Which measure is used to price a swap?

When we value the floating leg of a standard vanilla swap, we replace the expectation of the future floating rates by the forward rates known today. However my understanding is that the forward rate is equal to the expectation of a spot rate only under the corresponding maturity forward measure. So for example if we have a spot rate r(T,D) that is known at time T but paid at time T+D then the forward rate f(T,D) known at time 0 is equal to the expectation of r(T,D) only under the T+D forward measure.

Now I don't understand why for vanilla swaps, we are not concerned about that when we replace the future libor spot rates by their forward rates? Or are we implicitly assuming that each payment of the floating leg is priced under its own forward T_i+D measure for example (where T_i is the time we observe the Libor rate and T_i+D is the time we pay it)?

Thanks,

I want to propose a different answer here. I think mathematical expectation (under any measure) is not used in valuing an interest swap.

Years ago I used to explain swaps to beginners by speaking in terms of expectation (perhaps because that is how I learned it myself, although I am not sure). "We see in this example that the market expects future Libor to have these values ...". I stopped doing this when I realized that this is a misleading explanation.

A forward rate is not an expectation (except under some special assumptions, which we don't need to make in this case, so why make them). A forward is the current market price of transfering cash from one future period to another. The current value of a swap can be written in terms of various spot and forward rates so it can be calculated from market prices without using the Expectations operator.

In other words the value of a swap is found from the current market values of its components, not as an average of some random variables in a certain measure. The methodology is more like Arrow Debreu than Black Scholes Merton. The value of a thing is the sum of the values of its constituent parts. To use an American example: If Apple Pie consists of Apple Filling and Crust, then the price of Apple Pie for forward delivery one year hence is the sum of the forward price for Crust and the forward price for Apple Filling.

In my opinion expectations don't matter (though you can use them if you want) in deriving basic swap results.

• You can derive the value of a swap using risk neutral expectations, but I agree with you, it gives little insight into the product. Generally speaking, better to avoid using risk neutral expectations if there is a static hedging strategy, the latter is more insightful than the former. Jul 15 '20 at 19:18

Each payment is valued in its own forward measure. As price is discounted expectation of all cashflows (in the risk neutral measure), you can write it as sum of expectations of each cashflow. Then each cashflow is valued independently of the other at its respective forward measure, under which the payment float rate is a martingale. Thus, each cashflow can be valued at its forward rate.

Technical note: we change measure by individual cash flow (away from the common risk neutral measure - money market account numeraire):

$$\beta(t) \mathbf{E}_t\left[\beta(t_{i+1})^{-1}(L(t_i,t_i,t_{i+1}) -K)\right] = P(t,t_{i+1}) \mathbf{E}_t^{t_{i+1}}\left[(L(t_i,t_i,t_{i+1}) -K)\right]$$

and then use

$$\mathbf{E}_t^{t_{i+1}}\left[L(t_i,t_i,t_{i+1})\right] = L(t,t_i,t_{i+1})$$

So, here we do use arbitrage pricing theory (instead of basic arguments: model-less pricing of a FRA and then decomposing a swap in FRA's, as in above answers), but it's a widely accepted theory.