Lecture7_BinomialOptionPricing

# E p s u 1 p s d the should equal equilibrium mean s

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Unformatted text preview: parameters: no dividend (1) We choose the tree parameters p, u, and d to fit the We and real/empirical mean & volatility of the stock price changes: changes: Recall our original setting (no dividend, no no adjustment term): δ S /S = r δ t + σ ε √δ t and Sδ t = S exp(η), where η ~ N(rδ t, σ√δ t) ), σ√ The average stock price (i.e., p S u + (1– p ) S d) The should equal equilibrium mean S erδt (why S erδt? – the the return of holding the stock should equal the risk-free rate) rate) The volatility of stock price (i.e., the square root of The p S 2 u 2 + (1– p ) S 2 d 2 – [p S u + (1– p ) S d ]2) should should equal the equilibrium volatility S eσ√δ t 15-41 Equation of tree parameters: no dividend (2) Mean: S erδt = p S u + (1– p ) S d Mean: = > erδt = p u + (1– p ) d t Variance: S 2 e2 (σ √δ t)) = p S 2 u 2 + (1– p ) S 2 d 2 – [p S u + (1– p ) S d ]2 Two equations with three unknown (p, u, and d ) and So, to solve the problem, we let d =1/u and So, tedious derivations lead to the following solution for parameters: for 1. u = eσ√δt (i.e., d = e-σ√δt , the intuition is simple: the stock price will either go up or down by 1 st. dev.) stock 2. p = (erδt – d )/(u – d) 2. )/( 15-42 Equation of tree parameters: with dividend (1) We choose the tree parameters p, u, and d to fit the We and real/empirical mean & volatility of the stock price changes: changes: Recall our original setting (with dividend rate q, no with no adjustment term): δ S /S = (r − q )δ t + σ ε √δ t and Sδ t = S exp(η), where η ~ N((r − q ) δ t, σ√δ t) ), σ√ The average stock price (i.e., p S u + (1– p ) S d) The should equal equilibrium mean S e(r-q)δt (why S e(r-q)δt? – the return of holding e-qδt share of the stock (that will grow to 1 share) should equal the risk-free rate) grow The volatility of stock price (i.e., the square root of The p S 2 u 2 + (1– p ) S 2 d 2 – [p S u + (1– p ) S d ]2) should should equal the equilibrium volatility S eσ√δ t 15-43 Equation of tree parameters: with dividend (2) Mean: S e(r-q)δt = p S u + (1– p ) S d Mean: = > e(r-q)δt = p u + (1– p ) d t Variance: S 2 e2 (σ √δ t)) = p S 2 u 2 + (1– p ) S 2 d 2 – [p S u + (1– p ) S d ]2 Two equations with three unknown (p, u, and d ) and So, to solve the problem, we let d =1/u and So, tedious derivations lead to the following solution for parameters: for u = eσ√δt (i.e., d = e-σ√δt , the intuition is simple: the stock price will either go up or down by 1 st. dev.) stock p = (e(r-q) δt – d )/(u – d) )/( 15-44...
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## This document was uploaded on 03/03/2014.

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