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This paper is concerned with the study of insurance related derivatives on financial markets that are based on non-tradable underlyings, but are correlated with tradable assets. We calculate exponential utility-based indifference prices, and corresponding derivative hedges. We use the fact that they can be represented in terms of solutions of forward-backward stochastic differential equations (FBSDE) with quadratic growth generators. We derive the Markov property of such FBSDE and generalize results on the differentiability relative to the initial value of their forward components. In this case the optimal hedge can be represented by the price gradient multiplied with the correlation coefficient. This way we obtain a generalization of the classical ‘delta hedge’ in complete markets.
We consider Backward Stochastic Differential Equations (BSDEs) with generators that grow quadratically in the control variable. In a more abstract setting, we first allow both the terminal condition and the generator to depend on a vector parameter x. We give sufficient conditions for the solution pair of the BSDE to be differentiable in x. These results can be applied to systems of forward-backward SDE. If the terminal condition of the BSDE is given by a sufficiently smooth function of the terminal value of a forward SDE, then its solution pair is differentiable with respect tot the initial vector of the forward equation. Finally we prove sufficient conditions for solutions of quadratic BSDEs to be differentiable in the variational sense (Malliavin differentiable).
We extend some recent works by Delong and Imkeller concerning Backward
stochastic differential equations with time delayed generators (delay
BSDE). We provide sharper a priori estimates and show that the solution
of a delay BSDE is in $L^p$. We introduce decoupled systems of SDE and
delay BSDE (which we term delay FBSDE) and give sufficient conditions
for the variational differentiability of their solutions. We connect
these derivatives to the Malliavin derivatives of such delay FBSDE via
the usual representation formulas which in turn give access to several
path regularity results. In particular we prove an extension of the
$L2$-path regularity result for delay FBSDE.
We consider backward stochastic differential equations with drivers of quadratic growth (qgBSDE). We prove several statements concerning path regularity and stochastic smoothness of the solution processes of the qgBSDE, in particular we prove an extension of Zhang's path regularity theorem to the quadratic growth setting. We give explicit convergence rates for the difference between the solution of a qgBSDE and its truncation, filling an important gap in numerics for qgBSDE. We give an alternative proof of second order Malliavin differentiability for BSDE with drivers that are Lipschitz continuous (and differentiable), and then derive the same result for qgBSDE.
We consider the problem of numerical approximation for forward-backward stochastic
differential equations with drivers of quadratic growth (qgFBSDE). To illustrate the significance
of qgFBSDE, we discuss a problem of cross hedging of an insurance related financial
derivative using correlated assets. For the convergence of numerical approximation schemes for
such systems of stochastic equations, path regularity of the solution processes is instrumental.
We present a method based on the truncation of the driver, and explicitly exhibit error estimates
as functions of the truncation height. We discuss a reduction method to FBSDE with globally
Lipschitz continuous drivers, by using the Cole-Hopf exponential transformation. We finally
illustrate our numerical approximation methods by giving simulations for prices and optimal
hedges of simple insurance derivatives.
We consider backward stochastic differential equations with drivers of quadratic growth (qgBSDE). We prove several statements concerning path regularity and stochastic smooth-
ness of the solution processes of the qgBSDE, in particular we prove an extension of Zhang's path regularity theorem to the quadratic growth setting. We give explicit convergence rates for the difference between the solution of a qgBSDE and its truncation, filling an important gap in numerics for qgBSDE. We give an alternative proof of second order Malliavin differentiability for BSDE with drivers that are Lipschitz continuous (and differentiable), and
then derive the same result for qgBSDE.