Interpolating Discount Factors for Swap Valuation Between Coupon Dates
Summary
The document asks how to value a fixed-for-floating swap between coupon dates when available market rates are quoted at standard tenors. The responses distinguish simple interpolation from curve construction. One recommends bootstrapping a forecast curve when sufficient instruments are available; when they are not, it presents linear interpolation of log discount factors as a practical approximation. This produces constant continuously compounded rates between curve points and roughly linear interpolation of the floating rate.
The discussion cautions that interpolating discount factors directly is not generally a good choice. It also notes that annualized rates can be used with the relevant fraction of a year under the leg’s day-count convention, addressing the difference between ACT/365 and ACT/360. These are practical approximations rather than a full curve-building specification: the answer does not detail instrument selection, curve conventions, or the full separation of projection and discount curves used in modern swap valuation.
Key ideas
- Bootstrapping a curve is preferable when enough market instruments are available.
- Linear interpolation of log discount factors is offered as a practical approximation between curve tenors.
- Direct linear interpolation of discount factors is discouraged in the response.
- Apply annualized rates using the day-count fraction appropriate to the cash flow leg.
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Full text
# Pricing a Vanilla swap between coupons; What rates to use? # Pricing a Vanilla swap between coupons; What rates to use? Vanilla Swap question. Entered into a 5Y fixed for floating HUF swap. Fixed is annual coupons, Float is semi-annual coupons. 1 month later I want to price it. I set up my future values for Fixed coupons for the next 5Y and notional at the end, and my next [coupon + notional] for Float (the coupon is now in 5 months, and a Floating rate is valued at par right after it pays its coupon). I have the BUBOR rates. For my discount factors for my PV, do I use straight line interpolation of the rates? Or use the next interest rate? For example, with .39Y to go before the floating rate coupon, do I use the 0.5Y rate, the .25Y rate, or the interpolated (weighted average of rate and time) of both? Also under continuous compounding (e), since my Fixed leg is ACT/365 and BUBOR is ACT/360, do I have to multiply the BUBOR rate by (365/360) before getting my discount rate to make it equivalent? ## Answer by Attack68 (score 1) https://quant.stackexchange.com/a/36055 Linear interpolation of the discount factors is not a good idea. A better idea, in the absence of a full analysis, is to linearly interpolate the logarithm of the discount factors. You can use the 6M IBOR rate and other yearly tenor IRSs (versus 6M IBOR) to roughly bootstrap the 6M forecast curve. What linearly interpolating the logarithm of the discount factors does is to generate constant overnight (one-day) rates between your knot points. This results in an approximately linearly interpolated 6M rate between knot points. This is the standard method for most simple curves and even more advanced IRS trading desks utilise this form of interpolation in specific sections of their curve. ## Answer by Seroexcel (score 0) https://quant.stackexchange.com/a/25608 The best option would be to bootstrap a curve. But lacking shorter interest rates, this doesn't appear to be possible. Linear interpolation is the next best alternative. Since the BUBOR rates are all annualized, and so long as you're using the portion of the year for the given leg's day count convention, using the given rate (in this case the bootstrapped rate) should be fine.
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