Valuing a Fixed-Notional Equity Swaption with Bond Discount Factors

Summary

The document presents a proposed valuation expression for a swaption that gives its holder the right to enter an equity swap as a fixed-rate payer and equity receiver. It defines the swap rate from the discount factors for the swap's payment dates, then multiplies the positive excess of that rate over the strike by the sum of the discount factors. This resembles the payoff structure of a payer swaption on a fixed-rate leg.

The central question is how to derive the expression and whether the swap rate can stand in for equity return. The excerpt poses this question but offers no derivation or answer. In particular, it does not explain how the equity leg's return is modeled, how the swap's sign convention affects the payoff, or what assumptions make the displayed rate and discounting appropriate. The formula is therefore a starting point for analysis, not a complete pricing method; the cited source is identified but not reproduced.

Key ideas

• The proposed payoff is the positive difference between the swap rate and strike, multiplied by the discounted payment annuity.
• The swap rate is defined using bond discount factors over the swap's payment dates.
• The excerpt asks whether this rate can represent equity return but does not resolve the issue.
• A complete valuation requires assumptions about the equity leg, payoff conventions, and discounting that are absent here.

Tags

# Pricing Equity Swaptions


# Pricing Equity Swaptions












Consider a swaption to enter into a standard equity swap as a fixed-rate payer, equity receiver, in which the notional principal is fixed.

If the strike is K. the underlying swap starts at time 0=n and terminates at m.

Then the swap will have a value at the expiration of

$$SW(0,m)=Max(0,R-K) \sum_{i=1}^{m}B(0,i)$$

where $B(0, i)$ stand for zero-coupon bond with 1\$ face value that matures at $i$. And $R=\frac{1-B(0, m)}{\sum_{i=1}^{m}B(0,i) }$.

My question is how to derive this equation? (Since R is the swap rate, is it possible that swap rate can replace equity return?) Please give me some hints regarding this question. This result was from “The Pricing of Equity Swaps and Swaptions.” The Journal of Derivatives, 5 (Summer, 1998), page 29.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.