Calibration of Stock Betas from Skews of Implied Volatilities

Calibration of Stock Betas from Skews of Implied Volatilities Jean-Pierre Fouque University of California Santa Barbara Joint work with Eli Kollman (Ph.D. student at UCSB) Joint Seminar: Department of Systems Engineering and Engineering Management Department of Statistics March 19, 009 The Chinese University of Hong Kong 1

Capital Asset Pricing Model The original discrete time CAPM model defined the log price return on individual asset R a as a linear function of the risk free interest rate R f, the log return of the market R M, and a Gaussian error term: R a R f = β a (R M R f ) + ǫ a The beta coefficient β a was originally estimated using historical returns on the asset and market index, by a simple linear regression of asset returns on market returns. Fundamental flaw: it is inherently backward looking, and used in forward looking portfolio construction.

Previous Attempt to Forward Looking Betas Christoffersen, Jacobs, and Vainberg (008, McGill University, Canada) have attempted to extract the beta parameter from option prices on the underlying market and asset processes: β a = ( SKEWa SKEW M )1 3 ( V AR a V AR M )1, where V AR a (resp. V AR M ), and SKEW a (resp. SKEW M ) are the variance, and the risk-neutral skewness of returns of the asset (resp. of the market). Then, they use results from Carr and Madan (001) which relate these moments to options prices (Quad and Cubic) on the asset (resp. on the market). The advantage of this approach is that option prices are inherently forward looking on the underlying price processes. 3

Continuous Time CAPM The market price M t and an asset price X t evolve as follows: dm t M t = µdt + σ m dw (1) t, dx t X t = β dm t + σdw () t, M t for constant positive volatilities σ m and σ. In this model we assume independence between the Brownian motions driving the market and asset price processes: d W (1), W () t = 0, so that ( ) Cov dxt X t, dm t M t V ar dm t M t = = Cov Cov ( β dm t M t + σdw () t V ar dm t M ( ) t β dm t M t, dm t M t V ar dm t M t = β. ), dm t M t 4

Beta Estimation with Constant Volatility CAPM Observe that the evolution of X t is given by dx t X t = βµdt + βσ m dw (1) t + σdw () t, that is a geometric Brownian motion with volatility β σ m + σ Even if this quantity is known, along with the volatility σ m of the market process, one cannot disentangle β and σ. Then, one has to rely on historical returns data. This drawback, along with the fact that constant volatility does not generate skews, motivates us to introduce stochastic volatility in the model. 5

Stochastic Volatility in Continuous Time CAPM We introduce a stochastic volatility component to the market price process, that is we replace σ m by a stochastic process σ t = f(y t ): dm t M t = µdt + f(y t )dw (1) dx t X t = β dm t + σdw () t, M t t, dy t = 1 ǫ (m Y t)dt + ν ǫ dz t. In this model, the volatility process is driven by a mean-reverting OU process Y t with a large mean-reversion rate 1/ε and the invariant (long-run) distribution N(m, ν ). This model also implies stochastic volatility in the asset price through its dependence on the market return. It allows leverage: d W (1),Z t = ρ dt. However, we continue to assume independence between W () t and the other two Brownian motions W (1) t and Z t in order to preserve the interpretation of β. 6

Pricing Risk-Neutral Measure The market (or index) and the asset being both tradable, their discounted prices need to be martingales under a pricing risk-neutral measure. Setting with (W (1) t, W () t, W (3) t dm t M t = rdt + f(y t ) dx t X t = rdt + βf(y t ) Z t = ρ dw (1) t + 1 ρ dw (3) t, dy t = 1 ǫ (m Y t)dt ν [ ρ µ r ε + ν ( [ρ dw (1) ǫ t ) being three independent BMs, we write: ( dw (1) t + µ r ) f(y t ) dt, ( dw (1) t + µ r ) ( f(y t ) dt + σ dw () t + f(y t ) + ] 1 ρ γ(y t ) dt + µ r ) f(y t ) dt + ( 1 ρ (β 1)r σ ) dt, dw (3) t + γ(y t )dt) ]. 7

