On the Sensitivity of Option Prices with respect to the Risk Premium - PowerPoint PPT Presentation

On the Sensitivity of Option Prices with respect to the Risk Premium for Stochastic Volatility Models Jorge P . Zubelli jt work with C.Gomez Velez IMPA-Brazil. RICAM LINZ October 29, 2008 THANKS Hanna Pikkarainen & Karl Kunish & Wolfgang Runggaldier

Outline Intro on the Black-Scholes-Merton Model Stochastic Volatility Models Risk Premium and Incomplete Markets Fast Mean Reversion Stochastic Volatility Regimes The Calibration Problem for the Risk Premium The Malliavin Calculus Approach Conclusions

The Black-Scholes price model asset price dynamics dX t = µ X t dt + σ X t dW t (1) where W t is the Brownian Motion . The option price U ( x , t ) for x = X t solves the Black-Scholes eq.: ∂ t + σ 2 x 2 ∂ 2 U ∂ U ∂ x 2 + r ( x ∂ U ∂ x − U ) = 0 , 2 U ( x , T ; T , K , σ 2 ) = ( x − K ) + , Note: U = U ( x , t ; T , K , r , σ 2 ) .

Probabilistic Representation for U ( x , t ) From Feynman-Kac U ( x , t , T , K , σ 2 ) = e − r ( T − t ) E Q [( X T − K ) + | X t = x ] . where the expectation is w.r.t. the unique risk neutral measure Q determined by the model parameters { σ , r , µ } where the asset dynamics takes the form dX t = rX t dt + σ X t dW t .

Limitations of Classical Black-Scholes ◮ log-normality of asset prices is not verified by statistical tests ◮ option prices are subjet to the smile effects ◮ volatility of the prices tends fluctuate with time and revert to a mean value

Figure: Example of Data from IBOVESPA. Index × Vol

0.4 0.35 9 Feb, 2000 0.3 Skew 0.25 Implied Volatility 0.2 0.15 Excess kurtosis Historical Volatility 0.1 0.05 0 0.8 0.85 0.9 0.95 1 1.05 1.1 Moneyness K/x Figure: Implied Volatility - S& P 500 - 2/Fev/2000 - As a function of K / x (moneyness) Current index value: 1411.71 - 2 months for maturity. (From Fouque - IMA talk)

Figure: Implied Volatility Surface- (From Bruno Dupire - IMPA talk)

Smile effect : Empirical remark that calls having different strikes, but otherwise identical, have different implied volatilities. Before the 1987 crash the graph of I ( K ) (fix t , X , T ) had a U shape with a minimum close to K = X . Since 1987 it is a decreasing function in the range 95 % < K / X < 105 % then (for K >> X ) it bends upwards.

Smile effect : Empirical remark that calls having different strikes, but otherwise identical, have different implied volatilities. Before the 1987 crash the graph of I ( K ) (fix t , X , T ) had a U shape with a minimum close to K = X . Since 1987 it is a decreasing function in the range 95 % < K / X < 105 % then (for K >> X ) it bends upwards.

Smile effect : Empirical remark that calls having different strikes, but otherwise identical, have different implied volatilities. Before the 1987 crash the graph of I ( K ) (fix t , X , T ) had a U shape with a minimum close to K = X . Since 1987 it is a decreasing function in the range 95 % < K / X < 105 % then (for K >> X ) it bends upwards.

Stochastic Volatility Models dX t = µ X t dt + σ ( Y t ) X t dW 1 t , � (2) dY t = α ( m − Y t ) dt + β ( ρ dW 1 1 − ρ 2 dW 2 t + t ) , In the risk neutral measure the dynamics takes the form: dX t = rX t dt + σ ( Y t ) X t dW 1 ∗ t , � dY t = ( α ( m − Y t ) − β Λ( X t , Y t , t )) dt + β ( ρ dW 1 ∗ 1 − ρ 2 dW 2 ∗ + t ) , t (3) where Λ( t , x , y ) = ρ µ − r σ ( y ) +( 1 − ρ ) 1 / 2 γ ( x , y , t ) . Remark : In (3) we have a market price of volatility risk

Some references J-P .Fouque, G.Papanicolaou, and K-R.Sircar. Derivatives in financial markets with stochastic volatility . Cambridge University Press, Cambridge, 2000. J-P .Fouque, G.Papanicolaou, R.Sircar, and K.Solna. Multiscale stochastic volatility asymptotics. Multiscale Model. Simul. 2(1):22–42 (electronic), 2003. Y.Achdou, B.Franchi, and N.Tchou. A partial diferential equation connected to option pricing with stochastic volatility: regularity results and discretization. Math. Comp., 74(251), 2005. Y.Achdou and N.Tchou. Variational analysis for the Black and Scholes equation with stochastic volatility. M2AN Math. Model. Numer. Anal., 36(3), 2002.

The Pricing Equation for Stochastic Volatility Models The european call price U ( x , y , t , T , K ) satisfies U t + σ ( y ) 2 x 2 U xx − r ( xU x − U )+ ρβ x σ ( y ) U xy + β 2 2 U yy + 2 (4) ( α ( m − y ) − β Λ( y )) U y = 0 , U ( x , y , T ) = ( x − K ) + , where, � Λ( y ) = ρ µ − r 1 − ρ 2 γ ( y ) , and σ ( y ) + 0 ≤ t < T , x > 0 , y ∈ R . We will assume that γ = γ ( y ) .

