By Abraham Lioui

This can be a complicated textual content at the concept of ahead and futures markets which goals at offering readers with a complete wisdom of ways costs are validated and evolve over the years, what optimum suggestions you possibly can count on from the members, what characterizes such markets and what significant theoretical and useful ameliorations distinguish futures from ahead contracts. it may be of curiosity to scholars (majoring in finance with quantitative talents) teachers (both theoreticians and empiricists), practitioners, and regulators.

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**Extra info for Dynamic Asset Allocation with Forwards and Futures**

**Sample text**

Unfortunately, to assess whether the hedger would be better off using one type of derivatives or the other would require the exact knowledge of both the diffusion and the drift terms of the hedged portfolio value process. Suppose for illustrative purposes that the volatilities X(t,s) of the forward rates (see equation (3)) are either linear in maturity [X(t,s) = ( s - t ) c ] or exponential [z(t,s) = a" 1 (l-e" a(s " t) )aj. Then the futures and forward prices are known to differ by a deterministic term only and thus have the same instantaneous volatility19 [see Section III below for a numerical example].

Therefore, under Q, the drift of G(t,T) is nothing but the covariance between the forward price relative changes and its underlying bond price relative changes. We then can use the following well-known theorem. Consider a positive Ito process X(t), satisfying Novikov's condition - J ox(s)2dsFt 2 such that, under the measure Q: Then [Proof: Let Y(t) = X(T)exp(- JT|LLX(s)dsj. ] Consequently, since G(T,T) = S(T), we can write: G(t,T) = ] S(T)exp cov dG(u,T) dP(u,T)V " G(u,T) ' P(u,T) J (20) In the particular case where all variances and covariances are deterministic, we then have, using result (18): G(t,T) =H(t,T)exp cov fdG(u,T) dP(u/T)V , ^G(u,T) du (21) P(u,T)J The forward price thus is larger or smaller than its futures counterpart depending on the sign of the covariance between the forward price changes and the relevant zero-coupon price changes.

Using derivatives may increase the firm's market value, in particular for a firm using interest rate derivatives (which decreases its weighted average cost of capital). Since liquid futures contracts have relatively short maturities (less than one year), long term hedging requires rolling shorter term futures as time evolves. Neuberger (1999) examines this issue, performs tests on the crude oil futures market and concludes that the roll-over strategy is efficient. Finally, another aspect that will not be considered in this book is market making activity.