Paul Wilmott et al : the math of financial derivatives | 45 min redux

July 30, 2010
  • A classical piece (1995) – focus on continuous side of financial math (SDEs etc.)
  • Basic intro – options, real world data, interest rates, continuous compounding, present value
  • Random walk nature of financial time series – discrete vs continuous walk – SDE representation
  • We base derivative strategies around time series / stochastic process statistics rather than estimation
  • Enter Ito’s lemma – description based on Taylor series expansion : ignore stochastic nature of function, get Taylor expansion of function value change due to delta-change in single function parameter, get differential of sde-representation of function, big-O analysis and series approximation, get approximated value back to Taylor expansion and voila ! – we have the relation of small change in function of random variable to the small change in the variable itself
  • Consequence of Ito’ lemma – we can relate change in dependent variable (which we can’t observe directly) to the change in observed dependent variable (statistically – these two variables should be perfectly correlated)
  • Obvious application : determining change in option price based on change in stock price (in general – this can be abstracted to any similar problem)
  • We can generalize Ito’s lemma by introducing time-dependency of function (that is – multivariate dependency)
  • Finally – we can derive probability distributions of variables and use standard probability toolkit to relate value range of variables with appropriate probabilities
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