Fixed Point Algorithm
The fixed point algorithm begins by assuming that all banks pay their obligations in full. That is, we initialize the method by setting:
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We use the following fixed point function:
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While the magnitude norm of (p - the fixed point function of p) is greater than some small threshold, that is:
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the algorithm continues by recalculating p:
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Essentially, p is recalculated at each iteration and as p and the fixed point function of p get closer to each other, the method approaches the solution, p*. The fixed point function of p is a nondecreasing function. When started at p_0 =pBar, this method converges, eventually, to the maximum clearing payment solution. Thus, the fixed point mapping converges to the uniquie fixed point
which is our clearing vector, also called the payments vector.
Together the limited liability and absolute priority criteria imply that p* (within the interval [0, pBar]) is a clearing vector if and only if the following holds:
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where the first term represents what the node has and the second term represents what the node owes. Meaning, a clearing vector is the minimum of what a firm has and what it owes.
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Thus, the clearing vector p* is a fixed point of the mapping of the fixed point function from [0, pBar] to [0, pBar] defined by:
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where the lattice operation is defined as the following:
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Formally, after initializing pBar, e, and pi. We implement this algorithm in MATLAB as follows:
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