Market price of risk and risk-neutral measure γ(y t ) is a market price of volatility risk, and we defined the combined market price of risk: Setting Λ(Y t ) = ρ µ r f(y t ) + 1 ρ γ(y t ). dw (1) t = dw (1) t dw () t = dw () t + + µ r f(y t ) dt, (β 1)r σ dw (3) t = dw (3) t + γ(y t )dt, by Girsanov theorem, there is an equivalent probability IP (γ) such that (W (1) t, W () t, W (3) t ) are independent BMs under IP (γ), called the pricing equivalent martingale measure and determined by the market price of volatility risk γ. dt, 8

Dynamics under the risk-neutral measure Under IP (γ), the model becomes: dm t M t = rdt + f(y t )dw (1) t, dx t X t = rdt + βf(y t )dw (1) t + σdw () t, dy t = 1 ǫ (m Y t)dt ν ε Λ(Y t )dt + ν ǫ dz t, Z t = ρw (1) t + 1 ρ W (3) t. We take the point of view that by pricing options on the index M and on the particular asset X, the market is completing itself and indirectly choosing the market price of volatility risk γ. 9

Market Option Prices Let P M,ǫ denote the price of a European option written on the market index M, with maturity T and payoff h, evaluated at time t < T with current value M t = ξ. Then, we have P M,ǫ = IE (γ) { e r(t t) h(m T ) F t } = P M,ǫ (t, M t, Y t ), By the Feynman-Kac formula, the function P M,ǫ (t, ξ, y) satisfies the partial differential equation: where L ǫ P M,ǫ = 0, P M,ǫ (T, ξ, y) = h(ξ), L ǫ = 1 ǫ L 0 + 1 ǫ L 1 + L 10

Operator Notation L 0 = ν y + (m y) y L OU L 1 = ρν f(y)ξ ξ y ν Λ(y) y L = t + 1 f(y) ξ ξ + r(ξ ξ ) L BS(f(y)) Here L BS (σ) denotes the Black-Scholes operator with volatility parameter σ. The next step is to expand P M,ǫ in powers of ǫ P M,ǫ = P M 0 + ǫp M 1 + ǫp M + ǫ 3/ P M 3 + 11

Expanding Expansion of the solution ( 1 ǫ L 0 + 1 ǫ L 1 + L )(P M 0 + ǫp M 1 + ǫp M + ǫ 3/ P M 3 + ) = 0, one cancel the terms in 1/ε and 1/ ε by choosing P M 0 and P M 1 independent of y (observe that L 1 takes derivatives with respect y). The terms of order ε 0 lead to L 0 P M + L P M 0 = 0, which is a Poisson equation associated with L 0. The centering condition for this equation is L P M 0 = L P M 0 = 0, where denotes the averaging with respect to the invariant distribution of Y t with infinitesimal generator L 0. 1

Noting that Leading order term L = t + 1 σ ξ ξ + r(ξ ξ ) = L BS( σ), with σ = f, and imposing the terminal condition P0 M (T, ξ) = h(ξ), we deduce that P0 M is the Black-Scholes price of the option computed with the constant effective volatility σ. We also have P M = L 1 0 (L L )P M 0. so that the terms of order ε lead to L 0 P3 M + L 1 P M + L P1 M = 0, which is again a Poisson equation in P3 M which requires the solvability condition L 1 P M + L P1 M = 0. 13

Equation for the first correction L P M 1 + L 1 P M = L P M 1 L 1 L 1 0 (L L ) P M 0 = 0. Therefore P1 M is the solution to the Black-Scholes equation with constant volatility σ, with a zero terminal condition, and a source term given by L 1 L 1 0 (L L ) P0 M. In order to compute this source term we introduce a solution φ(y) of the Poisson equation L 0 φ(y) = f(y) f, so that L 1 L 1 0 (L L ) P M 0 = = ( 1 L 1 φ(y)ξ ξ = ρν φ f ξ ξ ( ξ P M 0 ξ ( ) 1 L 1 L 1 0 (f(y) f )ξ P M ξ 0 ) P0 M = 1 L 1φ ξ P0 M ξ ) ν φ Λ ξ P M 0 ξ 14

