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

#### kdmfe

The below topics are from Stochastic calculus for Finance II, Continuous time models by Steven E Shreve
(formula explained using LATEX code)

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.

#### Xerium

If my memory serves me correctly, you can. The D(t) is left outside to get the result we are looking for. D(t) is F_t-measurable so you can bring it in. If you look at the 3rd line of V_k,j on page 242, you can see that it is allowed to be moved in.

I recommend Fima's book over Shreve's if you're wanting to learn stochastic calculus. Shreve's is pretty outdated and the commentary isn't as helpful as fima's. There's also a lot of unnecessary bloat like interest-rate modelling and ATS (other books explain it better and are more comprehensive).

But your questions are better suited for the quant stackexchange website.

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#### kdmfe

quant stackexchange website. has blocked me from posting any more questions.
thank you bro for taking time to read my posts.

#### Xerium

quant stackexchange website. has blocked me from posting any more questions.
thank you bro for taking time to read my posts.
I made an edit that might be helpful. Make a new stack account and follow their rules. If you have lots of questions, try to find similar questions and apply similar logic

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