Option Pricing Under a Stressed-Beta Model Adam Tashman in collaboration with Jean-Pierre Fouque University of California, Santa Barbara Department of Statistics and Applied Probability Center for Research in Financial Mathematics and Statistics January 29, 2010 1
Capital Asset Pricing Model (CAPM) Discrete-time approach Excess return of asset R a R f is linear function of excess return of market R M and Gaussian error term: R a R f = β(r M R f ) + ɛ Beta coefficient estimated by regressing asset returns on market returns. 2
Difficulties with CAPM Some difficulties with this approach, including: 1) Relationship between asset returns, market returns not always linear 2) Estimation of β from history, but future may be quite different Ultimate goal of this research is to deal with both of these issues 3
Extending CAPM: Dynamic Beta Two main approaches: 1) Retain linearity, but beta changes over time; Ferson (1989), Ferson and Harvey (1991), Ferson and Harvey (1993), Ferson and Korajczyk (1995), Jagannathan and Wang (1996) 2) Nonlinear model, by way of state-switching mechanism; Fridman (1994), Akdeniz, L., Salih, A.A., and Caner (2003) ASC introduces threshold CAPM model. Our approach is related. 4
Estimating Implied Beta Different approach to estimating β: look to options market Forward-Looking Betas, 2006 P Christoffersen, K Jacobs, and G Vainberg Discrete-Time Model Calibration of Stock Betas from Skews of Implied Volatilities, 2009 J-P Fouque, E Kollman Continuous-Time Model, stochastic volatility environment 5
Example of Time-Dependent Beta Stock Industry Beta (2005-2006) Beta (2007-2008) AA Aluminum 1.75 2.23 GE Conglomerate 0.30 1.00 JNJ Pharmaceuticals -0.30 0.62 JPM Banking 0.54 0.72 WMT Retail 0.21 0.29 Larger β means greater sensitivity of stock returns relative to market returns 6
Regime-Switching Model We propose a model similar to CAPM, with a key difference: When market falls below level c, slope increases by δ, where δ > 0 Thus, beta is two-valued This simple approach keeps the mathematics tractable 7
Dynamics Under Physical Measure IP M t value of market at time t S t value of asset at time t dm t M t = µdt + σ m dw t Market Model; const vol, for now ds t S t = β(m t ) dm t M t + σdz t Asset Model β(m t ) = β + δ I {Mt <c} Brownian motions W t, Z t indep: d W, Z t = 0 8
Dynamics Under Physical Measure IP Substituting market equation into asset equation: ds t S t = β(m t )µdt + β(m t )σ m dw t + σdz t Asset dynamics depend on market level, market volatility σ m This is a geometric Brownian motion with volatility β 2 (M t )σ 2 m + σ 2 Note this is a stochastic volatility model 9
Dynamics Under Physical Measure IP Process preserves the definition of β: Cov V ar ( ds t S t ), dm t M t ( dm t M t ) = = ( Cov β(m t ) dm t M t V ar ( Cov β(m t ) dm t = β(m t ) V ar ) + σdz t, dm t M t ( ) dm t M t M t ), dm t M t ( dm t M t ) Since BM s indep 10
Dynamics Under Risk-Neutral Measure IP Market is complete (M and S both tradeable) Thus, unique Equivalent Martingale Measure IP defined as dip dip { = exp T t θ (1) dw s T t θ (2) dz s 1 2 T t } { (θ (1) ) 2 + (θ (2) ) 2} ds with θ (1) = µ r σ m θ (2) = r(β(m t) 1) σ 11
