Testing Whether a Direct Zero-Coupon Bond Model Is Arbitrage-Free

# Is this term structure model valid? (Modeling the Zerobonds directly)


# Is this term structure model valid? (Modeling the Zerobonds directly)












Let us define the dynamics of the discounted Zerobonds as

$$ \tilde{P}(t,T) = \int \sigma(t,T) dW_t + \tilde{P}(0,T)$$ Lets assume $\sigma(t,T)$ is s.t. $\tilde{P}(t,T) $ is a martingale and positive (and nonzero) $\mathbb{P} \ a.s.$ . Lets define a numeraire as follows:

$$ B(t) = \frac{1}{\tilde{P}(t,t)} $$ And finally define the dynamics of the actual Zerobonds by

$$P(t,T) = \tilde{P}(t,T) B(t)$$

Now by construction $P(t,t) =1 \forall t$ and there exists a numeraire (namely $B(t)$) s.t. the discounted Zerobonds are martingales.

I think $B(t)$ is a valid numeraire. It is by assumption strictly positive and I could show in the discrete case that it is an tradable asset (it can be replicated by trading in zerobonds). For the continuous case I have the following starting point

Lets assume the brownian motion is one dimensional. We need a selffinancing trading strategy $\phi$ in Zerobonds s.t.

$$ \sum\limits_{i=1}^{n} \phi_{i,t} P(t,t_{i}) = B(t)$$

which is equivalent to

$$ \sum\limits_{i=1}^{n} \phi_{i,t} \tilde{P}(t,t_{i}) = 1$$ This is where I am not sure in the continuous case. in the discrete case $\tilde{P}(t,t_{i}) $ results from $\tilde{P}(t-\Delta,t_{i}) + \sigma_{t,T}\varepsilon$ with $\varepsilon\sim \mathcal{N}(0,\Delta)$. Since $\varepsilon$ is equal for all Zerobonds we can find $\phi_{1t}$ and $\phi_{2t}$ s.t. on one hand $$ \phi_{1t}\sigma_{t,t_1}\varepsilon + \phi_{2t} \sigma_{t,t_2}\varepsilon=0 $$ (where I assumed that $\sigma_{t,T}$ is predictable) and on the other hand

$$\phi_{1t}\tilde{P}(t,t_{1}) + \phi_{2t}\tilde{P}(t,t_{2})=1 $$ These equation guarantee that the value of the portfolio matches the value of the numeraire and furthermore that the cost of setting up this portfolio at $t-\Delta $ equals:

$$\phi_{1t}P(t-\Delta,t_{1}) + \phi_{2t}P(t-\Delta,t_{2}) =B(t-\Delta)(\phi_{1t}\tilde{P}(t,t_{1}) + \phi_{2t}\tilde{P}(t,t_{2})) =B(t-\Delta) $$

Thus we have a selffinancing portfolio with initial cost $B(0)$ that replicates the $B(t)$ which makes it a numeraire.

So all in all this implies that this model is arbitrage-free.

Is this reasoning valid? If not where got I wrong? If yes, why haven't I seen this anywhere? Is it not practical to work directly with zerobonds?