Discounted Optimal Stopping Problems for Maxima of Geometric Brownian Motions With Switching Payoffs

We present closed-form solutions to some discounted optimal stopping problems for the running maximum of a geometric Brownian motion with payoffs switching according to the dynamics of a continuous-time Markov chain with two states. The proof is based on the reduction of the original problems to the equivalent free-boundary problems and the solution of the latter problems by means of the smooth-fit and normal-reflection conditions. We show that the optimal stopping boundaries are determined as the maximal solutions of the associated two-dimensional systems of first-order nonlinear ordinary differential equations. The obtained results are related to the valuation of real switching lookback options with fixed and floating sunk costs in the Black–Merton–Scholes model.

Hedging of Variable Annuities under Basis Risk

I study dynamic hedging for variable annuities under basis risk. Basis risk, which arises from the imperfect correlation between the underlying fund and the proxy asset used for hedging, has a highly negative impact on the hedging performance. In this paper, I model the financial market based on correlated geometric Brownian motions and analyze the risk management for a pool of stylized GMAB contracts. I investigate whether the choice of a suitable hedging strategy can help to reduce the risk for the insurance company. Comparing several cross-hedging strategies, I observe very similar hedging performances. Particularly, I find that well-established but complex strategies from mathematical finance do not outperform simple and naive approaches in the context studied. Diversification, however, could help to reduce the adverse impact of basis risk.

CREDIT DEFAULT SWAPS IN TWO-DIMENSIONAL MODELS WITH VARIOUS INFORMATIONS FLOWS

We study a credit risk model of a financial market in which the dynamics of intensity rates of two default times are described by linear combinations of three independent geometric Brownian motions. The dynamics of two default-free risky asset prices are modeled by two geometric Brownian motions which are dependent of the ones describing the default intensity rates. We obtain closed form expressions for the no-arbitrage prices of both risk-free and risky credit default swaps given the reference filtration initially and progressively enlarged by the two default times. The accessible default-free reference filtration is generated by the standard Brownian motions driving the model.

ON BOUNCING GEOMETRIC BROWNIAN MOTIONS

A pair of bouncing geometric Brownian motions (GBMs) is studied. The bouncing GBMs behave like GBMs except that, when they meet, they bounce off away from each other. The object of interest is the position process, which is defined as the position of the latest meeting point at each time. We study the distributions of the time and position of their meeting points, and show that the suitably scaled logarithmic position process converges weakly to a standard Brownian motion as the bounce size δ→0. We also establish the convergence of the bouncing GBMs to mutually reflected GBMs as δ→0. Finally, applying our model to limit order books, we derive a simple and effective prediction formula for trading prices.

ON MEAN–VARIANCE HEDGING UNDER PARTIAL OBSERVATIONS AND TERMINAL WEALTH CONSTRAINTS

In this paper, a mean-square minimization problem under terminal wealth constraint with partial observations is studied. The problem is naturally connected to the mean–variance hedging (MVH) problem under incomplete information. A new approach to solving this problem is proposed. The paper provides a solution when the underlying pricing process is a square-integrable semi-martingale. The proposed method for study is based on the martingale representation. In special cases, the Clark–Ocone representation can be used to obtain explicit solutions. The results and the method are illustrated and supported by examples with two correlated geometric Brownian motions.