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 November 14, 2009 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 ) + ǫ 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 I P M t value of market at time t S t value of asset at time t dM t = µdt + σ m dW t Market Model; const vol, for now M t dS t β ( M t ) dM t = + σdZ t Asset Model S t M t β ( M t ) = β + δ I { M t <c } Brownian motions W t , Z t indep: d � W, Z � t = 0 8
Dynamics Under Physical Measure I P Substituting market equation into asset equation: dS t = β ( M t ) µdt + β ( M t ) σ m dW t + σdZ t S t Asset dynamics depend on market level, market volatility σ m � β 2 ( M t ) σ 2 m + σ 2 This is a geometric Brownian motion with volatility Note this is a stochastic volatility model 9
Dynamics Under Physical Measure I P Process preserves the definition of β : � � � � dS t S t , dM t β ( M t ) dM t M t + σdZ t , dM t Cov Cov M t M t � � � � = dM t dM t V ar V ar M t M t � � β ( M t ) dM t M t , dM t Cov M t � � = Since BM’s indep dM t V ar M t = β ( M t ) 10
P ⋆ Dynamics Under Risk-Neutral Measure I Market is complete ( M and S both tradeable) P ⋆ defined as Thus, ∃ unique Equivalent Martingale Measure I � � � T � T � T � ( θ (1) ) 2 + ( θ (2) ) 2 � P ⋆ dI θ (2) dZ s − 1 θ (1) dW s − − P = exp ds dI 2 t t t with µ − r θ (1) = σ m r ( β ( M t ) − 1) θ (2) = σ 11
P ⋆ Dynamics Under Risk-Neutral Measure I dM t rdt + σ m dW ∗ = t M t dS t rdt + β ( M t ) σ m dW ∗ t + σdZ ∗ = t S t where dW t + µ − r dW ∗ = dt t σ m dZ t + r ( β ( M t ) − 1) dZ ∗ = dt t σ P ⋆ . By Girsanov’s Thm, W ∗ t , Z ∗ t are indep Brownian motions under I 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 ∗ E ⋆ � � e − r ( T − t ) h ( S T ) | M t = M, S t = S P ( t, M, S ) = I 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: � � r − σ 2 m dt + σ m dW ∗ dξ t = t 2 � � r − 1 2( β 2 ( e ξ t ) σ 2 m + σ 2 ) dt + β ( e ξ t ) σ m dW ∗ t + σdZ ∗ dX t = t 14
State Variables WLOG, let t = 0 In integral form, � � r − σ 2 t + σ m W ∗ m ξ t = ξ + t 2 Next, consider X at expiry (integrate from 0 to T ): � � � T r − σ 2 T − σ 2 m β 2 ( e ξ t ) dt X T = x + 2 2 0 � T β ( e ξ t ) dW ∗ t + σZ ∗ + σ m T 0 15
Working with X T e ξ t < c M t < c ⇒ ⇒ ξ t < log c β ( e ξ t ) = β + δ I { ξ t < log c } β ( M t ) = β + δ I { M t <c } ⇒ Using this definition for β ( e ξ t ) , X T becomes � � r − β 2 σ 2 m + σ 2 T + σ m βW ∗ T + σZ ∗ X T = x + T 2 � T � T ( δ 2 + 2 δβ ) σ 2 m I { ξ t < log c } dW ∗ − I { ξ t < log c } dt + σ m δ t 2 0 0 16
Occupation Time of Brownian Motion � T Expression for X T involves integral 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 � I P defined as � � d � I P T − 1 − θW ∗ 2 θ 2 T P ⋆ = exp dI � � r − σ 2 1 m θ = σ m 2 Under this measure, ξ t is a martingale with dynamics σ m d � dξ t = W t � � r − σ 2 t + 1 d � dW ∗ m W t = dt σ m 2 18
P ⋆ → � Changing Measure: I I P Since W ∗ and Z ∗ indep, Z ∗ not affected by change of measure Can replace Z ∗ with � Z Under � I P , x + A 1 T + σ m β � X T = W T � T σ � + Z T − A 2 I { ξ t < log c } dt 0 � T I { ξ t < log c } d � + σ m δ W t 0 where constants A 1 , A 2 defined as m ( β 2 − β ) + σ 2 r (1 − β ) − σ 2 A 1 = 2 δ ( δ + 2 β − 1) σ 2 m A 2 = 2 + δr 19
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 � � t ≥ 0 : � τ = inf { t ≥ 0 : ξ t = log c } = inf W t = ˜ c where c = log c − ξ ˜ σ m Density of first passage time of ξ t = ξ to level log c is � � c 2 | ˜ c | − ˜ √ p ( u ; ˜ c ) = 2 πu 3 exp , u > 0 2 u 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 ] [ τ ∧ T, T ] and τ ∧ T counts as occupation time If ξ t < log c , 21
Including First Passage Time Information Incorporating this information into X T yields x + A 1 T + σ m β � W T + σ � X T = Z T � T − A 2 ( τ ∧ T ) I { ˜ c> 0 } − A 2 c } dt I { � W t < ˜ τ ∧ T � T + σ m δ � c } d � W τ ∧ T I { ˜ c> 0 } + σ m δ W t I { � W t < ˜ τ ∧ T 22
Working with the Stochastic Integral c of � Stochastic integral can be re-expressed in terms of local time � L ˜ W at level ˜ c . Applying Tanaka’s formula to φ ( w ) = ( w − ˜ c ) I { w< ˜ c } between τ ∧ T and T , we get: � T c } d � W t = φ ( � W T ) − φ ( � W τ ∧ T ) + � T − � L ˜ c L ˜ c τ ∧ T . I { � W t < ˜ 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 terminal log-stock price Ψ + Case ξ > log c terminal log-stock price Ψ − Case ξ < log c 24
Consider Case ξ < log c as Example In this case, ˜ c > 0 and we have x + A 1 T + σ m β � W T + σ � X T = Z T � T c } dt + σ m δ � − A 2 ( τ ∧ T ) − A 2 W τ ∧ T I { � W t < ˜ τ ∧ T �� � � � � � � c } + � T − � L ˜ L ˜ c c + σ m δ W T − ˜ c c } − W τ ∧ T − ˜ c I { � I { � τ ∧ T W T < ˜ W τ ∧ T < ˜ Treat separately cases { τ < T } and { τ > T } 25
Case ξ < log c , contd. • On { τ > T } , we have: x + ( A 1 − A 2 ) T + σ m ( β + δ ) � W T + σ � X T = Z T T + ( � W T , � Ψ − =: Z T ) , where lower index T + stands for τ > T Distribution of X T is given by distn of independent Gaussian r.v. � Z T , 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 c − a )2 e − a 2 2 T − e − (2˜ � √ I P W T ∈ da, τ > T = da, a < ˜ c, 2 T 2 πT =: q T ( a ; ˜ c ) da 27
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