Short term climate events such as the sea surface temperature anomaly known as El Nino are financial risk sources leading to incomplete markets. To make such risk tradable, we use a market model in which a climate index provides an extra investment opinion. Given one possible market price of risk each agent can maximize the exponential utility from three sources of income: capital market, additional security, and individual risk exposure. Under an equilibrium condition the market price of risk is uniquely determined by a backward stochastic differential equation. We translate these stochastic equations into semi-linear partial differential equations for the simulation of which numerical schemes are available. We choose two simple models for sea surface temperature, and with ENSO risk exposed fisher and farmer and a nonh-exposed bank three toy agents. By simulating their optimal investment into the climat index we obtain first insight into the dynamics of the market.
We consider the problem of utility maximization for small traders on incomplete
financial markets. As opposed to most of the papers dealing with this
subject, the investors’ trading strategies we allow underly constraints described
by closed, but not necessarily convex, sets. The final wealths obtained by trading
under these constraints are identified as stochastic processes which usually are
supermartingales, and even martingales for particular strategies. These strategies
are seen to be optimal, and the corresponding value functions determined
simply by the initial values of the supermartingales. We separately treat the
cases of exponential, power and logarithmic utility.
We consider financial markets with agents exposed to an external source of
risk which cannot be hedged through investments on the capital market alone.
The sources of risk we think of may be weather and climate. Therefore we face
a typical example of an incomplete financial market. We design a model of a
market on which the external risk becomes tradable. In a first step we complete
the market by introducing an extra security which valuates the external risk
through a process parameter describing its market price. If this parameter is
fixed, risk has a price and every agent can maximize the expected exponential
utility with individual risk aversion obtained from his risk exposure on the one
hand and his investment into the financial market consisting of an exogenous set
of stocks and the insurance asset on the other hand. In the second step, the
market price of risk parameter has to be determined by a partial equilibrium
condition which just expresses the fact that in equilibrium the market is cleared
of the second security. This choice of market price of risk is performed in the
framework of nonlinear backwards stochastic differential equations.
Equilibrium trading of climate and weather risk and numerical simulation in a Markovian framework
(2004)
We consider financial markets with agents exposed to external sources of risk
caused for example by short term climate events such as the South Pacific sea
surface temperature anomalies widely known under the name El Nino. Since
such risks cannot be hedged through investments on the capital market alone,
we face a typical example of an incomplete financial market. In order to make
this risk tradable, we use a financial market model in which an additional insurance
asset provides another possibility of investment besides the usual capital
market. Given one of many possible market prices of risk each agent can maximize
his individual exponential utility from his income obtained from trading in
the capital market, the additional security, and his risk exposure function. Under
the equilibrium market clearing condition for the insurance security the market
price of risk is uniquely determined by a backward stochastic differential equation.
We translate these stochastic equations via the Feynman-Kac formalism
into semi-linear parabolic partial differential equations. Numerical schemes are
available by which these semilinear pde can be simulated. We choose two simple
qualitatively interesting models to describe sea surface temperature, and with
an ENSO risk exposed fisher and farmer and a climate risk neutral bank three
model agents with simple risk exposure functions. By simulating the expected
appreciation price of risk trading, the optimal utility of the agents as a function
of temperature, and their optimal investment into the risk trading security we
obtain first insight into the dynamics of such a market in simple situations.