Dynamics Under Risk-Neutral Measure IP dm t M t = rdt + σ m dw t ds t S t = rdt + β(m t )σ m dw t + σdz t where dw t = dw t + µ r σ m dt dzt = dz t + r(β(m t) 1) dt σ By Girsanov s Thm, W t, Z t are indep Brownian motions under IP. 12
Option Pricing P price of option with expiry T, payoff h(s T ) Option price at time t < T is function of t, M, and S (M,S) Markovian Option price discounted expected payoff under risk-neutral measure P { } P (t, M, S) = IE e r(t t) h(s T ) M t = M, S t = S 13
State Variables Define new state variables: X t = log S t, ξ t = log M t Initial conditions X 0 = x, ξ 0 = ξ Dynamics are: dξ t = dx t = ( ) r σ2 m dt + σ m dwt 2 ( r 1 ) 2 (β2 (e ξ t )σm 2 + σ 2 ) dt + β(e ξ t )σ m dw t + σdz t 14
State Variables WLOG, let t = 0 In integral form, ξ t = ξ + ( ) r σ2 m t + σ m Wt 2 Next, consider X at expiry (integrate from 0 to T ): ) T X T = x + (r σ2 T σ2 m β 2 (e ξ t )dt 2 2 + σ m T 0 β(e ξ t )dw t + σz T 0 15
Working with X T M t < c e ξ t < c ξ t < log c β(m t ) = β + δ I {Mt <c} β(e ξ t ) = β + δ I {ξt <log c} Using this definition for β(e ξ t ), X T becomes X T = x + (r β2 σm 2 + σ 2 ) T + σ m βwt + σzt 2 (δ 2 + 2δβ) σ2 m 2 T 0 I {ξt <log c}dt + σ m δ T 0 I {ξt <log c}dw t 16
Occupation Time of Brownian Motion Expression for X T involves integral T 0 I {ξ t <log c}dt This is occupation time of Brownian motion with drift To simplify calculation, apply Girsanov to remove drift from ξ 17
Occupation Time of Brownian Motion Consider new probability measure ĨP defined as dĩp { dip = exp θwt 1 } 2 θ2 T θ = 1 σ m ( ) r σ2 m 2 Under this measure, ξ t is a martingale with dynamics dξ t = σ m d W t d W t = dw t + 1 σ m ( ) r σ2 m dt 2 18
Changing Measure: IP ĨP Since W and Z indep, Z not affected by change of measure Can replace Z with Z Under ĨP, XT = x + A1T + σmβ WT + σ Z T A 2 T + σ m δ T where constants A 1, A 2 defined as 0 0 I {ξt <log c}dt I {ξt <log c}d W t A 1 = r(1 β) σ2 m(β 2 β) + σ 2 A 2 = δ(δ + 2β 1) σ2 m 2 + δr 19 2
First Passage Time Now that ξ t is driftless, easier to work with occupation time Run process until first time it hits level log c Denote this first passage time { τ = inf {t 0 : ξ t = log c} = inf t 0 : W } t = c where c = log c ξ σ m Density of first passage time of ξ t = ξ to level log c is p(u; c) = c ( ) exp c2, u > 0 2πu 3 2u 20
Including First Passage Time Information First passage time τ may happen after T, so need to be careful Can partition time horizon into two pieces: [0, τ T ] and [τ T, T ] If ξ t < log c, τ T counts as occupation time 21
Including First Passage Time Information Incorporating this information into X T yields X T = x + A 1 T + σ m β W T + σ Z T T A 2 (τ T ) I { c>0} A 2 +σ m δ W τ T I { c>0} + σ m δ τ T T τ T I { Wt < c} dt I { Wt < c} d W t 22
Working with the Stochastic Integral Stochastic integral can be re-expressed in terms of local time L c of W at level c. Applying Tanaka s formula to φ(w) = (w c)i {w< c} between τ T and T, we get: T τ T I { Wt < c} d W t = φ( W T ) φ( W τ T ) + L c T L c τ T. 23
Starting Level of Market: Three Cases Consider separately the three cases ξ = log c, ξ > log c, and ξ < log c (or equivalently c = 0, c < 0, c > 0) Notation for terminal log-stock price, given ξ Case ξ = log c terminal log-stock price Ψ 0 Case ξ > log c terminal log-stock price Ψ + Case ξ < log c terminal log-stock price Ψ 24
