Quantitative Finance

The space of call price functions has a natural noncommutative semigroup structure with an involution. A basic example is the Black--Scholes call price surface, from which an interesting inequality for Black--Scholes implied volatility is derived. The binary operation is compatible with the convex order, and therefore a one-parameter sub-semigroup gives rise to an arbitrage-free market model. It is shown that each such one-parameter semigroup corresponds to a unique log-concave probability density, providing a family of tractable call price surface parametrisations in the spirit of the Gatheral--Jacquier SVI surface. An explicit example is given to illustrate the idea. The key observation is an isomorphism linking an initial call price curve to the lift zonoid of the terminal price of the underlying asset.

In this paper we present a rigorously motivated pricing equation for derivatives, including general cash collateralization schemes, which is consistent with quoted market bond prices. Traditionally, there have been differences in how instruments with similar cash flow structures have been priced if their definition falls under that of a financial derivative versus if they correspond to bonds, leading to possibilities such as funding through derivatives transactions. Furthermore, the problem has not been solved with the recent introduction of Funding Valuation Adjustments in derivatives pricing, and in some cases has even been made worse. In contrast, our proposed equation is not only consistent with fixed income assets and liabilities, but is also symmetric, implying a well-defined exit price, independent of the entity performing the valuation. Also, we provide some practical proxies, such as first-order approximations or basing calculations of CVA and DVA on bond curves, rather than Credit Default Swaps.

Explicitly taking into account the risk incurred when borrowing at a shorter tenor versus lending at a longer tenor ("roll-over risk"), we construct a stochastic model framework for the term structure of interest rates in which a frequency basis (i.e. a spread applied to one leg of a swap to exchange one floating interest rate for another of a different tenor in the same currency) arises endogenously. This rollover risk consists of two components, a credit risk component due to the possibility of being downgraded and thus facing a higher credit spread when attempting to roll over short-term borrowing, and a component reflecting the (systemic) possibility of being unable to roll over short-term borrowing at the reference rate (e.g., LIBOR) due to an absence of liquidity in the market. The modelling framework is of "reduced form" in the sense that (similar to the credit risk literature) the source of credit risk is not modelled (nor is the source of liquidity risk). However, the framework has more structure than the literature seeking to simply model a different term structure of interest rates for each tenor frequency, since relationships between rates for all tenor frequencies are established based on the modelled roll-over risk. We proceed to consider a specific case within this framework, where the dynamics of interest rate and roll-over risk are driven by a multifactor Cox/Ingersoll/Ross-type process, show how such model can be calibrated to market data, and used for relative pricing of interest rate derivatives, including bespoke tenor frequencies not liquidly traded in the market.

In Bender and Dokuchaev (2013), we studied a control problem related to swing option pricing in a general non-Markovian setting. The main result there shows that the value process of this control problem can be uniquely characterized in terms of a first order backward SPDE and a pathwise differential inclusion. In the present paper we additionally assume that the cashflow process of the swing option is left-continuous in expectation (LCE). Under this assumption we show that the value process is continuously differentiable in the space variable that represents the volume which the holder of the option can still exercise until maturity. This gives rise to an existence and uniqueness result for the corresponding backward SPDE in a classical sense. We also explicitly represent the space derivative of the value process in terms of a nonstandard optimal stopping problem over a subset of predictable stopping times. This representation can be applied to derive a dual minimization problem in terms of martingales.

This paper is concerned with the following Markovian stochastic differential equation of mean-reversion type d R t =(θ+σα( R t ,t)) R t dt+σ R t d B t with an initial value R 0 = r 0 ∈R , where θ∈R and σ>0 are constants, and the mean correction function α:R×[0,∞)→α(x,t)∈R is twice continuously differentiable in x and continuously differentiable in t . We first derive that under the assumption of path independence of the density process of Girsanov transformation for the above stochastic differential equation, the mean correction function α satisfies a non-linear partial differential equation which is known as the viscous Burgers equation. We then develop a Galerkin type approximation scheme for the function α by utilizing truncation of discretised Fourier transformation to the viscous Burgers equation.

