By Marek Musiela

A brand new variation of a profitable, well-established booklet that offers the reader with a textual content taken with sensible instead of theoretical points of economic modelling encompasses a new bankruptcy dedicated to volatility hazard The subject matter of stochastic volatility reappears systematically and has been revised essentially, featuring a way more special analyses of interest-rate versions

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In other words, the terminal wealth associated with a dynamic portfolio comes exclusively from the initial investment and the capital gains generated by the trading process. We do not need to give here a more formal definition of self-financing property as it is clear that the following property is valid in any discrete-or continuous-time, arbitrage-free market. 32 1 Financial Derivatives Price monotonicity rule. In any model of an arbitrage-free market, if XT and YT are two European contingent claims, where XT ≥ YT , then πt (XT ) ≥ πt (YT ) for every t ∈ [0, T ], where we denote by πt (XT ) and πt (YT ) the arbitrage prices at time t of XT and YT respectively.

24) 6 Notice that in the present context, the knowledge of the settlement date T ∗ of a futures contract is not essential. It is implicitly assumed that T ∗ ≥ T , however. 6 Futures Call and Put Options 23 where the first term on the right-hand side represents gains (or losses) from the futures contract, and the second corresponds to a savings account (or loan). 24) reflects the fact that futures contracts are marked to market daily (that is, after each period in our model). f f A portfolio φ = (α0 , β0 ) is said to replicate the option when VT = CT , or more explicitly, if the equalities f VT (ω) = α0 (f u − f0 ) + (1 + r)β0 = C f u , if ω = ω1 , α0 (f d − f0 ) + (1 + r)β0 = C f d , if ω = ω2 , are satisfied.

1 Suppose that the spot market model M = (S, B, Φ) is arbitragefree. Let H stand for the rational price process of some attainable contingent claim X; more explicitly, H0 ∈ R and HT = X. Let us denote by ΦH the class of all portfolios in stock, bond and derivative security H . The extended market model (S, B, H, ΦH ) is arbitrage-free if and only if H0 = π0 (X). 5 Optimality of Replication Let us show that replication is, in a sense, an optimal way of hedging. Firstly, we say that a portfolio φ perfectly hedges against X if VT (φ) ≥ X, that is, whenever α0 S u + (1 + r)β0 ≥ X u , α0 S d + (1 + r)β0 ≥ X d .

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