Financial Engineering

4. Itô’s Calculus

4.1 Differentiation

In the previous chapter, we have observed that a sample Brownian path is nowhere differentiable. In other words, the differentiation dWtdt\frac{d W_t}{dt} does not exist in a classical calculus sense. However, while studying Brownian motions, or when using Brownian motion as a model, the situation of estimating the difference of a function f(Wt)f(W_t) over an infinitesimal interval [t,t+Δt][t, t + \Delta t] arises frequently.

Theorem 4.1 (Itô’s Lemma): Let f(t,x)f(t,x) be a differentiable function, and let XtX_t be a stochastic process satisfying  dXt=μt dt+σt dWt\dd X_t = \mu_t \dd t + \sigma_t \dd W_t for a Brownian motion WtW_t. Then  df(t,Xt)=(∂f∂t+μt∂f∂x+12σt2∂2f∂x2) dt+σt∂f∂x dWt \dd f(t,X_t) = \pa{\frac{\partial f}{\partial t} + \mu_t \frac{\partial f}{\partial x} + \frac{1}{2} \sigma_t^2 \frac{\partial^2 f}{\partial x^2}} \dd t + \sigma_t \frac{\partial f}{\partial x} \dd W_t
Proof: We expand f(t+Δt,x+Δx)f(t + \Delta t, x + \Delta x) using Taylor f(t+Δt,x+Δx)−f(t,x)=∂f∂tΔt+∂f∂xΔx+12∂2f∂t2(Δt)2+∂2f∂t∂xΔtΔx+12∂2f∂x2(Δx)2+…\begin{align*} & f(t + \Delta t, x + \Delta x) - f(t,x) \\ ={} & \frac{\partial f}{\partial t} \Delta t + \frac{\partial f}{\partial x} \Delta x + \frac{1}{2} \frac{\partial^2 f}{\partial t^2} (\Delta t)^2 + \frac{\partial^2 f}{\partial t \partial x} \Delta t \Delta x + \frac{1}{2} \frac{\partial^2 f}{\partial x^2} (\Delta x)^2 + \ldots \end{align*} For Δt→ dt\Delta t \to \dd t and Δx→ dx\Delta x \to \dd x, we have  df(t,x)=∂f∂t dt+∂f∂x dx+12∂2f∂t2 dt2+∂2f∂t∂x dt dx+12∂2f∂x2 dx2+…=∂f∂t dt+∂f∂x dx+12∂2f∂x2 dx2+o(dt)\begin{align*} \dd f(t,x) &= \frac{\partial f}{\partial t} \dd t + \frac{\partial f}{\partial x} \dd x + \frac{1}{2} \frac{\partial^2 f}{\partial t^2} \dd t^2 + \frac{\partial^2 f}{\partial t \partial x} \dd t \dd x + \frac{1}{2} \frac{\partial^2 f}{\partial x^2} \dd x^2 + \ldots \\ &= \frac{\partial f}{\partial t} \dd t + \frac{\partial f}{\partial x} \dd x + \frac{1}{2} \frac{\partial^2 f}{\partial x^2} \dd x^2 + o(\mathrm{d} t) \end{align*} As  dXt=μt dt+σt dWt\dd X_t = \mu_t \dd t + \sigma_t \dd W_t, we have  dXt2=(μt dt+σt dWt)2=μt2 dt2+2μtσt dt dWt+σt2 dWt2=σt2 dt+o(dt)\begin{align*} \dd X_t^2 &= (\mu_t \dd t + \sigma_t \dd W_t)^2 \\ &= \mu_t^2 \dd t^2 + 2 \mu_t \sigma_t \dd t \dd W_t + \sigma_t^2 \dd W_t^2 \\ &= \sigma_t^2 \dd t + o(\mathrm{d} t) \end{align*} where we used  dWt→0\dd W_t \to 0 as Brownian paths are continuous as well as the fact that the local quadratic variation  dWt2\dd W_t^2 is  dt\dd t. Therefore, omitting o(dt)o(\mathrm{d} t), we have  df(t,Xt)=∂f∂t dt+∂f∂x dXt+12∂2f∂x2 dXt2=∂f∂t dt+∂f∂x(μt dt+σt dWt)+12∂2f∂x2(σt2 dt)=(∂f∂t+μt∂f∂x+12σt2∂2f∂x2) dt+σt∂f∂x dWt\begin{align*} \dd f(t,X_t) &= \frac{\partial f}{\partial t} \dd t + \frac{\partial f}{\partial x} \dd X_t + \frac{1}{2} \frac{\partial^2 f}{\partial x^2} \dd X_t^2 \\ &= \frac{\partial f}{\partial t} \dd t + \frac{\partial f}{\partial x} \pa{\mu_t \dd t + \sigma_t \dd W_t} + \frac{1}{2} \frac{\partial^2 f}{\partial x^2} \pa{\sigma_t^2 \dd t} \\ &= \pa{\frac{\partial f}{\partial t} + \mu_t \frac{\partial f}{\partial x} + \frac{1}{2} \sigma_t^2 \frac{\partial^2 f}{\partial x^2}} \dd t + \sigma_t \frac{\partial f}{\partial x} \dd W_t \end{align*}
