Convexity of Payer and Receiver Swaptions
Summary
The document explains the sign of convexity for payer and receiver swaptions by starting with the underlying interest rate swap. A payer swap pays fixed and receives floating, so it benefits when rates rise. However, the gains from higher rates are discounted more heavily, which makes the payer swap’s value respond less strongly as rates continue to increase. A payer swaption grants the right to enter this swap, so its value inherits this behavior, particularly when the option is deeply in the money and behaves more like the underlying swap.
The receiver side works in reverse. A receiver swap benefits from lower rates and loses value when rates rise, but the higher rates also discount those losses more heavily. That cushions the decline and gives the receiver swap and swaption positive convexity in the explanation. The discussion emphasizes the effect of discounting on swap value, alongside the option’s changing moneyness. It is a qualitative account; it does not specify a valuation model, curve assumptions, or how convexity may vary across market conditions and swaption terms.
Key ideas
- A payer swap is positioned to benefit from rising rates but has negative convexity due to discounting.
- A payer swaption inherits the payer swap’s rate exposure when exercised or deeply in the money.
- A receiver swap benefits from falling rates and has positive convexity in the described argument.
- Discounting changes how strongly swap gains or losses affect value as rates move.
- The explanation is qualitative and does not provide a pricing model or term-specific estimates.
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# Positive convexity swaptions # Positive convexity swaptions Can I please understand why payer swaptions have positive convexity and receiver swaptions have negative convexity? I understand payer swaptions are akin to put options on bonds and put options have positive convexity but this doesn't really answer for me why payer swpations have positive convexity. Convexity means I gain more if interest rates go down than if interest rates go up. How do I gain more if interest rates go down in a payer swaptions. Payer swaptions give me limitless profit opportunities. I simply don't get this. Same issue with receiver swaptions. ## Answer by AdB (score 2) https://quant.stackexchange.com/a/43300 Start by considering the underlying interest rate swaps (IRSs). If you enter a payer swap (paying fixed, receiving floating rates), you are positioned for higher rates - i.e. you gain from a rate increase. However, when rates increase, you also discount your gains more! This is why payer swaps exhibit negative convexity: you still gain when rates increase, but you gain less due to discounting. Now to your question: convexity of payer swaption payoff. A payer swaption is an option to enter into a payer IRS at a future time. Since the payer IRS is positioned for higher rates, so is the payer swaption. The same argument now applies - you gain when rates increase, but you gain less due to discounting. Hence, a payer swaption exhibits negative convexity. Note also that as interest rates increase, the option becomes more and more in the money. When it is way in the money, i.e. delta is close to 1, the option will move just like the underlying, whose payoff is concave (negatively convex) in the underlying. The reverse argument is true for receiver swap(tions). They are positioned for lower rates. This means their value decreases when rates go up. However, this loss is discounted harder, which affects the value positively. In summary: their value goes down when rates increase, but it goes down less due to positive convexity.
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