Multiperiod models

• Finite sample space $\Omega = \{\omega_1, ..., \omega_n\} $ with prob $P(\{ \omega_i \}) \gt 0$ .
• filtration $\mathcal{F}_n$ which represents $T$ periods.
• $d$ assets, with prices $S_i(t)$ at time $t$ , $S_i(t) \in \mathcal{F}_t$
• asset 0 (numeraire), $S_0 (0) = 1$ $S_0 (t) \gt 0 \forall t, S_0 (t) = (1 + r)^t$

A trading strategy is a sequence of random vectors $(\phi_0(t), \phi_1(t), ..., \phi_d(t))$ for $t= 1, ..., T$ where $\phi_j(t) $ = amount of asset $j$ to hold from $t-1$ to $t$ , which is predictable ( $\phi_j(t) \in \mathcal{F}_{t-1}$ . This gives the portfolio to hold at all times.

• value of a strategy at time $t$ is $V_\phi (t) = \sum \phi(t) S_i(t)$
• change in value due to movements in price during period $t$ is $\sum \phi(t) (S(t) - S(t-1))$
• strategy is self-financing if $\phi(t) S(t) = \phi(t+1) S(t)$ – no money added or removed.
• gains process is $G_\phi (t) \sum^t_{s=1} \phi(s) (S(s) - S(s-1))$
• for self-financing strategies $V_\phi (t) = V_\phi (0) + G_\phi (t)$ .

Discounted values are $\tilde{V}_\phi (t) = \frac{V_\phi (t)}{S_0 (t)}$ , and similarly for prices and gains.

Lemma: A strategy $\phi$ is self-financing wrt prices $\iff$ it is self-financing with respect to discounted prices.

Lemma: A strategy $\phi$ is self-financing $\iff V_\phi (t) = V_\phi (0) + G_\phi (t) \iff \tilde{V}_\phi (t) = \tilde{V}_\phi (0) + \tilde{G}_\phi (t)$

Artbitrage

A trading strategy is an arbitrage opportunity if:

• it is self-financing
• $V_\phi (0) = 0$
• $V_\phi (T) \ge 0$ with $P(V_\phi (T) \ge 0) \gt 0$

(All arbitrage opportunities are local – use same techniques as single period models)

Martingale measures

A probability measure $Q$ is a martingale measure if all discounted asset prices $\tilde{S}_i (t)$ are martingales when $Q$ is used.

Theorem: The model is arbitrage free $\iff$ there is a martingale measure $Q$ equivalent to the original $P$ .

Proof:

$\Leftarrow$ : Let $\phi$ be a self-financing strategy. Then $\tilde{V}_\phi (t)$ is a martingale when $Q$ is used. \[ \tilde{V}_\phi (t) = \tilde{V}_\phi (0) + \sum^t_{s=1} \phi(s) \cdot \left( \tilde{S}(s) - \tilde{S}(s-1) \right) \] \[ = \tilde{V}_\phi (0) + \sum^d_{r=1} \sum^t_{s=1} \phi(s) \cdot \left( \tilde{S}_i(s) - \tilde{S}_d(s-1) \right) \] (the sum of a martingale is a martingale) so $E_Q [\tilde{V}_\phi (t+1) | \mathcal{F}_t] = \tilde{V}_\phi (t)$ and even $E_Q [\tilde{V}_\phi (T)] = \tilde{V}_\phi (0)$ .

For an arbitrage strategry $\tilde{V}_\phi (0) =0$ but $\tilde{V}_\phi (T) \ge 0$ with $P \gt 0$ and so $E_Q [\tilde{V}_\phi (T)] \gt 0$ , so if $Q$ exists then there can not be an arbitrage strategy.

$\Rightarrow$ : Suppose no arbitrage. Let $V = \{ \tilde{V}_\phi (T) : \phi $ is self-financing $V_\phi (0) = 0 \}$ (can think of $V$ as a linear subspace of $\mathcal{R}^N$ . Define $Q$ s.t $Q(\{ \omega_i \} ) \gt 0$ and $\sum^N_{i=1} Q(\{ \omega_i \} ) Y(\omega_i) =0$ , $\forall Y \in V$ . ie. $E_Q [Y] = 0$ , $\forall Y \in V$ , ie. $E_Q [\tilde{V}_\phi (T) ] = 0 \forall$ self-financing $\phi$

Now show every discounted asset price is a martingale under $Q$ . ie. $E_Q [ \tilde(S)_i (t+1) - \tilde{S}_i (t) | \mathcal{F}_t] = 0$ , $\forall i,t$ $E_Q [( \tilde(S)_i (t+1) - \tilde{S}_i (t)) \cdot 1_B ] = 0$ , $\forall B \in \mathcal{F}_t$

We need to show $(( \tilde(S)_i (t+1) - \tilde{S}_i (t)) \cdot 1_B = \tilde{V}_\phi (t)$ for some $\phi$

Take: $\phi(s) = 0$ , for $s = 1, ... , t$ $\phi(t) = ( - S_i(t), 0,0,...,0,1_i,0,0,...,0) \cdot 1_B$ $\phi(s) = ( S_i(t+1) - S_i(t), 0, ..., 0) \cdot 1_B$ for $s \gt t$

Replicating contingent claims

A contingent claim is a random variable on $\Omega$ . A contingent claim $X$ is attainable if there is some self-financing strategy $\phi$ with $V_\phi(T)= X$ .

Theorem: Suppose we have an arbitrage-free multi-period model. Let $X$ be a contingent claim. Then any strategy $\phi$ replicating $X$ has the same $V_\phi (t) \forall t$ , ie if $\phi, \psi$ are two strategies replicating $X$ , then $V_\phi (t) = V_\psi (t) \forall t$ .

Proof: Let $Q$ be an equivalent martingale measure. Then $\tilde{V}_\phi (t)$ and $\tilde{V}_\psi (t)$ are $Q$ -martingales.

• $E_Q [\tilde{V}_\phi (t) | \mathcal{F}_t] = \tilde{V}_\phi (t)$
• $\tilde{V}_\phi (t) = E_Q {[ \frac{X}{S_o (T)} | \mathcal{F} ]}$
• $V_\phi (t) = S_0 (t) \cdot E_Q {[ \frac{X}{S_o (T)} | \mathcal{F} ]}$
• similarly $V_\psi (t) = S_0 (t) \cdot E_Q {[ \frac{X}{S_o (T)} | \mathcal{F} ]}$

The arbitrage price process of an attainable contigent claim $X$ is $(V_t(t))^T_{t=0}$ for any $\phi$ replicating $X$ . It is denoted $(\pi_X (t))^T_{t=0}$ .

The risk-neutral valuation formula gives the value of claim $X$ at time $t$ ,

\[\pi_X (t) = S_0 (t) \cdot E_Q {[\frac{X}{S_0 (T)} | \mathcal{F}_t ]} \] In particular $\pi_X (0) = E {[ \frac{X}{S_0 (T)}]}$ . $\tilde{\pi}_X$ is a martingale under $Q$ , if X is attainable. This only works in $X$ is attainable, must check if any doubt.

Complete markets

A market is complete if any contingent claim $X$ is attainable.

Theorem: An arbitrage free multi-period model is complete $\iff$ it has a unique equivalent martingale measure. For such markets, can use risk-neutral valuation formula without checking that the claim is attainable.

Fundamental theorem of asset pricing

• arbitrage free $\iff$ equiv martingale measure exists
• complete $\iff$ equiv martingale measure unique

For scenario trees, having a unique solution implies $d+1$ branches at each point.