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arXiv2018

The randomised Heston model

We propose a randomised version of the Heston model-a widely used stochastic volatility model in mathematical finance-assuming that the starting point of the variance process is a random variable. In such a system, we study the small-and large-time behaviours of the implied volatility, and show that the proposed randomisation generates a short-maturity smile much steeper (`with explosion') than in the standard Heston model, thereby palliating the deficiency of classical stochastic volatility models in short time. We precisely quantify the speed of explosion of the smile for short maturities in terms of the right tail of the initial distribution, and in particular show that an explosion rate of~$t^γ$ ($γ\in[0,1/2]$) for the squared implied volatility--as observed on market data--can be obtained by a suitable choice of randomisation. The proofs are based on large deviations techniques and the theory of regular variations.

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arXiv2021

Option-Based Pricing of Wrong Way Risk for CVA

The two main issues for managing wrong way risk (WWR) for the credit valuation adjustment (CVA, i.e. WW-CVA) are calibration and hedging. Hence we start from a novel model-free worst-case approach based on static hedging of counterparty exposure with liquid options. We say "start from" because we demonstrate that a naive worst-case approach contains hidden unrealistic assumptions on the variance of the hazard rate (i.e. that it is infinite). We correct this by making it an explicit (finite) parameter and present an efficient method for solving the parametrized model optimizing the hedges. We also prove that WW-CVA is theoretically, but not practically, unbounded. The option-based hedges serve to significantly reduce (typically halve) practical WW-CVA. Thus we propose a realistic and practical option-based worst case CVA.

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arXiv2017

Covariance of random stock prices in the Stochastic Dividend Discount Model

Dividend discount models have been developed in a deterministic setting. Some authors (Hurley and Johnson, 1994 and 1998; Yao, 1997) have introduced randomness in terms of stochastic growth rates, delivering closed-form expressions for the expected value of stock prices. This paper extends such previous results by determining a formula for the covariance between random stock prices when the dividends' rates of growth are correlated. The formula is eventually applied to real market data.

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arXiv2016

Static vs adapted optimal execution strategies in two benchmark trading models

We consider the optimal solutions to the trade execution problem in the two different classes of i) fully adapted or adaptive and ii) deterministic or static strategies, comparing them. We do this in two different benchmark models. The first model is a discrete time framework with an information flow process, dealing with both permanent and temporary impact, minimizing the expected cost of the trade. The second model is a continuous time framework where the objective function is the sum of the expected cost and a value at risk (or expected shortfall) type risk criterion. Optimal adapted solutions are known in both frameworks from the original works of Bertsimas and Lo (1998) and Gatheral and Schied (2011). In this paper we derive the optimal static strategies for both benchmark models and we study quantitatively the improvement in optimality when moving from static strategies to fully adapted ones. We conclude that, in the benchmark models we study, the difference is not relevant, except for extreme unrealistic cases for the model or impact parameters. This indirectly confirms that in the similar framework of Almgren and Chriss (2000) one is fine deriving a static optimal solution, as done by those authors, as opposed to a fully adapted one, since the static solution happens to be tractable and known in closed form.

q-fin.PR↗

arXiv2020

Gated Neural Networks for Option Pricing: Rationality by Design

We propose a neural network approach to price EU call options that significantly outperforms some existing pricing models and comes with guarantees that its predictions are economically reasonable. To achieve this, we introduce a class of gated neural networks that automatically learn to divide-and-conquer the problem space for robust and accurate pricing. We then derive instantiations of these networks that are 'rational by design' in terms of naturally encoding a valid call option surface that enforces no arbitrage principles. This integration of human insight within data-driven learning provides significantly better generalisation in pricing performance due to the encoded inductive bias in the learning, guarantees sanity in the model's predictions, and provides econometrically useful byproduct such as risk neutral density.

q-fin.CP↗

arXiv2016

Discrete Sums of Geometric Brownian Motions, Annuities and Asian Options

The discrete sum of geometric Brownian motions plays an important role in modeling stochastic annuities in insurance. It also plays a pivotal role in the pricing of Asian options in mathematical finance. In this paper, we study the probability distributions of the infinite sum of geometric Brownian motions, the sum of geometric Brownian motions with geometric stopping time, and the finite sum of the geometric Brownian motions. These results are extended to the discrete sum of the exponential Lévy process. We derive tail asymptotics and compute numerically the asymptotic distribution function. We compare the results against the known results for the continuous time integral of the geometric Brownian motion up to an exponentially distributed time. The results are illustrated with numerical examples for life annuities with discrete payments, and Asian options.