First correction and market parameters The first correction term ε P M 1 L ( ε P M 1 ) + V M,ε ξ P M 0 ξ ( ε P M 1 )(T, ξ) = 0. solves the following problem: + V M,ε 3 ξ ( ξ P M ) 0 = 0, ξ ξ with V M,ǫ = ǫν φ Λ and V M,ǫ ǫρν 3 = φ f. In fact, the solution is given explicitly by ( ε P M 1 = (T t) V M,ε ξ P0 M ξ + V M,ε 3 ξ ξ One can then deduce the price approximation ( P M,ε = P0 M + (T t) V M,ε ξ P0 M ξ + V M,ε 3 ξ ξ ( ξ P M 0 ξ ( ξ P M 0 ξ )). )) + O(ε). 15

Parameter Reduction One of the inherent advantages of this approximation is parameter reduction. While the full stochastic volatility model requires the four parameters (ǫ, ν, ρ, m) and the two functions f and γ, our approximated option price requires only the three group parameters: The effective historical volatility σ The volatility level correction V M,ǫ due to the market price of volatility risk The skew parameter V M,ǫ 3 proportional to ρ We can further reduce to only two parameters by noting that V M,ǫ is associated with a second order derivative with respect to the current market price ξ. As such, it can be considered as a volatility level correction and absorbed into the volatility of the leading order Black-Scholes price. 16

Adjusted effective volatility We introduce the adjusted effective volatility σ M = σ + V M,ε, and we denote by P M the corresponding Black-Scholes option price. Next, we define the first order correction εp1 M solution to L BS (σ M )( ε P1 M ) + V M,ε 3 ξ (ξ P M ) 0 = 0, ξ ξ ( ε P M 1 )(T, ξ) = 0. It is indeed given explicitly by ε P M 1 = (T t)v M,ε 3 ξ ξ (ξ P M 0 ξ and one can show that the order of accuracy is preserved: P M,ε = P0 M + (T t)v M,ε 3 ξ (ξ P M ) 0 + O(ε) ξ ξ ), 17

Observe that and therefore Proof of order of accuracy L BS (σ M ) = L BS ( σ) + 1 L BS ( σ)(p M 0 (P M 0 P M 0 )(T, ξ) = 0. (V M,ε )ξ ξ, P0 M ) = V M,ε ξ P0 M ξ, Since the source term is O( ε) because of the V M,ε factor, the difference P0 M P0 M is also O( ε). Next we write P M,ε (P M 0 + ε P M 1 ) P M,ε (P M 0 + ε P M 1 ) + (P M 0 + ε P M 1 ) (P M 0 + ε P M 1 ), which, combined with the previous accuracy result, shows that the only quantity left to be controlled is the residual R (P M 0 + ε P M 1 ) (P M 0 + ε P M 1 ). 18

Proof of order of accuracy (continued) From the equations satisfied by P0 M, ε P1 M, P0 M, ε P1 M, it follows that L BS ( σ)(p M 0 + ε P M 1 ) + V M,ε ξ P M 0 ξ L BS (σ M )(P M 0 + ε P M 1 ) + V M,ε 3 ξ ξ ( ξ P0 M ξ + V M,ε 3 ξ ξ (ξ P0 M ) ξ = 0. ) = 0 Denoting by H ε = V M,ε ξ ξ + V M,ε 3 ξ ξ H ε = V M,ε 3 ξ ) (ξ, ξ ξ (ξ ), ξ one deduces that the residual R satisfies the equation: 19

