# Valuation of Derivatives and Portfolio Optimization

Portfolio optimization and the valuation of derivatives as well as structured products are main subjects of financial mathematics.

#### Portfolio Optimization

The main focus of portfolio optimization is the determination of an optimal investment strategy at the financial market. In fact, the investor needs to decide how many proportions of which security paper to hold in order to maximize the benefit out of the final assets in the investment horizon.

In order to face this challenge, we offer our clients the implementation of classical reference models such as:

• The Markowitz One Period Model in many different variants (Expectation value – Variance, Variance – Expectation value, client oriented extension to generic risk measures)
• The time continuous Merton Model for arbitrary many security papers/assets

Additionally, we provide extensions of these reference models, e.g.:

• Optimal investment under market-crash risk
• Optimal investment regarding transaction costs
• Realistic modeling of trading dates (discrete approximation of the Merton model, the “Relaxed Investor-Model” by Rogers)
• Optimal determination of Bond-Portfolios (even for bonds with default risk)

#### Valuation of Options

The main focus lies in the derivation of valuation formulas and the development of numeric algorithms for the calculation of prices for complex derivatives.

Derivatives are security papers with a value which depends on the development of a given commodity (e.g. stocks or interest rates). Within the trading scope of major banks option trad-ing plays an important role, especially in times of bad market conditions. In order to be able to offer attractive products with a limited risk of loss and at the same time profitable chance of winning banks employ derivatives with a very complex payout structure. On the one hand these certificates should assure investors not to experience loss (“Capital Guaranteed Products”) and on the other hand they should restrict the payment of banks to the investors in the case of winnings.

In order to value the derivatives, realistic market models are needed. Those models should be able to represent the market prices of standard products as well as to realistically model the price development of given security papers. Furthermore, efficient numerical methods are needed to valuate derivatives with complex payout profiles within these models.

We offer expertise in the field of stochastic (Heston Model) and local volatility models. In various projects with different economic partners we deployed systems to run valuations for different kind of derivatives or structured notes. Moreover, an extensive Heston-Framework has been developed which unites calibration methods and valuations of exotic options via Monte Carlo engines. In addition, an Excel connection exists to simplify the processing of market data.