# What is the replicating portfolio for a zero coupon bond?

Let $R(t)$ be an interest rate process and $D(t)=e^{-\int_0^t R(s)ds}$ and $B(t,T)$ the price at time t of a zero coupon bond maturing at T.

Can anyone explain why this formula holds? $D(t)B(t,T) = \widetilde{E}[D(T)|F(t)]$?

The following is my understanding. Please correct me if I am incorrect. To apply to argument of risk neutral pricing, one needs to first construct a portfolio of assets whose payoff matches the payoff of the derivative security under dynamic hedge. For example, the replicating portfolio of a European call is its underlying stock and risk free bond.

Similar, to price a zero coupon bond, one needs to find a replicating portfolio as well. If this is right, What is the replicating portfolio then?

You are not completely right. Risk-neutral pricing means that the price you'll quote for a given instrument at a given point in time will be the same as the capital you'd require at the same point in time to build a self-financing strategy which will deliver exactly the same future cash flows. The replication mechanism need not be dynamic, it may be static (see for instance static replication of a forward contract by "cash & carry" argument).

This concept is quite intuitive and easy to grasp. However, assuming the existence of a risk-free investment vehicle, best known as the risk-free asset, along with a complete market model defined under a probability measure $\mathbb{P}$, noting that the above claim is strictly equivalent to saying that "any self-financing strategy should emerge as a martingale under a probability measure $\tilde{\mathbb{P}}$ equivalent to $\mathbb{P}$ and associated to the risk-free asset numéraire" is far less intuitive.

There is a lot of theory behind this last statement (see fundamental theorems of asset pricing). I invite you to investigate on your own as this would require a book to answer. But if you want to be convinced, just go back to the binomial model set up and observe how by using a replication argument, one can get rid of historical probabilities and define a new set of probabilities under which discounted asset prices are martingales.

At the end of the day, if you have a complete market model, by absence of arbitrage, there should be a unique risk-neutral measure $\tilde{\mathbb{P}}$ under which: $$\frac{V_t}{B_t} \text{ is a } \tilde{\mathbb{P}}\text{-martingale} \tag{1}$$ for any self-financing strategy $V_t$ and $B_t = \exp(\int_0^t r(s) ds)$ the $t$-value of a risk-free cash account in which $1$ unit of currency has been invested at $t=0$.

• Investing in the ZC bond is a self-financing strategy: $V_t \to B(t,T)$
• What you have defined as $D_t$ is the inverse of what we just defined as $B_t$: $B_t \to 1/D(t)$
• Claiming $(1)$ by definition means that, $\forall t$ $$\frac{V_t}{B_t} = \mathbb{E}^{\tilde{\mathbb{P}}}\left[ \frac{V_T}{B_T} \vert \mathcal{F}_t \right]$$ which using the above notations writes: $$B(t,T) D(t) = \mathbb{E}^{\tilde{\mathbb{P}}}\left[ \underbrace{B(T,T)}_{=1} D(T) \vert \mathcal{F}_t \right]$$ hence $$B(t,T) = \mathbb{E}^{\tilde{\mathbb{P}}}\left[ 1 e^{-\int_t^T r(s) ds} \right]$$ where the first line is exactly the equality you gave and the second finds the following interpretation: the value of a contingent claim is the expectation of the discounted value of the future cash flows it shall pay, where the discount is performed at the risk-free rate.