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Extra info for Bloomberg Businessweek (2 July 2012)
BACKWARD AND FORWARD KOLMOGOROV EQUATIONS In this section no results will be derived per se. Some results, however, relevant to our topic will be stated. Given a generalized price process, dS = µ(S, t) dt + σ (S, t) dz(t), ST = St + t =T dS(t ), t =t with arbitrary drift function and local volatility function, we may ask the following questions: • Given that we start at stock price St at time t, what is the probability distribution over ST at some time T > t? , expectation values of functions of the path end-point or state variable)?
SN+1 ) say, this generates N samples of returns, and a sample of variances of returns ri = ln Si , ri = ln Si+1 − ln Si = ln Si+1 Si , 32 THE MODELS Mean = r = Variance = ( r − Volatility = √ N 1 N ri = i=1 1 N N (ri+1 − ri ), i=1 1 (N − 1) r )2 = N ( ri − r )2 , i=1 variance. The formulae are the same as above; just differing by a variable change. Again, clearly as N → ∞, and the time-step goes to zero (and for a particular security price, S, and particular time, t), we would expect to get Volatility = σ.
2 Plot of a path on the (x, y) plane given by x = cos(π t), y = sin(π t) for 0≤t≤1 25 Preliminary Mathematical Tools Then df = −2xy2 π sin(π t) + 2x2 yπ cos(π t). dt Specifically, along this path the value of this is incorrect. But the expression df = df dt is defined. For another path ∂f ∂f dx + dy ∂x ∂y encompasses all possible paths. Another intuitive meaning for 1-forms is that they are ‘‘really’’ defined under the integral, and we see that along a given path t=1 t=1 df = t=0 t=0 t=1 = t=0 ∂f ∂f dx + dy ∂x ∂y ∂f dx ∂f dy + ∂x dt ∂y dt dt = f (x(t = 1), y(t = 1)) − f (x(t = 0), y(t = 0)).