Learn / Risk & ALM
SABR
SABR is Stochastic Alpha Beta Rho. It is a description of a forward rate whose volatility also moves. The numbers below are a classroom reading of the four names. They are not a fit to quotes.
01
One Black volatility is a flat line
The Black swaption model takes one volatility and returns one price for one strike. If every strike used that same volatility, the plot of volatility against strike would be a horizontal line. A quoted smile is higher in the wings than at the forward rate, and often higher on one wing than the other. SABR is one story for how a rate and its volatility, moving together, can produce that shape.
02
The four names
- Alpha is the starting volatility of the forward rate. A classroom alpha of 20% means the rate’s volatility begins at 20%.
- Beta sets how that volatility scales with the level of the rate. A beta of 0.5 means volatility scales with the square root of the rate: a higher rate carries a higher volatility, but not in proportion to the rate itself.
- Rho is the correlation between the rate and its volatility. A rho of −0.3 means they tend to move in opposite directions. That tilt lifts the low-strike wing relative to the high-strike wing.
- Nu is the volatility of volatility. A nu of 0.40 means the volatility itself moves quickly. Faster movement fattens both wings.
03
Where it sits
SABR is what you read after the swaption volatility grid, when one cell is not enough and the strike axis bends. A path for the overnight rate itself, used when exercise can happen on several dates, is the Hull–White lesson.