# How to get forward rate fixing when valuating a swap

Suppose I want to value a (fwd) starting swap, that means I would like to calculate the fixed rate $$S_{\alpha, \beta}(t)$$. Note, I'm using Brigo's Notation here. We know that the discounted payoff of the cash flows are of the form (payer swap)

$$D(t,T_i) N \tau_i(L(T_{i-1},T_i)-S_{\alpha, \beta}(t))$$

where $$N$$ is the notional, $$\tau_i$$ the daycount between $$T_{i-1}$$ and $$T_i$$ and $$D(t,T_i)$$ the discount factor for maturity $$T_i$$.

By observing that the floating rate side can be written as a telescoping sum we can infer $$S_{\alpha,\beta}$$ from the NA condition that both legs need the same present value, that is

$$S_{\alpha,\beta}(t)=\frac{P(t,T_{\alpha})-P(t,T_\beta)}{\sum_{i=\alpha+1}^\beta P(t,T_i)}$$

Now, I have a very practical question. How are the forward fixing in reality constructed to evaluate such a swap? It seem to me that we have multiple unknows, the swap rate and the forward fixings. However, to evaluate all the present values of the cash flows, I need to know the $$L(T_{i-1},T_i)$$ at time $$t$$. Are there other instruments out there with very long maturity to get this forward fixings?

Example

Suppose I have a 1y spot starting swap which pays floating and fixed semi-annually. That means I have the cash flows for the fixed rate

$$K\frac{1}{2}N (P(0,6M)+P(0,1Y))$$

where we get the discount values for a discount curve, say OIS. The $$\frac{1}{2}$$ is the perfect half year between payments. On the other hand, for the floating I need

$$\frac{1}{2}N(L(0,6M)P(0,6M) + L(6M,1Y)P(0,1Y))$$

As said, if we use a different curve (OIS) for the discounting, i.e. $$P(0,\cdot)$$ I need to know the $$L(0,6M)$$ and $$L(6M, 1Y)$$. The former is the current fixing and known today. However, how do I get the next one? This gets more complicated for swaps with maturity say 30Y+.

• Hi; do you know how interest rate curve bootstrapping works? Forward fixings ("forward curve") are estimated from swap quotes observed today. Apr 4, 2022 at 8:58
• @Kermittfrog thanks for sharing this. I do know the method for like treasury bonds. however, here I have an additional question. Let's say I would like to know if a current swap seems cheap/expansive. If I understand your comment / paper right, the swap rate(s) imply the fwd fixing. But I would like to go the other way around, don't I? Apr 4, 2022 at 11:53
• Yes. Given the forward curve, you can calculate fair, i.e. market-consistent, (forward starting) swap rates as well and compare them to your quote. Apr 4, 2022 at 13:57
• @Kermittfrog but wasn't the forward curve created from the swap curve in the first place? You see my confusion? I really appreciate your help / patience Apr 4, 2022 at 14:56
• Take the swap curve. Produce a forward curve. Use the forward curve (plus discounting curve) and calculate the (future) present value of your forward starting leg. Find the fixed rate leg whose future present value of discounted cash flows equals the float leg future PV. That's your forward starting swap rate. Apr 4, 2022 at 15:15