Rainbow Options in Discrete Time, I
This chapter is central for our exposition. It contains the main results describing the applications of the basic risk-neutral evaluation formula for games, developed in the previous chapter, to pricing rainbow (or colored) options under various market co
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Rainbow Options in Discrete Time, I
13.1 Colored European Options as a Game Against Nature Recall that a European option is a contract between two parties where one party has the right, but not the obligation, to complete a transaction in the future (at a previously agreed amount, date, and price). More precisely, consider a financial market dealing with several securities: risk-free bonds (or bank accounts) and J common stocks, J = 1, 2 . . .. If J > 1, then the corresponding options are called colored or rainbow options (J-color option for a given J). Suppose the prices of the i , i ∈ {1, 2, . . ., J}, change in discrete moments units of these securities, Bm and Sm of time m = 1, 2, . . . according to the recurrent equations Bm+1 = ρ Bm , where the i i i , ρ ≥ 1 is an interest rate that remains unchanged over time, and Sm+1 = ξm+1 Sm i where ξm , i ∈ {1, 2, . . . , J}, are unknown sequences taking values in some fixed intervals Mi = [di , ui ] ⊂ R. This model generalizes the colored version of the classic Cox–Ross–Rubinstein (CRR) model in a natural way. In the latter, a sequence ξmi is confined to taking values only among two boundary points di , ui , and it is supposed to be random with some given distribution. In our model any value in the interval [di , ui ] is allowed, and no probabilistic assumptions are made. Hence it is often referred to as an interval model. An option’s type is specified by a given premium function f of J variables. The following examples are standard. Option delivering the best of J risky assets and cash: f (S1 , S2 , . . . , SJ ) = max(S1 , S2 , . . . , SJ , K);
(13.1)
Calls on the maximum of J risky assets: f (S1 , S2 , . . . , SJ ) = max(0, max(S1 , S2 , . . . , SJ ) − K);
(13.2)
Multiple-strike options: f (S1 , S2 , . . . , SJ ) = max(0, S1 − K1 , S2 − K2 , . . . , SJ − KJ );
P. Bernhard et al., The Interval Market Model in Mathematical Finance, Static & Dynamic Game Theory: Foundations & Applications, DOI 10.1007/978-0-8176-8388-7 13, © Springer Science+Business Media New York 2013
(13.3)
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13 Rainbow Options in Discrete Time, I
Portfolio options: f (S1 , S2 , . . . , SJ ) = max(0, n1 S1 + n2S2 + · · · + nJ SJ − K);
(13.4)
And spread options: f (S1 , S2 ) = max(0, (S2 − S1 ) − K).
(13.5)
Here, the S1 , S2 , . . . , SJ represent the expiration date values of the underlying assets (in principle unknown at the start), and K, K1 , . . . , KJ represent the (agreed from the beginning) strike prices. The presence of max in all these formulas reflects the basic assumption that the buyer is not obligated to exercise her right and would do it only in case of a positive gain. The investor is supposed to control the growth of her capital in the following way. Let Xm denote the investor’s capital at the time m = 1, 2, . . .. At each time m − 1 the investor determines her portfolio by choosing the number γmi of common stock of each kind to be held, so that the structure of the capital is represented by the formula Xm−1 =
J
j + ∑ γmj Sm−1
j=1
J
Xm−1 − ∑ γmj Sm−1 , j
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