Cap/Floor Volatility Surface Construction

1. Background & Definitions

In interest rate derivatives (Cap/Floor, Floorlet/Caplet) pricing and risk management, constructing an arbitrage-free, smooth volatility surface (Volatility Surface) that reflects market term structure and smile structure is crucial. Below we detail the step-by-step process, including the concepts of "PriceVol" and "YieldVol" and their applications.

• Cap: A series of Caplets with fixed tenors (e.g., 3M, 6M), paying max(L-K,0).
• Floor: Composed of Floorlets, paying max(K-L,0).
• Volatility Surface: σ(T,K) represents implied volatility for Cap/Floor expiry T and strike K.

Domestic markets commonly use a "dual-curve" framework:

1. OIS Discounting (FR007 OIS)
2. IBOR Forward (SHIBOR3M Forward)

2. Market Data Collection

1. ATM Cap Quotes
   • ATM implied volatilities for various expiries (1Y, 2Y,...)
2. OTM (Straddle/RR) Quotes (if available)
3. Short-end OIS/IBOR Instruments
   • For building discount and forward curves
4. OIS×IBOR Basis Swaps
   • Additional basis pricing data if discount and forward curves differ

3. Building Discount & Forward Curves

1. OIS Discount Curve
   • Instruments: OIS swaps, collateralized repos
   • Output: Discount factors D_OIS(0,T)
2. IBOR Forward Curve
   • Using FRA, IRS, OIS×IBOR basis swaps on OIS curve to bootstrap forward rates F_IBOR(0;T_{i-1},T_i)

4. PriceVol vs YieldVol

Two common measures of implied volatility in Cap/Floor markets:

1. PriceVol (Normal Vol)

   • Units: % or 10⁻²
   • Assumes option prices follow normal distribution
   • Pricing model: Bachelier (Normal)
   • Volatility parameter σ_N represents absolute price volatility

2. YieldVol (Lognormal Vol)

   • Units: Annualized dimensionless
   • Assumes forward rates F follow lognormal distribution
   • Pricing model: Black-76 (Lognormal)
   • Volatility parameter σ_L represents relative rate volatility

Conversion via Black/Bachelier pricing formulas:
C_Bachelier(F,K,σ_N) = C_Black(F,K,σ_L) ⇒ σ_N ↔ σ_L

Systems typically use a boolean flag (e.g., PriceVol) to determine model:

• PriceVol = True ⇒ Use Bachelier model
• PriceVol = False ⇒ Use Black-76 model

5. Cap → Caplet Volatility Stripping (Bootstrap)

5.1 Piecewise Pricing

CapPrice(0,T_N) = Σ Δ_i D_OIS(0,T_i) OptionPrice(F_i,K,σ_i)

• Δ_i: Day count fraction for period i
• OptionPrice: Bachelier or Black formula

5.2 Sequential Bootstrap

For each expiry T_i:

1. Use previously stripped σ_j, j<i
2. Solve for current implied vol using market Cap ATM price:
   • If PriceVol=True, invert Bachelier formula for σ_N,i
   • If PriceVol=False, invert Black formula for σ_L,i

6. Smile (Strike Dimension) Modeling

1. Single-period SABR Fitting
   • Fit SABR parameters (α,β,ρ,ν) to ATM + OTM quotes for expiry T_i
   • Generate σ(T_i,K) for any K using Hagan formula
2. Cross-term Smoothing
   • Spline or parametric (e.g., SSVI) interpolation for α(T), ρ(T), ν(T)
3. Arbitrage Checks
   • Ensure positive density: ∂²C/∂K² ≥ 0

7. Term Dimension Interpolation & Extension

1. Fixed Strike Interpolation
   • Spline/linear interpolation along T axis
2. Boundary Treatment
   • Short-end: Align σ(T→0) with spot volatility
   • Long-end: Typically flat or linear extrapolation