Quantitative Finance

This paper presents a new model for pricing financial derivatives subject to collateralization. It allows for collateral arrangements adhering to bankruptcy laws. As such, the model can back out the market price of a collateralized contract. This framework is very useful for valuing outstanding derivatives. Using a unique dataset, we find empirical evidence that credit risk alone is not overly important in determining credit-related spreads. Only accounting for both collateral posting and credit risk can sufficiently explain unsecured credit costs. This finding suggests that failure to properly account for collateralization may result in significant mispricing of derivatives. We also empirically gauge the impact of collateral agreements on risk measurements. Our findings indicate that there are important interactions between market and credit risk.

We develop a new nonparametric approach for estimating the risk-neutral density of asset prices and reformulate its estimation into a double-constrained optimization problem. We evaluate our approach using the S\&P 500 market option prices from 1996 to 2015. A comprehensive cross-validation study shows that our approach outperforms the existing nonparametric quartic B-spline and cubic spline methods, as well as the parametric method based on the Normal Inverse Gaussian distribution. As an application, we use the proposed density estimator to price long-term variance swaps, and the model-implied prices match reasonably well with those of the variance future downloaded from the CBOE website.

In complete markets, there are risky assets and a riskless asset. It is assumed that the riskless asset and the risky asset are traded continuously in time and that the market is frictionless. In this paper, we propose a new method for hedging derivatives assuming that a hedger should not always rely on trading existing assets that are used to form a linear portfolio comprised of the risky asset, the riskless asset, and standard derivatives, but rather should design a set of specific, most-suited financial instruments for the hedging problem. We introduce a sequence of new financial instruments best suited for hedging jump-diffusion and stochastic volatility market models. The new instruments we introduce are perpetual derivatives. More specifically, they are options with perpetual maturities. In a financial market where perpetual derivatives are introduced, there is a new set of partial and partial-integro differential equations for pricing derivatives. Our analysis demonstrates that the set of new financial instruments together with a risk measure called the tail-loss ratio measure defined by the new instrument's return series can be potentially used as an early warning system for a market crash.

In this paper we propose a closed-form approximation for the price of basket options under a multivariate Black-Scholes model, based on Taylor expansions and the calculation of mixed exponential-power moments of a Gaussian distribution. Our numerical results show that a second order expansion provides accurate prices of spread options with low computational costs, even for out-of-the-money contracts.

In this paper we study recent developments in the approximation of the spread option pricing. As the Kirkś Approximation is extremely flawed in the cases when the correlation is very high, we explore a recent development that allows approximating with simplicity and accuracy the option price. To assess the goodness of fit of the new method, we increase dramatically the number of simulations and scenarios to test the new method and compare it with the original Kirkś formula. The simulations confirmed that the Modified Kirkś Approximation method is extremely accurate, improving Kirkś approach for two-asset spread options.

A one-factor asset pricing model with an Ornstein--Uhlenbeck process as its state variable is studied under partial information: the mean-reverting level and the mean-reverting speed parameters are modeled as hidden/unobservable stochastic variables. No-arbitrage pricing formulas for derivative securities written on a liquid asset and exponential utility indifference pricing formulas for derivative securities written on an illiquid asset are presented. Moreover, a conditionally linear filtering result is introduced to compute the pricing/hedging formulas and the Bayesian estimators of the hidden variables.

In this paper we show how to hedge a zero coupon bond with a smaller amount of initial capital than required by the classical risk neutral paradigm, whose (trivial) hedging strategy does not suggest to invest in the risky assets. Long dated zero coupon bonds we derive, invest first primarily in risky securities and when approaching more and more the maturity date they increase also more and more the fraction invested in fixed income. The conventional wisdom of financial planners suggesting investor to invest in risky securities when they are young and mostly in fixed income when they approach retirement, is here made rigorous. The paper provides a strong warning for life insurers, pension fund managers and long term investors to take the possibility of less expensive products seriously to avoid the adverse consequences of the low interest rate regimes that many developed economies face.

During recent years the counterparty risk subject has received a growing attention because of the so called Basel Accord. In particular the Basel III Accord asks the banks to fulfill finer conditions concerning counterparty credit exposures arising from banks' derivatives, securities financing transactions, default and downgrade risks characterizing the Over The Counter (OTC) derivatives market, etc. Consequently the development of effective and more accurate measures of risk have been pushed, particularly focusing on the estimate of the future fair value of derivatives with respect to prescribed time horizon and fixed grid of time buckets. Standard methods used to treat the latter scenario are mainly based on ad hoc implementations of the classic Monte Carlo (MC) approach, which is characterized by a high computational time, strongly dependent on the number of considered assets. This is why many financial players moved to more enhanced Technologies, e.g., grid computing and Graphics Processing Units (GPUs) capabilities. In this paper we show how to implement the quantization technique, in order to accurately estimate both pricing and volatility values. Our approach is tested to produce effective results for the counterparty risk evaluation, with a big improvement concerning required time to run when compared to MC approach.

The paper reviews origins of the approach to pricing derivatives post-crisis by following three papers that have received wide acceptance from practitioners as the theoretical foundations for it - [Piterbarg 2010], [Burgard and Kjaer 2010] and [Burgard and Kjaer 2013]. The review reveals several conceptual and technical inconsistencies with the approaches taken in these papers. In particular, a key component of the approach - prescription of cost components to a risk-free money account, generates derivative prices that are not cleared by the markets that trade the derivative and its underlying securities. It also introduces several risk-free positions (accounts) that accrue at persistently non-zero spreads with respect to each other and the risk free rate. In the case of derivatives with counterparty default risk [Burgard and Kjaer 2013] introduces an approach referred to as semi-replication, which through the choice of cost components in the money account results in derivative prices that carry arbitrage opportunities in the form of holding portfolio of counterparty's bonds versus a derivative position with it. This paper derives no-arbitrage expressions for default-risky derivative contracts with and without collateral, avoiding these inconsistencies.

We develop a semi-analytic approach to the valuation of auto-callable structures with accrual features subject to barrier conditions. Our approach is based on recent studies of multi-assed binaries, present in the literature. We extend these studies to the case of time-dependent parameters. We compare numerically the semi-analytic approach and the day to day Monte Carlo approach and conclude that the semi-analytic approach is more advantageous for high precision valuation.