This paper studies the problem of hedging and pricing a European call option under proportional transaction costs, from two complementary perspectives. We first derive the optimal hedging strategy under CARA utility, following the stochastic control framework of Davis et al. (1993), characterising the no-transaction band via the Hamilton-Jacobi-Bellman Quasi-Variational Inequality (HJBQVI) and the Whalley-Wilmott asymptotic approximation. We then adopt a deep hedging approach, proposing two architectures that build on the No-Transaction Band Network of Imaki et al. (2023): NTBN-Delta, which makes delta-centring explicit, and WW-NTBN, which incorporates the Whalley-Wilmott formula as a structural prior on the bandwidth and replaces the hard clamp with a differentiable soft clamp. Numerical experiments show that WW-NTBN converges faster, matches the stochastic control no-transaction bands more closely, and generalises well across transaction cost regimes. We further apply both frameworks to the bull call spread, documenting the breakdown of price linearity under transaction costs.
We propose a formulation to construct new classes of financial price processes based on the insight that the key variable driving prices $P$ is the earning-over-price ratio $γ\simeq 1/P$, which we refer to as the earning yield and is analogous to the yield-to-maturity of an equivalent perpetual bond. This modeling strategy is illustrated with the choice for real-time $γ$ in the form of the Cox-Ingersoll-Ross (CIR) process, which allows us to derive analytically many stylised facts of financial prices and returns, such as the power law distribution of returns, transient super-exponential bubble behavior, and the fat-tailed distribution of prices before bubbles burst. Our model sheds new light on rationalizing the excess volatility and the equity premium puzzles. The model is calibrated to five well-known historical bubbles in the US and China stock markets via a quasi-maximum likelihood method with the L-BFGS-B optimization algorithm. Using $φ$-divergence statistics adapted to models prescribed in terms of stochastic differential equations, we show the superiority of the CIR process for $γ_t$ against three alternative models.
Introduced in the late 90s, the passport option gives its holder the right to trade in a market and receive any positive gain in the resulting traded account at maturity. Pricing the option amounts to solving a stochastic control problem that for $d>1$ risky assets remains an open problem. Even in a correlated Black-Scholes (BS) market with $d=2$ risky assets, no optimal trading strategy has been derived in closed form. In this paper, we derive a discrete-time solution for multi-dimensional BS markets with uncorrelated assets. Moreover, inspired by the success of deep reinforcement learning in, e.g., board games, we propose two machine learning-powered approaches to pricing general options on a portfolio value in general markets. These approaches prove to be successful for pricing the passport option in one-dimensional and multi-dimensional uncorrelated BS markets.
The current design space of derivatives in Decentralized Finance (DeFi) relies heavily on oracle systems. Replicating market makers (RMMs) provide a mechanism for converting specific payoff functions to an associated Constant Function Market Makers (CFMMs). We leverage RMMs to replicate the approximate payoff of a Black-Scholes covered call option. RMM-01 is the first implementation of an on-chain expiring option mechanism that relies on arbitrage rather than an external oracle for price. We provide frameworks for derivative instruments and structured products achievable on-chain without relying on oracles. We construct long and binary options and briefly discuss perpetual covered call strategies commonly referred to as "theta vaults." Moreover, we introduce a procedure to eliminate liquidation risk in lending markets. The results suggest that CFMMs are essential for structured product design with minimized trust dependencies.
We study the liquidity commonality impact of local and foreign institutional investment in the Australian equity market in the cross-section and over time. We find that commonality in liquidity is higher for large stocks compared to small stocks in the cross-section of stocks, and the spread between the two has increased over the past two decades. We show that this divergence can be explained by foreign institutional ownership. This finding suggests that foreign institutional investment contributes to an increase in the exposure of large stocks to unexpected liquidity events in the local market. We find a positive association between foreign institutional ownership and commonality in liquidity across all stocks, particularly in large and mid-cap stocks. Correlated trading by foreign institutions explains this association. However, local institutional ownership is positively related to the commonality in liquidity for large-cap stocks only.
Hassane Abba Mallam, Diakarya Barro, Yameogo WendKouni
et al.
In this article, we present an approach which allows to take into account the effect of extreme values in the modeling of financial asset returns and in the valorisation of associeted options. Specifically, the marginal distribution of assets returns is modeled by a mixture of two gaussiens distributions. Moreover, we model the joint dependence structure of the returns using an extremal copula which is suitable for our financial data. Applications are made on the Atos and Dassault Systems actions of the CAC40 index. Monte-Carlo method is used to compute the values of some equity options: the call on maximum, the call on minimum, the digital option and the spreads option with the basket (Atos, Dassault systems).
We propose a novel time discretization for the log-normal SABR model which is a popular stochastic volatility model that is widely used in financial practice. Our time discretization is a variant of the Euler-Maruyama scheme. We study its asymptotic properties in the limit of a large number of time steps under a certain asymptotic regime which includes the case of finite maturity, small vol-of-vol and large initial volatility with fixed product of vol-of-vol and initial volatility. We derive an almost sure limit and a large deviations result for the log-asset price in the limit of large number of time steps. We derive an exact representation of the implied volatility surface for arbitrary maturity and strike in this regime. Using this representation we obtain analytical expansions of the implied volatility for small maturity and extreme strikes, which reproduce at leading order known asymptotic results for the continuous time model.
