Daniel Alexandre Bloch’s research while affiliated with Sorbonne University and other places

While the question of the trillions of dollars required to both mitigate and adapt to climate change is more pressing now than ever, only a financial mechanism can scale up capital to such an extent. However, climate risk represents the inability to predict the global mean surface temperature (GMST) evolution and to anticipate its impact on the global economy. Financial solutions exist, called Climate Derivatives, that directly transfer climate risks from the product issuers to the long-term investors by directly modelling the dynamics of the GMST. The evaluation of climate risk requires a model incorporating the three main sources of uncertainties in climate prediction: 1) the model, 2) the scenarios (corresponding to the estimation of the future external forcing applied to the climate system) which are highly unpredictable, 3) the climate internal variability composed of different cyclic processes at different timescales. We choose a two-box model with efficacy factor to capture the deterministic part of the behaviour of the climate, model the forcings with a three dimensional polynomial function, and include stochastic processes representing additional risk. We obtain a bivariate Ornstein-Uhlenbeck process and calibrate the instantaneous volatility of the model to the historical observations of the GMST, capturing both the trend of the model and its internal variabilities. Since the uncertainty on the forcing scenarios corresponds to a qualitative and objective choice, we let the parties take a view on the scenarios. We then assume incomplete market and estimate the price of the climate derivatives by considering the indifference pricing method. We obtain a matrix of indifference prices representing all possible trades where the rows and columns correspond to the different views of the parties. We can then properly analyse climate risk and provide a financial mechanism which can be operated globally or locally, from regions to municipalities or cities.

The implied volatility surface being a mapping from Black-Scholes prices, necessary and sufficient conditions for the surface to be free from static arbitrage must be defined in terms of the properties and limits of the Black-Scholes formula. Acknowledging this argument, we develop a parametric interpolation and extrapolation of the implied volatility surface to prevent arbitrage both in space and time. Expressing the price surface in terms of moneyness and variance time rather than standard calendar time, we decompose the market option prices into weighted sums of strike shifted Black-Scholes counterparts, combined with a term structure interpolation of implied total variance. As a result, static arbitrage is satisfied by construction while the dynamics of the implied volatility is taken into consideration, allowing for proper dynamic risk management. This simple model, intended to be used by practitioners, allows an analytical computation of the Greeks, the Skew and the Curvature of the fitted implied volatility surface. At last, we generate meaningful stress scenarios for risk management purpose by stress testing the model parameters while preserving the necessary and sufficient conditions for the call price surface to be free from arbitrage.

The principal index of climate change that may result from increased atmospheric concentrations of carbon dioxide is the increase in global mean temperature. Internalising the climate variable, we revisit the existing schemes for fighting global warming and propose a positive action mechanism by introducing the Climate Credit Mechanism based on governments, regions or municipalities allotting tradable rewards not to pollute to local green projects and companies reducing their GHG emissions. While the carbon trading mechanism is a political global scheme focusing only on polluters, we provide incentives for not polluting through a viable local financial mechanism where local capital matches local interests. We illustrate the mechanism by showing how local governments could allot Climate Credits to long term investors in exchange for capital in the form of a state bond, in view of developing the renewable energies. Similarly, we show how local governments could allot Climate Credits to the land owners in order to preserve lands from being improperly hydraulically fractured in view of extracting shale gas. By supplying Climate Credits to land owners, local government directly supports water and health quality in the state, while promoting the development of renewable energies.

The only thing one can say about financial markets is that parsimonious information on option prices is available in time and space, and that we can only use the No-Dominance law (or stronger version of No-Arbitrage) to account for it. Thus, one requires a consistent model to assess relative value between them. We describe a single parametric model for the entire volatility surface with interpolation and extrapolation technique generating a smooth and robust implied volatility surface without arbitrage in space and time. Prices can now be generated such that the No-Dominance principle is preserved, and one can safely assess relative value between them. In order to perform statistical analysis of the relationships between points on the implied volatility surface (IVS), we are left with finding a way of modeling dynamically the agents rational anticipations. We assume that the volatility surface is dynamically modified according to the stock price realisations. Having related the stock price level to the implied volatility surface, we use their respective historic evolution to characterise the transition probabilities, that is, the conditional densities. A statistical technique is used to regress the observed implied smile against the realised stock level. Therefore, the current stock evolution directly influences its future increment which means that, given the stock price at a future time, the conditional density is known.

