Introduction to Stochastic Calculus for Finance: A New by Dieter Sondermann

By Dieter Sondermann

Even though there are lots of textbooks on stochastic calculus utilized to finance, this quantity earns its position with a pedagogical process. The textual content provides a short (but under no circumstances "dirty") highway to the instruments required for complex finance in non-stop time, together with alternative pricing via martingale tools, time period constitution versions in a HJM-framework and the Libor industry version. The reader will be conversant in hassle-free actual research and uncomplicated likelihood thought.

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Extra resources for Introduction to Stochastic Calculus for Finance: A New Didactic Approach (Lecture Notes in Economics and Mathematical Systems)

Example text

T = φt (ω) denotes the number of shares of the security held by an investor at time t in state ω. Adaptation to (Ft ) means that the investment decision can only be based on the information available at time t. Given the portfolio strategy φt , the value of the portfolio at time t is of the form Vt = φt Xt + ηt = V (Xt , t) (14) where ηt is a money account, yielding no interest. ) is called self-financing if, after an initial investment V0 = η0 , all changes in the value of the portfolio Vt are only due to the accumulated gains (or losses) resulting from price changes of Xt .

According to Itˆ o’s formula it follows 1 dΠ = dV − Vx dX = V˙ dt + Vxx σ 2 X 2 dt. 2 (16) But, whereas the return dXt (ω) depends on ω, the right-hand side of (16) does not, since Xt2 (ω) is fixed at t. e. d Π = r Π dt. (17) Combining (16) and (17) gives the PDE 1 V˙ + r X Vx + σ 2 X 2 Vxx = r · V 2 (Black-Scholes PDE). (18) This PDE holds for any derivative of the form F (XT ). A simple example is a forward contract on XT fixed at time t = 0 at price K. It is easy to check that F (t) = Xt − K e−r(T −t) = V (Xt , t) is the (unique) solution of (18) under the boundary constraint F (T ) = V (XT , T ) = XT − K.

5). Using only elementary facts of independent normally distributed random variables, it leads to the Dol´eans-Dade exponential as new density under a change of measure for the Brownian motion. Sect. 2 deals with the Girsanov transformation in general form, as can be found in Revuz-Yor (1991). The proofs are straightforward applications of tools developed in Chap. 2. This section, which at first sight looks rather abstract, is basic for the applications to finance in Chapters 4 and 5, where the general Girsanov transformation is repeatedly used.

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