Definition 5.1 (Itô Stochastic Integral)

Partition $[a,b]$ as $a=t_{0}<t_{1}<\ldots <t_{n-1}<t_{n}=b$ and set $\Delta t=t_{i+1}-t_{i}=(b-a)/n$, $\Delta W_{t_{i}}=W_{t_{i+1}}-W_{t_{i}}$. The Itô stochastic integral of $g$ is defined as

where $l.i.m.$ denotes mean square convergence. In other words, if $G_{n}=\sum _{i=0}^{n-1}g(t_{i})\Delta W_{t_{i}}$ and $\mathcal{I}=\int _{a}^{b}g(t)\, dW_{t}$ then $l.i.m._{n\to \infty }G_{n}=\mathcal{I}$ means that

Simple Itô integrals

Simple Itô stochastic integrals can be verified directly from the Definition 5.1. For example taking $g(t)=1$ gives

where the last line follows since $t_0=a$ and $t_n=b$ irrespective of the partition. Similarly, taking $a=0$ and $b=t$ gives

since $W_0=0$ and we used $\tau$ as a dummy variable for time since $t$ is the upper limit of the integral. Now take $g(t)=k$ for some constant $k$ to see that

Comments

• Solutions of Itô integrals are stochastic processes.

• Note that a general Itô stochastic integral of the form

  $$X_{t}=X_0+\int _{0}^{t}g(\tau )dW_{\tau }$$

  can be written in differential form as

  $$dX_{t}= g(t)dW_t \, .$$

  This latter form is called the stochastic differential representation of $X_t$. In the following section we will generalise this construct.

Definition 5.2

A stochastic process $X_{t}$ is said to satisfy an Itô stochastic differential equation (S.D.E.), if we may write

Comments

• The corresponding integral representation is

  $$X_{t}=X_{0}+\int _{0}^{t}\alpha (\tau ,X_{\tau })\, d\tau + \int _{0}^{t}\beta (\tau ,X_{\tau })\, dW_{\tau }\, .$$

• Note that the first integral above is the usual Riemann integral and the second is an Itô stochastic integral.

• The $\alpha$ term is known as the drift coefficient and $\beta ^2$ as the diffusion coefficient.

• If $\alpha (t,X_{t})$ and $\beta (t,X_{t})$ do not depend explicitly on $t$ then the S.D.E. is said to be time-homogeneous.

Simple S.D.E.s

• Suppose that $X_0=0$, $\alpha (t,X_{t})=0$ and $\beta (t,X_{t})=1$. Then

  $$dX_{t}=dW_{t}.$$

  Integrating both sides of the above between $0$ and $t$ gives

  $$\begin{eqnarray*} \int _{0}^{t}dX_{\tau } & =& \int _{0}^{t}dW_{\tau }\\ \Rightarrow X_{t}-X_{0}& =& W_t-W_0 \end{eqnarray*}$$

  for which we see that using $X_0=0$ and $W_0=0$ gives $X_t=W_t$, a standard Brownian motion process.

• Take $X_0=x_0$, $\alpha (t,X_{t})=a$ and $\beta (t,X_{t})=b$ where $x_0$, $a$ and $b$ are real numbers. The resulting S.D.E. is

  $$dX_{t}=a\, dt+b\, dW_{t}\, .$$

  Integrating both sides of the above between $0$ and $t$ gives

  $$\begin{eqnarray*} \int _{0}^{t}dX_{\tau } & =& \int _{0}^t a d\tau +b\int _{0}^{t}dW_{\tau }\\ \Rightarrow X_{t}-X_{0}& =& at+b(W_t-W_0) \end{eqnarray*}$$

  for which we obtain

  $$X_{t}=x_0+a\, t+b\, W_{t}$$

  and we recognise $X_t$ as a generalised Brownian motion process.

• We would like to be able to write down a S.D.E. for the stock price process $S_{t}$, assuming it follows a geometric B.M. We know that $S_{t}=\exp (X_{t})$ where $X_{t}$ is a generalised Brownian motion with initial condition $\log S_0$, drift $\mu -\sigma ^{2}/2$ and diffusion coefficient $\sigma ^2$ and we know $dX_{t}$. Hence, we just need a way of differentiating $\exp (X_{t})$ in a sensible way. Essentially, we need a chain rule for S.D.E.s. This chain rule is the Itô Formula.