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arXiv2016

Short Maturity Asian Options in Local Volatility Models

We present a rigorous study of the short maturity asymptotics for Asian options with continuous-time averaging, under the assumption that the underlying asset follows a local volatility model. The asymptotics for out-of-the-money, in-the-money, and at-the-money cases are derived, considering both fixed strike and floating strike Asian options. The asymptotics for the out-of-the-money case involves a non-trivial variational problem which is solved completely. We present an analytical approximation for Asian options prices, and demonstrate good numerical agreement of the asymptotic results with the results of Monte Carlo simulations and benchmark test cases in the Black-Scholes model for option parameters relevant in practical applications.

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arXiv2016

XVA at the Exercise Boundary

XVA is a material component of a trade valuation and hence it must impact the decision to exercise options within a given netting set. This is true for both unsecured trades and secured / cleared trades where KVA and MVA play a material role even if CVA and FVA do not. However, this effect has frequently been ignored in XVA models and indeed in exercise decisions made by option owners. This paper describes how XVA impacts the exercise decision and how this can be readily evaluated using regression techniques (Longstaff and Schwartz 2001). The paper then assesses the materiality of the impact of XVA at the exercise boundary on swaption examples.

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arXiv2016

Exponential functionals of Levy processes and variable annuity guaranteed benefits

Exponential functionals of Brownian motion have been extensively studied in financial and insurance mathematics due to their broad applications, for example, in the pricing of Asian options. The Black-Scholes model is appealing because of mathematical tractability, yet empirical evidence shows that geometric Brownian motion does not adequately capture features of market equity returns. One popular alternative for modeling equity returns consists in replacing the geometric Brownian motion by an exponential of a Levy process. In this paper we use this latter model to study variable annuity guaranteed benefits and to compute explicitly the distribution of certain exponential functionals.

q-fin.PR↗

arXiv2016

The Cross-section of Expected Returns on Penny Stocks: Are Low-hanging Fruits Not-so Sweet?

In this paper, we study the determinants of expected returns on the listed penny stocks from two perspectives. Traditionally financial economics literature has been devoted to study the macro and micro determinants of expected returns on stocks (Subrahmanyam, 2010). Very few research has been carried out on penny stocks (Liu, Rhee, & Zhang, 2011; Nofsinger & Verma, 2014). Our study is an effort to contribute more empirical evidence on penny stocks in the emerging market context. We see the return dynamics of penny stocks from corporate governance perspective. Issues such as shareholding patters are considered to be of much significance when it comes to understand the price movements. Using cross-sectional data on 167 penny stocks listed in the National Stock Exchange of India, we show that (i) Returns of portfolio of lower market-cap penny stocks are significantly different(higher) than that of higher market-cap penny stocks. (ii)Returns of portfolio lower P/E stocks are significantly different (higher) than that of higher P/E stocks. Similarly, returns of portfolio of lower P/B stocks are significantly different (higher) than that of higher P/B stocks, and returns of portfolio of lower priced penny stocks are significantly different(higher) than that of higher priced penny stocks. (iii) Trading volume differences due to alphabetism are insignificant. (iv)Differences in returns of portfolios based on beta and shareholding patterns are insignificant.

q-fin.PR↗

arXiv2018

Asset Pricing with Random Volatility

This paper proposes to model asset price dynamics with a mixture of diffusion processes where the instantaneous volatility of the underlying diffusion process contains a random vector. The marginal probability distributions of the proposed process can match exactly the risk-neutral distributions implied by both spot vanilla options and forward start options. We can also derive the explicit pricing formula for derivatives that have a closed-form solution under Generalized Geometric Brownian Motion.

q-fin.MF↗

arXiv2018

On exponential functionals of processes with independent increments

In this paper we study the exponential functionals of the processes $X$ with independent increments , namely $$I_t= \int _0^t\exp(-X_s)ds, _,\,\, t\geq 0,$$ and also $$I_{\infty}= \int _0^{\infty}\exp(-X_s)ds.$$ When $X$ is a semi-martingale with absolutely continuous characteristics, we derive recurrent integral equations for Mellin transform ${\bf E}( I_t^α)$, $α\in\mathbb{R}$, of the integral functional $I_t$. Then we apply these recurrent formulas to calculate the moments. We present also the corresponding results for the exponential functionals of Levy processes, which hold under less restrictive conditions then in the paper of Bertoin, Yor (2005). In particular, we obtain an explicit formula for the moments of $I_t$ and $I_{\infty}$, and we precise the exact number of finite moments of $I_{\infty}$.

math.PR↗

arXiv2020

Model-independent pricing with insider information: a Skorokhod embedding approach

