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This paper assumes that the underlying asset prices are lognormally distributed and drives necessary and sufficient conditions for the valuation of options using a Black-Scholes type methodology. It is shown that the price of a futures-style, market-to-market option is given by Black s formula if the pricing kernel is lognormally distributed. Assuming that this condition is fulfilled, it is then shown that the Black-Scholes formula prices a spot-settled contingent claim, if the interest-rate accumulation factor is lognormally distributed. Otherwise, the Black-Scholes formula holds if the product of the pricing kernel and the interest-rate accumulation factor is lognormally distributed.
The problem of fair pricing of contingent claims is well understood in the contex of an arbitrage free, complete financial market, with perfect information: the so-called arbitrage approach permits to construct a unique valuation operator compatible with observed price processes. In the more realistic context of partial information, the equilibrium analysis permits to construct a unique valuation operator which only depends on some particular price processes as well as on the dividends process. In this paper we present these two approaches and we explore their links and the conditions under which they are compatible ; In particular, we derive from the equilibrium conditions some links between the price processes paramaters and those of the dividend processes paramaters.
This text provides an accessible introduction to the classical mathematical underpinnings of modern finance. Professor Bjork concentrates on the probabilistic theory of continuous arbitrage pricing of financial derivatives.
Arbitrage pricing plays an important role in asset valuation. The most applications of arbitrage asset pricing theories are based on the law of one price or asymptotic arbitrage free markets. We provide some new results on arbitrage and especially the arbitrage pricing theory by distinguishing between the absence of arbitrage, the law of one price and the absence of riskless arbitrage. Then we find the implications of these conditions for arbitrage asset pricing. Since the three concepts of the absence of arbitrage imply that the linear functionals that give the mean and the cost of a portfolio are continuous, hence there exist unique portfolios that represent these functionals. We detect a positive distance between these portfolios and therefore between the functionals. Thus the law of one price and the absence of a riskless arbitrage opportunity lead to systematic mispricing if both the contingent claims and the assets are mispriced. The beta pricing literature usually makes strong assumptions to obtain exact asset pricing. This belongs to a debate over which factors have the best theoretical or empirical justification. In the light of our results it is more advisable to acknowledge that almost only approximate arbitrage asset pricing can be obtained. The introduction of risky arbitrage opportunities in the sense that there might be an arbitrage opportunity with positive probability but not with probability one requires the knowledge of the risk aversion of investors. Therefore exact asset pricing can only be obtained by equilibrium asset pricing models. Our results generalizes to other arbitrage asset pricing theories like the Black and Scholes option valuation model and even the Modigliani-Miller Theorem.