Itô Formula

Consider an S.D.E. of the form

Let $G(t,x)$ be a real valued function with continuous partial derivatives

then

Sketch proof (not examinable)

To give an indication of why this is the case, write

Now take a Taylor series expansion of $G(t+\Delta t,x+\Delta x)$ about $(t,x)$ to give

where $G_{tt}=\partial ^{2}G/\partial t^{2}$ and $G_{tx}=\partial ^{2}G/\partial t \partial x$. Now replace $\Delta x$ with $\alpha \Delta t+\beta \Delta W$ and $(\Delta x)^{2}$ by $\alpha ^{2}(\Delta t)^{2}+2\alpha \beta \Delta t\Delta W+\beta ^{2}(\Delta W)^{2}$ to obtain

Now approximate $(\Delta W)^{2}$ by $\Delta t$ (and note in fact that $\textrm{E}[(\Delta W^{2})]=\Delta t$) to write the preceding expression as

Itô’s formula then follows by letting $\Delta t\to 0$. The above sketch proof is not examinable.

Comments

• We can write down Itô’s formula for a function of Brownian motion $W_{t}$ by noting that $W_{t}$ satisfies an S.D.E. with $X_{t}=W_{t}$, $\alpha (t,X_{t})=0$ and $\beta (t,X_{t})=1$. Hence for a function $G(t,W_{t})$ we obtain

  $$\begin{equation}\label{ItoBM} dG(t,W_{t})=\left(G_{t}(t,W_{t})+\frac{1}{2}G_{xx}(t,W_{t})\right)\, dt+G_{x}(t,W_{t})\, dW_{t}\, .\tag{0}\end{equation}$$

  This is known as the special case of Itô formula and is appropriate for deriving the S.D.E. satisfied by a function of $t$ and $W_t$.

Example 5.1 (Two applications of Itô formula)

Deriving the S.D.E. satisfied by G.B.M.

Consider again the task of formulating geometric Brownian motion (to model stock price $S_{t}$) as an S.D.E. We can write $S_{t}=\exp (X_{t})$ where

and $X_0=\log (S_0)$. The S.D.E. satisfied by the $X_t$ process is then given by

Hence, we derive an S.D.E. for $S_{t}$ by applying Itô’s formula with $G(t,x)=e^x$. Identify

Hence

and applying Itô’s formula gives

Now, writing $S_{t}=e^{X_{t}}$, we obtain the usual S.D.E. for geometric Brownian motion as

Solving the S.D.E. satisfied by G.B.M.

We can solve the above S.D.E. explicitly (in the sense that we can obtain a closed form expression for $S_{t}$). Take the S.D.E. satisfied by G.B.M. and write

which suggests that the solution involves $\log (S_t)$. We can check this hunch via Itô formula. Apply the Itô formula to $\log (S_{t})$ by taking $G(t,x)=\log (x)$ and identify

Hence,

Our hunch is just about correct, except for the extra term involving $\sigma ^2$ appearing on the RHS. Nevertheless, we may still proceed by integrating both sides of the preceding equation to give

which is exactly what we would expect.

Example 5.2

Using Itô’s formula, solve

Solution

We would expect the solution to the above integral to contain a $W_{t}^{2}$ term. Therefore we find $d(W_{t}^{2})$ via Itô’s formula for functions of Brownian motion, given by equation (0).

Set $G(t,x)=x^{2}$ and calculate $$G_{t}(t,W_{t})=0\, ,\qquad G_{x}(t,W_{t})=2W_{t}\, ,\qquad G_{xx}(t,W_{t})=2\, .$$ Hence, we obtain $$\begin{eqnarray*} d(W_{t}^{2})& =& dt+2W_{t}\, dW_{t}\\ \Rightarrow W_{t}\, dW_{t}& =& \frac{1}{2}d(W^{2}_{t})-\frac{1}{2}dt\\ \Rightarrow \int _{0}^{t}W_{\tau }\, dW_{\tau }& =& \frac{1}{2}\int _{0}^{t}d(W^{2}_{\tau })-\frac{1}{2}\int _{0}^{t}d\tau \\ & =& \frac{1}{2}W_{t}^{2}-\frac{1}{2}t\, . \end{eqnarray*}$$

Example 5.3

Show that $Y_{t}=t^{3}+t^{2}W_{t}^{4}$ is an Itô process by writing it in the form

for suitable choices of the processes $\alpha _{t}$ and $\beta _{t}$.

