Zero-Coupon Bond Pricing and Replication
Summary
This document asks how risk-neutral zero-coupon bond pricing relates to replication. It presents the discounted expected payoff formula for a bond that pays one unit at maturity and compares it with the replication argument used to motivate pricing contingent claims. The central question is whether the bond formula assumes a riskless money market and what practical replication might look like.
The text provides no proposed replication strategy or worked market example; it is a question framed around a textbook passage. It distinguishes the pricing definition from the replication justification for option pricing, but does not resolve whether or how the bond can be replicated. Practical replication would depend on the available instruments and assumptions about rates, credit, and markets, which the document explicitly leaves open.
Key ideas
- The document contrasts risk-neutral pricing of a zero-coupon bond with replication-based option pricing.
- The bond is described as paying a fixed face amount at a specified maturity, with no interim payments.
- It asks whether the discounted expectation formula presumes access to a riskless money market.
- It does not provide an answer or a concrete replication method.
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Full text
# Replicating a bond
# Replicating a bond
In Shreve's Stochastic Calculus for Finance Volume II, section 6.5, page 273, Shreve talks about pricing a zero-coupon bond.
> A zero-coupon bond is a contract promising to pay a certain "face" amound, which we take to be $1$, at a fixed maturity date $T$. Prior to that, the bond makes no payments. The risk-neutral pricing formula (5.2.30) says that the discounted price of this bond should be a martingale under the risk-neutral measure. In other words, for $0\leq > t \leq T$, the price of the bond $B(t, T)$ should satisfy $$D(t)B(t, T) = \widetilde{\mathbb{E}}[D(t) | > \mathcal{F}(t)]\text{.}\tag{6.5.2}$$ (Note that $B(T, T)=1$.) This gives us the zero-coupon bond pricing formula $$D(t)B(t, T) = \widetilde{\mathbb{E}}[e^{-\int_t^T R(s) \mathrm{d}s} > | \mathcal{F}(t)]\text{,}\tag{6.5.3}$$ which we take as a definition.
The justification for equation (5.2.30) was based on a replication strategy. The price of the option comes from creating a portfolio process $X(t)$ that trades in the underlying stock and money market to replicate the option value $V(t)$.
> $$D(t)V(t) = \widetilde{\mathbb{E}}[D(T)V(T) | \mathcal{F}(t)], 0\leq > t \leq T\text{.}\tag{5.2.30}$$
So I have two versions of my question:
- What replication strategy is implied in Shreve's book by taking definition (6.5.3)? Are we assuming that there is a riskless money market?
- If Shreve's book does not imply a replication strategy, what are the simple replication strategies used in practice? (perhaps ignoring complicated details like credit rating and foreign money markets)Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.