Cox-Ross-Rubinstein model

This is a $binomial$ model. Numeraire is bank account with $S_0 (t) = (1+r)^t$ . Other account is stock, with:

\[ S_1 (t+1) = { \{ \array{ u s_1(t) & prob p \\

Number of periods, $T$ , gives size of $\Omega$ , as $2^T$ . Number of possible values of $S_1(T) = T + 1$ . Let $Z_T = \frac{S_1 (t)}{S_1(t-1)}$ for $t = 1,..., T$

Arbitrage-free and complete

If $d \lt 1+r \lt u$ , the CRR model is arbitrage free and complete with EMM given by:

• $Q(Z_t = u) = q = \frac{(1+r) - d}{u-d}$
• $Q(Z_t = u) = 1-q = \frac{u - (1+r)}{u-d}$

Otherwise the model contains arbitrage.

Proof:

Look at one branch-point, compute EMM:

If:

• $d \lt 1+r \lt u$ then $0 \lt q \lt 1$
• $1+r = d$ or $u$ then measure not equivalent
• $1+r \lt d$ or $1 +r \gt u$ solution is not a probability measure

Pricing some contingent claims

An important class of claims are given by $X = f( S_1(T))$ for some function $f$ (payoff is some function of final value).

European call

: $f(S) = (S - K)_T$ for some constant “strike price” $K$ (Want to buy at T for no more than K, payoff is difference in price above K)

European put

: $f(S) = (K - S)_T$ for some constant “strike price” $K$ (Want to sell at T for no less than K, payoff is difference in price below K)

Collar

Cash or nothing

Path dependent options (‘exotic options’)

Barrier option

eg. “knockout option”, “down and out” (similarly “up and out”, “up and in”, “down and in”

Lookback option

Calims where the payoff involves $\min_t S_i(t)$ or $\max_t S_i(t)$ . Eg. $[ \max_t S_i (t) ] - S_i(T)$ , eliminates regret at having sold at a bad time

Asian options

Based on average price over time: $X = -\frac{1}{T} \sum_{t=0}^{T} S_i(t) + S_i(T)$ . Gives the efects of having sold at the average prices if held in conjunction with 1 share to be sold at time $T$ .

American put

Holder must make a decision when to claim, eg. American put. $X_\tau = [K - S_i(\tau)]_t (1+r)^{T-\tau}$ . Like Euro, excpe that holder must claim the payoff at time $\tau$ (which the holder decides) called the exercise time. $\tau$ may be any rv in $\{ 0, 1, ..., T\}$ satisfying $\{ \tau \gt t\} \in \mathcal{F}_t$ (non-anticipation).

For a given strategy $\tau$ being followed the payoff $X_\tau$ is a random varibale so is attainable. Hence it has arbitrage price process $\pi_{X_\tau} (t) = (1+r)^{t-\tau} E_Q [ X_\tau | \mathcal{F}_t]$ .

Market value assumes the holder is using the bst strategy $\tau$ (ie. the one which maximises the value of the put). So we have an optimisation problem:

\[\max _{X_\tau} (0) s.t. \tau : \Omega \to \{0, 1, ..., T\}$ is a rv with $\{ \tau \gt t \} \in \mathcal{F}_t \forall t \]

Only finitely many possible $\tau$ so an optimal strategy exists.

Consider a node $\nu$ at time $t$ . Should we exercise at $\nu$ ? (can assume no exercise on any path leading to $\nu$ ).

Let $\lambda(\nu) =$ value of $\tilde{\pi}_{X_\tau}$ at $\nu$ when $\tau$ is don’t exercise before time $t$ , then folllow a strategy to maximise $\tilde{\pi}_{X_\tau}$ (if $t=T$ , $\lambda(\nu) = [K - S_1(\nu)]_t[1+r]^-T)$ .

If $t \lt T$ and we exercise now, then we get $[K - S_1 (\nu)]_t (1+r)^-t$ If $t \lt T$ and we wait, then we get $q \lambda(\nu^+) + (1-q)\lambda(\nu^-)$

So, $\lambda(\nu) = max \{ [K- S_1(\nu)]_t (1+r)^-t, q \lambda{\nu^+} + (1-q)\lambda(\nu^-)\}$ .

If $t=0$ then $\lambda(\nu) = \tilde{\pi}_{X_\tau} (0)$ for the optimal strategy that achieves it.

So this calculation lets us find the arbitrage value of the options and the optimal strategy that achieves it.

American call

Like Euro, but payoff can be claimed at time $\tau$ . $X_\tau = [S_i(\tau) - K]_t (1+r)^{T-\tau}$