Figure: Solution of the Heston Model (Using the software PREMIA)

Interpretation of the risk premium γ � µ − r � � � � x σ ( y ) ∂ U ∂ x + βρ∂ U 1 − ρ 2 ∂ U dU ( t , X t , Y t ) = + rU + γβ dt σ ( y ) ∂ y ∂ y � � � x σ ( y ) ∂ U ∂ x + βρ∂ U 1 − ρ 2 ∂ U dW ∗ ∂ y dW ∗ + 1 , t + β 2 , t ∂ y Note: ◮ Small changes in the vol-vol β leads to an infinitesimal change on the price amplified by a factor γ . ◮ If ρ 2 = 1 it does not appear (but eq. degenerates). ◮ Even if ρ = 0 it is present. ◮ The risk premium has to be determined from market data Find the functional derivative of the price w.r.t γ . MAIN PROBLEM:

Interpretation of the risk premium γ � µ − r � � � � x σ ( y ) ∂ U ∂ x + βρ∂ U 1 − ρ 2 ∂ U dU ( t , X t , Y t ) = + rU + γβ dt σ ( y ) ∂ y ∂ y � � � x σ ( y ) ∂ U ∂ x + βρ∂ U 1 − ρ 2 ∂ U dW ∗ ∂ y dW ∗ + 1 , t + β 2 , t ∂ y Note: ◮ Small changes in the vol-vol β leads to an infinitesimal change on the price amplified by a factor γ . ◮ If ρ 2 = 1 it does not appear (but eq. degenerates). ◮ Even if ρ = 0 it is present. ◮ The risk premium has to be determined from market data Find the functional derivative of the price w.r.t γ . MAIN PROBLEM:

Different Directions... 1. Asymptotic Approach: Consider the asymptotic behavior when some of the parameters goes to infinity... Fouque, Papanicolaou, Sircar,Solna (See the book by FPS). 2. IP: Inverse Problems Identification of the Risk Premium. We initiated a study along this direcion in a recent PhD thesis of my student Cesar Gomez Considered the map: Γ : Λ �− → P ( t , x , y ; T , K )

Different Directions... 1. Asymptotic Approach: Consider the asymptotic behavior when some of the parameters goes to infinity... Fouque, Papanicolaou, Sircar,Solna (See the book by FPS). 2. IP: Inverse Problems Identification of the Risk Premium. We initiated a study along this direcion in a recent PhD thesis of my student Cesar Gomez Considered the map: Γ : Λ �− → P ( t , x , y ; T , K )

Different Directions... 1. Asymptotic Approach: Consider the asymptotic behavior when some of the parameters goes to infinity... Fouque, Papanicolaou, Sircar,Solna (See the book by FPS). 2. IP: Inverse Problems Identification of the Risk Premium. We initiated a study along this direcion in a recent PhD thesis of my student Cesar Gomez Considered the map: Γ : Λ �− → P ( t , x , y ; T , K )

Asymptotic Approach The Equation in Dimensionless Variables f = √ We work with the dimensionless variables � t =rt, � rf , � x = x / K , and drop the hats altogether. 2 ( f ( y )) 2 x 2 ∂ 2 P ∂ x 2 + ρν xf ( y ) ∂ 2 P ∂ x ∂ y + ν 2 ∂ 2 P ∂ P ∂ t + 1 ∂ y 2 + 2 � � � ∂ P � x ∂ P ε − 1 ( m − y ) − ν Λ( t , x , y ) ∂ x − P + ∂ y = 0 , (5) P ( T , x , y ) = h ( x ) , with ε = r α − 1 ν = r − 1 / 2 β . T = rT E , and (6) ν : Called the adimensional vol-vol

Interpretation 1. ε = r / α is the adimensional inverse of the rate of mean-reversion 2. Small ε means FAST MEAN-REVERSION 3. ν = r − 1 / 2 β is the adimensional vol-vol 4. Caveat: Fouque-Papanicolaou-Sircar (FPS ) work with the vol-vol √ ν = νε 1 / 2 / � 2 If the volatility is fast mean-reverting, then the market will see, to leading order, an effective constant volatility plus small corrections: This is modeled by subsuming that the mean reversion time ε := r / α is small as compared to the other time scale

PDEs and the Problem Asymptotic Behavior of the Solutions to the Equation: 2 ( f ( y )) 2 x 2 ∂ 2 P ∂ x 2 + ρν xf ( y ) ∂ 2 P ∂ x ∂ y + ν 2 ∂ 2 P ∂ P ∂ t + 1 ∂ y 2 + 2 � � � ∂ P � x ∂ P ε − 1 ( m − y ) − ν Λ( t , x , y ) ∂ x − P + ∂ y = 0 , (7) P ( t = T , x , y ) = h ( x ) (Final Condition.) under the regimes: 1. ν 2 ∼ ε − 1 (Fouque, Papanicolaou, Sircar) 2. ν 2 ≪ ε − 1 (M. Souza & JPZ)

PDEs and the Problem Asymptotic Behavior of the Solutions to the Equation: 2 ( f ( y )) 2 x 2 ∂ 2 P ∂ x 2 + ρν xf ( y ) ∂ 2 P ∂ x ∂ y + ν 2 ∂ 2 P ∂ P ∂ t + 1 ∂ y 2 + 2 � � � ∂ P � x ∂ P ε − 1 ( m − y ) − ν Λ( t , x , y ) ∂ x − P + ∂ y = 0 , (7) P ( t = T , x , y ) = h ( x ) (Final Condition.) under the regimes: 1. ν 2 ∼ ε − 1 (Fouque, Papanicolaou, Sircar) 2. ν 2 ≪ ε − 1 (M. Souza & JPZ)