Valuing a Swap with Floating Rates Set on the Payment Date
Summary
The note studies a modified interest rate swap whose floating payment at a period end uses a rate set on that same date, rather than the usual rate fixed at the preceding date. It asks how to value the floating leg and determine the corresponding fair fixed rate. The question proposes a lognormal rate process and a valuation that includes a variance adjustment.
The answer focuses on the expected discounted floating payment under the maturity forward measure. It rewrites the payment using the bond price and evaluates first and second moments of the rate-related term. Under the stated diffusion assumption, the second moment carries an exponential factor involving integrated volatility squared, which produces the variance adjustment. The response supplies only this component of the derivation; it does not independently develop the full swap valuation or discuss whether the assumed rate dynamics are realistic.
Key ideas
- The modified floating leg uses a rate observed on the payment date, unlike standard advance-set floating coupons.
- The valuation turns the payment into expressions involving inverse powers of the maturity bond price.
- The rate expectation under the forward measure uses first and second moments.
- A lognormal diffusion produces a volatility-dependent adjustment to the second moment.
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# Floating swap payoff with rate determined on current instead of previous date
# Floating swap payoff with rate determined on current instead of previous date
I am attempting to determine the payoffs a modified swap, in which the floating payments at a time $T_k$ are made on the current date (i.e. $L(T_k,T_{k+1})\equiv L_{k+1}(T_k)$) rather than at the previous date $T_{k-1}$ (i.e. the usual $L(T_{k-1},T_k)\equiv L_k(T_{k-1})$). I want to prove that that the total time-$t$ payoff (with $\delta_k=T_k-T_{k-1}$) from the floating leg is $$\sum_{k=1}^n\delta_kL_{k+1}(T_k)[1+\delta_kL_{k+1}(T_k)]P(t,T_{k+1}),$$ where $\delta_k$ and therefore assuming $\mathrm{d}L_{k+1}(t)=\sigma_{k+1}(t)L_{k+1}(t)\mathrm{d}W_{k+1}(t)$, the fair value of the fixed rate should be $$R=\frac{\sum_{k=1}^n\left[\delta_kL_{k+1}(t)+\delta_k^2L_{k+1}(t)^2\exp\left(\int_t^{T_k}\sigma_{k+1}(s)^2\mathrm{d}s\right)\right]P(t,T_{k+1})}{\sum_{k=1}^n\delta_kP(t,T_k)}.$$ I was thinking that $L_{k+1}(T_k)$ is not a $\mathbb{Q}^{T_k}$-martingale, but a $\mathbb{Q}^{T_{k+1}}$-martingale, and so changing our numéraire pair from $(P(\cdot,T_k),\mathbb{Q}^{T_k})$ to $(P(\cdot,T_{k+1}),\mathbb{Q}^{T_{k+1}})$ should help, but I'm stuck with a floating time-$T_k$ payoff of $$P(t,T_{k+1})\frac{P(t,T_{k+1})}{P(t,T_k)}\mathbb{E}^{\mathbb{Q}^{T_{k+1}}}\left[\frac{\delta_kL(T_k,T_{k+1})}{P(T_k,T_{k+1})}\Bigg|\mathcal{F}_t\right],$$ which doesn't seem helpful at the slightest.
## Answer by ir7 (score 1)
https://quant.stackexchange.com/a/69260
I'll only address the calculation of $$ E_t^T\left[ \delta L(S,T)P(S,T)^{-1} \right]. $$
$$\delta L(S,T)P(S,T)^{-1} = P(S,T)^{-2} - P(S,T)^{-1}$$
$$ E_t^T\left[ P(S,T)^{-1}\right] = E_t^T\left[ 1+\delta L(S,T)\right] = 1+ \delta L(t,S,T)$$
$$ E_t^T\left[ P(S,T)^{-2}\right] = E_t^T\left[ (1+ \delta L(S,T))^{2}\right] $$ $$= 1+ 2\delta L(t,S,T) + \delta^2 L(t,S,T)^2 \exp\left(\int_t^S \sigma(u)^2 du \right) $$
when $$ dL(u,S,T) = \sigma(u)L(u,S,T)dW_u $$
under $Q^T$ measure.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.