Taming Risks: Financial Models and Numerics
by Jiri Hoogland and Dimitri Neumann
The increasing complexity of the financial world and the speed at which markets respond to world-events requires both good models for the dynamics of the financial markets as well as proper means to use these models at the high speed required in present-day trading and risk-management. Research at CWI focuses on the development of models for high-frequency data and applications in option-pricing, and tools to allow fast evaluation of complex simulations required for option-pricing and risk-management.
The modeling of equity price movements already started in 1900 with the work of Bachelier, who modeled asset prices as Brownian motion. The seminal papers by Merton, Black, and Scholes, in which they derived option prices on assets, modeled as geometric Brownian motions, spurred the enormous growth of the financial industry with a wide variety of (very) complex financial instruments, such as options and swaps. These derivatives can be used to fine-tune the balance between risk and profit in portfolios. Wrong use of them may lead to large losses. This is where risk-management comes in. It quantifies potentially hazardous positions in outstanding contracts over some time-horizon.
Option pricing requires complex mathematics. It is of utmost importance to try to simplify and clarify the fundamental concepts and mathematics required as this may eventually lead to simpler, less error-prone, and faster computations. We have derived a new formulation of the option-pricing theory of Merton, Black, and Scholes, which leads to simpler formulae and potentially better numerical algorithms.
Brownian motion is widely used to model asset-prices. High-frequency data clearly shows a deviation from Brownian motion, especially in the tails of the distributions. Large price-jumps occur in practice more often than in a Brownian motion world. Thus also big losses occur more frequently. It is therefore important to take this into account by more accurate modeling of the asset-price movements. This leads to power-laws, Levy-distributions, etc.
Apart from options on financial instruments like stocks, there exist options on physical objects. Examples are options to buy real estate, options to exploit an oil-well within a certain period of time, or options to buy electricity. Like ordinary options, these options should have a price. However, the writer of such an option (the one who receives the money) usually cannot hedge his risk sufficiently. The market is incomplete, in contrast with the assumptions in the Black-Scholes model. In order to attach a price to such an option, it is necessary to quantify the residual risk to the writer. Both parties can then negotiate how much money should be paid to compensate for this risk. We explore ways to partially hedge in incomplete markets.
A relatively new phenomenon in the financial market has been the introduction of credit risk derivatives. These are instruments which can be used to hedge against the risk of default of a debitor. It is obvious that this kind of risk requires a different modeling approach. The effect of default of a firm is a sudden jump in the value of the firm and its liabilities, and should be described by a jump process (for example, a Poisson-process). In practice, it is difficult to estimate the chance of default of some firm, given the information which is available. For larger firms, credit-worthiness is assessed by rating agencies like Standard and Poors. We are looking at methods to estimate and model the default risk of groups of smaller firms, using limited information.
The mathematics underlying financial derivatives has become quite formidable. Sometimes prices and related properties of options can be computed using analytical techniques, often one has to rely on numerical schemes to find approximations. This has to be done very fast. The efficient evaluation of option prices, greeks, and portfolio risk-management is very important.
Many options depend on the prices of different assets. Often they allow the owner of the option to exercise the option at any moment up to the maturity of the (so-called) American-style option. The computation of prices of these options is very difficult. Analytically it seems to be impossible. Also numerically they are a tough nut to crack. For more than three underlying assets it becomes very hard to use tree or PDE methods. In that case Monte Carlo methods may provide a solution. The catch is that this is not done easily for American-style options. We are constructing methods which indirectly estimate American-style option prices on multiple assets using Monte Carlo techniques.
Monte Carlo methods are very versatile as their performance is independent of the number of underlying dynamic variables. They can be compared to gambling with dice in a casino many, many times, hence the name. Even if the number of assets becomes large, the amount of time required to compute the price stays approximately the same. Still the financial industry demands more speedy solutions, ie faster simulation methods. A potential candidate is the so-called Quasi-Monte Carlo method. The name stems from the fact that one gambles with hindsight (prepared dice), hence the ‘Quasi’. It promises a much faster computation of the option-price. The problems one has to tackle are the generation of the required quasi-random variates (the dice) and the computation of the numerical error made. We try to find methods to devise optimal quasi-random number generators. Furthermore we look for simple rules-of-thumb which allow for the proper use of Quasi-Monte Carlo methods.
For more information see
http://dbs.cwi.nl:8080/cwwwi/owa/cwwwi.print_themes?ID=15
Please contact:
Jiri Hoogland or Dimitri Neumann - CWI
Tel: +31 20 5924102
E-mail: {jiri, neumann}@cwi.nl
