Volatility Time Scales and. Perturbations

Volatility Time Scales and Perturbations Jean-Pierre Fouque NC State University, soon UC Santa Barbara Collaborators: George Papanicolaou Stanford University Ronnie Sircar Princeton University Knut Solna UC Irvine Papers: http://www.math.ncsu.edu/ fouque 1 Bloomberg, New York October 26, 25

Main Idea Use perturbation techniques to correct constant volatility models in order to capture the effects of stochastic volatility Applications to: Equity, Fixed Income, and Credit Markets 2

Equity Perturbations around Black-Scholes to account for: Distributions of Returns (under physical measure IP) Volatility Time Scales Smile/Skew in Implied Volatilities (under risk neutral measure IP ) 3

REQUIRED QUALITIES Parametrization of the Implied Volatility Surface I(t ; T, K) Universal Parsimonous Parameters: Model Independence Stability in Time: Predictive Power Easy Calibration: Practical Implementation Compatibility with Price Dynamics: Applicability to Pricing other Derivatives and Hedging 4

Extensive Data Ananlysis = Two-Scale Stochastic Volatility Models Under the physical measure IP: ε << T << 1/δ dx t = µx t dt + f(y t, Z t )X t dw () t dy t = 1 ε (m Y t)dt + ν 2 ε dw (1) t dz t = δ c(z t )dt + δ g(z t )dw (2) t d < W (), W (1) > t = ρ 1 dt d < W (), W (2) > t = ρ 2 dt 5

Two-Scale Stochastic Volatility Models ε << T << 1/δ Under the risk neutral measure IP chosen by the market: dx t = rx t dt + f(y t, Z t )X t dw () t ( 1 dy t = ε (m Y t) ν ) 2 Λ(Y t, Z t ) ε ( dz t = δ c(z t ) ) δ g(z t )Γ(Y t, Z t ) dt + ν 2 ε dw (1) t dt + δ g(z t )dw (2) t d < W (), W (1) > t = ρ 1 dt d < W (), W (2) > t = ρ 2 dt Λ and Γ: market prices of volatility risk 6

Pricing Equation { } P ε,δ (t, x, y, z) = IE e r(t t) h(x T ) X t = x, Y t = y, Z t = z ( 1 ε L + 1 ε L 1 + L 2 + δm 1 + δm 2 + P ε,δ (T, x, y, z) = h(x) ) δ ε M 3 P ε,δ = L = (m y) y y 2 L 1 = ν 2 (ρ 1 fx 2 x y Λ ) y L 2 = t + 1 ( 2 f2 x 2 2 x + r x ) 2 x + ν2 2 M 1 = gγ z + ρ 2gfx 2 x z M 2 = c z + g2 2 2 z 2 M 3 = ν 2 ρ 12 g 2 y z 7

European Options Approximations Combination of singular and regular perturbations = ( P ε,δ (t, x, y, z) P BS (t,x;t, σ) + (T t) V δ P BS σ + V 1 δ x 2 P BS x σ +(T t) ( V ε 2 x 2 2 P BS x 2 + V ε 3 x x Leading order Black-Scholes price P BS (t,x; σ(z)): L BS (σ(z))p BS = P BS (T, x; σ(z)) = h(x) at the z-dependent effective volatility σ(z): σ 2 (z) = f 2 (, z) ( x 2 2 P BS x 2 where the brackets denote the average with respect to the invariant distribution N(m f, ν 2 f ). ) )) 8

The small parameters (V δ, V1 δ, V2 ε, V3 ε ) are given by V δ = ν s δ Λ s σ 2 2 = ν f ε φ Λ f 2 y V ε V δ 1 = ρ 2 ν s δ 2 f σ V3 ε ν f ε = ρ 1 f φ 2 y σ = d σ/dz, and φ(y, z) is a solution of the Poisson equation L φ(y, z) = f 2 (y, z) σ 2 (z). Accuracy: Smooth payoffs: error = O(ε + δ) Calls (kinks): error = O(ε log ε + δ) Digitals (jumps): error = O(ε 2/3 log ε + δ) 9