Consider Case ξ < log c as Example In this case, c > 0 and we have X T = x + A 1 T + σ m β W T + σ Z T T A 2 (τ T ) A 2 I { Wt < c} dt + σ mδ W τ T τ T ) ) ] +σ m δ [( WT c I ( Wτ T { WT < c} c I { Wτ T < c} + L c T L c τ T Treat separately cases {τ < T } and {τ > T } 25
Case ξ < log c, contd. On {τ > T }, we have: X T = x + (A 1 A 2 )T + σ m (β + δ) W T + σ Z T =: Ψ T + ( W T, Z T ), where lower index T + stands for τ > T Distribution of X T is given by distn of independent Gaussian r.v. ZT, and conditional distn of W T given {τ > T }. 26
Case ξ < log c, contd. Conditional distn of W T given {τ > T }: From Karatzas and Shreve, one easily obtains: } ( 1 IP { WT da, τ > T = e a2 2T e (2 c a)2 2T 2πT =: q T (a; c) da ) da, a < c, 27
Case ξ < log c, contd. On {τ = u} with u T, we have W u = c, and X T = x + (A 1 A 2 )T + σ m (β + δ) c + σ m β( W T W u ) + σ Z T T +A 2 I { Wt W u >0} dt u +σ m δ [( WT W ) ] u I { WT W u <0} + L c T L c u Distn of X T given by distn of Z T and indep triplet ( BT u, L 0 T u, Γ+ T u) Triplet comprised of value, local time at 0, and occupation time of positive half-space, at time T u, of standard Brownian motion B. 28
Case ξ < log c, contd. In distribution: X T = x + (A 1 A 2 )T + σ m (β + δ) c + σ m B T u ( β + δ I{BT u <0}) + σ ZT +A 2 Γ + T u + σ mδl 0 T u =: Ψ T (B T u, L 0 T u, Γ + T u, Z T ). Distn of triplet ( B T u, L 0 T u, Γ+ T u) developed in paper by Karatzas and Shreve. 29
Karatzas-Shreve Triplet (1984) IP { WT da, L 0 T db, Γ } + T dγ 2p(T γ; b) p(γ; a + b) if a > 0, b > 0, 0 < γ < T, = 2p(γ; b) p(t γ; a + b) if a < 0, b > 0, 0 < γ < T, where p(u; ) is first passage time density 30
Back to Option Pricing Formula Given final expression for X T, option price at time t = 0 is P 0 = IE { e rt h(s T ) } { = ĨE e rt h(e X T ) dip } { dĩp } = ĨE e rt h(e X T )e θ W T 1 2 θ2 T = e rt e 1 2 θ2 T ĨE {h(e X T )e θ W T } 31
Option Pricing Formula, contd. Decompose expectation on {τ T } and {τ > T }, Denote by n T (z) the N (0, T ) density, Define the following convolution relation involving the K-S triplet: T γ 0 = g(a, b, γ; T u)p(u; c)du 2p(γ; a + b) p(t γ; b + c ) if a > 0 2p(γ; b) p(t γ; a + b + c ) if a < 0 =: G(a, b, γ; T ) 32
Option Pricing Formula, contd. The option pricing formula becomes P 0 = e (r+ 1 2 θ2 )T where [ e θ c ( + D ± = T 0 0 h(e Ψ± T (a,b,γ,z) )e θa G(a, b, γ; T ) da db dγ n T (z)dz )] h(e Ψ ± T + (a,z) )e θa q T (a; c)da n T (z)dz D ± (, c) if c > 0 ( c, ) if c < 0 33
Note About Market Stochastic Volatility (SV) Assumption of constant market volatility σ m not realistic Let market volatility be driven by fast mean-reverting factor Introducing market SV in model has effect on asset price dynamics To leading order, these prices are given by risk-neutral dynamics with σ m replaced by adjusted effective volatility σ (see Fouque, Kollman (2009) for details) One could derive a formula for first-order correction, but formula is quite complicated and numerically involved 34
Market Implied Volatilities Following Fouque, Papanicolaou, Sircar (2000) and Fouque, Kollman (2009), introduce Log-Moneyness to Maturity Ratio (LM M R) LMMR = log(k/x) T and for calibration purposes, we use affine LMMR formula I b + a ɛ LMMR with intercept b and slope a ɛ to be fitted to skew of options data Then estimate adjusted effective volatility as ( ) σ b + a ɛ r b 2 2 35
