Re: [Quantlib-users] Using instantaneous forward rates

[Arkadiy N. <ark...@gm...>]

Thank you for the clarification, that’s exactly what I was looking for - I probably could have looked it up in the code, but I am not particularly efficient in navigating C++ code, so really appreciate your responding quickly!

Sent from my iPhone

> On Mar 11, 2021, at 4:24 AM, Luigi Ballabio <lui...@gm...> wrote:
> 
> 
> Hello,
>     the forwards are instantaneous — meaning they span an infinitesimal time, not 0.001, so you can calculate the 1M and 3M rates as integrals.
> For the flat region over 1M, you'll get exp(- rate * 1M) = 1M DF.  For the 3M, exp(I) = 3M DF where I is the integral between 0 and 3M; it's the area of the rectangle below the flat part plus the area of the trapezoid below the linear part, so you have simple formulas for both.
> 
> Alternatively, once you bootstrapped the curve, you can ask it the implied deposit rates by calling
> 
>     curve->forwardRate(d1, d2, day_count, Simple);
> 
> where d1 is the beginning of the deposit (usually spot, not today), d2 = d1 + 1M or 3M (taking holidays into account), and day_count is the day-count convention of the deposit.
> 
> Hope this helps,
>     Luigi
> 
> 
> 
>> On Wed, Mar 10, 2021 at 4:30 PM Arkadiy Naumov <ark...@gm...> wrote:
>> Have a question for the community, which is a bit on obscure side. When using piecewise yield curve with interpolation on forward rates, how should I use instantaneous forward rates to make sure the original instruments (those set up by ratehelpers) are priced accurately?
>> 
>> Here is what I mean:
>> 
>> suppose all I have is two deposit helpers for 1M and 3M and I chose linear interpolation.
>> Instantaneous forward rates (which span 0.001 fraction of a year) are flat prior to 1M. If these were daily rates, I would be looking for the rate that satisfies exp(-rate/365)^31 = 1M DF (1M DF is implied by 1M deposit rate). How do I modify this for the instantaneous forward? Is it exp(-rate*0.001)^85 (this is approximate - .001 is about 2.74 times smaller than 1 day of 365-day year)
>> 
>> And then to get the rates filled out between 1M and 3M, is it 
>> PRODUCT (exp(-(1M_NodeRate + k*n) *0.001)) = 3M DF ?
>> 
>> Where k is the slope (it is linear interpolation after all) and n is the number of times 0.001 fits between 1M and 3M
>> 
>> Which would mean I can calculate k analytically
>> 
>> Thank you in advance! I am going to go ahead and test my theory out anyway, but if it’s incorrect, I need the advice anyway!
>> 
>> Sent from my iPhone
>> 
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