By Marek Musiela
A new version of a profitable, well-established booklet that gives the reader with a textual content excited about useful instead of theoretical elements of economic modelling
Includes a brand new bankruptcy dedicated to volatility risk
The subject of stochastic volatility reappears systematically and has been revised essentially, proposing a way more specific analyses of interest-rate models
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Additional info for Martingale Methods in Financial Modelling
Hence, the arbitrage price of portfolio A at time t dominates the price of portfolio B - that is Ct + Ke-r(T-t) >_ S t. 'it E [O,Tj. 37). 39) may be verified by means of similar arguments. To check that Cf = Ct, we consider the following portfolios: portfolio A - one American call option and K e-r(T-t) of cash; and portfolio B - one share of stock. If the call option is exercised at some date t* E [t, Tj, then the value of portfolio A at time t* equals St- - K + Ke-r(T-t-) < St-, while the value of B is St-.
2). 1. Assume that the one-period, two-state security market model (8, B, qJ) is arbitrage-free. Then the forward price at time 0 for the settlement date T of one share of stock 8 equals Fs(O, T) = (1 + r)80' Proof. 2. 33), we get = lEn" (f- 1 X) = IE p • (8;') - f- 1K = 8 0 - f- 1K = 0, 1 + r. 1 with the above proposition, we conclude that in a one-period model of a spot market, the futures and forward prices of financial assets for the same settlement date are equal. Remarks. It seems instructive to consider a slightly more general model of a one-period market.
As we will see in what follows, the arbitrage price of any European or American contingent claim in the binomial model of a financial market is independent of the choice of the probability of upward and downward movements of the stock price at any particular node. Indeed, it is uniquely determined by the assumed values of the stock price - that is, on the particular behavior of the sample paths of the stock price process. Put another way, the valuation results that we are going to derive will appear to be distribution-free, meaning that they do not depend on the choice of the subjective (or actual) probability measure P.
Martingale Methods in Financial Modelling by Marek Musiela