By Ferrari Brockhaus Olive, Oliver Brockhaus, Andrew Ferraris, Christopher Gallus, Douglas Long, Reiner Martin, Marcus Overhaus
This reference textual content offers unique, practice-based research of modelling and hedging fairness derivatives. It includes an research of chance idea and stochastic calculus and a close dialogue of sensible software program implementation concerns.
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Additional info for Modelling and Hedging Equity Derivatives
The stochastic integral can be interpreted as the cumulative gains and losses incurred from trading asset i according to strategy ϕi. This is clear if ϕi is 1(t,s], ie just a buy-and-hold strategy, as ; in general, one can approximate a complex trading strategy by a sequence of linear combinations of buy-and-hold strategies, which justifies regarding loss, even for complex strategies. as the cumulative profit and We can now introduce the concept of arbitrage: an arbitrage opportunity is a self-financing trading strategy that generates wealth from nothing.
From a mathematical point of view, we aim at presenting Clark's formula. For a rigorous account of the subject, we refer the reader to the books by Ikeda and Watanabe  and Nualart , as well as the papers by Ocone  on integral representation, Ocone and Karatzas  on the Clark formula, and Colwell, Elliott and Kopp  on applications to hedging. The first two examples below are taken from a lecture given by FÃ¶llmer . Assume that F is differentiable at function DF(ω) on Ω such that Â < previous page < previous page in the sense that there exists a linear continuous page_33 page_34 next page > next page > Page 34 It is known that linear continuous functions on Ω may be identified with signed finite measures on [0, T] in the sense that If h ℘ Ω can be written as an integral 50 51 then, by applying Fubini's theorem, we have We introduce the set of functions h ℘ Ω satisfying the additional condition that The following result is due to Bismut .
Thus S/Y with respect to Q is a geometric Brownian motion with drift rf - d, and hence The quanto option has the price Â < previous page < previous page page_30 page_31 next page > next page > Page 31 in domestic currency. Using the Girsanov transformation and Q above, we obtain because S with respect to Q has the drift rd - d + ρσSσX. Foreign Approach We consider the contract from the standpoint of a foreign investor. For a foreign investor, SX and X are tradable assets, whereas S cannot be traded directly.