What Could Be Causing the Discrepancy in My Fixed Income Mathematics Solution?

In summary: The only discrepancy is in the range of the integral in the last term, which changes from 0 to s in your solution to 0 to t in the textbook's solution. This may be due to a typo or a different approach used by the author. It is recommended to double check your solution and confirm with the textbook or your instructor for clarification.
  • #1
bdw1386
1
0
This problem comes from a book on fixed income finance. The solution is provided, so I gave it a shot and had a slight discrepancy. Probably just due to my rustiness, so hopefully it's an easy one for you guys.

Homework Statement



Differentiate the following expression with respect to t:

[tex]
exp[\int_{0}^{t}d\tau \lambda(\tau)]P(t)+R\int_0^tds(-\frac{dP(s)}{ds})exp[\int_0^sd\tau\lambda(\tau)]
[/tex]

Homework Equations



N/A

The Attempt at a Solution



Using the product rule and the FTC on both terms:
[tex]
exp[\int_{0}^{t}d\tau \lambda(\tau)]\frac{dP(t)}{dt}+P(t)exp[\int_{0}^{t}d\tau \lambda(\tau)]\lambda(t)
+
R\int_0^tds(-\frac{dP(s)}{ds})exp[\int_0^sd\tau\lambda(\tau)](0)
+
R(-\frac{dP(t)}{dt})exp[\int_0^sd\tau\lambda(\tau)]
[/tex]

The third term falls out, so we get:
[tex]
exp[\int_{0}^{t}d\tau \lambda(\tau)]\frac{dP(t)}{dt}+P(t)exp[\int_{0}^{t}d\tau \lambda(\tau)]\lambda(t)
+
R(-\frac{dP(t)}{dt})exp[\int_0^sd\tau\lambda(\tau)]
[/tex]

The textbook matches my solution exactly EXCEPT that the integral in the last term goes from 0 to t, rather than 0 to s. I couldn't figure out why the s changed to a t. The final exp[.] expression comes from the product rule and is simply copied from the original equation:

[tex]
x=R\int_0^tds(-\frac{dP(s)}{ds})
[/tex]
[tex]
y=exp[\int_0^sd\tau\lambda(\tau)]
[/tex]
[tex]
\frac{d}{dt}[xy] = x\frac{dy}{dt}+y\frac{dx}{dt} = y\frac{dx}{dt}
[/tex]
because [itex]\frac{dy}{dt}[/itex] = 0.

What am I missing?
 
Physics news on Phys.org
  • #2
Your work here appears to be correct.
 

FAQ: What Could Be Causing the Discrepancy in My Fixed Income Mathematics Solution?

What is fixed income mathematics?

Fixed income mathematics is a branch of mathematics that deals with the valuation and analysis of financial instruments that have a fixed rate of return, such as bonds, loans, and mortgages.

What are the key concepts in fixed income mathematics?

The key concepts in fixed income mathematics include present value, yield, duration, convexity, and interest rate risk.

How is fixed income mathematics used in finance?

Fixed income mathematics is used in finance to value fixed income securities, determine their risk and return profiles, and make investment decisions based on these calculations.

What skills are required to understand fixed income mathematics?

A strong understanding of basic mathematical concepts, such as algebra, calculus, and statistics, is necessary to understand fixed income mathematics. Additionally, knowledge of financial markets and instruments is helpful.

What are the limitations of fixed income mathematics?

Fixed income mathematics assumes that interest rates and cash flows will remain constant over the life of a security, which may not always be the case in real-world scenarios. Additionally, it does not take into account external factors, such as economic events or market conditions, that may impact the performance of fixed income securities.

Back
Top