STOCHASTIC CALCULUS

4.4 STOCHASTIC CALCULUS

Financial and computational mathematics use stochastic processes extensively and thus we are called to manipulate equations of this sort. To do so, we mostly use Ito’s stochastic calculus. The ideas of this calculus are simple and are based on the recognition that the magnitudes of second-order terms of asset prices are not negligible. Many texts deal with the rules of stochastic calculus, including Arnold (1974), Bensoussan (1982, 1985), Bismut (1976), Cox and Miller (1965), Elliot (1982), Ito (1961), Ito and McKean (1967), Malliaris and Brock (1982) and my own (Tapiero, 1988, 1998). For this reason, we shall consider here these rules in an intuitive manner and emphasize their application. Further, for simplicity, functions of time such as x(t) and y(t) are written by x and y except when the time specification differs.

93 The essential feature of Ito’s calculus is Ito’s Lemma. It is equivalent to the

STOCHASTIC CALCULUS

total differential rule in deterministic calculus. Explicitly, state that a functional relationship y = F(x, t), continuous in x and time t, expresses the value of some

economic variable y measured in terms of another x (for example, an option price measured in terms of the underlying stock price on which the option is written, the value of a bond measured as a function of the underlying stochastic

y in calculus, resulting from an application of Taylor series expansion of F(x, t), provides the following relationship:

y=

t+

Of course, having higher-order terms in the Taylor series development yields: ∂ F 1 ∂ 2 F 2 ∂ F 1 ∂ 2 F 2 2 ∂ F

y= ∂ t t+ ∂ t 2 + ∂ x x+ ∂ x 2 +

2 2 ∂ t∂x

2 are non-negligible (since they are also of ∂ F ∂ F 1 ∂ 2 F 2

y= ∂ t t+ ∂ x x+

This is essentially Ito’s differential rule (also known as Ito’s Lemma), as we shall see below for continuous time and continuous state stochastic processes.

4.4.1 Ito’s Lemma

Let y = F(x, t) be a continuous, twice differentiable function in x and t, or ∂ F/∂t, ∂ F/∂ x, ∂ 2 F/∂ x 2 and let {x(t), t ≥ 0} be defined in terms of a stochastic

differential equation with drift f (x, t) and volatility (diffusion) σ (x, t), dx = f (x, t) dt + σ (x, t) dw, x(0) = x 0 , 0≤t≤T

then:

(dx) dF = 2 .

∂ t dt + ∂ x dx + 2 ∂ x 2

Or ∂ F ∂ F 1 ∂ 2 F 2

dF = ∂ t dt + ∂ x [ f (x, t) dt + σ (x, t) dw] + 2 ∂ x 2 [ f (x, t) dt + σ (x, t) dw] Neglecting terms of higher order than dt, we obtain Ito’s Lemma: ∂ F 1 2 2 ∂ F ∂ F

dF = +

f (x, t) + σ (x, t) 2 dt + σ (x, t) dw ∂ t

94 PROBABILITY AND FINANCE

This rule is a ‘work horse’ of mathematical finance in continuous time. Note in particular, that when the function F(.) is not linear, the volatility affects the process drift. Applications to this effect will be considered subsequently. General- izing to multivariate processes is straightforward. For example, for a two-variable

process, y = F(x 1 , x 2 , t ) where {x 1 (t), x 2 (t); t ≥ 0} are two stochastic processes while F admits first- and second-order partial derivatives, then the stochastic total

differential yields: ∂ F ∂ F 1 ∂ 2 F ∂ F 1 ∂ 2 F

∂ x 1 x 2 in which case we introduce the appropriate processes {x 1 (t), x 2 (t); t ≥ 0} and

maintain all terms of order dt. For example, define y = x 1 x 2 , then for this case: ∂ F ∂ F ∂ 2 F ∂ F ∂ 2 F ∂ 2 F

∂ x 1 ∂ x 2 1 ∂ x 2 ∂ x 2 2 ∂ x 1 x 2 which means that: dF = x 2 dx 1 +x 1 dx 2 + dx 1 dx 2 Other examples will be highlighted through application in this and subsequent

chapters. Below, a number of applications in economics and finance are consid- ered.