Note: The stochastic process Xt=μt+σWtX_t = \mu t + \sigma W_t is known as the Brownian motion with drift μ\mu and volatility σ\sigma.
Proposition 4.2 (Itô’s Lemma for Brownian Motion): Let Xt=WtX_t = W_t, then Itô’s Lemma simplifies to  df(t,Xt)=∂f∂t dt+∂f∂x dWt+12∂2f∂x2 dt \dd f(t,X_t) = \frac{\partial f}{\partial t} \dd t + \frac{\partial f}{\partial x} \dd W_t + \frac{1}{2} \frac{\partial^2 f}{\partial x^2} \dd t
Example (Univariate quadratic function): Consider the function f(t,x)=12x2f(t,x) = \frac{1}{2} x^2 and Xt=WtX_t = W_t. Then  df(t,Xt)=∂f∂t dt+∂f∂x dWt+12∂2f∂x2 dt=Wt dWt+12 dt\begin{align*} \dd f(t,X_t) &= \frac{\partial f}{\partial t} \dd t + \frac{\partial f}{\partial x} \dd W_t + \frac{1}{2} \frac{\partial^2 f}{\partial x^2} \dd t \\ &= W_t \dd W_t + \frac{1}{2} \dd t \end{align*} Note how this contradicts classical derivation rules, i.e.  df(Wt) dWt≠Wt\frac{\dd f(W_t)}{\dd W_t} \neq W_t.
Example (Multivariate quadratic function): Consider the function f(t,x)=t2+x2f(t,x) = t^2 + x^2. Then  df(t,Xt)=(∂f∂t+μt∂f∂x+12σt2∂2f∂x2) dt+σt∂f∂x dWt=(2t+2μtXt+σt2) dt+2σtXt dWt\begin{align*} \dd f(t,X_t) &= \pa{\frac{\partial f}{\partial t} + \mu_t \frac{\partial f}{\partial x} + \frac{1}{2} \sigma_t^2 \frac{\partial^2 f}{\partial x^2}} \dd t + \sigma_t \frac{\partial f}{\partial x} \dd W_t \\ &= \pa{2t + 2\mu_t X_t + \sigma_t^2} \dd t + 2\sigma_t X_t \dd W_t \end{align*}
Example (Exponential function): Consider the function f(t,x)=exp⁡(Xt)f(t,x) = \exp{X_t}. Then  df(t,Xt)=(∂f∂t+μt∂f∂x+12σt2∂2f∂x2) dt+σt∂f∂x dWt=(μt+12σt2)exp⁡(Xt) dt+σtexp⁡(Xt) dWt=(μt dt+12σt2 dt+σt dWt)exp⁡(Xt)\begin{align*} \dd f(t,X_t) &= \pa{\frac{\partial f}{\partial t} + \mu_t \frac{\partial f}{\partial x} + \frac{1}{2} \sigma_t^2 \frac{\partial^2 f}{\partial x^2}} \dd t + \sigma_t \frac{\partial f}{\partial x} \dd W_t \\ &= \pa{\mu_t + \frac{1}{2} \sigma_t^2} \exp{X_t} \dd t + \sigma_t \exp{X_t} \dd W_t \\ &= \pa{\mu_t \dd t + \frac{1}{2} \sigma_t^2 \dd t + \sigma_t \dd W_t} \exp{X_t} \end{align*} Fixing σt=σ\sigma_t = \sigma and μt=μ−12σ2\mu_t = \mu - \frac{1}{2} \sigma^2 we obtain  df(t,Xt)=(μ dt+σ dWt)f(t,Xt)\dd f(t,X_t) = \pa{\mu \dd t + \sigma \dd W_t} f(t, X_t) where Xt=(μ−12σ2)t+σWtX_t = \pa{\mu - \frac{1}{2} \sigma^2} t + \sigma W_t.
Note: Given XtX_t from the last example, we define the geometric Brownian motion St=s0⋅exp⁡(Xt)=s0⋅exp⁡((μ−12σ2)t+σWt) S_t = s_0 \cdot \exp{X_t} = s_0 \cdot \exp{\pa{\mu - \frac{1}{2} \sigma^2} t + \sigma W_t} where s0∈Rs_0 \in \R is the constant initial value. Note that StS_t solves the stochastic differential equation of the return price model  dStSt=μ dt+σ dWt\frac{\dd S_t}{S_t} = \mu \dd t + \sigma \dd W_t introduced in the previous chapter.

