BLACK-SCHOLES VISITS GHANA STOCK EXCHANGE PART II

Black-Scholes visited GSE in the first part of this write-up on 22 Jun 2009. But the respected opinion of others in the field made it difficult for BS to do any proper simulation. This simulation was to be in this second part but it is still deferred to another update of this topic. For now we consider the conditions necessary for BS to land properly on GSE corridors.

These series of topics are necessary not for their book value but because it is a contribution to the development of a world class GSE. Already, it is reported that GSE is in the process of a major transformation, viz. introduction of derivative securities on GSE. Additionally, the financial market is now trading with very complex hedging strategies, so this series is my contribution to the discussion.

BS is actually based on the geometric Brownian motion model for stock prices. Stock price changes are random and follow the path walked by a drunkard or of a pollen grain suspended in liquid. Interestingly one variable not necessary for valuation using BS is the mean returns. And for the insight that it is not necessary to use any risk premium in valuation, the inventors of BSM were awarded the Nobel Price. Black died earlier so the award was shared by Scholes and Merton.

BS is suitable for European-style exercise terms. But most derivative securities the world over are traded in American-style exercise terms. This is possibly due to their flexibility. The American-style is rather limited by the early exercise problem. The application of BS to such instruments then led to a decomposition of an American put price as the sum of the European put (using BS) plus an early exercise premium.

Under BS, markets are efficient. Markets are not to be consistently predictable. The Bank of Ghana Financial market Department in 2001 discovered that GSE stock prices followed an inverse pattern and so became unattractive. This is obvious since these were prices for pure stock. Stock price changes, are usually known to follow a random pattern. Certainly if the price follows an inverse pattern then it is not profitable and as well predictable. So BS and Ito process generalizations break down. A popular belief is that “it is a sign of a well-functioning stock market that movements in stock prices are unpredictable”.

In another report, GSE annual real returns followed an undulating pattern since 1991, falling every two years and rising the following two years. In this report it was shown that the stock market is more sensitive to inflation than the Treasury bill market. It rightly stated that lower inflation is expected to result in higher real returns on stocks. Another popular belief is that, the larger the fluctuations (i.e. higher values of vola) the harder it is to predict the future value of an asset.

Interest rates and volatility are supposed to remain constant and known in the BS model. The risk-free rate in Ghana is often quoted on Government Bonds. In the report of BOG Fin Markets Dept in 2001, standard deviation on stocks for a ten-year period was 53% for nominal returns and 41.25% for real returns on stocks.

In the BS model, no commissions are charged. On GSE a small commission is paid. On GSE, listed companies pay high dividend because they earn high profits after tax. This requires BS to be modified. Merton's formula is rather required.

Finally, BS requires that changes in log-returns are normally distributed. Interestingly, the prices of stocks cannot be simulated using risk-neutral pricing technique, what is actually to be simulated with risk-neutral pricing technique is the changes in the prices of stocks. The simulated average change is used to discount the theoretical value of the stock prices. I made it easier for myself by looking at the drift and diffusion terms as equivalent to capital cost rate and its fluctuation. This means the BS drift plus or minus diffusion represent the capital cost rate to be used to discount the stock price. This procedure yields the lower as well as upper bounds of the price. If so then BS can be used to price anything including the price of electricity in Ghana.

To use the BS, I defined the boundary conditions for GSE stock prices, as the continuously discounted value of the current stock price with respect to a change in time. Solving for the function of the stock price process, I discovered it does not satisfy the BSM partial differential equation. As such my defined boundary condition cannot be the price of a derivative. This is already known.

For now to use the BSM to valuate the stock price on GSE, I assume that BSM can be used just like GBM or even CAPM or any adapted interest rate process or money market process to define the path taken by the price change process.

This discussion of the visit of BS on GSE is to be taken to a forum on yahoo groups. Interested Ghanaian Quants or to-be-Quants are invited to write to me to enable them contribute to the discussion in a more efficient manner.

Paul A. Agbodza
(Atikpui via Ho).