P ε 1(t, x, z) = (T t) solves L BS ( σ)p ε 1 + Corrections Equations ( V ε 2 x 2 2 P BS x 2 ( V ε 2 x 2 2 P BS x 2 + V ε 3 x x + V ε 3 x x ( )) x 2 2 P BS x 2 ( )) x 2 2 P BS x 2 =, P ε 1(T, x, z) = ( ) P1 δ (t, x, z) = (T t) V δ P BS σ + V 1 δ x 2 P BS x σ solves L BS ( σ)p δ 1 + 2 ( V δ P BS σ ) + V 1 δ x 2 P BS x σ =, P δ 1 (T, x) = (for European options: P BS σ = (T t)σx 2 2 P BS x 2 ) 1

Term Structure of Implied Volatility I + I ε 1 + I δ 1 = σ + [b ε + b δ (T t)] + [a ε + a δ (T t)] log(k/x) T t where the parameters ( σ, a ε, a δ, b ε, b δ ) depend on z and are related to the group parameters (V δ, V1 δ, V2 ε, V3 ε ) by a ε = V 3 ε σ 3, b ε = V 2 ε σ V 3 ε σ2 (r σ 3 2 ) a δ = V 1 δ σ 2, b δ = V δ 1 δ σ2 (r σ 2 2 ), 11

.5.1 α=a ε +a δ τ.15.2.25.3.35.2.4.6.8 1 1.2 1.4.26 β=σ+b ε +b δ τ.24.22.2.4.6.8 1 1.2 1.4 τ Term-structures fits 12

.38 LMMR Fit to Residual.36.34.32 δ adjusted Implied Volatility.3.28.26.24.22.2.18 2.5 2 1.5 1.5.5 LMMR δ-adjusted implied volatility I b δ τ a δ (LM) as a function of LMMR. The circles are from S&P 5 data, and the line R + a ε (LMMR) shows the fit using the estimated parameters. 13

A slow volatility factor is needed.5 Pure LMMR Fit.45.4 Implied Volatility.35.3.25.2.15 2.5 2 1.5 1.5.5 LMMR Implied volatility as a function of LMMR. The circles are from S&P 5 data, and the line a(lmmr) + b shows the fit using maturities up to two years. 14

A fast volatility factor is needed.4 LM Fit to Residual.35 τ adjusted Implied Volatility.3.25.2.15.35.3.25.2.15.1.5.5 LM The circles are from S&P 5 data, and the line a δ (LM) + σ shows the fit using the estimated parameters from only a slow factor fit. 15

.5 τ=43 days.35 71 days.35 16 days.4.3.3 Implied Volatility.3.2.1.25.2.15.1.25.2.15.1.5.1 5 5 LMMR 2 2 LMMR 1 1 LMMR τ=197 days.2 288 days.24 379 days Implied Volatility.25.2.15.5.5 LMMR.195.19.185.18.5.5 LMMR.22.2.18.16.2.2 LMMR Figure 1: S&P 5 Implied Volatility data on June 5, 23 and fits to the affine LMMR approximation for six different maturities. 16

.5.1 m + m 1 τ.15.2.25.1.2.3.4.5.6.7.8.9 1 1.1 τ(yrs).194.193 b + b 1 τ.192.191.19.189.188.1.2.3.4.5.6.7.8.9 1 1.1 τ(yrs) Figure 2: S&P 5 Implied Volatility data on June 5, 23 and fits to the two-scales asymptotic theory. The bottom (rep. top) figure shows the linear regression of b (resp. a) with respect to time to maturity τ = T t. 17

Higher order terms in ε, δ and εδ I 4 a j (τ) (LM) j + 1 τ Φ t, j= where τ denotes the time-to maturity T t, LM denotes the moneyness log(k/s), and Φ t is a rapidly changing component that varies with the fast volatility factor 18

