By Dr Hans-Peter Deutsch (auth.)

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Extra info for Derivatives and Internal Models

Example text

Then the amount of time passing between steps is gw. , proportional to W  w. • Which real parameter should be modeled by a random walk? At rst glance, we might take the market price of a risk factor. 8 can be decomposed into individual steps as follows : V5  V0 = (V0  V1 )  (V2  V1 ) + (V3  V2 ) + (V4  V3 ) + (V5  V4 )> or more generally, Vq  V0 = q X gVl with gVl = Vl  Vl1 l=1 If the market price itself were a random walk, then as a result of the self similarity property, the individual steps gVl would also be random walks.

The square of this product is considerably smaller, namely U(W  w)2 = 0=00000625. , all non-linear terms are simply neglected. 4: [1 + U(W  w)]1  1  U(W  w) + {z } | linear terms (U(W  w))2 ± · · · {z } | higher order terms are neglected! 1 The expansion used here is (1 + {)31 = 1 3 { + {2 3 {3 + {4 3 {5 ± · · · . Such series expansions can be found in any book of mathematical formulas. The result is now obtained by substituting U(W 3 w) for {. 20 3. 14% contin. 3: Interest rates for the same discount factor based on dierent day count and compounding conventions.

2: The general discount factor for the time span from w to W= 1 (but greater than 0). 2. The reader is urged to become familiar with this notation as it will be used throughout the book. , U = U(w> W ). 2 rather than EU(w>W ) (w> W ) on the understanding that the interest rate U refers to the rate corresponding to the times specied in the argument of the discount factor. If, as is occasionally the case, the interest rate is not dependent on the times (as in some option pricing models), we sometimes adopt the convention of denoting the constant interest rate by u and the corresponding discount factor by Eu (w> W ).