Pricing and hedging of derivatives in illiquid markets Pierre Patie - PowerPoint PPT Presentation

Pricing and hedging of derivatives in illiquid markets Pierre Patie RiskLab ETH Z¨ urich Joint work with R. Frey (Swiss Banking Institute) Frankfurt Math Finance Colloquium email: patie@math.ethz.ch homepage: http:/ /www.math.ethz.ch/˜patie RiskLab homepage: http:/ /www.risklab.ch

Pricing and hedging of derivatives in illiquid markets Outline I Model Description II Perfect Option Replication III Numerical Results IV Hedge Simulation - Tracking Error V Pricing Rule for Individual Claims VI Implied Parameters VII Conclusion � 2001 (RiskLab) c 1

The Model (1) The Market Riskless money market account B with price normalized to B t ≡ 1. ⊲ Market for B perfectly liquid. Risky asset with price process S . Market for S can be illiquid. ⊲ Asset Price Dynamics Let ( S t , t ≥ 0), defined on some filtered proba- bility space (Ω , ( F t ) t , P ), be the solution of the following SDE: dS t = σS t − dW t + ρS t − dα t , � �� � effect of hedging where for 0 ≤ t ≤ T , we assume that the large trader holds α t shares of S , ρ ≥ 0 is a liquidity coefficient, σ is a given reference volatility, ( W t , t ≥ 0) is a Brownian motion on (Ω , ( F t ) t ) , P ). � 2001 (RiskLab) c 2

The Model (2) Remarks ρ = 0 ⇒ standard Black-Scholes (BS) case where market ⊲ are assumed to be perfectly liquid (frictionless). ρ large ⇒ market illiquid. ⊲ 1 ρS t is the market depth at time t . ⊲ Possible extensions: ⊲ ρ ( . ) can be function of the asset price or stochastic (as σ ). � 2001 (RiskLab) c 3

Example: Stop-Loss Contract Scenario: large trader holds K shares and protects them by a stop-loss contract with trigger S , i.e., he automatically sells his shares at τ := inf { t > 0 , S t < S } . Market perfectly liquid ⇒ value of his position always ≥ V := KS . ⊲ What happens in our setup ? ⊲  K for t ≤ τ,  Strategy equals α t := 0 for t ≥ τ.  The asset price at τ equals S τ = S τ − (1 − ρK ) = S (1 − ρK ) , and we have for value of the position at time τ : V τ = KS τ = KS − ρSK 2 < V . = ⇒ Stop-loss yields imperfect protection! � 2001 (RiskLab) c 4

Market Volatility – Feedback-Effects from Hedging Class of strategies considered: α t = Φ( t, S t ) for some smooth function Φ : [0 , T ] × R + → R with derivative Φ S satisfying ρS Φ S ( t, S ) < 1. Proposition: If large trader uses strategy Φ( t, S t ) the asset price follows diffusion of the form: = v ( t, S t ) S t dW t + b ( t, S t ) S t dt, (1) dS t where σ v ( t, S ) = 1 − ρS Φ S ( t, S ) , � � σ 2 S 2 Φ SS ρ b ( t, S ) = Φ t ( t, S ) + . 2(1 − ρS Φ S ( t, S )) 2 1 − ρS Φ S ( t, S ) Remarks: Volatility depends on Φ S , i.e. on ”Gamma”. ⊲ Volatility is increased if Φ S > 0, it decreased if Φ S < 0. ⊲ � 2001 (RiskLab) c 5

Hedging of Derivatives – Basic Concepts Used Hedger uses strategy ( α t , β t ) ⇒ stock price S t ( α ). Mark to market value: V M = α t S t ( α ) + β t . t � T Value of a self-financing strategy: V M = V M + 0 α s − dS s ( α ) . 0 T Definition: consider a derivative with payoff h ( S T ) and a self-financing hedging strategy ( α t , β t ). The tracking error e M T of this strategy equals � � T � e M V M T = h ( S T ( α )) − + 0 α s − dS s ( α ) (2) , 0 e M measures loss (profit) from hedging. T Remark: one can prove that if the large trader uses the Black-Scholes strategy e M is always positive. T � 2001 (RiskLab) c 6