.5 5 June, 23: S&P 5 Options, 15 days to maturity.5 5 June, 23: S&P 5 Options, 71 days to maturity.45.45.4.4 Implied Volatility.35.3 Implied Volatility.35.3.25.25.2.2.15.75.8.85.9.95 1 1.5 1.1 1.15.15.3.4.5.6.7.8.9 1 1.1 1.2 1.3 Log Moneyness + 1 Log Moneyness + 1.28 5 June, 23: S&P 5 Options, 197 days to maturity.23 5 June, 23: S&P 5 Options, 379 days to maturity.26.22.24.21 Implied Volatility.22.2 Implied Volatility.2.19.18.18.17.16.16.14.75.8.85.9.95 1 1.5 1.1 1.15 1.2 Log Moneyness + 1.15.8.9 1 1.1 1.2 1.3 1.4 1.5 Log Moneyness + 1 Figure 3: S&P 5 Implied Volatility data on June 5, 23 and quartic fits to the asymptotic theory for four maturities. 19

4 8 3 6 a 4 2 a 3 4 1 2.5 1 1.5 2.5 1 1.5 2 τ (yrs) τ(yrs) 5 4.1 a 2 3 2 1 a 1.2.3.4 1.5 1 1.5 2.5.5 1 1.5 2 τ (yrs) Figure 4: S&P 5 Term-Structure Fit using second order approximation. Data from June 5, 23. 2

1 25 8 2 6 15 a 4 4 a 3 1 2 5.5 1 1.5 τ (yrs.).5 1 1.5 τ a 2 12 1 8 6 4 2.5 1 1.5 τ a 1.1.2.3.4.5.6.7.5 1 1.5 τ Figure 5: S&P 5 Term-Structure Fit. Data from every trading day in May 23. 21

Parameter Reduction and Direct Calibration ( ) L BS ( σ) P 1 + Q 1 + + 2 ( V 2 x 2 2 P BS x 2 ( V P BS σ + V 3 x x + V 1x 2 P BS x σ ( x 2 2 P BS x 2 ) = )) Set σ = σ 2 + 2V 2. At the same order, the correction is: ( PBS (T t) V σ + V 1x 2 PBS x σ + V 3x ( x 2 2 P )) BS x x 2 I b + τb δ + ( a ε + τa δ) LMMR b = σ + V ( 3 2σ 1 2r ) σ 2 b δ = V + V ( 1 1 2r ) 2 σ 2, a ε = V 3 σ 3, a δ = V 1 σ 2 22

Accuracy of Approximation For European options with smooth payoffs h(x): P ε,δ = PBS + P 1 + Q 1 + O(ε + δ) For European calls or puts, h(x) continuous piecewise smooth: P ε,δ = PBS + P 1 + Q 1 + O(ε log ε + δ) For European digital option, h(x) = Q1 {x>k} discontinuous: P ε,δ = PBS + P 1 + Q 1 + O(ε 2 3 log ε + δ) 23

Exotic Derivatives (Binary, Barrier, Asian,...) Calibrate σ, V, V 1 and V 3 on the implied volatility surface Solve the corresponding problem with constant volatility σ = P = P BS (σ ) Use V, V 1 and V 3 to compute the source ( PBS 2 V σ + V 1x 2 PBS ) + V 3 x ( x 2 2 P ) BS x σ x x 2 Get the correction by solving the SAME PROBLEM with zero boundary conditions and the source. 24

American Options Calibrate σ, V, V 1 and V 3 on the implied volatility surface Solve the corresponding problem with constant volatility σ = P and the free boundary x (t) Use V, V 1 and V 3 to compute the source P 2 (V σ + V 1x 2 P ) + V 3 x (x 2 2 P ) x σ x x 2 Get the correction by solving the corresponding problem with fixed boundary x (t), zero boundary conditions and the source. 25

