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Pricing

• Swap value is zero at initiation, so can calculate price
• Replicate swap with underlying assets and price those

Key insights for pricing:

• Value of a floating rate security is par value at the start and at all coupon reset dates
• To the borrower, an interest rate swap is equivalent to borrowing at a fixed rate.
• Therefore, value of fixed payments is equivalent to the price of a coupon bond (see this section of Fixed Income Investments). So the fixed rate is:
$r_{fs} = \frac{1 - P_0(0,t_n)}{\textstyle \sum_{i=1}^n P_0(0,t_i)} \quad \mbox{- or - } \quad r \textstyle \sum_{i=1}^n P_0(0,t_i) + P(0,t_n) = 1$
• P0 is the present value factor based on the interest rate applicable to that period (0 to tn) which can be:
• a single rate
• a series of chain-linked periodic rates)

Valuing

Value $V_t = \frac{S_t}{S_0} - P_0(0,t_n) - r_{FS}(\sum_{i=1}^n P_0(0,t_i))$

where
PV factors are as for valuing interest rate swaps
New term structure for time t is used to calculate PV factors

$\frac{1+x^2}{y-4}$
$\frac{1+c^{e-y}}{(1+t)^n}$