In today's era of big data, deep learning and artificial intelligence have formed the backbone for cryptocurrency portfolio optimization. Researchers have investigated various state of the art machine learning models to predict Bitcoin price and volatility. Machine learning models like recurrent neural network (RNN) and long short-term memory (LSTM) have been shown to perform better than traditional time series models in cryptocurrency price prediction. However, very few studies have applied sequence models with robust feature engineering to predict future pricing. in this study, we investigate a framework with a set of advanced machine learning methods with a fixed set of exogenous and endogenous factors to predict daily Bitcoin prices. We study and compare different approaches using the root mean squared error (RMSE). Experimental results show that gated recurring unit (GRU) model with recurrent dropout performs better better than popular existing models. We also show that simple trading strategies, when implemented with our proposed GRU model and with proper learning, can lead to financial gain.

Collateralization with daily margining has become a new standard in the post-crisis market. Although there appeared vast literature on a so-called multi-curve framework, a complete picture of a multi-currency setup with cross-currency basis can be rarely found since our initial attempts. This work gives its extension regarding a general framework of interest rates in a fully collateralized market. It gives a new formulation of the currency funding spread which is better suited for the general dependence. In the last half, it develops a discretization of the HJM framework with a fixed tenor structure, which makes it implementable as a traditional Market Model.

In the over-the-counter market in derivatives, we sometimes see large numbers of traders taking the same position and risk. When there is this kind of concentration in the market, the position impacts the pricings of all other derivatives and changes the behaviour of the underlying volatility in a nonlinear way. We model this effect using Heston's stochastic volatility model modified to take into account the impact. The impact can be incorporated into the model using a special product called a market driver, potentially with a large face value, affecting the underlying volatility itself. We derive a revised version of Heston's partial differential equation which is to be satisfied by arbitrary derivatives products in the market. This enables us to obtain valuations that reflect the actual market and helps traders identify the risks and hold appropriate assets to correctly hedge against the impact of the market driver.

The European market clearing problem is characterized by a set of heterogeneous orders and rules that force the implementation of heuristic and iterative solving methods. In particular, curtailable block orders and the uniform purchase price (UPP) pose serious difficulties. A block is an order that spans over multiple hours, and can be either fully accepted or fully rejected. The UPP prescribes that all consumers pay a common price, i.e., the UPP, in all the zones, while producers receive zonal prices, which can differ from one zone to another. The market clearing problem in the presence of both the UPP and block orders is a major open issue in the European context. The UPP scheme leads to a non-linear optimization problem involving both primal and dual variables, whereas block orders introduce multi-temporal constraints and binary variables into the problem. As a consequence, the market clearing problem in the presence of both blocks and the UPP can be regarded as a non-linear integer programming problem involving both primal and dual variables with complementary and multi-temporal constraints. The aim of this paper is to present a non-iterative and heuristic-free approach for solving the market clearing problem in the presence of both curtailable block orders and the UPP. The solution is exact, with no approximation up to the level of resolution of current market data. By resorting to an equivalent UPP formulation, the proposed approach results in a mixed-integer linear program, which is built starting from a non-linear integer bilevel programming problem. Numerical results using real market data are reported to show the effectiveness of the proposed approach. The model has been implemented in Python, and the code is freely available on a public repository.

In this paper we introduce a new approach to model-free path-dependent option pricing. We first introduce a general duality result for linear optimisation problems over signed measures introduced in [3] and show how the the problem of model-free option pricing can be formulated in the new framework. We then introduce a model to solve the problem numerically when the only information provided is the market data of vanilla call or put option prices. Compared to the common approaches in the literature, e.g. [4], the model does not require the marginal distributions of the stock price for different maturities. Though the experiments are carried out for simple path-dependent options on a single stock, the model is easy to generalise for multi-asset framework.