Stochastic Integrals and the Zero-Coupon Bond Derivation

Summary

The document is a beginner’s question about steps in a stochastic derivation of a zero-coupon bond formula. It asks why the drift term μ(t) is identified with the expectation of the short rate r(t), how one displayed identity follows, and how a later equation is derived. The accepted response says that the stochastic integral has expectation zero under the relevant assumptions, which explains the expectation relationship. It recommends separating the expressions at two times to see the algebraic step, then reviewing quadratic variation and the distribution of stochastic integrals for the remaining derivation.

The response is a roadmap rather than a complete derivation: the equations and referenced picture are not included in the document, and no page reference from the requested textbook is supplied. The expectation claim relies on conditions that make the stochastic integral a martingale with zero expectation. Readers therefore need the original model assumptions and surrounding equations to verify the argument and connect it fully to bond pricing.

Key ideas

• A stochastic integral has zero expectation under suitable assumptions, supporting the stated expectation relationship.
• The algebraic step can be examined by writing the short rate expressions at the two times separately.
• Quadratic variation and stochastic integral distributions are suggested topics for understanding the later derivation.
• The answer does not provide the full derivation or the requested textbook page reference.

Tags

# I want to know stochastic derivation of zero coupon bond formula


# I want to know stochastic derivation of zero coupon bond formula












I'm elementary level of stochastic calculus. In the above picture, from equation (11) to (12) I don't know what is the clue of $μ(t)$ is the expectation of $r(t)$ and how from this identity we can get equation (12)

Also, I totally don't know how to derive equation (13).

I have a reference book that shreve's stochastic calculus for finance. And I want to know where page I need to refer to understand this derivation.

Thank you.

## Answer by Yoda And Friends (score 0, accepted)

https://quant.stackexchange.com/a/70088

You get that $E[r(t)] = \mu(t)$ because the expectation (under some assumption, true in this case) of a stochastic integral is zero.

You get (12) by simple algebra (try writing r(s) and r(t) separately...).

This may help: https://math.stackexchange.com/questions/3972254/expectation-of-stochastic-integral-martingale

Regarding (13), I would advise to search "Quadratic variation" and "distribution of stochastic integral"...

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