Introduction Hedging EIAs Numerical Results Conclusion References Hedging Equity-Linked Products Under Stochastic Volatility Models Anne MacKay, ASA Department of Mathematics and Statistics Concordia University, Montreal August 13, 2011 Joint work with Dr. Patrice Gaillardetz, Concordia University, Montreal Dr. Etienne Marceau, Universit´ e Laval, Qu´ ebec Research funded by the Natural Sciences and Engineering Research Council of Canada (NSERC) and by the Fonds qu´ eb´ ecois de la recherche sur la nature et technologie (FQRNT) Anne MacKay – Concordia University Hedging Equity-Linked Products Under Stochastic Volatility
Introduction Hedging EIAs Numerical Results Conclusion References Outline of the Presentation: 1 Introduction Heston Model Equity-Linked Products 2 Hedging EIAs Hedging Strategies Hedging Errors 3 Numerical Results Assumptions Black-Scholes Hedging Strategies Heston Hedging Strategies 4 Conclusion Anne MacKay – Concordia University Hedging Equity-Linked Products Under Stochastic Volatility
Introduction Hedging EIAs Heston Model Numerical Results Equity-Linked Products Conclusion References Heston Model Introduced by Heston (1993) Better at describing the high peaks and heavy tails of the empirical distribution of log-returns Stock index price dynamics under the physical measure given by dS t = µ S t dt + √ v t S t dZ (1) , t dv t = κ ′ ( θ ′ − v t ) dt + σ √ v t dZ (2) , t where µ , κ ′ , θ ′ , σ are constants and � dZ (1) dZ (2) � = ρ dt . t t Market price of volatility risk is given by λ Risk-neutral parameters κ = κ ′ + λ and θ = κ ′ θ ′ κ ′ + λ Anne MacKay – Concordia University Hedging Equity-Linked Products Under Stochastic Volatility
Introduction Hedging EIAs Heston Model Numerical Results Equity-Linked Products Conclusion References Price of a European call option under the Heston Model Price of a European call option given by C H ( x t , v t , τ ) = Ke − r τ ( e x t P 1 ( x t , v t , τ ) − P 0 ( x t , v t , τ )) where x t = log( e r ( T − t ) S t ), τ = T − t and K P j ( x t , v t , τ ) = 1 2+ � ∞ � exp( iux t + C j ( u , τ ) θ + D j ( u , τ ) v t ) � 1 Re du , π iu 0 for j = 0 , 1, with C j ( u , τ ) and D j ( u , τ ) functions of u , τ , κ , θ , σ and ρ . Anne MacKay – Concordia University Hedging Equity-Linked Products Under Stochastic Volatility
Introduction Hedging EIAs Heston Model Numerical Results Equity-Linked Products Conclusion References Equity-Linked Products Insurance policies that offer participation in financial market while protecting the initial investment May offer other types of benefits Two main categories: Variable Annuities Equity-Indexed Annuities (EIAs) Anne MacKay – Concordia University Hedging Equity-Linked Products Under Stochastic Volatility
Introduction Hedging EIAs Heston Model Numerical Results Equity-Linked Products Conclusion References Review of Literature First studied under the Black-Scholes model by Brennan and Schwartz (1976) and Boyle and Schwartz (1977) Hardy (2003) discusses product design and pricing techniques Tiong (2000) and Lee (2003) present closed-form expressions for the price of the financial guarantees embedded in EIAs Lin and Tan (2003) prices EIAs under stochastic interest rate models Lin et al. (2009) uses a regime-switching model to value EIAs Anne MacKay – Concordia University Hedging Equity-Linked Products Under Stochastic Volatility
Introduction Hedging EIAs Heston Model Numerical Results Equity-Linked Products Conclusion References Equity-Indexed Annuities First sold in 1995 by Keyport Life Premium invested for 5 to 15 years Guaranteed return on initial investment Additional return based on the performance of a stock index Additional return may be reduced or capped Actual return of the EIA depends on its design (point-to-point, annual reset, ...) Anne MacKay – Concordia University Hedging Equity-Linked Products Under Stochastic Volatility
Introduction Hedging EIAs Heston Model Numerical Results Equity-Linked Products Conclusion References Point-to-Point EIA Payoff based on the value of the index at inception and at maturity of the contract Participation rate α in additional return, 0 < α ≤ 1 Participation rate ̺ in guaranteed return g , 0 < ̺ ≤ 1 Payoff: � � S T � � B PTP ( S T , T ) = max , ̺ (1 + g ) T 1 + α − 1 (1) S 0 Anne MacKay – Concordia University Hedging Equity-Linked Products Under Stochastic Volatility
