risk that is not captured by existing models. These risks driving the spreads are closely related to the risks affecting the funding of banks participating in benchmark interest rate panels, specifically “roll-over“ risk, this being the risk of not being able to refinance borrowing at the benchmark interest rate. We explicitly model funding liquidity and credit risk, as these are the two components of roll-over risk, developing first a model framework and then considering a specific instance of this framework based on affine term structure models. Subsequently, another specific instance of the model of roll-over risk is constructed using polynomial processes. Instead of pricing options through closed-form expressions for conditional moments with respect to observed process, the price of a zero-coupon bond is expressed as a polynomial of a finite degree in the sense of Cheng & Tehranchi (2015). A formula for discrete-tenor benchmark interest rates (e.g., LIBOR) under roll-over risk is constructed, which depends on the quotient of polynomial processes. It is shown how such a model can be calibrated to market data for the discount factor bootstrapped from the overnight index swap (OIS) rate curve. This is followed by a chapter in which a numerical method for the valuation of financial derivatives with a two-dimensional underlying risk is considered, in particular as applied to the problem of pricing spread options. As is common, analytically closed-form solutions for pricing these payoffs are unavailable, and numerical pricing methods turn out to be non-trivial. We price spread options in a model where asset prices are driven by a multivariate normal inverse Gaussian (NIG) process. We consider a pricing problem in the fixed-income market, specifically, on cross-currency interest rate spreads and on LIBOR-OIS spreads. The final contribution in this dissertation tackles regime switching in a rough-volatility Heston model, which incorporates two important features. The regime switching is motivated by fundamental economic changes, and a Markov chain to model the switches in the long-term mean of the volatility is proposed. The rough behaviour is a more local property and is motivated by the stylized fact that volatility is less regular than a standard Brownian motion. The instantaneous volatility process is endowed with a kernel that induces rough behaviour in the model. Pricing formulae are derived and implemented for call and put options using the Fourier-inversion formula of Gil-Pelaez (1951).