Solution

We use the special case of Itô’s formula (0) with $G(t,x)=t^{3}+t^{2}x^{4}$. Hence $$G_{t}(t,W_{t})=3t^{2}+2tW_{t}^{4}\, ,\, \, G_{x}(t,W_{t})=4t^{2}W_{t}^{3}\, ,\, \, G_{xx}(t,W_{t})=12t^{2}W_{t}^{2}$$ and we get $$d(t^{3}+t^{2}W_{t}^{4})=dY_{t}=\left(3t^{2}+2tW_{t}^{4}+6t^{2}W_{t}^{2}\right)\, dt+4t^{2}W_{t}^{3}\, dW_{t}$$ which is in the form required.

Comment

• Taking $\Delta t$ smaller and smaller gives increased accuracy at greater computational expense.

Assessing the accuracy of the Euler-Maruyama scheme (not examinable)

Suppose that we’re interested in the accuracy of the numerical scheme at some (maturity) time $T$. Let $X_T^{\Delta t}$ denote the Euler-Maruyama approximation of $X_T$ at time $T$, based on a discretisation (step-size) of $\Delta t$.

The numerical scheme is strongly convergent if

The numerical scheme is weakly convergent if

for polynomials $g(\cdot )$.

The Euler-Maruyama scheme is strongly and weakly convergent subject to suitable conditions on the drift and diffusion coefficient. We may also wonder how good the approximation is for a particular $\Delta t$. The numerical scheme is strongly convergent with order $\delta$ if

where the constant $C_T$ depends on $T$ and the considered SDE. The numerical scheme is weakly convergent with order $\delta$ if

where $C_T^g$ depends on $T$, $g$ and the considered SDE. If a numerical scheme is convergent with order $\delta$ and we make the step size $k$ times smaller, the approximation error will decrease by a factor $k^\delta$.

The Euler-Maruyama scheme is weakly convergent with order 1 and strongly convergent with order 0.5. Therefore, the order equal 1 means that if we want to decrease the error 100 times, we have to make the step 100 times smaller. The order equal to 0.5 means that if we want to decrease the error 100 times, we have to make the step $100^2 = 10000$ times smaller. Consequently, the Euler-Maruyama scheme is useful for pricing options whose payoff is not path-dependent. Where the payoff is path-dependent, a higher order scheme (e.g. the Milstein scheme) should be used.

Rendleman and Bartter Model

In the Rendleman and Bartter model, the short rate is governed by the S.D.E.

This means that $r_{t}$ follows a geometric Brownian motion and is the same type of process that we assumed for stock price in Section 1. This assumption is a good starting point; for example, the model ensures that interest rates are positive. However, some key properties are also lacking, such as mean reversion.

Definition 5.3 (mean reversion)

When interest rates are high, the economy tends to slow down and there is a low demand for funds from borrowers. As a result, rates go down. When rates are low, there is a high demand from borrowers and rates tend to rise. This is known as mean reversion. The effect is negative drift when $r_{t}$ is high and positive drift when $r_{t}$ is low. Overall, interest rates appear to be pulled back to some long-run average level over time.

Comment

The Rendleman and Bartter model does not incorporate mean reversion since plainly,

for $\mu$ positive. If $\mu$ is negative,

Neither scenario is compatible with the definition of mean reversion.

Vasicek Model

Here, the short rate is modelled by

where $a,b$ and $\sigma$ are positive real numbers. Note that the model incorporates mean reversion and $r_{t}$ is pulled to a level $b$ at rate $a$. The process is also known as a Gaussian Ornstein-Uhlenbeck process.

To see that the Vasicek model is mean reverting, we apply Itô’s formula with $G(t,x)=xe^{at}$. We obtain

Hence,

Thus,

To proceed, we need (the first part) of the following result for square-integrable functions $g(t)$.

Result 5.1

• (i)

  $\textrm{E}\left(\int _{0}^{t}g(\tau )dW_{\tau }\right)=0$

• (ii)

  $\textrm{E}\left(\left[\int _{0}^{t}g(\tau )dW_{\tau }\right]^{2}\right)=\int _{0}^{t}g^{2}(\tau )d\tau$.

