Pricing a Vanilla Swap (Receive Fixed / Pay Floating) in Simple Terms

"Pricing" a vanilla interest rate swap involves determining the "swap rate," which is the fixed rate that equates the present value (PV) of all expected future floating cash flows with the PV of all future fixed cash flows. This ensures the swap's value is zero at inception, aligning with the fundamental principle of no-arbitrage (NA). This principle dictates that a derivative's price must prevent either party from being unfairly advantaged or disadvantaged.

The relationship can be expressed as:

\[ \text{Swap Value} = \text{Value of Fixed-Rate Bond (VP)} - \text{Value of Floating-Rate Bond (VFL)} \]

Key Analogies

- Receiving the fixed rate is akin to investing in a fixed-rate bond (\(VP\)), i.e., being the lender.
- Paying the floating rate is akin to issuing a floating-rate bond (\(VFL\)), i.e., being the borrower.

Floating Leg Value and Fixed Leg Value ( V F L VFL and V P VP) Floating Leg Value ( V F L VFL)

\[ VFL = 1 \]

This is because, at each coupon payment date, the floating rate resets, and the bond's value reverts to its par value. Essentially, it is equivalent to issuing a new bond at each reset date at the prevailing floating rate, with a repayment value of 1.

Fixed Leg Value ( V P VP)

The value of the fixed-rate bond is calculated as:

\[ VP = \frac{C}{(1 + t_1)} + \frac{C}{(1 + t_2)^2} + \dots + \frac{C}{(1 + t_n)^n} + \frac{P}{(1 + t_n)^n} \]

Where:

• \(C\): The coupon rate of the fixed-rate bond.
• \(P\): The principal or face value (typically 1).
• \(t_1, t_2, \dots, t_n\): Discount factors based on spot rates for each period.

The discount factor (\(DF\)) for each period is given by:

\[ DF = \frac{1}{(1 + t_i)^i} \]

Solving for the Fixed Coupon Rate ( C C)

The fixed coupon rate (\(C\)) is determined such that the bond is priced at par value (1) and redeemed at par at maturity (\(P = 1\)).

\[ C = \frac{1 - DF_n}{DF_1 + DF_2 + \dots + DF_n} \]

Where D F n DF n ​ is the discount factor for the final period n n. Example Calculation Swap Parameters

- Swap Term: 3 years
- Fixed Coupon Payments: Annually
- Spot Rates:

• Year 1: \(5\%\)
• Year 2: \(5.5\%\)
• Year 3: \(6\%\)

Step-by-Step Calculation 1. Calculate Discount Factors ( D F DF):

\[ DF_1 = \frac{1}{1 + 0.05} = 0.9524, \quad DF_2 = \frac{1}{(1 + 0.055)^2} = 0.8984, \quad DF_3 = \frac{1}{(1 + 0.06)^3} = 0.8396 \]

2. Floating Leg Value ( V F L VFL):

\[ VFL = 1 \]

3. Fixed Leg Value ( V P VP):

Substituting the discount factors:

\[ VP = (C \cdot 0.9524) + (C \cdot 0.8984) + (C \cdot 0.8396) + (1 \cdot 0.8396) \]

4. Equating Fixed and Floating Leg Values ( V P = V F L VP=VFL):

\[ (C \cdot 0.9524) + (C \cdot 0.8984) + (C \cdot 0.8396) + 0.8396 = 1 \]

5. Solve for C C:

\[ C \cdot (0.9524 + 0.8984 + 0.8396) + 0.8396 = 1 \] \[ C \cdot 2.6904 + 0.8396 = 1 \] \[ C \cdot 2.6904 = 1 - 0.8396 = 0.1604 \] \[ C = \frac{0.1604}{2.6904} \approx 0.0596 \]