Wednesday, May 23, 2012

drift + random walk -- baby steps

Update - a discrete random walk assumes the step size (in log space) is normally distributed.

When we enhance a granular binomial tree to be even higher granularity, the interval between 2 tree levels (sampling interval) becomes infinitesimal and we can use the standard calculus notation of " dt ". Black-Scholes differential equation becomes
\frac{dS}{S} = \mu \,dt+\sigma \,dW\,
Note t is a Timespan, not a Datetime. I call it TTL, time-to-live or time-to-maturity.
Note the LHS denominator is the spot price, not "t". The expression (dS/S) measures an stock's percentage return as _holding_period_ becomes infinitesimal. Basic calculus[1] gives us the integral of the LHS

    integral(1/S * dS) = ln(S)

Ignoring the dW part, integrating the right-hand-side gives some linear function of t [0]. Therefore under zero volatility, stock price is an exponential function of t [2]. Therefore Drift is exponential -- continuous compounding.

[0] note the "variable-of-integration" is S on the left hand side but " t " on the right-hand-side. This was a bit confusing to me.

Integrating the dW part is harder. Actually, since the dW (unlike drift) is inherently random, I doubt we can simply get the integral and predict the S at any value of t. Instead, we hope to derive the pdf of S at any point t in the future. Let me repeat the implicit but fundamental assumption -- the value of S at a given t is a random variable but has a pdf. This randomness comes from the dW, not the drift.

Once we have a pdf of S(t), expiration value of an European call is tractable. Since the terminal value is a hockey-stick payoff function, we multiply the pdf by a piecewise linear function, and find area under the curve. See other blog posts.

A note on the sigma in
\frac{dS}{S} = \mu \,dt+\sigma \,dW\,
BS assumes sigma to be constant. When sigma itself moves up and down following a random motion, we have a stochastic volatility model. A simplified non-constant sigma model is the local-volatility model, popular in investment banking.

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http://www.linkedin.com/in/tanbin