Relationship between Asset Volatilities and Spread Volatility - II
Oct 20, 2007
Continued from the previous article..............
The discrepancy outlined in the previous article is purely mathematical and has to do with how we model spreads. The reason for the discrepancy in the percentage spread volatility and the dollar spread volatility in the above example is because of the way we look at spread in the context of a Brownian motion (or random walk).
One is tempted to ask: How does a "spread" actually behave? Is it simply the difference in the two asset prices (values) or is it an asset itself? The key question is can we treat spread as an asset. To a trader or any financial markets practitioner, the answer is a resounding "yes". Spread is definitely an asset and we trade this asset all the time. A CMS spread or a 2/10 Treasury Note spread is an asset, just like, say, 6 month USD LIBOR or a 2 yr Treasury Bond. Of course, if you look at the closed form pricing rationale behind spread options (Margrabe's formula) it implies the option to exchange one asset for the other. Nevertheless, traders and investors are quite often looking at spreads as assets.
If a spread is an asset, just like, say a 6 month LIBOR, and if we want to compute return volatility of the spread by using the natural log of today's price divided yesterday's price as the basis for computing geometric returns (as in the above example) then the asset price equation should be given by a geometric brownian motion:
Where, is the spread, is the drift (should be in a riskless world), is the volatility of the spread (and here it is the return or the percentage volatility) and is a Wiener process. This equation models the rate of change of the asset (the spread is the asset here) and hence it is in fact modelling the return process. Thus the volatility in the second term is the return or the percentage volatility. Since the return process is being modeled as ?we use the natural log of one period's price divided by the previous period's price as the proxy for return.
However, there is big problem if we try to model spreads in the above framework.
The problem is, that a spread as an asset cannot be modeled as a geometric brownian motion (or random walk). In a geometric brownian motion the asset price (value) cannot be negative. But in the financial markets spreads can be, and quite often are, negative. Thus we should use an arithmetic brownian motion to model any spread as shown below:
This is the correct model for spreads. Spreads follow a normal distribution and not a lognormal distribution and they can be negative. This means that in the case of spreads we are actually modelling the price difference as opposed to the rate of change of the asset (as in the geometric brownian motion model). Therefore, the volatility is no longer the return or percentage volatility but the price (or dollar) volatility. It is the standard deviation of the price series rather than the return series.?
In fact, this is the only way that we can reconcile the volatility of the spread as an asset and the volatility of a spread, constructed as a portfolio of one long and one short asset. That is why when we use the portfolio volatility formula on the price levels to compute volatility both volatilities - the volatility of the spread as an asset and the volatility of the spread as a portfolio of one long and one short asset - match exactly.
If we use such an arithmetic brownian motion model (normal model) then we can compute the returns as today's price minus yesterday's price divided by yesterday's price. And the volatility of these returns will become the proxy for percentage volatility.
If we use this method of computation in the previous example, then we will find that the volatility computed from the difference in assets returns (i.e. using a portfolio volatility for spread where we are long an asset and short the other) and the volatility computed from the returns of the actual spread will come much closer to each other; however they will not become exactly equal. Of course, here the asset returns - in the portfolio - should also be computed as normal returns (today's price minus yesterday's price divided by yesterday's price) instead of lognormal returns.
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