Note that part (ii) is known as the Itô isometry. Justification of this result is left as an exercise. Combining the two results and noting that Itô integrals involve combinations of normal random variables leads to part (iii) that

• (iii)
  $$\int _{0}^{t}g(\tau )dW_{\tau } \sim N\left(0\, ,\, \int _{0}^{t}g^{2}(\tau )d\tau \right).$$

Now, returning to the Vasicek model and using the fact that $\textrm{E}\left(\int _{0}^{t}g(\tau )dW_{\tau }\right)=0$, we have

which is just $b$ in the limit as $t\to \infty$. The Vasicek model is mean reverting.

As an interesting exercise, we may also find the variance of $r_t$. We have using (ii) of Result 5.1 that

Part (iii) of Result 5.2 then gives

This distribution can be sampled recursively to give a skeleton path of the process (that is, a realisation of the process at discrete times). Two such realisations can be seen in Figure 2. The effect of mean reversion is apparent, as is the effect of $a$, since the larger the value of $a$ results in stronger reverting behaviour. It should also be clear that the Vasicek model allows for negative interest rates, which is an undesirable property.

Cox, Ingersoll and Ross Model

In the model proposed by Vasicek, the short term rate can become negative. Whilst this is not considered impossible by some (in practical terms), others consider it to be unsatisfactory. Cox, Ingersoll and Ross propose an alternative model where rates are always nonnegative,

where $a,b$ and $\sigma$ are positive real numbers. The process is mean reverting and the standard deviation of the change in interest rate in a short time period is proportional to $\sqrt{r_{t}}$. Consequently, as short rate increases, so does the standard deviation. Although the S.D.E. can be solved analytically, this task is beyond the scope of the course. Therefore, skeleton paths were generated using the Euler-Maruyama scheme and plotted in Figure 3.

Finally, note that the process is non-negative. To see this intuitively (rather than formally), note that the infinitesimal variance $\sigma ^2 r_t$ tends to zero as the process approaches zero (from above). We expect this to be the case for non-negative processes, otherwise fluctuations around zero due to the non-negligible stochastic noise term near zero will send the process negative.

Example 5.4

Suppose that stock price follows a S.V. model with parameters $q=0.0$, $a=0.2$, $b=0.1$ and $\sigma =0.02$. Suppose further that the initial price of stock is $\pounds 40$, the risk free interest rate is $r=0.05$, and we wish to find the fair price of the Asian call option with strike price $K=50$ and maturity $T=1$ year.

We estimate the fair price via Monte Carlo by performing the following steps:

1. Simulate daily prices $S^{d}_{0},S^{d}_{1},\ldots ,S^{d}_{252}$ in the risk-neutral world. For the S.V. model, this means setting $\mu =r$.

2. Calculate the payoff at time $T=1$:

   $$\left(\sum _{i=1}^{252}\frac{S^{d}_{i}}{252}-K\right)^{+} \, .$$

3. Repeat steps 1 and 2 to get $N$ sample values of the payoff.

4. Calculate the mean of these sample payoffs to get an estimate of the expected payoff.

5. Discount the expected payoff at the risk-free interest rate ($r=0.05$ here) to get an estimate of the value of the option.

The following $\mathsf{R\; }$function performs the algorithm:

sv=function(T=1,dt=1/252,s0=c(50,0.1),r=0.05,a=0.2,b=0.1,sigma=0.02,k=50,N=1000) 
{
 n=T/dt
 sdt=sqrt(dt)
 s=matrix(0,ncol=2,nrow=n+1)
 payoff=vector("numeric",len=N)
 for(j in 1:N)
 { 
  s[1,]=s0
  for(i in 2:(n+1))
  {
   s[i,2]=s[i-1,2]+a*(b-s[i-1,2])*dt+sigma*sqrt(s[i-1,2])*rnorm(1,0,sdt)
  }
  for(i in 2:(n+1))
  {
   s[i,1]=s[i-1,1]+r*s[i-1,1]*dt+sqrt(s[i-1,2])*s[i-1,1]*rnorm(1,0,sdt)
  }
  payoff[j]=max(mean(s[,1])-k,0)
 }
 exp(-r*T)*mean(payoff)
}

A single execution of this function gave an estimate of $\pounds 3.80$ for the fair price.