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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)).

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