We describe a single parametric model for the entire volatility surface with interpolation and extrapolation technique generating a smooth and robust implied volatility surface without arbitrage in space and time. It is used for marking option prices on indices and single stocks as well as for computing analytically a proper local volatility with smooth risk-neutral density. Greeks and stress scenarios are calculated analytically in the parametric model without recalibration of the model parameters. We perform a simple expansion of the parametric model obtaining an analytic representation of its implied volatility surface along its cone of diffusion. In view of adding control to the generated volatility surface, we modify the model by adding three new parameters producing, in an independent way, a parallel shift, skew shift and curvature shift of that surface along its cone of diffusion. These parameters can be used manually to modify the entire shape of the volatility surface, and can also be used to generate analytically the new local volatility surface when computing the vegas of an option. Then, in view of defining the best possible volatility surface for non-liquid stocks where only few brokers quotes exist, we describe a method combining historical model parameters of the implied volatility surface together with parameters from other liquid stocks observed on the market.

Given the large number of processes affecting global temperature in the climate system and the fact that these processes operate on different time scales, one cannot assume that a single time scale characterises either the interannual variability of climate or the response of climate to radiative forcing imposed over decades to centuries. Therefore, when modelling climate, we must consider different regimes for the time scale depending on the external forcing. We do this by introducing uncertainty to the characteristic time and to the sensitivity of climate to the stock of greenhouse gases in the atmosphere. We also make allowance for the possibility of non-linear diffusion term representing the weather and add jumps in global temperature. As a result, we improve a simple model introduced in earlier work by presenting three different jump-diffusion models for solving the estimated time constant problem and accounting for non-linear climate forcings. Copyright © 2012 Wilmott Magazine Ltd.

To capture the extra returns embedded in the tails of the market distributions, the literature focused on adding stochastic processes to the diffusion coefficient of the asset prices or even jumps to the asset prices as the drift was forced to match the risk-free rate. However, if we assume that part of the information contained in the implied volatility surface is incomplete and results from matching supply and demand, we loose the notion of risk-neutral measure and can modify the drift to match the market risk premium. Similarly to the commodity markets where the holder of the spot is compensated for holding one unit of inventory in case of shortage, we are going to compensate the holder of the spot in the equity market for the risk of a large downward jump. Assuming a stochastic convenience yield, we create a self-financing portfolio involving the delivery of one unit of the stock at maturity of the forward contract, and use it to compute the model dependent forward price. Specifying an Ornstein-Uhlenbeck process for the convenience yield, we derive the dynamics of the forward price, define the convenience yield measure and apply it to compute analytically the price of a call option when the volatility of the stock price is a deterministic function of time. We then extend the approach to the case where the instantaneous volatility is a deterministic function of time as well as the diffused underlying, quantify the extra bias in the local volatility with stochastic convenience yield, and solve it semi-analytically. Finally, we consider the case where the instantaneous volatility is a stochastic process external to the stock price, compute European option prices by deriving the characteristic function of the logarithm of the forward price both analytically and numerically, and use Malliavin calculus to derive approximation to these prices.

According to the World Economic and Social Survey 2009, the review of available estimates of mitigation and adaptation costs suggests that additional annual total investments in developing countries could be upwards of $1 trillion per year. Given the limited ability of developing countries to finance domestic mitigation, it will require large scale financing from developed countries to allow developing countries to meet the costs of climate change. However, the global sovereign debt crisis is threatening an already fragile recovery, and the cost of extra debt could lead some countries to default. The problem is most acute in Europe, where the domestic considerations of each of the 17 partners in the euro zone routinely block progress on a final solution to keep Greece, Ireland, Portugal and some other peripheral members of the euro zone from going bankrupt. Although the scale of financing required to achieve mitigation and adaptation is high, studies such as the Stern Review 2006, and Parry et al. 2009 have demonstrated that the costs of inaction by far outweigh the costs of action. Hence, identifying new and additional sources of finance to support adaptation and the transition to low-carbon development in developing countries is a crucial component of addressing global climate change. The financial system being interconnected, we introduce climate derivatives by directly modeling the annual global mean temperature, which is the triggering variable in reducing greenhouse gas emissions worldwide, and we propose a financial mechanism that directly transfer climate risks from the product issuers to the long-term investors. Climate derivatives provide a hedge to protect governments and sea-front developers against financial disruption in the aftermath of adverse climate events. They can also be used to favour the growth and development of both developed and developing countries, instead of penalising them with extra debt accounting for uncertain climate change.