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.
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.