Why a Bond Forward Expectation Identity Does Not Hold Generally

Summary

The document asks whether a conditional expectation identity involving two zero-coupon bond prices and a tradable asset’s future value holds under any probability measure. The proposed intuition is that the product of bond prices across adjacent maturities should behave like the price of the longer-maturity bond, at least in expectation.

An answer rejects the identity in general by choosing the shorter-period bond price itself as the future asset value: this makes the left side depend on a squared payoff, introducing convexity that is absent from the right side. A second response argues that validity for every tradable asset would force the random bond-price ratio to equal a deterministic value, which is generally inconsistent with a stochastic future bond price. The discussion gives a useful warning against treating bond-price products as interchangeable with longer-maturity prices, but it is not a complete derivation of pricing-measure relationships or arbitrage conditions.

Key ideas

• The proposed conditional expectation equality is not valid for every tradable asset in general.
• Using a bond price as the asset payoff illustrates how the left side can contain a squared, convex payoff.
• A universal equality would impose restrictive conditions on a future bond price that is stochastic.
• The discussion does not fully derive the relevant no-arbitrage pricing relationships.

Tags

# Bond forward arbitrage relationships


# Bond forward arbitrage relationships












I am trying to see if the following statement is true or not and I would really appreciate your help.

The statement is as follows:

$\forall $ Tradable Asset $V(t)$, $$ E[\frac{P(t,T_{i})P(T_{i},T_{i+1})}{P(t,T_{i+1})}V(T_i)|F_t] = E[V(T_i)|F_t]$$ Where the expectency is taken under any probability measure (not necessarily Risk neutral) although a solution with the Risk neutral measure is also more than welcome.

My intuition is that $P(t,T_{i})P(T_{i},T_{i+1}) \approx P(t,T_{i+1})$ especially under expectencies.

PS: $T(t,T_i)$ is the $T_i$ zero coupon bond price at time t.

Many thanks

## Answer by dm63 (score 2, accepted)

https://quant.stackexchange.com/a/72030

I don’t see how this can be true in general. For example, if $V(T_i)=P(T_i,T_(i+1))$ then the LHS would be a squared payoff with convexity, whereas the RHS is linear.

## Answer by Xman (score 0)

https://quant.stackexchange.com/a/72037

I finally managed to find the answer! The statement is False! Because if

$\forall $ Tradable Asset $V(t)$, $$ E[\frac{P(t,T_{i})P(T_{i},T_{i+1})}{P(t,T_{i+1})}V(T_i)|F_t] = E[V(T_i)|F_t]$$

Then

$\forall $ Tradable Asset $V(t)$, $$ E[(\frac{P(t,T_{i})P(T_{i},T_{i+1})}{P(t,T_{i+1})}-1)V(T_i)|F_t] = 0$$

Therfore almost surely $$ \frac{P(t,T_{i})P(T_{i},T_{i+1})}{P(t,T_{i+1})}-1= 0$$

This means that $$P(T_{i},T_{i+1}) = \frac{P(t,T_{i+1})}{P(t,T_{i})}$$ Which is false because the left hand side is stochastic whereas the right hand side is deterministic...

Thanks everyone

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