Numerical Results and Calibration 36
Asset Skews of Implied Volatilities Using Stressed-Beta model, price European call option Use following parameter settings: c S 0 r β σ σ T 1000 100 0.01 1.0 0.30 0.01 1.0 K = 70, 71,..., 150 to build implied volatility curves 37
Figure 1: Implied Volatility Skew vs. δ (M 0 = c) 0.48 0.46 0.44 δ=0.7 δ=0.5 δ=0.3 Implied Volatility (%) 0.42 0.4 0.38 0.36 0.34 0.32 0.3 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 K/S 38
Figure 2: Implied Volatility Versus Starting Market (δ = 0.5) 0.46 Implied Volatility (%) 0.44 0.42 0.4 0.38 0.36 0.34 M0=500 M0=900 M0=1000 M0=1100 M0=2000 0.32 0.3 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 K/S 39
Calibration to Data: Amgen Consider Amgen call options with October 2009 expiry Strikes: Take options with LMMR between 1 and 1, using closing mid-prices as of May 26, 2009 For simplicity, asset-specific volatility σ = 0 Market volatility σ estimated from call option data on S&P 500 Index (closest expiry Sep09) From affine LMMR, σ = 0.2549 40
Figure 3: Affine LMMR Fit to S&P 500 Index Options Implied Volatility (%) 0.29 0.28 0.27 0.26 0.25 0.24 0.23 0.22 0.21 0.2 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 LMMR 41
Calibration to Data: Amgen, contd. Need c, β, and δ Select params which min SSE between option model prices, market prices For context, closing level of S&P 500 Index as of May 26, 2009 was 910.33 Estimated parameters: ĉ = 925, ˆβ = 1.17, and ˆδ = 0.65. So market is below threshold 42
Figure 4: Volatility Skews for Amgen Call Options 0.5 0.48 0.46 0.44 market model Implied Volatility (%) 0.42 0.4 0.38 0.36 0.34 0.32 0.3 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 K/S 43
References Akdeniz, L., Salih, A.A., and Caner, M.: Time-Varying Betas Help in Asset Pricing: The Threshold CAPM. Studies in Nonlinear Dynamics and Econometrics, 6 (2003). Ferson, W.E.: Changes in Expected Security Returns, Risk, and the Level of Interest Rates. Journal of Finance, 44(5), 1191-1214 (1989). Ferson, W.E., and Harvey, C.R.: The Variation of Economic Risk Premiums. Journal of Political Economy, 99(2), 385-415 (1991). 44
References, contd. Ferson, W.E., and Harvey, C.R.: The Risk and Predictability of International Equity Returns. Review of Financial Studies, 6(3), 527-566 (1993). Ferson, W.E., and Korajczyk, R.A.: Do Arbitrage Pricing Models Explain the Predictability of Stock Returns? Journal of Business, 68(3), 309-349 (1995). Fouque, J.-P., Papanicolaou, G., and Sircar, R.: Derivatives in Financial Markets with Stochastic Volatility. Cambridge University Press (2000). 45
References, contd. Fouque, J.-P., and Kollman, E.: Calibration of Stock Betas from Skews of Implied Volatilities. Submitted (2009). Fridman, M.: A Two State Capital Asset Pricing Model. IMA Preprint Series #1221 (1994). Jagannathan, R., and Wang, Z.: The Conditional CAPM and the Cross-Section of Expected Returns. Journal of Finance, 51(1), 3-53 (1996). Karatzas, I., and Shreve, S.E.: Trivariate Density of Brownian Motion, its Local and Occupation Times, with Application to Stochastic Control. Annals of Probability, 12(3), 819-828 (1984). 46
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