4.2 Integration

We aim to provide an intuitive understanding of integration in the context of stochastic processes.

Recap (L2L^2-norm of a random variable): The L2L^2-norm of a random variable YY is defined as ∥Y∥=E[X2]\norm{Y} = \sqrt{\mathbb{E}[X^2]}.

In similar fashion, we define the L2L^2-norm of a process {Xt}t∈[0,T]\cb{X_t}_{t \in [0,T]}.

Definition 4.3 (L2L^2-norm of a stochastic process): The L2L^2-norm of a process {Xt}t∈[0,T]\cb{X_t}_{t \in [0,T]} is defined as ∥X∥=∫0TE[Xt2] dt\norm{X} = \sqrt{\int_0^T \mathbb{E}\bk{X_t^2} \dd t}.

We define the concept of a random step function to approximate a stochastic process.

Definition 4.4 (Random step function): A stochastic process {Xt}t∈[0,T]\cb{X_t}_{t \in [0,T]} can be approximated by any degree of accuracy by a random step function {Xtk(n)}k∈{0,…,n}\cb{X_{t_k}^{(n)}}_{k \in \cb{0, \ldots, n}} with intervals of length Tn\frac{T}{n}.
Note: Approximated by any degree of accuracy means that the sequence {X(n)}n∈N\cb{X^{(n)}}_{n \in \N} is Cauchy, i.e it converges to XX in the L2L^2-norm.

With the random step function definition, we can define the discrete stochastic integral.

Definition 4.5 (Discrete stochastic integral): Let {Xt}t∈[0,T]\cb{X_t}_{t \in [0,T]} be a stochastic process with ∥X∥<∞\norm{X} < \infty. The discrete stochastic integral is defined as I(X(n))=∑k=0n−1Xtk(n)(Wtk+1−Wtk) I\pa{X^{(n)}} = \sum_{k=0}^{n-1} X_{t_{k}}^{(n)} \pa{W_{t_{k+1}} - W_{t_k}}

With that and with convergence in L2L^2-norm, we can define the Itô integral.

Definition 4.6 (Itô integral): Let {Xt}t∈[0,T]\cb{X_t}_{t \in [0,T]} be a stochastic process. The Itô stochastic integral I(X)=∫0TXt dWt I(X) = \int_0^T X_t \dd W_t is defined as the limit in L2L^2-norm of the discrete stochastic integral I(X(n))I(X^{(n)}), i.e. I(X)=L2lim⁡n→∞I(X(n))I(X) \lneq{2} \lim_{n \to \infty} I\pa{X^{(n)}}.
Note: We note that the stochastic integral depends on the choice of the random step function evaluation. To show that, let Xtk(n)=WτkX_{t_k}^{(n)} = W_{\tau_k} with τk=(1−α)tk+αtk+1\tau_k = (1 - \alpha) t_k + \alpha t_{k+1}. Then one can prove that ∫0TWt dWt=12WT2+(α−12)T \int_0^T W_t \dd W_t = \frac{1}{2} W_T^2 + \pa{\alpha - \frac{1}{2}} T For the Itô integral we have α=0\alpha = 0. This integral is non-anticipating and therefore works well in a financial setting.
Proposition 4.7 (Itô Integral for separable integrand): If the integrand  dYt\dd Y_t is separable, i.e.  dYt=g(t) dt+h(Wt) dWt\dd Y_t = g(t) \dd t + h(W_t) \dd W_t for a Brownian motion WtW_t, then ∫tatb dYt=G(tb)−G(ta)+H(Wtb)−H(Wta) \int_{t_a}^{t_b} \dd Y_t = G(t_b) - G(t_a) + H(W_{t_b}) - H(W_{t_a}) where G(t)=∫g(t) dtG(t) = \int g(t) \dd t and H(x)=∫h(x) dxH(x) = \int h(x) \dd x.
Example (Itô integral of Brownina motion): We want to figure out the Itô integral of a Brownian motion from t0t_0 to TT, i.e. ∫t0TWt dWt\int_{t_0}^T W_t \dd W_t, i.e.  dYt=Wt dWt\dd Y_t = W_t \dd W_t. We define f(t,x)=x2f(t,x) = x^2 and Xt=WtX_t = W_t and note using Itô’s Lemma for differentiation that  dWt2= df(t,Xt)=2Wt dWt+ dt  ⟹   Wt dWt=12 dWt2−12 dt\begin{align*} & \dd W_t^2 = \dd f(t,X_t) = 2 W_t \dd W_t + \dd t \\ \implies ~ & W_t \dd W_t = \frac{1}{2} \dd W_t^2 - \frac{1}{2} \dd t \end{align*} Hence the integrand is separable and we conclude ∫0TWt dWt=12(WT2−W02)−12(T−t0) \int_0^T W_t \dd W_t = \frac{1}{2} \pa{W_T^2 - W_0^2} - \frac{1}{2} (T - t_0)

We take a look at the properties of the Itô integral.

