Value at risk and self-similarity

The concept of Value at Risk measures the "risk" of a portfolio and is a statement of the following form: With probability q the potential loss will not exceed the Value at Risk figure. It is in widespread use within the banking industry. It is common to derive the Value at Risk figure of d days from the one of one–day by multiplying with √d. Obviously, this formula is right, if the changes in the value of the portfolio are normally distributed with stationary and independent increments. However, this formula is no longer valid, if arbitrary distributions are assumed. For example, if the distributions of the changes in the value of the portfolio are self–similar with Hurst coefficient H, the Value at Risk figure of one–day has to be multiplied by dH in order to get the Value at Risk figure for d days. This paper investigates to which extent this formula (of multiplying by √d) can be applied for all financial time series. Moreover, it will be studied how much the risk can be over– or underestimated, if the above formula is used. The scaling law coefficient and the Hurst exponent are calculated for various financial time series for several quantiles

Crash hedging strategies and worst–case scenario portfolio optimization

Crash hedging strategies are derived as solutions of non–linear differential equations which itself are consequences of an equilibrium strategy which make the investor indifferent to uncertain (down) jumps. This is done in the situation where the investor has a logarithmic utility and where the market coefficients after a possible crash may change. It is scrutinized when and in which sense the crash hedging strategy is optimal. The situation of an investor with incomplete information is considered as well. Finally, introducing the crash horizon, an implied volatility is derived

Worst-case scenario portfolio optimization: a new stochastic control approach

We consider the determination of portfolio processes yielding the highest worst-case bound for the expected utility from final wealth if the stock price may have uncertain (down) jumps. The optimal portfolios are derived as solutions of non-linear differential equations which itself are consequences of a Bellman principle for worst-case bounds. A particular application of our setting is to model crash scenarios where both the number and the height of the crash are uncertain but bounded. Also the situation of changing market coefficients after a possible crash is analyzed

On the uniqueness of unbounded viscosity solutions arising in an optimal terminal wealth problem with transaction costs

We study the uniqueness of viscosity solutions of a Hamilton-Jacobi-Bellman equation which arises in a portfolio optimization problem in which an investor maximizes expected utility of terminal wealth in the presence of proportional transaction costs. Our main contribution is that the comparison theorem can be applied to prove the uniqueness of the value function in the portfolio optimization problem for logarithmic and power utility

Worst-case optimal investment with a random number of crashes

We study a portfolio optimization problem in a market which is under the threat of crashes. At random times, the investor receives a warning that a crash in the risky asset might occur. We construct a strategy which renders the investor indifferent about an immediate crash of maximum size and no crash at all. We then verify that this strategy outperforms every other trading strategy using a direct comparison approach. We conclude with numerical examples and calculating the costs of hedging against crashes

Pricing and hedging of Asian options: Quasi-explicit solutions via Malliavin calculus

We use Malliavin calculus and the Clark-Ocone formula to derive the hedging strategy of an arithmetic Asian Call option in general terms. Furthermore we derive an expression for the density of the integral over time of a geometric Brownian motion, which allows us to express hedging strategy and price of the Asian option as an analytic expression. Numerical computations which are based on this expression are provided

CRASH HEDGING STRATEGIES AND WORST-CASE SCENARIO PORTFOLIO OPTIMIZATION

Crash hedging strategies are derived as solutions of non-linear differential equations which itself are consequences of an equilibrium strategy which make the investor indifferent to uncertain (down) jumps. This is done in the situation where the investor has a logarithmic utility and where the market coefficients after a possible crash may change. It is scrutinized when and in which sense the crash hedging strategy is optimal. The situation of an investor with incomplete information is considered as well. Finally, introducing the crash horizon, an implied volatility is derived.Optimal portfolios, crash modelling, worst-case scenario, changing market coefficients, implied volatility, crash horizon

Crash Hedging Strategies and Optimal Portfolios

In traditional portfolio optimization under the threat of a crash the investment horizon or time to maturity is neglected. Developing the so-called crash hedging strategies (which are portfolio strategies which make an investor indifferent to the occurrence of an uncertain (down) jumps of the price of the risky asset) the time to maturity turns out to be essential. The crash hedging strategies are derived as solutions of non-linear differential equations which itself are consequences of an equilibrium strategy. Hereby the situation of changing market coefficients after a possible crash is considered for the case of logarithmic utility as well as for the case of general utility functions. A benefit-cost analysis of the crash hedging strategy is done as well as a comparison of the crash hedging strategy with the optimal portfolio strategies given in traditional crash models. Moreover, it will be shown that the crash hedging strategies optimize the worst-case bound for the expected utility from final wealth subject to some restrictions. Another application is to model crash hedging strategies in situations where both the number and the height of the crash are uncertain but bounded. Taking the additional information of the probability of a possible crash happening into account leads to the development of the q-quantile crash hedging strategy.Crash Hedging Strategien und Optimale Portfolio