First Conclusions A short time-scale of order few days is present in volatility dynamics It cannot be ignored in option pricing and hedging It can be dealt with by using singular perturbation methods It is efficient as a parametrization tool for the term structure of implied volatilities when combined with a regular perturbation 26

Fixed Income Perturbations around Vasicek (for instance) to account for: Volatility Time Scales Fit to Yield Curves Reference: Stochastic Volatility Corrections for Interest Rate Derivatives Mathematical Finance 14(2), April 24 27

Constant Volatility Vasicek Model Under the physical probability IP: d r t = a( r r t )dt + σd W t Under the risk-neutral pricing probability IP : d r t = a(r r t )dt + σd W t with a constant market price of interest rate risk λ: r = r λ σ a 28

Bonds Prices { Λ(t, T) = IE e T t r s ds F t } { = IE e T t r s ds r t } = P(t, r t ; T) Vasicek PDE: P t + 1 2 σ2 2 P x 2 + a(r x) P x x P = with the terminal condition P(t, x; T) = 1. Introduce the time-to-maturity τ = T t and seek a solution of the form: P(T τ, x; T) = A(τ)e B(τ)x by solving linear ODE s with A() = 1 and B() =. 29

Affine Yields B(τ) = 1 e aτ a { 1 e A(τ) = exp [R aτ τ R a + σ2 ( 1 e aτ ) ]} 2 4a 3 with Yield Curve: R = r σ2 2a 2 = r λ σ a σ2 2a 2 R(t, τ) = 1 τ log (Λ(t, t + τ)) = B(τ) r t + log A(τ) = R (R r t ) 1 e aτ aτ + σ2 4a 3 τ ( 1 e aτ ) 2 3

1.8 BOND PRICES.6.4.2 5 1 15 2 25 3 MATURITY.95.9 YIELD.85.8.75.7 5 1 15 2 25 3 MATURITY Figure 6: Bond prices (top) and cblue Yield curve (bottom) in the Vasicek model with a = 1, r =.1 and σ =.1. Maturity τ runs from to 3 years. R =.95 and the initial rate is x =.7. 31

Bond Options Prices Example: a Call Option with strike K and maturity T written on a zero-coupon bond with maturity T > T. The payoff h(λ(t, T)) = (Λ(T, T) K) + is a function of r T since Λ(T, T) = P(T, r T ; T) Call Option Price: { C(t, x; T, T ) = IE e T t } r s ds h (Λ(T, T)) r t = x solution of Vasicek PDE with terminal condition at t = T : C(T, x; T, T ) = ( P(T, x; T) K) + C(t, x; T, T ) = P(t, x; T)N(h 1 ) K P(t, x; T )N(h 2 ) 32

Stochastic Volatility Vasicek Models Under the physical measure: dr t = a(r r t )dt + f(y t )dw t where f is a positive function of a mean-reverting volatility driving process Y t. Example: Y t is an OU process: dy t = α(m Y t )dt + ν 2αdẐ t where Ẑt is a Brownian motion possibly correlated to the Brownian motion W t driving the short rate: Ẑ t = ρw t + 1 ρ 2 Z t (W t, Z t ) independent Brownian motions. 33

Stochastic Volatility Vasicek Pricing Models Under the risk-neutral pricing probability IP (λ,γ) : dr t = (a(r r t ) λ(y t )f(y t )) dt + f(y t )dwt ( dy t = α(m Y t ) ν 2α [ρλ(y t ) + γ(y t ) ]) 1 ρ 2 dt +ν ( 2α ρdwt + ) 1 ρ 2 dzt for bounded market prices of risk λ(y) and γ(y). Under fast mean-reversion: α is large 34

Bond Pricing P(t, x, y; T) = IE (λ,γ) {e T t } r s ds r t = x, Y t = y P t + 1 2 f(y)2 2 P x + (a(r 2 x) λ(y)f(y)) P x xp ( ) + α ν 2 2 P + (m y) P y2 y + ν ( 2α ρf(y) 2 P [ x y ρλ(y) + γ(y) ] ) P 1 ρ 2 y = with the terminal condition P(T, x, y; T) = 1 for every x and y. Expand : P ε = P + εp 1 + εp 2 + ε εp 3 + ε = 1/α 35