L BS ( σ)(r) = H ε P M 0 ( ) L BS (σ M ) V M,ε ξ ξ (P0 M + ε P1 M ) = H ε P M 0 + H ε P M 0 + V M,ε = H ε (P M 0 P M 0 ) + V M,ε = O(ε), ξ M (P ξ ξ M (P ξ 1 ) 0 + ε P M 0 P M 0 + ε P M where we have used in the last equality that H ε = O( ε), V M,ε = O( ε), P0 M P0 M = O( ε), and ε P1 M = O( ε). Since R vanishes at the terminal time T, we deduce R = O(ε) which concludes the proof. The new approximation has now only two parameters to be calibrated σ M and V M,ǫ 3, while preserving the accuracy of approximation. This parameter reduction is essential in the forward-looking calibration procedure presented next. 1 ) 0

Asset Option Approximation Let P a,ǫ denote the price of a European option written on the asset X, with maturity T and payoff h, evaluated at time t < T with current value X t = x. Then, we have P a,ǫ = IE (γ) { e r(t t) h(x T ) F t } = P a,ǫ (t, X t, Y t ). By the Feynman-Kac formula, the function P a,ǫ (t, x, y) satisfies the partial differential equation: where L a,ǫ P a,ǫ = 0, P a,ǫ (T, x, y) = h(x), 1

L a,ǫ = 1 ǫ L 0 + 1 ǫ L a 1 + L a, with L 0 = ν y + (m y) y, L a 1 = ρν βf(y)x x y ν Λ(y) y, L a = t + 1 ( β f(y) + σ ) x x + r(x x ) L BS( β f(y) + σ ). Observe that the only differences with options on the market index is the factor β in L a 1, and the modified square volatility β f(y) + σ in L a. it is easy to see that the only modifications in the approximation are:

1. σ is replaced by σ a = β σ + σ. V M,ε is replaced by V a,ε = β V M,ε = V a,ǫ = β ǫν φ Λ 3. V M,ε 3 is replaced by V a,ε 3 = β 3 V M,ε 3 = V a,ǫ 3 = β3 ǫρν φ f 4. σ M is replaced by σ a = β σ + σ + V a,ε 5. The option price approximation becomes P a,ε = P a 0 + (T t)v a,ε 3 x x (x P a 0 x ) + O(ε), where P a 0 is the Black-Scholes price with volatility σ a 6. Only the parameters V a,ε 3 and σ a need to be calibrated 3

Beta Estimation From the expressions for V M,ε 3 and V a,ε 3, one deduces that V a,ε 3 = β 3 V M,ε 3. It is then natural to propose the following estimator for β: β = ( V a,ǫ 3 V M,ǫ 3 Therefore in order to estimate the market beta parameter in a forward looking fashion using the implied skew parameters from option prices we must calibrate our two parameters V a,ǫ 3 and V M,ǫ 3. Next we show how to do that by using the implied volatility surfaces from options data. ) 1 3. 4

Calibration Method We know that a first order approximation of an option price (on the market or the individual asset) with time to maturity τ = T t, and in the presence of fast mean-reverting stochastic volatility, takes the following form: P ǫ P BS + τv ǫ 3 x x ( x P BS x ), where P BS is the Black-Scholes price with volatility σ. The European call option price PBS with current price x, time to maturity τ, and strike price K is given by the Black-Scholes formula P BS = xn(d 1) Ke rτ N(d ), where N is the cumulative standard normal distribution and d 1, = log(x/k) + (r ± 1 σ )τ σ τ. 5

Recall the relationship between Vega and Gamma for plain vanilla European options: P BS σ = τσ x P BS x, and rewrite our price approximation as P ǫ PBS + V 3 ǫ σ x x ( P BS σ ). Using the definition of the implied volatility P BS (I) = P ε, and expanding the implied volatility as I = σ + ǫi 1 + ǫi +, we obtain: P BS (σ ) + ǫi 1 P BS (σ ) σ + = PBS + V 3 ǫ σ x x ( P BS σ ) +. 6

