A derivative asset is a security whose terminal pay-off depends entirely on the price of one or more underlying assets. Call and put options on a stock or stock index are simple examples. since in 1973 F. Black and M. Scholes published their famous opinion pricing formula, a rapidly expanding literature has dealt with the valuation of derivates. Common to almost all pricing theories is that they are built on non-arbitrage arguments. Based on some assumptions on the structure of the underlying's price process, a self-financing trading strategy is constructed that yields exactly the same terminal pay-off as the derivative. This is referred to as "duplicating" the derivative's pay-off.
From practitioners working in securities markets we learned that duplicating strategies are not only seen as a theoretical concept for pricing exchange traded derivative securities. they are also widely used in order to hedge the risk incurred by selling tailor-made derivatives "over-the-counter". But these strategies typically call for selling the underlying asset after its price has declined and vice versa. Their implementations therefore is likely to destabilize prices. In fact, it has been identified by the Presidential Task Force on Market Mechanisms and many other analysts as having triggered the stock market crash in October 1987.
This observation and the rapidly increasing popularity of derivative securities called for a closer look at how the underlying's price is affected by the implementation of duplicating strategies. We derive an explicit expression for the transformation of market volatility under the influence of such strategies. it turns out that, even when the pricing theory postulates constant volatility which varies wit the underlying's price. Nevertheless, we can show that such simple strategies are still sufficient to cover completely the risk incurred by selling derivative contracts.
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