We consider option pricing using a discrete-time Markov switching stochastic volatility with co-jump model, which can model volatility clustering and varying mean-reversion speeds of volatility. For pricing European options, we develop a computationally efficient method for obtaining the probability distribution of average integrated variance (AIV), which is key to option pricing under stochastic-volatility-type models. Building upon the efficiency of the European option pricing approach, we are able to price an American-style option, by converting its pricing into the pricing of a portfolio of European options. Our work also provides constructive guidance for analyzing derivatives based on variance, e.g., the variance swap. Numerical results indicate our methods can be implemented very efficiently and accurately.
Elisa Alos, Fabio Antonelli, Alessandro Ramponi
et al.
In this work we want to provide a general principle to evaluate the CVA (Credit Value Adjustment) for a vulnerable option, that is an option subject to some default event, concerning the solvability of the issuer. CVA is needed to evaluate correctly the contract and it is particularly important in presence of WWR (Wrong Way Risk), when a credit deterioration determines an increase of the claim's price. In particular, we are interested in evaluating the CVA in stochastic volatility models for the underlying's price (which often fit quite well the market's prices) when admitting correlation with the default event. By cunningly using Ito's calculus, we provide a general representation formula applicable to some popular models such as SABR, Hull \& White and Heston, which explicitly shows the correction in CVA due to the processes correlation. Later, we specialize this formula and construct its approximation for the three selected models. Lastly, we run a numerical study to test the formula's accuracy, comparing our results with Monte Carlo simulations.
Credit Value Adjustment (CVA) is the difference between the value of the default-free and credit-risky derivative portfolio, which can be regarded as the cost of the credit hedge. Default probabilities are therefore needed, as input parameters to the valuation. When liquid CDS are available, then implied probabilities of default can be derived and used. However, in small markets, like the Nordic region of Europe, there are practically no CDS to use. We study the following problem: given that no liquid contracts written on the default event are available, choose a model for the default time and estimate the model parameters. We use the minimum variance hedge to show that we should use the real-world probabilities, first in a discrete time setting and later in the continuous time setting. We also argue that this approach should fulfil the requirements of IFRS 13, which means it could be used in accounting as well. We also present a method that can be used to estimate the real-world probabilities of default, making maximal use of market information (IFRS requirement).
Fabio Bellini, Pablo Koch-Medina, Cosimo Munari
et al.
We establish general versions of a variety of results for quasiconvex, lower-semicontinuous, and law-invariant functionals. Our results extend well-known results from the literature to a large class of spaces of random variables. We sometimes obtain sharper versions, even for the well-studied case of bounded random variables. Our approach builds on two fundamental structural results for law-invariant functionals: the equivalence of law invariance and Schur convexity, i.e., monotonicity with respect to the convex stochastic order, and the fact that a law-invariant functional is fully determined by its behaviour on bounded random variables. We show how to apply these results to provide a unifying perspective on the literature on law-invariant functionals, with special emphasis on quantile-based representations, including Kusuoka representations, dilatation monotonicity, and infimal convolutions.
To convert standard Brownian motion $Z$ into a positive process, Geometric Brownian motion (GBM) $e^{βZ_t}, β>0$ is widely used. We generalize this positive process by introducing an asymmetry parameter $ α\geq 0$ which describes the instantaneous volatility whenever the process reaches a new low. For our new process, $β$ is the instantaneous volatility as prices become arbitrarily high. Our generalization preserves the positivity, constant proportional drift, and tractability of GBM, while expressing the instantaneous volatility as a randomly weighted $L^2$ mean of $α$ and $β$. The running minimum and relative drawup of this process are also analytically tractable. Letting $α= β$, our positive process reduces to Geometric Brownian motion. By adding a jump to default to the new process, we introduce a non-negative martingale with the same tractabilities. Assuming a security's dynamics are driven by these processes in risk neutral measure, we price several derivatives including vanilla, barrier and lookback options.
Abstract: We examine, for −1<q<1, q-Gaussian processes, i.e. families of operators (non-commutative random variables) – where the at fulfill the q-commutation relations for some covariance function – equipped with the vacuum expectation state. We show that there is a q-analogue of the Gaussian functor of second quantization behind these processes and that this structure can be used to translate questions on q-Gaussian processes into corresponding (and much simpler) questions in the underlying Hilbert space. In particular, we use this idea to show that a large class of q-Gaussian processes possesses a non-commutative kind of Markov property, which ensures that there exist classical versions of these non-commutative processes. This answers an old question of Frisch and Bourret [FB].
Since the steady rise in human cases which started in 2007, Q fever has become a major public health problem in the Netherlands with 2,357 human cases notified in the year 2009. Ongoing research confirms that abortion waves on dairy goat farms are the primary source of infection for humans, primarily affecting people living close (under 5 km) to such a dairy goat farm. To reverse the trend of the last three years, drastic measures have been implemented, including the large-scale culling of pregnant goats on infected farms.