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

This preview shows page 4 - 6 out of 7 pages.

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
Image of page 4

Subscribe to view the full document.

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 .
Image of page 5
Image of page 6
  • Fall '11
  • COULON
  • Variance, Probability theory, Trigraph, Credit default swap, Wiener process

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern

Ask Expert Tutors You can ask 0 bonus questions You can ask 0 questions (0 expire soon) You can ask 0 questions (will expire )
Answers in as fast as 15 minutes