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1.5.1 Drift and volatility of market prices

[ SLIDE price history - drift - volatility - forecast || VIDEO modem - LAN - DSL ]

Have a look first at (1.5.1#fig.1, top), which shows the price of the Cisco share quoted on NASDAQ every trading day between 1994 and 2004.

Figure 1.5.1#fig.1: The upper plot shows the historical price (adjusted for splits) of the Cisco share during 10 years as a function of the trading day. On the bottom, the corresponding volatility calculated using the EWMA/ $ \lambda=0.94$ model (blue line) is displayed in a comparison with the volatility measured by the Chicago Board of Exchange (green circles).
\includegraphics[width=9cm]{figs/csco.eps}
After a prolonged period of exponential growth from USD 1 in Jul 94 to USD 80 in Dec 1999, the price drops by $ \sim$ 70% following a sector-wide correction of technology shares during the year 2000. The repercussions from the attack Sep 11, 2001 on the world trade center (WTC) are also visible, but led only to a temporary $ \sim$ 25% drop in the share price.

Even if it possible to measure a 50-100% annual drift in the share price up to the year 2000, this does not reflect the real growth of the company and was clearly not sustainable. Rather than using spot prices, drifts should be estimated using more fundamental analyses (such as the number of employees and customers) keeping in mind that, in the long term, it is hard to beat the 7-11% growth observed over a century in the American stock market.

How does the volatility in (1.5.1#fig.1, bottom), updated after every trading day using only information from the past, reflect the financial risk that can be judged a posteriori? To answer this question, note first that the long term average volatility of around 40% per annum does not really depend on the actual price of the share: the volatility only shows that typical gains or losses of at least 40% can be expected during any year under consideration. The volatility jumps to even higher values immediately AFTER every significant change in the share price, both on the way up and on the way down: a large movement of the price reflects the uncertainty of the investors, who are unsure if the amplitude of the change is exaggerated or if it should be even larger. For example, the volatility was large ($ \sim$ 100%) at the end of the year 2000 during the whole period when the price kept falling, but it was also large after the WTC attack when the prices recovered within only a couple of weeks. Clearly, the volatility cannot be used to forecast whether a spot price will rise or fall, but gives a good idea by how much the price may move in either direction: this is indeed the measure of risk we are seeking. In a word of caution, note that the volatility of a spot price is not a value that can be directly observed: the next section will show how different models produce different values, so that it can be misleading to use data from the Internet without knowing how it has been calculated.

Figure 1.5.1#fig.2: Forecast price of a share during 10 years in a simulation starting at $ S_0=1$ with an annual drift $ \bar{\mu}=0.15$ and a volatility $ \bar{\sigma}=0.2$ . Thirty possible realizations have been plotted (in green) with no particular one highlighted (in blue) to illustrate how the price spreads in time around $ S_\mathrm{mean}=S_0\exp[(\mu-\sigma^2/2)t])$ , but remain within the interval $ \exp[\log(S_\mathrm{mean})\pm 1.96\sigma\sqrt{t}]$ where 95% of all the realizations are found (in red).
\includegraphics[width=9cm]{figs/price.eps}

Even if there is no guarantee that a performance from the past will be repeated in the future, the historical values of a price are often used in Monte-Carlo simulations (experiments in sect.1.2) to forecast possible realizations assuming that the drift and the volatility will not change with time. Figure 1.5.1#fig.2 shows how the cloud of possible realizations evolves from the initial spot price $ S_0=1$ and broadly follows a mean value that grows exponentially in time at the drift rate. Each trajectory, however, is different and broadly spreads out with the square root of time and the volatility.

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