18.1 Numerical Procedures Chapter 18 Options, Futures, and Other Derivatives, 5th edition C 2002 by John C Hull
18.1 Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull Numerical Procedures Chapter 18
18.2 Binomial trees Binomial trees are frequently used to approximate the movements in the price of a stock or other asset In each small interval of time the stock price is assumed to move up by a proportional amount u or to move down by a proportional amount d Options, Futures, and Other Derivatives, 5th edition C 2002 by John C Hull
18.2 Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull Binomial Trees • Binomial trees are frequently used to approximate the movements in the price of a stock or other asset • In each small interval of time the stock price is assumed to move up by a proportional amount u or to move down by a proportional amount d
18.3 Movements in time。t (Figure 18.1) su Options, Futures, and Other Derivatives, 5th edition C 2002 by John C Hull
18.3 Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull Movements in Time dt (Figure 18.1) Su Sd S
18.4 1. Tree Parameters for a Nondividend paving stock o We choose the tree parameters p, u, and d so that the tree gives correct values for the mean standard deviation of the stock price changes in a risk-neutral world erot= pu+(1-pd 2t=p2+(1-p)d2-[pu+(1-p)d]2 a further condition often imposed isu=1/d Options, Futures, and Other Derivatives, 5th edition C 2002 by John C Hull
18.4 Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 1. Tree Parameters for a Nondividend Paying Stock • We choose the tree parameters p, u, and d so that the tree gives correct values for the mean & standard deviation of the stock price changes in a risk-neutral world e r dt = pu + (1– p )d s 2dt = pu 2 + (1– p )d 2 – [pu + (1– p )d ] 2 • A further condition often imposed is u = 1/ d
18.5 2. Tree Parameters for a Nondividend paying stock (Equations 18.4 to 18.7) When St is small, a solution to the equations is st G√St r ot Options, Futures, and Other Derivatives, 5th edition C 2002 by John C Hull
18.5 Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 2. Tree Parameters for a Nondividend Paying Stock (Equations 18.4 to 18.7) When dt is small, a solution to the equations is r t t t a e u d a d p d e u e d −s d s d = − − = = =