Strategy, Value and Risk by Jamie Rogers

Author:Jamie Rogers
Language: eng
Format: epub
ISBN: 9783030219789
Publisher: Springer International Publishing

(5.31)

(5.32)

(5.33)

Solving Eqs. (5.31–5.33) yields the following explicit expressions for the transitional probabilities:

(5.34)

(5.35)

(5.36)

The single period trinomial process in Fig. 5.7 can be extended to form a trinomial tree. Figure 5.8 depicts such a tree.

Fig. 5.8A trinomial tree model of an asset price

Let i denote the number of the time step and j, the level of the asset price relative to the initial asset price in the tree. If Si,j denotes the level of the asset price at node (i,j), then t = ti = i∆t, and an asset price level of Sexp(j∆x). Once the tree has been constructed, the spot price is known at every time and every state of the world consistent with the original assumptions about its behaviour process, and the tree can be used to derive prices for a wide range of derivatives.

The procedure is illustrated with reference to pricing a European and American call option with a strike price K on the spot price. The value of an option is represented at node (i,j) by Ci,j. In order to value an option, the tree is constructed as representing the evolution of the spot price from the current date out to the maturity date of the option. Let time step N correspond to the maturity date in terms of the number of time steps in the tree, that is, T = N∆t. The values of the option at maturity are determined by the values of the spot price in the tree at time step N and the strike price of the option:

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