Leading Order Term P t + 1 2 σ2 x 2 2 P x 2 + a (r x) P x xp = Effective volatility σ 2 = f 2 and r = r λf /a The zero order term P (t, x) is the Vasicek bond price P (T τ, x; T) = P(T τ, x; T) = A(τ)e B(τ)x computed with the constant parameters (a, r, σ). 36

The Correction P 1 = εp 1 The correction P 1 solves the source problem: ( L V asicek (a, r, σ) P 1 = V 1 x + V 2 with the zero terminal condition P 1 (T, x) =. 2 x 2 + V 3 It involves the constant quantities, small of order 1/ α V 3 = ν 2α ρ fφ V 2 = ν (ρ λφ + ) 1 ρ 2 γφ 2α V 1 = ν 2 α ( ρ λψ + ) 1 ρ 2 γψ 3 ) P x 3 2 νρ α fψ 37

The Correction P 1 : explicit computation Using the variable τ = T t and the explicit form P = Ae Bx : P 1 τ = 1 2 σ2 2 P 1 x 2 + â(r x) P 1 x x P 1 +A(τ)e B(τ)x ( V 3 B(τ) 3 V 2 B(τ) 2 + V 1 B(τ) ) We seek a solution of the form P 1 (T τ, x; T) = D(τ)A(τ)e B(τ)x with the condition D() = so that P 1 (T, x; T) = We get: and D(τ) = V 3 â 3 V 2 â 2 D = V 3 B 3 V 2 B 2 + V 1 B (τ B(τ) 12âB(τ)2 13â2 B(τ) 3 ) (τ B(τ) 12âB(τ)2 ) + V 1 â (τ B(τ)) 38

Summary The corrected bond price is given by P(T τ, x, y; T) P (T τ, x; T) + P 1 (T τ, x; T) = A(τ) (1 + D(τ))e B(τ)x where D is a small factor of order 1/ α. The error P ε (t, x, y; T) ( P (t, x : T) + P ) 1 (t, x; T) is of order 1/α. Corrections for bond options prices are also obtained. 39

1.8 BOND PRICES.6.4.2 5 1 15 2 25 3 MATURITY.95.9.85 YIELD.8.75.7.65 5 1 15 2 25 3 MATURITY Figure 7: Top: bond prices and corrected bond prices (dotted curve). Bottom: yield curve and corrected yield curve (dotted curve) in the simulated Vasicek model (constant and stochastic volatility) with: a = 1, r =.1 and σ =.1 as in Figure 3. Correction: V 3 = 1/ α (ρ ), α = 1 3 and λ = γ = implying V 1 = and V 2 =. Maturity τ runs from to 3 years and the initial rate is x =.7. 4

Model Parameters Rate of mean-reversion of short-rate: a Long-run mean under IP: r Specific volatility distribution: f( ) Correction Parameters a r Mean volatility σ Rate of mean-reversion of volatility : α Group parameter V 1 Mean-level of (Y t ): m Group parameter V 2 V-vol : β Group parameter V 3 Correlation: ρ Interest-rate risk premium: λ( ) Volatility risk premium: γ( ) 41

.62 Vasicek with stochastic volatility correction.6 bond yield.58.56.54 1 2 3 4 5 6 7 years to maturity.62 CIR with jumps.6 bond yield.58.56.54 1 2 3 4 5 6 7 years to maturity Figure 8: Snapshot of the yield curve fit with the stochastic volatility corrected Vasicek model (top) and with the single factor CIR model and down jumps (bottom) for September 6, 1998. 42

