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 B & S Merton Binomial Trinomial Finite difference Monte Carlo European Option European American Option American Option Options on Bonds & 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 = 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
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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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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
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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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