Fixed Income Derivatives
Change of Numeraire
CMS Spread Options
Constant Maturity Swap (CMS) & Convexity Adjustment Lab
Price Constant Maturity Swaps (CMS), calculate numeraire-change convexity adjustments, evaluate CMS caps and floors, and structure CMS curve spread options (e.g., 10Y − 2Y steepeners).
Adjusted CMS Rate: E^{T_p}[S(T)] = S_0 + \Delta_{\text{conv}} |
Convexity Adjustment: \Delta_{\text{conv}} \approx \frac{1}{2} S_0^2 \sigma_S^2 T \cdot \left| \frac{G''(S_0)}{G'(S_0)} \right| |
CMS Spread: \text{Spread} = S_{\text{CMS, 10Y}} - S_{\text{CMS, 2Y}}
Market Presets:
CMS Contract Parameters
$
%
%
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CMS Pricing & Convexity Summary
CMS CouponVanilla Fwd Swap Rate
4.00%
Annuity Measure S₀
Convexity Adjustment
+18.4 bps
+0.184% Rate Shift
Adjusted CMS Rate
4.184%
Payment Measure E^{T_p}
Total Expected PV ($)
$434,812
Discounted Cash Flow
CMS Option & Risk Metrics
CMS Caplet Value:
$108,420 (43.4 bps)
CMS Floorlet Value:
$174,210 (69.7 bps)
Caplet / Floorlet Parity Error:
0.0 bps ($0.00)
CMS Swap Annuity PVBP:
7.842 yrs ($196.1k/bp)
Payment Discount Factor P(0, T_p):
0.8290
Vega Sensitivity (per 1% Vol):
$8,245
Non-linear convexity expansion as swaption implied volatility increases from 10% to 45%.
Term structure of forward swap rates versus convexity-adjusted CMS rates across CMS 2Y to 30Y tenors.
Expected coupon cash flow ($) comparison between standard forward swap fixing and CMS with convexity adjustment.
Convexity Adjustment: Forward Swap Rate S vs Volatility σ (bps)
Adjusted CMS Rate: Fixing Date T vs Swap Tenor (CMS %)
CMS & Convexity Adjustment Mastery Quiz
1. Why does a Constant Maturity Swap (CMS) rate require a positive convexity adjustment?
2. How does an increase in implied swaption volatility affect the CMS rate?
3. What does a CMS 10Y − 2Y Curve Steepener option profit from?
Constant Maturity Swap (CMS) Engineering & FAQ
A Constant Maturity Swap (CMS) is an interest rate swap where one leg pays or receives a floating rate tied to a constant maturity long-term swap rate (such as the 10-year, 5-year, or 30-year swap rate) reset periodically (e.g. quarterly or semi-annually), rather than a short-term money market index like SOFR or EURIBOR.
A convexity adjustment is required because the natural pricing measure for a swap rate is the swap annuity measure, whereas a CMS coupon is paid on a single payment date under the forward money market measure. Because the swap annuity is a non-linear, negatively sloped function of the swap rate, higher swap rates decrease the discount annuity, causing the expected rate under the payment measure to be higher than the forward swap rate:
E^T[S(T)] = S_0 + Convexity Adjustment.
The convexity adjustment is directly proportional to the square of volatility (variance sigma^2) and the time to payment T. As swaption implied volatility increases or the fixing date is further in the future, the convexity adjustment increases significantly, widening the premium of the adjusted CMS rate over the vanilla forward swap rate.
A CMS Spread Option is a derivative whose payoff depends on the difference between two swap rates of different tenors, typically the 10-year swap rate minus the 2-year swap rate (CMS 10Y - CMS 2Y). A steepener option pays
max(CMS 10Y - CMS 2Y - Strike, 0), allowing investors or corporate borrowers to hedge or speculate on yield curve steepening.
Patrick Hagan (2003) demonstrated that any European CMS claim can be priced without model bias by statically replicating its non-linear payoff profile using a portfolio of vanilla swaptions across a continuum of strikes, integrating over receiver swaptions for strikes below the forward rate and payer swaptions for strikes above.