《金融期货与期权》（英文版） Chapter 18 Numerical Procedures

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 = − − = = =

18.6 The Complete Tree (Figure 18.2) Sod Options, Futures, and Other Derivatives, 5th edition C 2002 by John C Hull

18.6 Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull The Complete Tree (Figure 18.2) S0 S0u S0d S0 S0 S0u 2 S0d 2 S0u 2 S0u 3 S0u 4 S0d 2 S0u S0d S0d 4 S0d 3

18.7 Backwards induction We know the value of the option at the final nodes We work back through the tree using risk-neutral valuation to calculate the value of the option at each node, testing for early exercise when appropriate Options, Futures, and Other Derivatives, 5th edition C 2002 by John C Hull

18.7 Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull Backwards Induction • We know the value of the option at the final nodes • We work back through the tree using risk-neutral valuation to calculate the value of the option at each node, testing for early exercise when appropriate

188 Example: Put Option S0=50;X=50;r=10%;σ=40%; T=5 months =04167 δt=1 month=00833 The parameters imply ll=1.1224:d=0.8909 a=1.0084;p=0.5076 Options, Futures, and Other Derivatives, 5th edition C 2002 by John C Hull

18.8 Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull Example: Put Option S0 = 50; X = 50; r =10%; s = 40%; T = 5 months = 0.4167; dt = 1 month = 0.0833 The parameters imply u = 1.1224; d = 0.8909; a = 1.0084; p = 0.5076

18.9 Example(continued) igure 18.3 8907 9.35 70.70 70.70 0.00 0.00 064 56.1 56.1 56.1 2.16 1.30 0.00 50.00 50.00 50.00 449 3.77 266 44.55 44.55 44.55 6.96 6.38 5.45 3969 39.69 10.36 10.31 35.36 35.36 14.64 14.64 3150 1850 28.07 21.93 Options, Futures, and Other Derivatives, 5th edition C 2002 by John C Hull

18.9 Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull Example (continued) Figure 18.3 89.07 0.00 79.35 0.00 70.70 70.70 0.00 0.00 62.99 62.99 0.64 0.00 56.12 56.12 56.12 2.16 1.30 0.00 50.00 50.00 50.00 4.49 3.77 2.66 44.55 44.55 44.55 6.96 6.38 5.45 39.69 39.69 10.36 10.31 35.36 35.36 14.64 14.64 31.50 18.50 28.07 21.93

18.10 Calculation of delta Delta is calculated from the nodes at time st 2.16-6.96 △ 0.41 56.12-44.55 Options, Futures, and Other Derivatives, 5th edition C 2002 by John C Hull

18.10 Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull Calculation of Delta Delta is calculated from the nodes at time dt 0.41 56.12 44.55 2.16 6.96 = − − −  =