This thesis is comprised of four self-contained research papers taking us through a journey into the realms of quantitative finance. We join the ranks of practitioners and academic researchers who work on pricing of financial derivatives. A derivative is a financial instrument that derives its value from the value of some underlying quantity or asset, for instance, the stock price of a listed company. This means that a derivative does not have any intrinsic value in itself but should be priced relative to the value of the underlying and other derivatives traded in the market. The importance of sound pricing of such products is stressed by the sheer size of the derivatives market. Looking at just over-the-counter (OTC) products, the market value of outstanding derivatives equaled $24.7 trillion by the end of 2012 with a whopping $632.6 trillion in notional value (BIS, 2013). The research papers consider different aspects of derivatives pricing; yet, a common aim is to solve problems of a quantitative nature but with a clear financial grounding. Mathematics is an invaluable tool in this endeavor, but the story -- we hope -- is one of finance.

In the first two chapters we take a step into the risk anatomy of an equity option. Viewed as a package of risk exposures to the underlying stock, the option’s fair value must naturally price in all inherent risks and the magnitudes of the risk premiums will depend on the assumed stochastic dynamics of the underlying stock. Nevertheless, this notion is somewhat lost in modern derivatives pricing. The hunger for ever more sophisticated mathematical models to capture the informational content of option prices has called for the use of heavy numerical artillery, even to price plain vanilla options. However, numerical methods act as a black box that fails to reveal the link between the risks that actually drive option prices, the structural properties of the option pricing model, and the ultimate effects on the generated surface of option prices. To address this issue, in the first chapter, we expand the true option price around the classical Black-Scholes price. In particular, we show how the true price of an option is determined by a series of premiums on higher-order risks which are not priced under the Black-Scholes model assumptions. In practice, however, options are typically quoted in terms of implied volatilities. So, in the second chapter, we extend our analysis and explore how option risks and the analytical features of the underlying model are translated into implied volatilities.

In the third chapter, we study options on realized variance and, in particular, the impact of jump distributions on the implied volatility of variance. Despite the common notion to include jumps in the variance dynamics, an examination of the distribution of such jumps and its financial implications appear – to the best of our knowledge -- disturbingly absent from the literature. The chapter shows that the particular distribution of jumps does have a profound impact on the shape of the implied volatility smile of variance. Some jump distributions imply a behavior of the volatility of variance that is clearly at odds with the upward-sloping smile observed in variance markets such as, for example, options on CBOE's VIX index. Evidently, correctly grasping the distribution of jumps in variance is of paramount importance for trading and risk management of volatility derivatives. This chapter attempts to fill this gap.

Finally, in the last chapter we take a step back from the classical pricing paradigm and address risks of the derivatives business which used to be neglected. Prior to the recent financial crisis, dealers tended to ignore the credit risk of high-quality rated counterparties, but the lesson of history has shown that this was a particular dangerous assumption. In addition, as banks became reluctant to lend to each other with the crisis rumbling through the Western economies, the spread between the rate on overnight indexed swaps (OISs) and the LIBOR rate blew up making it apparent that LIBOR is contaminated by credit risk. To keep up with these sudden market changes, dealers today make a number of adjustments when they book derivatives trades, especially, in the OTC market. The credit valuation adjustment (CVA) corrects the price for the expected costs to the dealer due to the possibility that the counterparty may default, while the so-called debit valuation adjustment (DVA) is a correction for the expected benefits to the dealer due to his own default risk. The latter adjustment has the perverse effect that the dealer can book a profit as his default risk increases.

Most controversially, however, dealers often adjust the price for the costs of funding the trade. In the industry, this practice is known as a funding valuation adjustment (FVA). When a derivatives desk executes a deal with a client, it backs the trade by hedging it with other dealers in the market. This involves borrowing or lending cash and other assets. Classical derivatives pricing theory rests on the assumption that one can borrow and lend at a unique risk-free rate of interest. This assumption may have been reasonable prior to the crisis with banks funding their hedging strategies at LIBOR. But with drastically increasing spreads, LIBOR is clearly an imperfect proxy of the risk-free rate. In this final chapter we attempt to develop a general pricing framework for valuation of derivatives deals under the presence of complexities such as collateralization, counterparty credit risk and, in particular, funding costs.