Credit Perturbations around Merton/Black-Cox (in the context of the structural approach for instance) to account for: Volatility Time Scales in Default Times Fit to Yield Spreads References: Stochastic Volatility Effects on Defaultable Bonds Submitted 24 Stochastic Volatility Effects on Default Correlations In Preparation 43

Defaultable Bonds In the first passage structural approach, the payoff of a defaultable zero-coupon bond written on a risky asset X is h(x) = 1 {inf s T X s >B}. By no-arbitrage, the value of the bond is P B (t, T) = IE { e r(t t) 1 {inf s T X s >B} F t } = 1 {inf s t X s >B}e r(t t) IE { 1 {inft s T X s >B} F t }, Using the predictable stopping time τ t = inf{s t, X s B}: IE { 1 {inft s T X s >B} F t } = IP {τ t > T F t }. This defaultable zero-coupon bond is in fact a binary down-an-out barrier option where the barrier level and the strike price coincide. 44

Constant Volatility: Merton s Approach dx t = rx t dt + σx t dwt ( X t = X exp (r 1 2 σ2 )t + σwt ). In the Merton s approach, default occurs if X T < B: Defaultable bond = European digital option u d (t, x) = IE { e rτ 1 {XT >B} X t = x } = e rτ IP {X T > B X t = x} = e rτ N(d 2 (τ)) with the usual notation τ = T t and the distance to default: log ( ) ) x B + (r σ2 2 τ d 2 (τ) = σ τ 45

Constant Volatility: Black-Cox Approach IE { } 1 {inft s T X s >B} F t { ) = IP inf ((r σ2 t s T 2 )(s t) + σ(w s Wt ) > log ( B x ) } X t = x computed using distribution of minimum, or using PDE s: IE { e r(t t) 1 {inft s T X s >B} F t } = u(t, X t ) where u(t, x) is the solution of the following problem which is to be solved for x > B. L BS (σ)u = on x > B, t < T u(t, B) = for any t T u(t, x) = 1 for x > B, 46

Constant Volatility: Barrier Options Using the European digital pricing function u d (t, x) L BS (σ)u d = on x >, t < T u d (T, x) = 1 for x > B, and otherwise By the method of images one has: u(t, x) = u d (t, x) where we denote ( x B ) 1 2r σ 2 u d ( t, B2 x = e r(t t) ( N(d + 2 (T t)) ( x B d ± 2 (τ) = ± log ( x B ) ) + (r σ2 2 σ τ ) 1 2r σ 2 N(d 2 (T t)) ) ) τ 47

Yield Spreads Curve The yield spread Y (, T) at time zero is defined by e Y (,T)T = P B (, T) P(, T), where P(, T) is the default free zero-coupon bond price given here, in the case of constant interest rate r, by P(, T) = e rt, and P B (, T) = u(, x), leading to the formula Y (, T) = 1 T log (N (d 2 (T)) ( x B ) 1 2r σ 2 N ( d 2 (T))) 48

45 4 35 Yield spread in basis points 3 25 2 15 1 5 1 1 1 1 1 1 2 Time to maturity in years Figure 9: The figure shows the sensitivity of the yield spread curve to the volatility level. The ratio of the initial value to the default level x/b is set to 1.3, the interest rate r is 6% and the curves increase with the values of σ: 1%, 11%, 12% and 13% (time to maturity in unit of years, plotted on the log scale; the yield spread is quoted in basis points) 49

45 4 35 Yield spread in basis points 3 25 2 15 1 5 1 1 1 1 1 1 2 Time to maturity in years Figure 1: This figure shows the sensitivity of the yield spread to the leverage level. The volatility level is set to 1%, the interest rate is 6%. The curves increases with the decreasing ratios x/b: (1.3, 1.275, 1.25, 1.225, 1.2). 5

