Pricing CMS Range Accruals Across Risk-Neutral and Forward Measures

Summary

The document explains how to value a CMS indexed range accrual when simulated zero-coupon rates follow a short-rate model under the risk-neutral measure, while the desired probability is expressed under the payment-date forward measure. The payoff event is whether a future swap rate lies within specified bounds.

Under the payment-date forward measure, estimate the probability by averaging the in-range indicator across paths. Under the risk-neutral measure, discount that indicator by the savings account at payment to obtain the payoff’s present value, then divide by the initial discount factor to payment to recover the corresponding forward-measure expectation. This expresses the measure conversion through numeraire valuation rather than an explicit pathwise convexity adjustment. The answer is conceptual and gives no simulation results or details about discretization, model calibration, or variance reduction.

Key ideas

• Under the payment-date forward measure, the range accrual probability is the average of the in-range indicator.
• Under the risk-neutral measure, discount the indicator payoff by the savings account value at payment.
• Divide the risk-neutral present value by the initial payment-date discount factor to obtain the forward-measure expectation.
• The response does not provide implementation details or numerical validation.

Tags

# CMS convexity adjustment in a range accrual Monte Carlo


# CMS convexity adjustment in a range accrual Monte Carlo












I'm trying to price a CMS indexed range accrual using Monte Carlo simulations. Let's say i have n trajectories of ZC rates using G2++ model under risk neutral measure. My question is how do i take into account the convexity adjustment and compute it using monte carlo simulation? What i'm trying to compute is the expectation of the indicator function of my 10 Y swap rate at time $T_i$ being between $K1$ and $K2$ under the forward measure $T_p$ (the payment measure) at time t: $$ \mathbb{E_t}^{Q^{Tp}}(\mathbb{1}_{K_{min} <S^{i,i+10Y}(T_i)<K_{max}})$$ however my Monte carlo trajectories are under the risk neutral measure.

Thank you in advance

## Answer by Antoine Conze (score 2, accepted)

https://quant.stackexchange.com/a/34239

If you have done your simulation under the payment date forward measure then you only need to take the expectation of the indicator of the swap rate being between $K_1$ and $K_2$.

If you have done your simulation under the risk neutral measure (which is associated with the savings account as numeraire) then you take the expectation of the indicator of the swap rate being between $K_1$ and $K_2$ divided by the savings account value on the payment date (your numeraire) to obtain the PV of the payoff. Should you want the expectation of the indicator under the payment date forward measure, you only have to divide the PV with the initial discount to payment date.

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