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Quantitative Finance · المسرد

ما معنى Recombining tree؟

Definition 2.3 Derivatives and Volatility · الفصل 2 — The Binomial Model

A recombining tree is a tree in which an up move followed by a down move reaches the same node as the reverse: after kk steps the share is at one of the k+1k+1 values Sujdk−jSu^jd^{k-j}, j=0,…,kj=0,\dots,k, instead of 2k2^k. Pricing on it costs O(n2)O(n^2) operations and O(n)O(n) memory.

The whiteboard tree: share price (top line) and American put value (bottom line) at every node, u=1.1, d=0.9, 2% per step, strike 100. At 90 and at 81 the put is worth more exercised (10 and 19) than held (9.196 and 17.039), and the tree takes the larger value. Data: the tutorial.
Figure 2.1. The whiteboard tree: share price (top line) and American put value (bottom line) at every node, u=1.1u=1.1, d=0.9d=0.9, 2% per step, strike 100. At 90 and at 81 the put is worth more exercised (10 and 19) than held (9.196 and 17.039), and the tree takes the larger value. Data: the tutorial.

أمثلة

Example 2.5 (The whiteboard call)

With u=1.1u=1.1, d=0.9d=0.9, R=1.02R=1.02: p=(1.02−0.9)/0.2=0.6p=(1.02-0.9)/0.2=0.6. After three steps the share is at 133.1, 108.9, 89.1 or 72.9, and the call pays 33.1, 8.9, 0 or 0. Its value is (0.63×33.1+3×0.62×0.4×8.9)/1.023=10.3603(0.6^3\times33.1+3\times0.6^2\times0.4\times8.9)/1.02^3=10.3603. At the first node the replicating portfolio holds Δ=0.6240\Delta=0.6240 share and a deposit of −52.04-52.04; put–call parity holds on the tree, C−P=100−100/1.023=5.7678C-P=100-100/1.02^3=5.7678.

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