The replication of derivatives with a portfolio consisting of underlyings and a bank account as, for example, in Equation 11.3, can of course also be used to hedge the derivative’s risk resulting from the stochastic movement of its underlying (or conversely a derivative could be used to hedge such a portfolio). This is accomplished by going short in the portfolio and long in the derivative or vice versa. This idea can be extended to hedging against influences other than the underlying price, for example, changes in the volatility, interest rate, etc. Such concepts of safeguarding against a risk factor have already made their appearance in arbitrage arguments in previous chapters and will be presented in their general form in this chapter. In addition to the fundamental Assumptions 1, 2, 3, 4 and 5 from Section 5, continuous trading will also be assumed below, i.e. Assumption 6. We will allow the underlying to perform a general Ito process1 of the Form 3.16 and assume that it pays a dividend yield q.
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