Oberseminar Finanz- und Versicherungsmathematik

We study the use of neural networks as nonparametric estimation tools for the hedging of options. To this end, we design a network, named HedgeNet, that directly outputs a hedging strategy given relevant features as input. This network is trained to minimise the hedging error instead of the pricing error. Applied to end-of-day and tick prices of S&P 500 and Euro Stoxx 50 options, the network is able to reduce the mean squared hedging error of the Black-Scholes benchmark significantly. We illustrate, however, that a similar benefit arises by a simple linear regression model that incorporates the leverage effect. (Joint work with Weiguan Wang).

Liquidity risk is typically added exogenously to a market price process. This is conceptually unsatisfying. We build a model, which integrates liquidity risk into the market price process. In particular, we add a liquidity (jump) component to the standard geometric Brownian motion and show that this approach models market prices better than without the liquidity component. Since long positions have to be liquidated at the bid price, we model bid and ask price individually. We verify our model with 50 million bond price data. We suggest that this model should underlie long positions in risk management approaches such as VaR (Value at Risk), ES (Expected Shortfall) and EVT (Extreme Value Theory). The talk is based on a joint work with Robert Engle and Anna van Elst.

We develop a simple and flexible technique to price executive stock options (ESOs) with vesting periods and liquidation penalties. The vesting period implies that the ESO is activated when a designed performance measure triggers a prespecified barrier. The performance measure is usually an accounting figure, such as the ROE or the EBITDA, normally correlated with the stock price. Once the option is activated, the holder has the right to buy the stock whenever she wants during the residual life of the option. The bivariate strutucture of the ESO, whose payoff depends jointly on the performance measure and the stock, makes usual lattice techniques difficult to apply. We first reduce the ESO to a compound forward-starting American call option on the stock. We then show how to evaluate the ESO option by means of an intuitive hybrid method that uses simulation to determine the bivariate distribution of the foward-starting date of the option and the corresponding price of the stock, and lattice techniques to retrieve the initial value of the activated call option.
Liquidation penalties are common in ESOs, aiming at lowering the chances of selling the ESOs and the underlying company shares. We show that the presence of even mild liquidation penalties triggers the existence of optimal exercise opportunities for the ESOs that are absent when the option can be fully liquidated.

Joint with M. De Donno and Alessandro Sbuelz

We propose a simple lifecycle strategy entailing contributions made during accumulation being invested entirely into a risky portfolio until pre-specified ‘switch age’ and then entirely into a risk-free portfolio after the switch age, followed by withdrawing during decumulation from both portfolios based on annuitization rates that vary with age according to remaining life expectancy. First, we show analytically that the strategy is optimal for range of investors with HARA risk preferences, and derive the dynamics of the investment strategy. Second, we show numerically that the proposed strategy delivers limited loss of utility versus an optimal solution for investors with CRRA preferences and low risk aversion, while significantly outperforming strategies commonly used in practice. The proposed strategy offers an attractive alternative for use in practical settings as it is simple to follow and removes the need for portfolio rebalancing.

We study two aspects of making optimal investment decisions for pension investors in the savings phase. First, we explore the impact of an investor’s perception towards inflation risk on their investment strategy. We find that mis-specifying inflation risk reduces the expected utility of the risk averse investors, and more risk averse investors face larger reductions. For investors who adopt terminal wealth constraints (e.g. minimum guarantee), ignoring inflation results in real wealth not adhering to the real constraints. The conclusion is that investors ignore inflation at their peril. Secondly, we compare the retirement outcomes derived from the risk averse and loss averse utility functions. We use a numerical dynamic programming approach and a model that includes ongoing pension contributions to savings, prohibits short-selling and borrowings, and, when applicable, includes wealth constraints. We find that the loss averse utility function, without wealth constraints, naturally results in a more favorable retirement income distribution that peaks at the investor's chosen income goal with some level of robustness. We conclude that the investor can benefit from adopting a loss aversion-derived optimal investment strategy to target a sufficient level of income at retirement.