Challenge: Yields at Short Maturities As stated by Eom et.al. (empirical analysis 21), the challenge for theoretical pricing models is to raise the average predicted spread relative to crude models such as the constant volatility model, without overstating the risks associated with volatility or leverage. Several approaches (within structural models) have been proposed that aims at the modeling in this regard. These include Introduction of jumps (Zhou,...) Stochastic interest rate (Longstaff-Schwartz,...) Imperfect information (on X t ) (Duffie-Lando,...) Imperfect information (on B) (Giesecke) 51

Stochastic Volatility Models where we assume that dx t = µx t dt + f(y t )X t dw () t dy t = α(m Y t )dt + ν 2α dw (1) t f non-decreasing, < c 1 f c 2 Invariant distribution of Y : N(m, ν 2 ) independent of α α > is the rate of mean reversion of Y The standard Brownian motions W () and W (1) are correlated d W (), W (1) = ρ 1 dt t 52

Stochastic Volatility Models under IP In order to price defaultable bonds under this model for the underlying we rewrite it under a risk neutral measure IP, chosen by the market through the market price of volatility risk Λ 1, as follows dx t = rx t dt + f(y t )X t dw () t, ( dy t = α(m Y t ) ν ) 2αΛ 1 (Y t ) dt + ν 2α dw (1) t. Here W () and W (1) are standard Brownian motions under IP correlated as W () and W (1). We assume that the market price of volatility risk Λ 1 is bounded and a function of y only. 53

Yield spread in basis points 3 2 1 2 4 6 8 1 12 14 16 18 2 3 2 1 SV path.25.15 1 1 1 1 1 1 2.2.1.5 2 4 6 8 1 12 14 16 18 2 Time to maturity in years Figure 11: Uncorrelated slowly mean-reverting stochastic volatility: α =.5 and ρ 1 =. 54

Yield spread in basis points 3 2 1 2 4 6 8 1 12 14 16 18 2 3 2 1 SV path.25.15 1 1 1 1 1 1 2.2.1.5 2 4 6 8 1 12 14 16 18 2 Time to maturity in years Figure 12: Correlated slowly mean-reverting stochastic volatility: α =.5 and ρ 1 =.5. 55

Yield spread in basis points 4 3 2 1 2 4 6 8 1 12 14 16 18 2 SV path 4 3 2 1.15.5 1 1 1 1 1 1 2.2.1 2 4 6 8 1 12 14 16 18 2 Time to maturity in years Figure 13: Uncorrelated stochastic volatility: α =.5 and ρ 1 =. 56

Yield spread in basis points 4 3 2 1 2 4 6 8 1 12 14 16 18 2 SV path 4 3 2 1 1 1 1 1 1 1 2.4.3.2.1 2 4 6 8 1 12 14 16 18 2 Time to maturity in years Figure 14: Correlated stochastic volatility: α =.5 and ρ 1 =.5. 57

Yield spread in basis points 3 2 1 2 4 6 8 1 12 14 16 18 2 3 2 1 SV path 1 1 1 1 1 1 2.8.6.4.2 2 4 6 8 1 12 14 16 18 2 Time to maturity in years Figure 15: Uncorrelated fast mean-reverting stochastic volatility: α = 1 and ρ 1 =. 58

Yield spread in basis points 3 2 1 2 4 6 8 1 12 14 16 18 2 3 2 1 SV path 1 1 1 1 1 1 2.8.6.4.2 2 4 6 8 1 12 14 16 18 2 Time to maturity in years Figure 16: Correlated fast mean-reverting stochastic volatility: α = 1 and ρ 1 =.5. 59

Yield spread in basis points 6 4 2 5 1 15 2 6 4 2 1 1 1 1 1 1 2.4 SV path.2 5 1 15 2 Time to maturity in years Figure 17: Highly correlated fast mean-reverting stochastic volatility: α = 1 and ρ 1 =.5. 6

Yield spread in basis points 8 6 4 2 2 4 6 8 1 12 14 16 18 2 SV path 8 6 4 2 1 1 1 1 1 1 2.8.6.4.2 2 4 6 8 1 12 14 16 18 2 Time to maturity in years Figure 18: High leverage correlated fast mean-reverting stochastic volatility: x/b = 1.2, α = 1 and ρ 1 =.5. 61

