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
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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 Merton Binomial Trinomial Finite difference Monte Carlo European Option European American Option American Option Options on Interest Rates Analytical Numerical
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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
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Geometric Brownian motion model 2200 S/S = μ t + σ z 2200 S = S t + S z = S t + S ε √∆ t If t "small" (continuous model) dS = 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 = - + ) 1 ( t S Se d S q u qS t = - - + 2 2 2 2 2 2 2 ) ( ) 1 (
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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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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
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Derivatives09-3 - Modeling the Stochastic Process for Derivative Analysis

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