In this paper, we consider the pricing and hedging of a financial derivative for an insider trader, in a model-independent setting. In particular, we suppose that the insider wants to act in a way which is independent of any modelling assumptions, but that she observes market information in the form of the prices of vanilla call options on the asset. We also assume that both the insider's information, which takes the form of a set of impossible paths, and the payoff of the derivative are time-invariant. This setup allows us to adapt recent work of Beiglboeck, Cox and Huesmann (2016) to prove duality results and a monotonicity principle, which enables us to determine geometric properties of the optimal models. Moreover, we show that this setup is powerful, in that we are able to find analytic and numerical solutions to certain pricing and hedging problems.

q-fin.MF↗

arXiv2017

Understanding the Non-Convergence of Agricultural Futures via Stochastic Storage Costs and Timing Options

This paper studies the market phenomenon of non-convergence between futures and spot prices in the grains market. We postulate that the positive basis observed at maturity stems from the futures holder's timing options to exercise the shipping certificate delivery item and subsequently liquidate the physical grain. In our proposed approach, we incorporate stochastic spot price and storage cost, and solve an optimal double stopping problem to give the optimal strategies to exercise and liquidate the grain. Our new models for stochastic storage rates lead to explicit no-arbitrage prices for the shipping certificate and associated futures contract. We calibrate our models to empirical futures data during the periods of observed non-convergence, and illustrate the premium generated by the shipping certificate.

q-fin.TR↗

arXiv2020

Pricing variance swaps with stochastic volatility and stochastic interest rate under full correlation structure

This paper considers the case of pricing discretely-sampled variance swaps under the class of equity-interest rate hybridization. Our modeling framework consists of the equity which follows the dynamics of the Heston stochastic volatility model, and the stochastic interest rate is driven by the Cox-Ingersoll-Ross (CIR) process with full correlation structure imposed among the state variables. This full correlation structure possess the limitation to have fully analytical pricing formula for hybrid models of variance swaps, due to the non-affinity property embedded in the model itself. We address this issue by obtaining an efficient semi-closed form pricing formula of variance swaps for an approximation of the hybrid model via the derivation of characteristic functions. Subsequently, we implement numerical experiments to evaluate the accuracy of our pricing formula. Our findings confirmed that the impact of the correlation between the underlying and the interest rate is significant for pricing discretely-sampled variance swaps.

q-fin.PR↗

arXiv2016

Loading Pricing of Catastrophe Bonds and Other Long-Dated, Insurance-Type Contracts

Catastrophe risk is a major threat faced by individuals, companies, and entire economies. Catastrophe (CAT) bonds have emerged as a method to offset this risk and a corresponding literature has developed that attempts to provide a market-consistent pricing methodology for these and other long-dated, insurance-type contracts. This paper aims to unify and generalize several of the widely-used pricing approaches for long-dated contracts with a focus on stylized CAT bonds and market-consistent valuation. It proposes a loading pricing concept that combines the theoretically possible minimal price of a contract with its formally obtained risk neutral price, without creating economically meaningful arbitrage. A loading degree controls how much influence the formally obtained risk neutral price has on the market price. A key finding is that this loading degree has to be constant for a minimally fluctuating contract, and is an important, measurable characteristic for prices of long-dated contracts. Loading pricing allows long-dated, insurance-type contracts to be priced less expensively and with higher return on investment than under classical pricing approaches. Loading pricing enables insurance companies to accumulate systematically reserves needed to manage its risk of ruin in a market consistent manner.

q-fin.MF↗

arXiv2016

Distributional Mellin calculus in $\mathbb{C}^n$, with applications to option pricing

We discuss several aspects of Mellin transform, including distributional Mellin transform and inversion of multiple Mellin-Barnes integrals in $\mathbb{C}^n$ and its connection to residue expansion or evaluation of Laplace integrals. These mathematical concepts are demonstrated on several option-pricing models. This includes European option models such as Black-Scholes or fractional-diffusion models, as well as evaluation of quantities related to the optimal exercise price of American options.

q-fin.PR↗

arXiv2016

Regularization and analytic option pricing under $α$-stable distribution of arbitrary asymmetry

We consider a non-Gaussian option pricing model, into which the underlying log-price is assumed to be driven by an $α$-stable distribution. We remove the a priori divergence of the model by introducing a Mellin regularization for the Lévy propagator. Using distributional and $\mathbb{C}^n$ tools, we derive an analytic closed formula for the option price, valid for any stability $α\in]1,2]$ and any asymmetry. This formula is very efficient and recovers previous cases (Black-Scholes, Carr-Wu); we calibrate the formula on market datas, make numerical tests, and discuss its many interesting properties.

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