Barrier Options under Stochastic Volatility u(t, x, y) = e r(t t) IE { h(x T )1 {inft s T X s >B} X t = x, Y t = y }, P B (t, T) = 1 {inf s t X s >B}u(t, X t, Y t ). The function u(t, x, y) satisfies for x B the problem ( t + L X,Y r ) u = on x > B, t < T u(t, B) = for any t T u(t, x) = h(x) for x > B where L X,Y is the infinitestimal generator of the process (X, Y ) under IP. 62

Leading Order Term under Stochastic Volatility In the regime α large, as in the European case, u(t, x, y) is approximated by u (t, x) which solves the constant volatility problem L BS (σ )u = u (t, B) = on x > B, t < T for any t T u (T, x) = h(x) for x > B where σ is the corrected effective volatility. 63

Stochastic Volatility Correction Define the correction u 1(t, x) by L BS (σ )u 1 = V 3 x x u 1(t, B) = ( ) x 2 2 u x 2 on x > B, t < T for any t T u 1(T, x) = for x > B. Remarkably, the small parameter V 3 is the same as in the European case (calibrated to implied volatilities). 64

Define Computation of the Correction v 1(t, x) = u 1(t, x) (T t)v 3 x x so that v 1(t, x) solves the simpler problem L BS (σ )v 1 = v 1(t, B) = g(t) ( x 2 2 u ), x 2 on x > B, t < T for any t T v1(t, x) = for x > B ( )) g(t) = V 3 (T t) lim x B (x x x 2 2 u x 2 To summarize we have u(t, x, y) u (t, x) + (T t)v 3 x x with explicit computation in the case h(x) = 1. ( x 2 2 u ) x 2 + v 1(t, x) 65

25 Term structure of yield 2 15 1 5 2 4 6 8 1 12 14 16 18 2 25 2 15 1 5 1 1 1 1 1 Time to maturity in years Figure 19: The price approximation for σ =.12,r =.,V 3 =.3, x/b = 1.2. 66

Slow Factor Correction The first correction u (z) 1 (t, x) solves the problem L BS ( σ(z))u (z) 1 = 2 ( V (z) u BS σ + V 1(z)x x ( ubs σ )) on x > B, t < T, u (z) 1 (t, B) = for t T, u (z) 1 (T, x) = for x > B, where u BS is evaluated at (t, x, σ(z)), and V (z) and V 1 (z) are small parameters of order δ, functions of the model parameters, and depending on the current level z of the slow factor. 67

7 Fits to Ford Yields Spreads, 12/9/4 6 Yield Spreads (%) 5 4 3 2 Black Cox Stochastic Volatility Data 1 2 4 6 8 1 Time to maturity Figure 2: Black-Cox and two-factor stochastic volatility fits to Ford yield spread data. The short rate is fixed at r =.25. The fitted Black-Cox parameters are σ =.35 and x/b = 2.875. The fitted stochastic volatility parameters are σ =.385, corresponding to R 2 =.129, R 3 =.12, R 1 =.16 and R =.8. 68

1 Add R 3 1 Add R Yield Spreads (%) 5 5 1 5 5 1 1 Add R 1 1 Add R 2 σ * Yield Spreads (%) 5 5 1 Time to maturity 5 5 1 Time to maturity 69

6 Fit to IBM Yield Spreads 12/1/4 5 Yield Spread (%) 4 3 2 Black Cox Stochastic Volatility Data 1 5 1 15 2 25 Time to maturity (years) Figure 21: Black-Cox and two-factor stochastic volatility fits to IBM yield spread data. The short rate is fixed at r =.25. The fitted Black-Cox parameters are σ =.35 and x/b = 3. The fitted stochastic volatility parameters are σ =.36, corresponding to R 2 =.355, R 3 =.112, R 1 =.13 and R =.45. 7