Securities Pricing with Information-Sensitive Discounting

In this paper incomplete-information models are developed for the pricing of securities in a stochastic interest rate setting. In particu- lar we consider credit-risky assets that may include random recovery upon default. The market filtration is generated by a collection of information processes associated with economic factors, on which in- terest rates depend, and information processes associated with mar- ket factors used to model the cash flows of the securities. We use information-sensitive pricing kernels to give rise to stochastic interest rates. Semi-analytical expressions for the price of credit-risky bonds are derived, and a number of recovery models are constructed which take into account the perceived state of the economy at the time of default. The price of European-style call bond options is deduced, and it is shown how examples of hybrid securities, like inflation-linked credit-risky bonds, can be valued. Finally, a cumulative information process is employed to develop pricing kernels that respond to the amount of aggregate debt of an economy.Asset pricing, incomplete information, stochastic interest rates, credit risk, recovery models, credit-inflation hybrid securities, information-sensitive pricing kernels

Modulated Information Flows in Financial Markets

We model continuous-time information flows generated by a number of information sources that switch on and off at random times. By modulating a multi-dimensional L\'evy random bridge over a random point field, our framework relates the discovery of relevant new information sources to jumps in conditional expectation martingales. In the canonical Brownian random bridge case, we show that the underlying measure-valued process follows jump-diffusion dynamics, where the jumps are governed by information switches. The dynamic representation gives rise to a set of stochastically-linked Brownian motions on random time intervals that capture evolving information states, as well as to a state-dependent stochastic volatility evolution with jumps. The nature of information flows usually exhibits complex behaviour, however, we maintain analytic tractability by introducing what we term the effective and complementary information processes, which dynamically incorporate active and inactive information, respectively. As an application, we price a financial vanilla option, which we prove is expressed by a weighted sum of option values based on the possible state configurations at expiry. This result may be viewed as an information-based analogue of Merton's option price, but where jump-diffusion arises endogenously. The proposed information flows also lend themselves to the quantification of asymmetric informational advantage among competitive agents, a feature we analyse by notions of information geometry.Comment: 27 pages, 1 figur

Simultaneous Trading in 'Lit' and Dark Pools

We consider an optimal trading problem over a finite period of time during which an investor has access to both a standard exchange and a dark pool. We take the exchange to be an order-driven market and propose a continuous-time setup for the best bid price and the market spread, both modelled by L\'evy processes. Effects on the best bid price arising from the arrival of limit buy orders at more favourable prices, the incoming market sell orders potentially walking the book, and deriving from the cancellations of limit sell orders at the best ask price are incorporated in the proposed price dynamics. A permanent impact that occurs when 'lit' pool trades cannot be avoided is built in, and an instantaneous impact that models the slippage, to which all 'lit' exchange trades are subject, is also considered. We assume that the trading price in the dark pool is the mid-price and that no fees are due for posting orders. We allow for partial trade executions in the dark pool, and we find the optimal trading strategy in both venues. Since the mid-price is taken from the exchange, the dynamics of the limit order book also affects the optimal allocation of shares in the dark pool. We propose a general objective function and we show that, subject to suitable technical conditions, the value function can be characterised by the unique continuous viscosity solution to the associated partial integro differential equation. We present two explicit examples of the price and the spread models, and derive the associated optimal trading strategy numerically. We discuss the various degrees of the agent's risk aversion and further show that roundtrips, i.e. posting the remaining inventory in the dark pool at every point in time, are not necessarily beneficial

Stochastic modelling with randomized Markov bridges

We consider the filtering problem of estimating a hidden random variable X by noisy observations. The noisy observation process is constructed by a randomized Markov bridge (RMB) (Zt)t∈[0,T] of which terminal value is set to ZT=X. That is, at the terminal time T, the noise of the bridge process vanishes and the hidden random variable X is revealed. We derive the explicit filtering formula, governing the dynamics of the conditional probability process, for a general RMB. It turns out that the conditional probability is given by a function of current time t, the current observation Zt, the initial observation Z0, and the a priori distribution ν of X at t = 0. As an example for an RMB, we explicitly construct the skew-normal randomized diffusion bridge and show how it can be utilized to extend well-known commodity pricing models and how one may propose novel stochastic price models for financial instruments linked to greenhouse gas emissions

