CRR in real-life

Represent a time $T$ by $n$ discrete steps of length $h$ . Use $t \in [0, T]$ for real-time, the time at step $k$ is $t = kh$ .

Interest

Usually continuously compounded interest, $e^rt$ after time $t$ . If $r_0$ is the interest rate per step in the CRR model, then $1 + r_0 = e^rh$ .

Asset prices

Many asset prices behave (approximately) like CRR model:

• behaviour in disjoint time intervals is independent
• price growth over any fixed interval ~ log-normal. $\log {( \frac{S_i(t)}{S_i(0)} )} ~ N(\mu_t, \sigma^2_t)$ .

Calculating $u$ and $d$

Under the martingale measure we want:

• $E {[ \frac{S_i(t)}{e^rt} ]} = S_1(0)$
• $E {[ \frac{S_i(t)}{S_i(0)} ]} = e^rt$
• $E {[ e^{\mu_t + \sigma_t^2 Z} ]} = e^rt$ , where $Z ~ N(0,1)$
• $e^\mu_t E[ e^\sigma_t^2 Z] = e^rt$
• $e^{\mu_t + \frac{1}{2} \sigma_t^2 } = e^rt$ , as $E {[ e^{\lambda Z} ]} = e^{ \frac{1}{2} \lambda^2}$
• $\mu_t = rt - \frac{1}{2} \sigma_t^2$

$S_1(t) = S_1(0) exp(rt - \frac{1}{2} \sigma^2_t + \sigma_t Z)$

What is $\sigma_t$ ?

Note that after $k$ periods of length $t$ :

• $S_i(kt) = S_1(0) exp( rt - \frac{1}{2} \sigma_t^2 Z_1) \cdot exp( rt - \frac{1}{2} \sigma_t^2 Z_1) \cdot ...$
• $ = S_1(0) exp( rkt - \frac{k}{2} \sigma^2_t + \sigma_t (Z_1 + ... + Z_k))$
• $ = S_1(0) exp( rkt - \frac{k}{2} \sigma^2_t + \sigma_t \sqrt{k} Z)$ .

So $\sigma_kt = \sqrt{k} \sigma_t$ , ie $\sigma_k \propto \sqrt{t}$ . $\sigma_t = \sigma \sqrt{t}$ .

$\sigma$ is called the *volatility parameter of the asset. Typical values for $\sigma$ are $0.1 - 0.4 year^{-1/2}$ .

CRR model

We need to approximate the distribuition of $S_1(0) exp( (r-\frac{1}{2}\sigma^2_t)h + \sigma \sqrt{t} Z)$ by the discrete distribution:

• possible values: $S_i(0) u^j d^{n-j}$ , $j= 0, ..., n$
• with probabilities: $\choose{n}{j} q^j (1-q)^{n-j}$

Choose $u, d$ so that means and variances match:

• $\frac{n}{2} (\log u - \log d ) = (r- \frac{1}{2} \sigma^2)T$
• $nq(1-q)(\log u - \log d )^2 = \sigma^2 T$

or since $T= nh$ .

• $\frac{1}{2} (\log u - \log d ) = (r- \frac{1}{2} \sigma^2)h$
• $q(1-q)(\log(u) - \log d )^2 = \sigma^2 h$

To satisfy first equation, let:

• $\log u = (r - \frac{1}{2}\sigma^2) + \alpha$
• $\log d = (r - \frac{1}{2}\sigma^2) - \alpha$

And note $q = \frac{1+ r - d}{u-d} = \frac{e^rt -d}{u-d}$ , so $\alpha$ must satisfy:

\[ \frac{{( e^{\sigma^2 \frac{h}{2}} - e^{-\alpha} )} {( e^{-\alpha} - e^{\sigma^2 \frac{h}{2}} )}}{ {( e^\alpha - e^{-\alpha} )}^2} 4 \alpha^2 = \sigma^2 h \]

This can be solved numerically, or use CRR approximation, $\alpha = \sigma \sqrt{h}$ , which is very good when $\sigma$ is small (which it usually is).