Conditional Expectations in Risk-Neutral Zero-Coupon Bond Pricing

Summary

The document raises a question about the risk-neutral valuation of a zero-coupon bond when interest rates vary over time. It compares a bond-pricing expression involving the ratio of discount factors with a general risk-neutral pricing formula, asking whether the discounting over the period before the valuation date can be canceled inside the conditional expectation.

The key mathematical issue is that the discount factor at the valuation time is already known given the information available then, so it can be moved through a conditional expectation. However, the future discounting from the valuation date to maturity depends on the future rate path and remains conditional on current information. The document itself presents a proposed manipulation that drops this conditioning, but that step is not justified merely because the time interval begins at the valuation date. This is a question rather than a complete answer, so it offers no numerical example or empirical evidence; its value is in highlighting the distinction between known past discounting and uncertain future discounting.

Key ideas

• Risk-neutral bond prices depend on expected future discounting conditional on current information.
• A discount factor measurable at the valuation date can be moved outside its conditional expectation.
• Future short rates over the remaining life of a bond may still depend on the current information set.
• Canceling a shared past interval does not remove uncertainty in the future rate path.
• The document poses the issue but does not provide a resolved derivation or example.

Tags

# Pricing of Zero Coupon bond under Risk-neutral pricing measure


# Pricing of Zero Coupon bond under Risk-neutral pricing measure












Pg 242 Topic 5.6.2: Futures contract

Risk-neutral pricing of a zero-coupon bond is given by the below formulae:

$$ B(t,T) \, = \,\frac{1}{D(t)}. \tilde E~[D(T)\mid F(t)], 0\,\leq \,t\,\leq\,T\,\leq\,\bar T $$

I understand here the interest rates are not constant, but is either deterministic or stochastic.

The interest rate path of D(t), it is either a subset or adaptable from the path of D(T).

If my understanding is correct, why cannot we nullify the common path for the period [0,t].

If we take D(t) inside the E-tilda then the following will be the steps:

$$ B(t,T) \, = \, \tilde E~[ exp\{ -\int_{0}^{t} R(u)\,du+ \int_{0}^{T} R(u)\,du\}\mid F(t)]$$ $$ B(t,T) \, = \, \tilde E~[ exp\{ \int_{t}^{T} R(u)\,du\} \mid F(t)] $$

Since it is now full expectation, or the period [t,T] is not dependent on Filtration (t), we can write the above as

$$ B(t,T) \, = \, \tilde E~[ exp\{ \int_{t}^{T} R(u)\,du\}] $$

If I turn to page Page 218 Topic 5.2.4 Pricing under the Risk-Neutral measure.

I compare Equation 5.2.30 and 5.2.31 In Equation 5.2.31, the D(t) is taken inside E-tilda

$$ V(T) \, = \, \tilde E~[ exp\{ -\int_{t}^{T} R(u)du\}.V(T)\mid F(t)]$$

My doubt is, if it could be taken inside E-tilda in Equation 5.2.31 why I cannot consider D(t) in E-tilda in Topic 5.6.2 for Zero-coupon bond.

Kindly if anyone can help clarify this difference in approach.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.