In particular, the Humboldt Current along the South American coast may be affec- ted. It normally transports cold water northward. The effects of this change of the sea surface temperature on marine life are tremendous. The trade wind shift disrupts the upwelling of oxygen and nutrient rich cold water, one of the basic conditions for dense concentrations of marine life.

Let us give some numbers first for the local effect on the Peruvian fishing industry. As in most developing countries in the tropics with economies depending largely on few branches for example in food pro- duction, the sensitivity to climatic fluctuations is very high.

The consequences of this distortion of ocean currents due to changes in the SOI are much more global than one may conjecture at first glance. The change of the tropical Pacific sea surface temperatures induced by the fluctuation of trade winds affects the atmosphere in turn directly by causing convection. Dense tropical rain clouds are created which, besides increasing the amount of precipitation the western hemisphere receives, distort the atmos- pheric air flow in altitudes of 5—10 km above sea level.

The climatic effects of ENSO create globally and even locally groups of possible agents on markets which are affected in a very different, sometimes even complementary, way. Let us illustrate this by giving some examples. Another pair of groups of economic agents with complementary interests is given by farmers and fishers even in the same national economy.

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For ex- ample, rice and cotton, two of the primary crops grown in Northern Peru, are highly sensi- tive to the quantities and timing of rainfall. In our case this is the random process describing the sea surface temperature in the Southern Pacific. It is usual- ly modeled as a one-dimensional stochastic process.

There are around 15 reduced models for ENSO, of which we briefly sketch 3. For example, the stochastic differential equation model by Barcilon, Fang and Wang describes a nonlinear interaction between two physical quantities: the thermocline depth in some area of the South Pacific, i. The system turns out to be an autonomous nonlinear stochastic oscillator which in some parameter regimes acts as a stochastically perturbed bistable differential equation with an intrinsically defined periodicity.

Typical trajectories of the temperature component K relevant for our purposes show stochastic bi-stable behavior see Figure 5. Most of the time they fluctuate in the vicinity of one or the other of two meta-stable equi- libria, interrupted by spontaneous rapid transitions between the domains of attraction of these states. The transitions follow some randomly periodic pattern, which actually is the reason why they are considered as depicting the qualitative behavior of the temperature evolution of the Pacific sea surface characteristic for ENSO.

Equation SDE driven by a Brownian motion.

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It describes the motion of a state variable traveling through a bi-stable potential landscape, with an explicit periodic dependence of the potential shape creating a non-autonomous stochastic system. This equation is obtained from an empirical orthogonal function development fitted to time series of observations over several decades of recent climate history.

This fitting shows that 10—20 orthogonal functions are needed, which leads to the dimensionality of the linear Ornstein-Uhlenbeck type stochastic equation. It appears to be the reduced model most frequently used for ENSO predictions. Mathematically it creates a diffusion with nontrivial rotation numbers implying random periodicity for the sea surface temperature variable K which can be gen- erated through the vector X. To retain the qualitative features of the development of K for simulation purposes, we may describe it as a simple mean-reverting linear SDE with an additional deterministic periodic forcing.

A third, very simple but equally interesting qualitative example was discussed by Suarez et al. The delay effect in this equation may also be interpreted alternatively. After setting up a simple financial market model to deal with distributing ENSO risk in a way considered as optimal by the individual agents interested in trading it, we do numeri- cal simulations of their optimal investment strategies. The climate component we use in these simulations, besides mean-reverting Ornstein-Uhlenbeck processes, will be the above mentioned one-dimensional non-autonomous stochastic hopping between two meta-stable states of a diffusion travelling in a simple potential landscape with two wells, the relative depth of which alternates periodically.

This model constitutes the paradigm of stochastic resonance. See Herrmann et al. The main advantage of this model is that the market is ideally composed of agents with complementary interests. In fact, climate events usually are em- bedded into nonlinear models, for which the frequently used linear prediction mechanisms have considerable shortcomings. At this place, we should mention that our model is able to deal with any risk source located in the exterior of a classical stock market.

Our agents are typically exposed to a climate process K described by one of the simple models discussed in the previous section. For our simulations we shall use the phenome- nological randomly periodic bi-stable temperature process. The agents composing our market are allowed to have three sources of income, and trade continuously within a time interval [0, T].

Think of W 1 as describing the uncertainty inside a usual stock market, while W 2 is the driving uncertainty in the cli- mate process K. Firstly, the agents can trade on a financial market.

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In the simplest possible case which we are going to sketch, the financial market consists of a bond and a risky asset. We take the bond as a trivial constant process with interest rate 1, and assume that the stock price pro- cess X is of the type of a geometric Brownian motion. In the simplest case, we only think of shuffling climate risk within the market of affected agents. For our simulations in this set- ting, we consider a toy market with two or three toy agents, represented by ENSO affected farmer farming company and fisher fishing company , plus eventually a bank not directly exposed to climate.

Consequently, three typical types of qualitative risk exposure will be considered. Recall our model climate process K showing randomly periodic bistable behavior by hopping between a low K 1 and a high K 2 meta-stable state. The former one corresponds to usual conditions for sea surface temperature of the Southern Pacific, while the second one represents ENSO conditions. So the fisher may have his temperature of op- timal income near the lower equilibrium, while the farmer might profit more from higher precipitation rates at the higher temperature equilibrium.

This in particular means that the fisher profits from temperature values under which the farmer suffers most, and vice ver- sa. The exposure of the bank is taken to be independent of K. More formally, we arrive at the following simple exposure models. The rice farmer is taken to have an exposure of the same type as the fisher. The optimal income is just obtained at K 2, the second meta-stable point of a bi-stable process K.

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As an additional agent, we consider a bank b whose profits only come from its portfolio management from investment on the financial market, and which participates in the clima- te risk share only to diversify its portfolio. Due to the presence of the uncertainty factor W 2 which is independent of the uncertainty factor W1 governing the stock market, the risks represented by H a cannot be hedged on the stock market alone: we face a typical incomplete market situation.

We deal with this prob- lem by a technique of market completion. We introduce and add to the market a security Y, through which climate risk becomes tradable and which therefore acts as the third source of income. Agents active on the market may buy or sell individual amounts of this climate index according to their random risk exposures.

The price process for this climate index of course has to be driven by the climate uncertainty. Of course, this price is not fixed from the beginning, but has to be determined by the dynamics of the market, i. We can proceed in two steps to achieve such an equilibrium. This will lead him to an individual optimal investment strategy.

We describe the preferences by an expo- nential utility function with individual risk aversion, and assume that the agent optimizes this utility from terminal wealth obtained through his three sources of income. Hence in our model the external risk dy- namically determines in a unique way the market price of risk of the climate index Y via the risky incomes, the preferences and the partial market clearing condition.

This basically means that the wealth processes, resulting from investment along these strategies, are well defined. Now a tends to maximize his terminal wealth measured by exponential utility. As shown in Hu et al. It is shown in Hu et al. Conversely, if the market has a partial equilibrium, is is given by a unique solution of the BSDE 7.

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