By definition P BS (σ ) = PBS, so that ǫi1 = V ǫ 3 σ ( P BS σ ) 1 x x ( P BS σ ). Using the explicit computation of the Vega P BS σ = x τ e d 1 /, π and consequently x x ( P BS σ ) = ( 1 d 1 σ τ ) P BS σ, we deduce by using the definition of d 1: ǫi1 = V 3 ǫ ( ) σ 1 d 1 σ = V 3 ǫ τ σ ( 1 r σ ) + V 3 ǫ log(k/x) σ 3. τ 7

Define Log-Moneyness to Maturity Ratio (LMMR) LMMR = log(k/x) τ we obtain the affine LMMR formula I σ + ǫi 1 = b + a ǫ LMMR, with the intercept b and the slope a ε to be fitted to the skew of options data, and related to our model parameters σ and V3 ε by: b = σ + V 3 ǫ σ a ǫ = V 3 ǫ σ 3., ( 1 r ) σ, 8

Calibration Formulas for V ε 3 We know that b and σ differ from a quantity of order ε. Therefore by replacing σ by b in the relation V3 ǫ = a ǫ σ 3, the order of accuracy for V3 ε is still ε since a ε is also of order ε. Consequently we deduce V ǫ 3 = a ǫ σ 3 a ǫ b 3 V ε 3. It is indeed also possible to extract σ as follows. b = σ + aε σ ( 1 r ) ) = σ a (r ε σ. σ Using again the argument that b and σ differ from a quantity of order ε and a ε is also of order ε, by replacing σ by b in the last term in the relation above, the order of accuracy is still ε, and we conclude that σ b + a ǫ (r b ) σ. 9

Beta Calibration Defining the market fitted parameters as a M,ǫ and b M and the asset parameters as a a,ǫ and b a, we obtain our main formula: ˆβ = V a,ǫ 3 V M,ǫ 3 1/3 = ( a a,ǫ a M,ǫ ) 1/3 ( b a b M where b a + a a,ǫ LMMR (resp. b M + a M,ǫ LMMR) is the linear fit to the skew of implied volatilities for call options on the individual asset (resp. on the market index). Observe the similarity with the formula ), β a = ( SKEWa SKEW M )1 3 ( V AR a V AR M )1, used by Christoffersen, Jacobs, and Vainberg (008). 30

In the following figure: LMMR fit examples Implied volatilities of June 17, 009 maturity options for the S&P 500 and Amgen, plotted against the option s Log-Moneyness to Maturity Ratio (LMMR). These are for February 18, 009 option prices. The blue line is the affine fit of implied volatilities on LMMR by which the V 3 parameter is fit. The parameters fit for each series are S&P 500 Fit: a M,ǫ = 0.11 and b M = 0.48 V M,ǫ 3 = 0.0095 Amgen Fit: a a,ǫ = 0.18 and b a = 0.434 V a,ǫ 3 = 0.010 The beta estimate for Amgen on that day is then 1.03 31

LMMR fits: S&P500 and Amgen 0.55 S&P 500 0.55 AMGN 0.5 0.5 Implied Vol 0.45 0.4 Implied Vol 0.45 0.4 0.35 0.35 0.3 1 0.5 0 0.5 1 LMMR 0.3 1 0.5 0 0.5 1 LMMR 3

Forward and Backward Looking Betas In the following figure: The solid blue line is the forward looking beta (y-axis) calibrated on June 17, 009 expiration call options over the course of 10 market days (x-axis) from February 9, 009 to February 3, 009. The dashed red line is the corresponding historical beta calibrated on a series of historical prices of the same length as the time to maturity of the options. 33

4 AA 4 AMGN 4 AMZN 0 4 5 10 ATI 0 4 5 10 CEG 0 4 5 10 GE 0 4 5 10 GOOG 0 4 5 10 GS 0 4 5 10 IBM 0 4 5 10 PEP 0 4 5 10 XOM 0 5 10 0 5 10 0 5 10 34

THANKS FOR YOUR ATTENTION 35