Introduction Hedging EIAs Heston Model Numerical Results Equity-Linked Products Conclusion References Pricing Point-to-Point EIAs � K − 1+ α Let K = ̺ (1 + g ) T and L = S 0 � . α Re-write (1) as: B PTP ( S T , T ) = K + α max( S T − L , 0) . S 0 Under the no-arbitrage assumption, we have that P t ( S t , τ ) = Ke − r τ + α C ( S t , L , τ ) , S 0 where P t ( S t , τ ) is the price at time t of the point-to-point EIA of maturity T and C ( S t , L , T ) is the price at time t of a European call option of strike L and maturity T . Anne MacKay – Concordia University Hedging Equity-Linked Products Under Stochastic Volatility
Introduction Hedging EIAs Hedging Strategies Numerical Results Hedging Errors Conclusion References The Greeks ∆: Sensitivity to changes in the price of the underlying Γ: Sensitivity of the delta to changes in the price of the underlying V : Sensitivity to changes in the volatility For the European call option in the Heston model: P 1 + ∂ P 1 − e − x t ∂ P 0 ∆ H = C , t ∂ x t ∂ x t − ∂ 2 P 1 � ∂ 2 P 0 1 �� ∂ P 1 � �� − ∂ P 0 Γ H − e − x t = C , t ∂ x 2 ∂ x 2 S t ∂ x t ∂ x t t t � � e x t ∂ P 1 − ∂ P 0 V H Ke − r τ = C , t ∂ v t ∂ v t Anne MacKay – Concordia University Hedging Equity-Linked Products Under Stochastic Volatility
Introduction Hedging EIAs Hedging Strategies Numerical Results Hedging Errors Conclusion References Delta Hedging Protects the insurer against small changes in index prices. Based on following replicating portfolio H ∆ t = ∆ P , t S t + ξ t , with ξ t is an amount invested in a risk-free asset. ξ t chosen so that H ∆ t = P t ( S t , τ ). Strategy is self-financing when applied in continuous time. Anne MacKay – Concordia University Hedging Equity-Linked Products Under Stochastic Volatility
Introduction Hedging EIAs Hedging Strategies Numerical Results Hedging Errors Conclusion References Gamma Hedging Improves the delta hedging strategy when it is applied in discrete time Based on following replicating portfolio H Γ t = α Γ τ ) + α Γ 1 , t C ( S t , L , ¯ 2 , t S t + ξ t , with ξ t is an amount invested in a risk-free asset and Γ P , t α Γ = 1 , t Γ C , t α Γ ∆ P , t − α Γ = 1 , t ∆ C , t . 2 , t To hedge EIAs, use calls with the longest maturity possible. Anne MacKay – Concordia University Hedging Equity-Linked Products Under Stochastic Volatility
Introduction Hedging EIAs Hedging Strategies Numerical Results Hedging Errors Conclusion References Vega Hedging Protects the insurer against small changes in both index prices and volatility. Based on following replicating portfolio H V t = α V τ ) + α V 1 , t C ( S t , L , ¯ 2 , t S t + ξ t , with ξ t is an amount invested in a risk-free asset and V P , t α V = 1 , t V C , t α V ∆ P , t − α V = 1 , t ∆ C , t . 2 , t To hedge EIAs, use calls with the longest maturity possible. Anne MacKay – Concordia University Hedging Equity-Linked Products Under Stochastic Volatility
Introduction Hedging EIAs Hedging Strategies Numerical Results Hedging Errors Conclusion References Hedging Errors Due to the discretization of the hedging process Occur when rebalancing the replicating portfolio Hedging error at time t defined by HE t = P t ( S t , τ ) − H t − Total discounted hedging error given by mT − ir � m HE i PV ( HE ) = e i =1 if rebalancing occurs m times a year at equal time intervals. Used to assess the performance of the hedging strategy. Anne MacKay – Concordia University Hedging Equity-Linked Products Under Stochastic Volatility
Introduction Hedging EIAs Assumptions Numerical Results Black-Scholes Hedging Strategies Conclusion Heston Hedging Strategies References Assumptions 10-year maturity point-to-point EIA with g = 0 and ̺ = 1. Participation rate α chosen so that the price of the EIA is 1. Risk-free rate r = 0 . 02. Black-Scholes parameters: µ BS = 0 . 0636 and σ BS = 0 . 19. Heston parameters: κ = 5 . 1793, θ = 0 . 0178, σ = 0 . 1309, v 0 = 0 . 0286, ρ = − 0 . 7025. Index prices follow Heston model with different volatility risk premia λ . Anne MacKay – Concordia University Hedging Equity-Linked Products Under Stochastic Volatility
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