# Abstracts

**Optimality and robustness of rule-based trigger strategies**

*Ruediger Kiesel (University Duisburg-Essen, Germany)*

Thursday June 5, 14:00-14:30 | session 8.2 | Portfolio Optimization | room CD

The focus of the paper is on the performance of proportional insurance (PPI) strategies under transaction costs which impede continuous time trading. Instead of deriving optimal trade and no trade regions, we restrict ourselves to tractable rule based trigger strategies. A meaningful objective which triggers the portfolio rebalancing allows then the derivation of the growth optimal strategy in quasi-closed form, i.e. in a Black and Scholes model setup and proportional transaction costs. We consider finite and infinite investment horizons and compare the performance of the growth optimal 'rule-based' trigger strategy with the one of the overall optimal strategy. In addition, we consider deviations from the Black and Scholes model setup. In particular, we give robustness results for stochastic volatility but also account of gap risk, i.e. the risk that the guarantee is not honored.

**Robust Long-Term Interest Rate Risk Hedging in Incomplete Bond Markets**

*Sally Shen (Maastricht University, The Netherlands)*

Thursday June 5, 14:30-15:00 | session 8.2 | Portfolio Optimization | room CD

Pricing and hedging long dated liabilities faces two challenges. On one hand, the market for long maturity financial instrument is missing. The longest government bonds even in developed financial markets (such as US and Germany) has maturities no longer than 30 year. However, pension liabilities are well defined future cash flow obligations with more than 50 years of maturities. It is regulated by Solvency II that pension and insurance companies should follow a ``market-consistent'' principle to value their liabilities. Here comes the puzzle: how to implement ``market - consistent '' valuation on those commitments with maturities longer than 30 years; or how to discount the far future cash flows with the absence of a complete bond market?

On the other hand, extrapolating term structure model is exposed to parameter misspecification. Any valuation of a long dated liability with missing bond market must be model based. Conditional on a model, we could derive a term structure for all maturities. However, there are a large number of models that perfectly fit bond prices up to maturity of 30 years but yet imply different prices for the non-traded-period bonds.

Despite much work on the term structure models, very few studies consider the impact of model uncertainty on long dated liability valuation. In this paper, we propose a robust optimal hedging policy that minimize the hedging error of long dated liabilities in the presence of parameter uncertainty and missing bond market. Our replicating portfolio is robust to model misspecification in the sense that the investment policy is less sensitive to the choice of models.

We solve a dynamic robust optimization problem that maximin agent's utility. On one hand, agent allocates her instantaneous wealth between a short-term and a median-term bond market so as to minimize the expected shortfall of a long maturity commitment. On the other hand, Mother nature perturbs the estimate parameters in order to maximize the expected shortfall given the decision of the agent. The equilibrium portfolio is therefore, robust against model ambiguity. We use GMM approach to estimate the one-factor affine term structure model. Then we employ Least Square Monte Carlo method to solve the backward stochastic differential equation numerically.

We find that both naive and robust optimal portfolios depend on the hedging horizon and current funding ratio. The robust policy suggests to take more risk when the current funding ratio is low, but is another way around when funding ratio is sufficiently high. The robust yield curve derived through the minimum assets required to eliminate shortfall risk is lower than the naive one. For long-term investor, the robust policy is likely to outperform the naive one when bond premia is over estimated.

**Optimal Investment in a Black–Scholes Model with a Bubble**

*Sebastian Herrmann (ETH Zürich, Switzerland)*

Thursday June 5, 15:00-15:30 | session 8.2 | Portfolio Optimization | room CD

Financial bubbles seem to exhibit a strong upward trend followed by a sharp decline when the bubble bursts. Starting from this descriptive perspective, we propose an extension of the Black–Scholes model which accommodates for this effect. In addition to the standard instantaneous expected return in the Black–Scholes model, we allow for a time-dependent positive excess return which is compensated for by a negative jump at a random time representing the bursting of the bubble. Since the excess return and the distribution of the random time may be chosen almost arbitrarily, the model is flexible enough to display different types of bubbles, including the ones in the sense of Protter (2013). Moreover, the model is tractable enough to allow for (semi-)explicit calculations and may thus be used as a toy example to study qualitative effects of financial bubbles.

We study the problem of maximising expected utility from terminal wealth for a power utility investor. Using the convex duality approach, we determine the optimal strategy and the corresponding certainty equivalent up to the solution to an integral equation (or a first-order ODE). A decomposition of the optimal strategy into the myopic and hedging demand allows to analyse the effects of the stochastic investment opportunities. On the one hand, investors with relative risk aversion below 1 speculate on an early bursting of the bubble in the sense that their optimal strategy lies below their myopic demand prior to the bursting of the bubble; the optimal strategy might even involve short-selling. On the other hand, investors with relative risk aversion above 1 hedge against a late bursting of the bubble in the sense that their optimal strategy lies above their myopic demand; the optimal strategy might even lie above the Merton proportion under extreme circumstances.

Numerical examples reveal how the optimal strategy and its myopic and hedging demand depend on the model parameters. In particular, it is shown that the optimal strategy is not fundamentally different when the stock price process is a strict local martingale (as opposed to a true martingale) under a certain class of equivalent local martingale measures including the dual minimiser of the utility maximisation problem. In addition, we illustrate the welfare loss compared the the classical Black–Scholes model and its dependence on the model parameters.

**Life-Cycle Asset Allocation under Stochastic Interest-Rates and Stock-Return Predictability**

*Thomas Leirvik (Yale University and University of Nordland, USA/Norway)*

Thursday June 5, 15:30-16:00 | session 8.2 | Portfolio Optimization | room CD

I analyze the optimal allocation of wealth to cash, bonds, and stocks when the interest rate is stochastic and the stock index has a time-varying mean. I find that, under certain economic conditions, the investor may optimally increase investments in stocks and bonds at the same time, which is due to the dynamic trading policies and the correlation between the asset classes. I also find that in different economic regimes, short-term investors have very different investment policies than long-term investors. Thus, dynamic asset allocation with nonzero bond-stock correlation helps explain why, during extreme market conditions such as the recent financial crisis, some investors sold all types of assets short, whereas other investors considered it an unprecedented buying opportunity. In order to avoid the dimensionality problem of the partial differential equation that corresponds to the framework I use, I will apply a method that is similar to former results in the literature, see for example Zariphopolou (1999, Mathematical Methods for Operations Research) and Benth and Karlsen (2001, International Journal of Applied and Theoretical Finance), but is slightly more complex, as I consider a portfolio of both cash, bonds, and stocks with a nonzero correlation structure. Despite this, I give an analytical solution to the indirect utility function as well as the optimal trading policy, proved via a verification theorem.