Alessandra Cretarola

BSDEs under partial information and financial applications

In this paper we provide existence and uniqueness results for the solution of BSDEs driven by a general square-integrable martingale under partial information. We discuss some special cases where the solution to a BSDE under restricted information can be derived by that related to a problem of a BSDE under full information. In particular, we provide a suitable version of the Follmer–Schweizer decomposition of a square-integrable random variable working under partial information and we use this achievement to investigate the local risk-minimization approach for a semimartingale financial market model.

Local Risk-Minimization under the Benchmark Approach

We study the pricing and hedging of derivatives in incomplete financial markets by considering the local risk-minimization method in the context of the benchmark approach, which will be called benchmarked local risk-minimization. We show that the proposed benchmarked local risk-minimization allows to handle under extremely weak assumptions a much richer modeling world than the classical methodology.

GKW representation theorem and linear BSDEs under restricted information. An application to risk-minimization.

In this paper we provide Galtchouk-Kunita-Watanabe representation results in the case where there are restrictions on the available information. This allows to prove existence and uniqueness for linear backward stochastic differential equations driven by a general cadlag martingale under partial information. Furthermore, we discuss an application to risk-minimization where we extend the results of Follmer and Sondermann (1986) to the partial information framework and we show how our result fits in the approach of Schweizer (1994).This paper focuses on the valuation and hedging of gas storage facilities, using a spot-based valuation framework coupled with a financial hedging strategy implemented with futures contracts. The first novelty consist in proposing a model that unifies the dynamics of the futures curve and the spot price, which accounts for the main stylized facts of the US natural gas market, such as seasonality and presence of price spikes. The second aspect of the paper is related to the quantification of model uncertainty related to the spot dynamics.

Optimal consumption policies in illiquid markets

We investigate optimal consumption policies in the liquidity risk model introduced by Pham and Tankov (Math. Finance 18:613–627, 2008). Our main result is to derive smoothness C1 results for the value functions of the portfolio/consumption choice problem. As an important consequence, we can prove the existence of the optimal control (portfolio/consumption strategy) which we characterize both in feedback form in terms of the derivatives of the value functions and as the solution of a second-order ODE. Finally, numerical illustrations of the behavior of optimal consumption strategies between two trading dates are given.

A benchmark approach to risk-minimization under partial information

The goal of this paper is to investigate (locally) risk-minimizing hedging strategies under the benchmark approach in a financial semimartingale market model where there are restrictions on the available information. More precisely, we characterize the optimal strategy as the integrand appearing in the Galtchouk–Kunita–Watanabe decomposition of the benchmarked contingent claim under partial information and provide its description in terms of the integrand in the classical Galtchouk–Kunita–Watanabe decomposition under full information via dual predictable projections. Finally we show how these results can be applied to unit-linked life insurance contracts.

Hedging of unit-linked life insurance contracts with unobservable mortality hazard rate via local risk-minimization

In this paper we investigate the local risk-minimization approach for a combined financial-insurance model where there are restrictions on the information available to the insurance company. In particular we assume that, at any time, the insurance company may observe the number of deaths from a specific portfolio of insured individuals but not the mortality hazard rate. We consider a financial market driven by a general semimartingale and we aim to hedge unit-linked life insurance contracts via the local risk-minimization approach under partial information. The Follmer–Schweizer decomposition of the insurance claim and explicit formulas for the optimal strategy for pure endowment and term insurance contracts are provided in terms of the projection of the survival process on the information flow. Moreover, in a Markovian framework, this leads to a filtering problem with point process observations.

The F\"ollmer-Schweizer decomposition under incomplete information

In this paper we study the Föllmer–Schweizer decomposition of a square integrable random variable with respect to a given semimartingale S under restricted information. Thanks to the relationship between this decomposition and that of the projection of with respect to the given information flow, we characterize the integrand appearing in the Föllmer–Schweizer decomposition under partial information in the general case where is not necessarily adapted to the available information level. For partially observable Markovian models where the dynamics of S depends on an unobservable stochastic factor X, we show how to compute the decomposition by means of filtering problems involving functions defined on an infinite-dimensional space. Moreover, in the case of a partially observed jump-diffusion model where X is described by a pure jump process taking values in a finite dimensional space, we compute explicitly the integrand in the Föllmer–Schweizer decomposition by working with finite dimensional filters. Finally, we use our achievements in a financial application where we compute the optimal hedging strategy under restricted information for a European put option and provide a comparison with that under complete information.

A Continuous Time Model for Bitcoin Price Dynamics

This chapter illustrates a continuous time model for the dynamics of Bitcoin price, which depends on an attention or sentiment factor. The model is proven arbitrage-free under mild conditions and a quasi-closed pricing formula for European style derivatives is provided.

A Confidence-Based Model for Asset and Derivative Prices in the Bitcoin Market

We endorse the idea, suggested in recent literature, that BitCoin prices are influenced by sentiment and confidence about the underlying technology; as a consequence, an excitement about the BitCoin system may propagate to BitCoin prices causing a Bubble effect, the presence of which is documented in several papers about the cryptocurrency. In this paper we develop a bivariate model in continuous time to describe the price dynamics of one BitCoin as well as the behavior of a second factor affecting the price itself, which we name confidence indicator. The two dynamics are possibly correlated and we also take into account a delay between the confidence indicator and its delivered effect on the BitCoin price. Statistical properties of the suggested model are investigated and its arbitrage-free property is shown. Further, based on risk-neutral evaluation, a quasi-closed formula is derived for European style derivatives on the BitCoin. A short numerical application is finally provided.

A sentiment-based model for the BitCoin: theory, estimation and option pricing

In recent literature it is claimed that BitCoin price behaves more likely to a volatile stock asset than a currency and that changes in its price are influenced by sentiment about the BitCoin system itself; in Kristoufek [10] the author analyses transaction based as well as popularity based potential drivers of the BitCoin price finding positive evidence. Here, we endorse this finding and consider a bivariate model in continuous time to describe the price dynamics of one BitCoin as well as a second factor, affecting the price itself, which represents a sentiment indicator. We prove that the suggested model is arbitrage-free under a mild condition and, based on risk-neutral evaluation, we obtain a closed formula to approximate the price of European style derivatives on the BitCoin. By applying the same approximation technique to the joint likelihood of a discrete sample of the bivariate process, we are also able to fit the model to market data. This is done by using both the Volume and the number of Google searches as possible proxies for the sentiment factor. Further, the performance of the pricing formula is assessed on a sample of market option prices obtained by the website deribit.com.