Proposition 4.8 (Isometry of the Itô integral): Let {Xt}t∈[0,T]\cb{X_t}_{t \in [0,T]} be a stochastic process. Then E[(∫0TXt dWt)2]=∫0TE[Xt2] dt \E\bk{\pa{\int_0^T X_t \dd W_t}^2} = \int_0^T \E\bk{X_t^2} \dd t or equivalently written in norms ∥I(X)∥=∥X∥\norm{I(X)} = \norm{X}.
Proposition 4.9 (Properties of the Itô integral): Let {Xt}t∈[0,T]\cb{X_{t}}_{t \in [0,T]} and {X~t}t∈[0,T]\{\tilde{X}_{t}\}_{t \in [0,T]} be stochastic processes for which the Itô integral is well-defined. The following properties hold:
  1. linearity: ∫0TαXt+βX~t dWt=α∫0TXt dWt+β∫0TX~t dWt\int_0^T \alpha X_{t} + \beta \tilde{X}_{t} \dd W_t = \alpha \int_0^T X_{t} \dd W_t + \beta \int_0^T \tilde{X}_{t} \dd W_t
  2. expectation: E[∫0TXt dWt]=0\E\bk{\int_0^T X_t \dd W_t} = 0
  3. variance: Var⁡[∫0TXt dWt]=∫0TE[Xt2] dt\Var\bk{\int_0^T X_t \dd W_t} = \int_0^T \E\bk{X_t^2} \dd t
  4. martingale property: E[∫0TXt dWt∣Fu]=∫0uXt dWt\E\bk{\int_0^T X_t \dd W_t \mid F_u} = \int_0^u X_t \dd W_t for u∈[0,T]u \in [0,T]
  5. product property: E[∫0TXt dWt ∫0TX~t dWt]=∫0TE[XtX~t] dt\E\bk{\int_0^T X_t \dd W_t ~ \int_0^T \tilde{X}_t \dd W_t} = \int_0^T \E\bk{X_t \tilde{X}_t} \dd t

4.3 Generalization

Theorem 4.10 (Itô’s lemma for multidimensional processes): Let f:RN→Rf: \mathbb{R}^N \to \mathbb{R} be a function in C2\mathcal{C}^2 and {Xt}t={(Xt(1),…,Xt(N))}t\cb{\boldsymbol{X}_t}_{t} = \cb{\pa{X_t^{(1)}, \ldots, X_t^{(N)}}}_{t} be an NN-dimensional stochastic process. Then  df(Xt)=∑i=1N∂f∂xi dXt(i)+12∑i=1N∑j=1N∂2f∂xi∂xj d[X(i),X(j)]t \dd f(\boldsymbol{X}_t)=\sum_{i=1}^N \frac{\partial f}{\partial x_i} \dd X_t^{(i)}+\frac{1}{2} \sum_{i=1}^N \sum_{j=1}^N \frac{\partial^2 f}{\partial x_i \partial x_j} \dd [X^{(i)}, X^{(j)}]_t or equivalently f(Xtb)=f(Xta)+∑i=1N∫tatb∂f∂xi(Xs) dXs(i)+12∑i=1N∑j=1N∫tatb∂2f∂xi∂xj(Xs) d[X(i),X(j)]s f(\boldsymbol{X}_{t_b}) = f(\boldsymbol{X}_{t_a}) + \sum_{i=1}^N \int_{t_a}^{t_b} \frac{\partial f}{\partial x_i}(\boldsymbol{X}_s) \dd X_s^{(i)} + \frac{1}{2} \sum_{i=1}^N \sum_{j=1}^N \int_{t_a}^{t_b} \frac{\partial^2 f}{\partial x_i \partial x_j}(\boldsymbol{X}_s) \dd [X^{(i)}, X^{(j)}]_s
Proposition 4.11 (Itô’s product rule): Let XtX_t and YtY_t be two stochastic processes. Then  d(XtYt)=Xt dYt+Yt dXt+ d[X,Y]t \dd (X_t Y_t) = X_t \dd Y_t + Y_t \dd X_t + \dd [X,Y]_t