Patrick S. Hagan

Equivalent Black volatilities

We consider European calls and puts on an asset whose forward price F(t) obeys dF(t)=α(t)A(F)dW(t,) under the forward measure. By using singular perturbation techniques, we obtain explicit algebraic formulas for the implied volatility σ B in terms of todays forward price F 0 ≡ F(0), the strike K of the option, and the time to expiry tex . The price of any call or put can then be calculated simply by substituting this implied volatility into Blacks formula. For example, for a power law (constant elasticity of variance) model dF(t)=aFβ dW(t) we obtain σ B = a/f aυ 1− β {1 + (1−β)(2+β)/24 (F 0 − K/f aυ)2 + (1 − β)2/24 a 2 tex /f aυ 2−2β +…} where f aυ = ½(F 0 + K). Our formula for the implied volatility is not exact. However, we show that the error is insignificant, rarely approaching 1/1000 of the time value of the option. We also present more accurate (albeit more complicated) formulas which can be used for the implied volatility.

Probability Distribution in the SABR Model of Stochastic Volatility

We study the SABR model of stochastic volatility (Wilmott Mag, 2003 [10]). This model is essentially an extension of the local volatility model (Risk 7(1):18–20 [4], Risk 7(2):32–39, 1994 [6]), in which a suitable volatility parameter is assumed to be stochastic. The SABR model admits a large variety of shapes of volatility smiles, and it performs remarkably well in the swaptions and caps/floors markets. We refine the results of (Wilmott Mag, 2003 [10]) by constructing an accurate and efficient asymptotic form of the probability distribution of forwards. Furthermore, we discuss the impact of boundary conditions at zero forward on the volatility smile. Our analysis is based on a WKB type expansion for the heat kernel of a perturbed Laplace-Beltrami operator on a suitable hyperbolic Riemannian manifold.

the distribution of exit times for weakly colored noise

AbstractWe analyze the exit time (first passage time) problem for the Ornstein-Uhlenbeck model of Brownian motion. Specifically, consider the positionX(t) of a particle whose velocity is an Ornstein-Uhlenbeck process with amplitudeσ/ρ and correlation time ε2,

Half-range analysis of a counter-current separator

dX/dt = \sigma Z/\varepsilon , dZ/dt = - Z/\varepsilon ^2 + 2^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \xi (t)/\varepsilon

LIBOR market model with SABR style stochastic volatility

whereξ(t) is Gaussian white noise. Let the exit timetex be the first time the particle escapes an interval −At} by directly solving the Fokker-Planck equation. In brief, after taking a Laplace transform, we use singular perturbation methods to reduce the Fokker-Planck equation to a boundary layer problem. This boundary layer problem turns out to be a half-range expansion problem, which we solve via complex variable techniques. This yields the Laplace transform ofF(t) to within a transcendentally smallO(e−A/εσ +e−B/εσ error. We then obtainF(t) by inverting the transform order by order in ε. In particular, by lettingB→∞ we obtain the solution to Wang and Uhlenbecks unsolved problem b; throughO(ε2σ2/A1) this solution is

Markov interest rate models

F(t) = Erf\left\{ {\frac{{A + \varepsilon \sigma \alpha + \varepsilon \sigma z_0 }}{{2\sigma (t - \varepsilon ^2 \kappa )^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} }}} \right\} + ... for \frac{t}{{\varepsilon ^2 }} > > 1