Module 7 · Derivatives

Pricing and Valuation of Interest-Rate and Other Swaps

EN: Swap as a series of forwards / equivalent bonds; finding the swap rate.
VN: Swap như chuỗi forward hoặc 2 trái phiếu tương đương.

1. Plain-Vanilla Interest-Rate Swap Concept

One party pays a fixed rate and receives floating; the counterparty does the reverse. Notional principal is not exchanged.

2. Swap as Two Bonds Core

\[ V_{\text{pay-fixed swap}} = B_{\text{floating}} - B_{\text{fixed}} \]

Components

B_floating Value of a hypothetical floating-rate bond (resets to par at each reset date).
B_fixed Value of a fixed-rate bond paying the swap fixed rate.

At inception, value = 0 → swap fixed rate is set such that B_floating = B_fixed.

Practice problem

At inception of a fixed-for-floating swap, what is the value to the pay-fixed side?

Show solution

At inception, swap is set such that B_floating = B_fixed.

Value = B_floating − B_fixed = 0.

Value = 0 (set so initial value is zero).

3. Swap Fixed Rate Core

\[ \text{Swap rate} = \frac{1 - PV(\text{final})}{\sum_{t=1}^{N} PV_t} \]

Or via discount factors

\[ s_{\text{swap}} = \frac{1 - DF_N}{\sum_{t=1}^{N} DF_t} \]

DF_t = discount factor for cash flow at time t.

Practice problem

DF1=0.97, DF2=0.93, DF3=0.88. Compute 3-year swap rate.

Show solution

Sum DF = 0.97 + 0.93 + 0.88 = 2.78

Rate = (1 − 0.88)/2.78 = 0.12/2.78

≈ 4.32%

4. Swap as Series of Forwards Core

Each settlement of an interest-rate swap can be viewed as a forward rate agreement (FRA). The fixed swap rate is a weighted average of the forward rates over the life of the swap.

Practice problem Practice

Practice problem

1-year discount factor = 0.96, 2-year = 0.91. Compute the 2-year swap fixed rate (annual settlements).

Show solution

Swap rate = (1 − DF_N) / Σ DF_t

= (1 − 0.91) / (0.96 + 0.91)

= 0.09 / 1.87

≈ 4.81%