Black-76 Swaption Pricing with Annuity Scaling and Theta
Summary
The document presents Black-76 formulas for European payer and receiver swaptions, scaling the option payoff by the present value of the underlying swap’s fixed-leg annuity. It defines the forward swap rate, strike, expiry, volatility, compounding frequency, and a discount factor, then gives the standard Black variables used to calculate option value. It also attempts to derive payer swaption theta by differentiating the discounted Black expression with respect to expiry.
The material is a pricing question rather than a complete validated treatment. The receiver formula appears to repeat the same normal cumulative distribution term where the opposite-tail term is expected, and the displayed theta expression may have sign or tail inconsistencies. It asks for delta and gamma but supplies no derivations for them. The formulas’ use depends on consistent rate, annuity, and discounting conventions, which the brief setup does not fully specify.
Key ideas
- A swaption’s Black-76 value is the option value on the forward swap rate multiplied by the fixed-leg annuity and discount factor.
- The payer payoff uses the forward rate above the strike, while a receiver payoff uses the opposite direction.
- The document differentiates the payer pricing expression with respect to expiry to formulate theta.
- The receiver expression and theta derivation should be checked carefully before implementation.
- Delta and gamma are raised as questions but are not derived in the document.
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Full text
# Black-76 Model for Swaption Price and Greeks
# Black-76 Model for Swaption Price and Greeks
I'm in the early stages of developing a swaption pricing model.
Suppose $t_1$ is the tenor of the swap rate in years, $F$ is the forward rate of the underlying swap, $X$ is the strke rate of the swaption, $r$ is the risk-free rate, $T$ is the swaption expiration (term) in years, $\sigma$ is the volatility of the forward-starting swap rate and $m$ is the compounding per year in swap rate.
As I understand, the Black-76 model for the price of a European payer swaption is
$$P_{PS}= \frac{1-(1+\frac{F}{m})^{-t_1m}}{F}\cdot e^{-rT}[F\Phi(d_1)-X\Phi(d_2)],$$
where
$$d_1=\frac{\ln(\frac{F}{X})+ \frac{\sigma^2T}{2}}{\sigma\sqrt{T}}\quad\text{and}\quad d_2 = d_1-\sigma\sqrt{T}.$$
Equivalently, for a receiver swaption, the price is given by the formula
$$P_{RS}= \frac{1-(1+\frac{F}{m})^{-t_1m}}{F}\cdot e^{-rT}[X\Phi(-d_2)-F\Phi(-d_2)].$$
This is like the original formulae in Black's model except for the additional term $\frac{1-(1+\frac{F}{m})^{-t_1m}}{F}$(source). In additional to validating that these are indeed the correct pricing formulae, I'd like to derive formula for two greeks in particular: theta ($\Theta$) and gamma ($\Gamma$).
Theta
$$\begin{align} \Theta_{PS} =\frac{\partial P_{PS}}{\partial T} = \Bigg[\frac{1-(1+\frac{F}{m})^{-t_1m}}{F}\Bigg]\cdot\frac{\partial}{\partial T}\{e^{-rT}[F\Phi(d_1)-X\Phi(d_2)]\}= \frac{1-(1+\frac{F}{m})^{-t_1m}}{F}\cdot\Bigg[-\frac{Fe^{-rT}\phi(d_1)\sigma}{2\sqrt{T}}-rFe^{-rT}\Phi(-d_1)+rXe^{-rT}\Phi(-d_2)\Bigg] \end{align}$$
where the term in the square parentheses in the standard formula for the theta of a put option under Black's model. $\Theta_{RS}$ derived analogously.
Does anyone know a source for the delta and gamma of a swaption under Black model?
Many thanksShown 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.