n n Δ t j n n 1 1 n so there are 2 n 1 nodes at that time The corresponding X

# N n δ t j n n 1 1 n so there are 2 n 1 nodes at that

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n = n Δ t , j = n, n + 1 , · · · , 0 , 1 , · · · , n , so there are 2 n + 1 nodes at that time. The corresponding X values are X n,j = j Δ X . The difference with the binomial tree that approximated geometric Brownian motion that we studied before is that the tree probabilities change with level, since mean-reversion implies a higher probability of a downward move higher up the tree than lower down. Generically, the trinomial tree is as in Figure 15, and there are three (risk-neutral) probabilities ( q u , q m , q d ) corresponding to the up, middle and down branches to be found. These will depend on location j . a115 a115 a115 a115 a115 a115 a115 a115 a115 a115 a115 a115 a115 a115 a115 a115 a115 a115 a115 a115 a115 a115 a115 a115 a115 a115 a115 a115 a115 a115 a115 a115 a115 a115 a115 a115 q u q m q d Figure 15: Generic Trinomial Tree For most of the tree, the branching is of the form (a) in Figure 16, with possibilities, up by dX , stay-the-same or down by dX . Over the time period Δ t , the probabilities are chosen to match the mean and variance of dX t . From (49), IE Q { dX t } = aX t dt , which leads to 77

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a115 a115 a115 a115 a0 a0 a0 a0 a0 a0 a115 a115 a115 a115 a115 a115 a115 a115 ( a ) ( b ) ( c ) Figure 16: Different trinomial branches the condition q u Δ X + q m · 0 + q d ( Δ X ) = a ( j Δ X t. From var Q { dX t } = σ 2 dt , we have q u X ) 2 + q d X ) 2 a 2 ( j Δ X ) 2 t ) 2 = σ 2 Δ t. The third condition comes from the probabilities adding to one: q u + q m + q d = 1 . Recall we have chosen Δ X = σ t , so these are three equations for the three probabilities. For j not too large and not too negative, we can find solutions that are probabilities (between zero and one) given by q u = 1 6 + 1 2 ( a 2 j 2 t ) 2 aj Δ t ) (50) q m = 2 3 a 2 j 2 t ) 2 q d = 1 6 + 1 2 ( a 2 j 2 t ) 2 + aj Δ t ) The recommendation in Hull is to switch to the branching (c) in Figure 16 when j j max := ceilingleftbigg 0 . 184 a Δ t ceilingrightbigg .
• Fall '11
• COULON
• Variance, Probability theory, Trigraph, Credit default swap, Wiener process

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