Derivatives09-3 - Modeling the Stochastic Process for Derivative Analysis

# Derivatives09-3 - Modeling the Stochastic Process for Derivative Analysis

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Session 3. Modeling the Stochastic Process for Derivative Analysis

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Model of the behavior of spot price Geometric Brownian motion continuous time, continuous stock prices Binomial discrete time, discrete stock prices approximation of geometric Brownian motion
Creation of synthetic option Geometric Brownian motion requires advanced calculus (Ito’s lemna) Binomial (already covered) based on elementary algebra

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Options: the family tree Black Merton Scholes (1973) Analytical models Numerical models Analytical approximation models Term structure models B & S Merton Binomial Trinomial Finite difference Monte Carlo European Option European American Option American Option Options on Bonds & Interest Rates Analytical Numerical
Modelling stock price behaviour Consider a small time interval t : S = S t + t - S t 2 components of S: drift : E( S ) = μ S t [ μ = expected return (per year)] volatility : S / S = E( S/S) + random variable (rv) Expected value E(rv) = 0 Variance proportional to t Var(rv) = σ ² t Standard deviation = σ √∆ t rv = Normal (0, σ √∆ t) = σ × Normal (0, √∆ t) = σ × z z : Normal (0, √∆ t) = σ × ε × √∆ t ε : Normal(0,1) 2200 z independent of past values (Markov process)

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Geometric Brownian motion illustrated Geometric Brownian motion -100.00 -50.00 0.00 50.00 100.00 150.00 200.00 250.00 300.00 350.00 400.00 0 8 16 24 32 40 48 56 64 72 80 88 96 104 112 120 128 136 144 152 160 168 176 184 192 200 208 216 224 232 240 248 256 Drift Random shocks Stock price
Geometric Brownian motion model 2200 S/S = μ t + σ z 2200 S = μ S t + σ S z = μ S t + σ S ε √∆ t If t "small" (continuous model) dS = μ S dt + σ S dz

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Binomial representation of the geometric Brownian u , d and q are choosen to reproduce the drift and the volatility of the underlying process: Drift: Volatility: Cox, Ross, Rubinstein’s solution: S uS dS q 1- q t e u = σ u d 1 = d u d e q t - - = μ t Se Sd q qSu = - + μ ) 1 ( t S Se d S q u qS t = - - + 2 2 2 2 2 2 2 ) ( ) 1 ( σ μ
Binomial process: Example dS = 0.15 S dt + 0.30 S dz ( μ = 15%, σ = 30%) Consider a binomial representation with t = 0.5 u = 1.2363, d = 0.8089, q = 0.6293 Time 0 0.5 1 1.5 2 2.5 28,883 23,362 18,897 18,897 15,285 15,285 12,363 12,363 12,363 10,000 10,000 10,000 8,089 8,089 8,089 6,543 6,543 5,292 5,292 4,280 3,462

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Call Option Valuation:Single period model, no payout Time step = t Riskless interest rate = r Stock price evolution uS S dS No arbitrage: d<e r t < u 1-period call option C u = Max(0, uS-X ) C u =? C d = Max(0, dS-X ) q 1- q q 1- q
Option valuation: Basic idea Basic idea underlying the analysis of derivative securities Can be decomposed into basic components 2200 possibility of creating a synthetic identical security by combining: - Underlying asset - Borrowing / lending

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Synthetic call option Buy δ shares Borrow B at the interest rate r per period Choose δ and B to reproduce payoff of call option δ u S - B e r t = C u δ d S - B e r t = C d Solution: Call value C = δ S - B dS uS C C d u - - = δ t r d u e d u uC dC B - - = ) (
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