Term Risk in Interest Rate Markets

Using a stylised financial system along with a systemic perspective thereof, the definition of an aggregated banking system that is default-free but vulnerable to liquidity risks is enabled. Within this setup, a consistent mathematical modelling framework for term interest rate systems is derived that enables the pricing and valuation of associated linear derivative instruments. It is then demonstrated that term rates may not be synthetically replicated, in general, which in turn enables the extraction and explanation of the genesis of term risk. These findings provide: (i) a rigorous understanding of the incomplete market paradigm that encapsulates inter-bank term rates and the risk management processes involved therein; and (ii) quantitative theoretical evidence against global interest rate reform proposals advocating for the replacement of term Libor (London inter-bank offered rate) reference rates with overnight rate-based alternatives

Real-Time Risk Management: An AAD-PDE Approach

We apply adjoint algorithmic differentiation (AAD) to the risk management of securities when their price dynamics are given by partial differential equations (PDE). We show how AAD can be applied to forward and backward PDEs in a straightforward manner. In the context of one-factor models for interest rates or default intensities, we show how price sensitivities are computed reliably and orders of magnitude faster than with a standard finite-difference approach. This significantly increased efficiency is obtained by combining (i) the adjoint forward PDE for calibrating model parameters, (ii) the adjoint backward PDE for derivatives pricing, and (iii) the implicit function theorem to avoid iterating the calibration procedure

AAD and least-square Monte Carlo: fast Bermudan-style options and XVA Greeks

We show how Adjoint Algorithmic Differentiation (AAD) can be used to calculate price sensitivities in regression-based Monte Carlo methods reliably and orders of magnitude faster than with standard finite-difference approaches. We present the AAD version of the celebrated least-square algorithms of Tsitsiklis and Van Roy (2001) and Longstaff and Schwartz (2001). By discussing in detail examples of practical relevance, we demonstrate how accounting for the contributions associated with the regression functions is crucial to obtain accurate estimates of the Greeks, especially in XVA applications

Heat Kernel Framework for Asset Pricing in Finite Time

A heat kernel approach is proposed for the development of a general, flexible, and mathematically tractable asset pricing framework in finite time. The pricing kernel, giving rise to the price system in an incomplete market, is modelled by weighted heat kernels which are driven by multivariate Markov processes and which provide enough degrees of freedom in order to calibrate to relevant data, e.g. to the term structure of bond prices. It is shown how, for a class of models, the prices of bonds, caplets, and swaptions can be computed in closed form. The dynamical equations for the price processes are derived, and explicit formulae are obtained for the short rate of interest, the risk premium, and for the stochastic volatility of prices. Several of the closed-form asset price models presented in this paper are driven by combinations of Markovian jump processes with different probability laws. Such models provide a rich basis for consistent applications in several sectors of a financial market including equity, fixed-income, commodities, and insurance. The flexible, multidimensional and multivariate structure, on which the asset price models are constructed, lends itself well to the transparent modelling of dependence across asset classes. As an illustration, the impact on prices by spiralling debt, a typical feature of a financial crisis, is modelled explicitly, and contagion effects are readily observed in the dynamics of asset returns

Rational models for inflation-linked derivatives

We construct models for the pricing and risk management of inflation-linked derivatives. The models are rational in the sense that linear payoffs written on the consumer price index have prices that are rational functions of the state variables. The nominal pricing kernel is constructed in a multiplicative manner that allows for closed-form pricing of vanilla inflation products suchlike zero-coupon swaps, year-on-year swaps, caps and floors, and the exotic limited-price-index swap. We study the conditions necessary for the multiplicative nominal pricing kernel to give rise to short rate models for the nominal interest rate process. The proposed class of pricing kernel models retains the attractive features of a nominal multicurve interest rate model, such as closed-form pricing of nominal swaptions, and it isolates the so-called inflation convexity-adjustment term arising from the covariance between the underlying stochastic drivers. We conclude with examples of how the model can be calibrated to EUR data. Read More: https://epubs.siam.org/doi/10.1137/18M123576