Converting a Futures Convexity Adjustment to Simple Rates

Summary

The document considers how to express a futures-versus-forward convexity adjustment derived under the Ho-Lee short-rate model when rates are quoted with simple rather than continuous compounding. The stated continuous-rate relation subtracts a term proportional to short-rate variance and the two dates defining the forward period. The question proposes applying the adjustment directly to simple rates, raising a concern that the conversion is incorrect.

The response converts each simple rate into its continuously compounded equivalent using the logarithm of its accumulation factor, applies the convexity adjustment in continuous-rate terms, and thereby relates the two simple rates. This avoids treating the rate conversion as annual compounding or subtracting the adjustment directly from a simple-rate accumulation expression. The treatment is limited to the stated model and convention; the source contains a typographical inconsistency in one displayed time difference, so implementations should check the accrual-period definition and units carefully.

Key ideas

• The stated Ho-Lee adjustment is formulated for continuously compounded forward and futures rates.
• A simple rate converts to its continuous equivalent through the logarithm of its accumulation factor divided by the accrual period.
• Apply the convexity adjustment after converting both quoted simple rates to continuous-rate form.
• Check time-period notation and units because the response contains an apparent typographical inconsistency.

Tags

# From continuous compounding to simple compounding - convexity adjustment


# From continuous compounding to simple compounding - convexity adjustment












I have derived the convexity adjustment expression for futures rates using the Ho-Lee model, to arrive at the following: $$ ForwardRate = FuturesRate - \frac{1}{2}\sigma^2T_1T_2 $$ where $T_1$ refers to the time when the forward rate starts, $T_2$ when it finishes and $\sigma$ refers to the volatility of the short rate process.

I have derived the above expression in continuous time assuming continuous compounding, but my futures rate is a simply compounded rate. Is the following conversion to simple compounding correct? $$ \left(1 + ForwardRate\times(T_2-T_1)\right)^{(T_2-T_1)} = \left(1 + FuturesRate\times(T_2-T_1)\right)^{(T_2-T_1)} - \frac{1}{2}\sigma^2T_1T_2 $$

I am under the impression I'm terribly wrong!

## Answer by Ami44 (score 1, accepted)

https://quant.stackexchange.com/a/32371

Since the simple interest $r_{s}$ and the continuous compounded interest $r_{c}$ are connected by $$(1 + r_{s} \cdot (T_{2}-T_{1})) = e^{r_{c} \cdot (T_{2}-T_{1})}$$ it follows for the continuous compounded interest: $$r_{c} = \frac{1}{T_{2}-T_{1}} \cdot \ln{(1+r_{s} \cdot (T_{1}-T{2}))}$$ your convexity formula becomes than:

$$ \frac{1}{T_{2}-T_{1}} \cdot \ln{(1 + ForwardRate \cdot (T_{2}-T_{1}))} = \frac{1}{T_{2}-T_{1}} \cdot \ln{(1+FutureRate \cdot (T_{2}-T_{1}))} - \frac{1}{2}\sigma^{2}T_{1}T{2} $$ This is your formula with $ForwardRate$ und $FutureRate$ expressed as simple interest.

In your calculation you seem to think of annual compounding.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.