Computational Finance: a Brief Overview

Financial derivative securities, such as options and futures, can be viewed as a form of insurance. These instruments are routinely used by large corporations to hedge currency fluctuations, uncertain energy costs and commodity price volatility. Of course, like any powerful tool, derivatives can also be misused. An example of this situation occurred when Enron Corporation used financial derivatives to mask balance sheet difficulties.

Many financial contracts contain embedded options. As a result, individual investors are often unaware that they frequently buy and sell options. Some examples of these embedded options include mortgage prepayment privileges, equity linked GICs, and fixed rate natural gas home heating contracts.

Populations in the developing world are getting older and living longer, meaning more people will need to live on invested capital. Consequently, management of risks in investment portfolios will become increasingly important for individual investors.

Conventional wisdom states that investment in a diversified portfolio of equities has a low risk for a long term investor. However, in a recent article ("Irrational Optimism," Financial Analysts Journal, E. Simson, P.Marsh, M. Staunton, vol. 60 (2004) 15-25) an extensive analysis of historical data of equity returns was carried out. Projecting this information forward, the authors conclude that the probability of a negative real return over a twenty year period, for an investor holding a diversified portfolio, is about 14 per cent. In fact, most individuals in defined contribution pension plans have poorly diversified portfolios. Making more realistic assumptions for defined contribution pension plans, the authors find that the probability of a negative real return over twenty years is about 25 per cent.

Incorrect accounting for risk can be blamed for the so called pension surpluses in many corporate pension plans. For example, during the late 1990's many pension plans were declared to be in a surplus position. This was based on extrapolating into the future the extraordinary market returns which occurred during the internet bubble. These accounting surpluses were then used to pay CEO bonuses, and to increase corporate earnings. However, basic financial theory states that any market return over the risk free (i.e. government bond) rate, must be accompanied by some risk (i.e. there is some chance you will lose).

In other words, by assuming a high market return for a pension plan portfolio, risk was transferred from the corporation to the holders of the pension plans. If the company experiences a shortfall in the pension plan, and ultimately declares bankruptcy, then the pensioners bear the risk. Of course, the CEO who was awarded the large bonus for discovering the pension plan surplus has retired comfortably by the time of the bankruptcy. In the article "Risk Transfer in Public Pension Plans," (J. Gold, in "The Pension Challenge: risk transfers and retirement income security," edited by O. Mitchell, K, Smetters, Oxford University Press, 2003), Gold demonstrates that if the increased risk of a higher return assumption is accounted for correctly, then these pension plan surpluses disappear.

How can we correctly value risk transfers? How much should an investor be willing to pay to insure against risk? This a problem in derivatives valuation.

A derivative contract is based on an underlying asset. The standard model for the underlying asset price movement assumes that prices evolve according to a random walk with a drift. It is possible for an option seller to set up a hedging portfolio, which is then dynamically rebalanced in response to changes in the underlying asset price. Then, regardless of the random movement of the asset price, the seller of the option is able to pay out the value of the this contract at expiry. The cost of setting up this dynamic portfolio is given by the solution of a deterministic partial differential equation (PDE).

However, there is strong evidence that the normal market behavior assumed by the standard model is punctuated by occasional large jumps or drops in prices (e.g. the Iraq war). Taking these large random jumps into account leads to a partial integro differential equation (PIDE) for the value of an option. If we assume that the market has these occasional large jumps, the standard methods for hedging do not work. The design of efficient hedging strategies which will protect against "jump risk" is a challenging aspect of current research.