Perfect Option Replication Problem: can we replicate a derivative perfectly (i.e. e M = 0) if we T adapt the strategy? This is a fixed-point problem: volatility structure used in computing the hedge must be the one resulting from hedging activity. Proposition: suppose that the smooth function u ( t, S ) solves the parabolic partial differential equation (PDE) σ 2 u t + 1 2 S 2 = 0 , (1 − ρSu SS ) 2 u SS u ( T, S ) = h ( S ) . Then Φ( t, S t ) := u S ( t, S t ) is a replicating strategy, u ( t, S t ) is the hedge cost. � 2001 (RiskLab) c 7

Numerical Solution (1) To avoid problems with the volatility range, we considered ⊲ the modified operator σ 2 2 S 2 max { δ 0 , u t + 1 (1 − min { δ 1 ,ρSu SS } ) 2 } u SS . To solve the nonlinear PDE we proceed as follows: ⊲ ≫ time and space discretization by finite differences methods, ≫ implicit scheme for space derivatives approximation, (unconditionally stable scheme) ≫ we solve the resulting nonlinear system by using the Newton method for each time step. � 2001 (RiskLab) c 8

European Call Prices 60 rho = 0 rho = 0.05 rho = 0.1 rho = 0.2 rho = 0.4 50 40 Option Prices 30 20 10 0 70 80 90 100 110 120 130 140 150 160 Stock Prices Hedge cost of European call u ( S, T ) for various values of ρ (Strike = 100, σ = 0 . 4, T − t = 0 . 25 years). � 2001 (RiskLab) c 9

European Call Greeks 1 rho = 0 0.05 rho = 0 rho = 0.05 rho = 0.05 rho = 0.1 rho = 0.1 rho = 0.2 rho = 0.2 rho = 0.4 rho = 0.4 0.8 0.04 0.6 0.03 Gamma Delta 0.4 0.02 0.2 0.01 0 0 60 80 100 120 140 160 60 80 100 120 140 160 Stock Prices Stock Prices Hedge ratio u S and Gamma u SS for an European call for various values of ρ (Strike = 100, σ = 0 . 4, T − t = 0 . 25 years). � 2001 (RiskLab) c 10

Call Spread Prices rho = 0 10 rho = 0.05 rho = 0.1 rho = 0.2 8 Options Prices 6 4 2 0 60 80 100 120 140 160 Stock Prices Hedge cost of Call Spread u ( S, T ) for various values of ρ (Strike 1 = 100, Strike 2 = 110, σ = 0 . 4, T − t = 0 . 25 years). � 2001 (RiskLab) c 11

Call Spread Greeks 0.25 rho = 0 rho = 0 0.01 rho = 0.05 rho = 0.05 rho = 0.1 rho = 0.1 rho = 0.2 rho = 0.2 0.2 0.005 0.15 Gamma Delta 0 0.1 -0.005 0.05 0 -0.01 60 80 100 120 140 160 60 80 100 120 140 160 Stock Prices Stock Prices Hedge ratio u S and Gamma u SS for a Call Spread for various values of ρ (Strike 1 = 100, Strike 2 = 110, σ = 0 . 4, T − t = 0 . 25 years). � 2001 (RiskLab) c 12

Numerical Solution (2) First order approximation (Papanicolaou and Sircar (1999)) First, we denote by L BS the Black-Scholes operator: L BS C := C t + 1 2 σ 2 S 2 C SS + r ( SC S − C ) . For small ρ ( ρ << 1), we construct a regular perturbation series: C ( S, t, ρ ) = C BS ( S, t ) + ρC ( S, t ) + O ( ρ 2 ) , where L BS C BS = 0 , and L BS C = − σ 2 S 3 ( C BS SS ) 2 . Therefore we can approximate the solution of the non-linear PDE by computing successively solution of two Black-Scholes linear PDE. We compared prices obtained with the direct solver and the approximation for European call options. � 2001 (RiskLab) c 13

European Call Prices - Linear Approximation 45 rho = 0 Linear Approximation - rho = 0.05 40 rho = 0.05 35 30 Option Prices 25 20 15 10 5 0 70 80 90 100 110 120 130 140 Stock Prices Hedge cost of European call u ( S, T ) for various values of ρ (Strike = 100, σ = 0 . 4, T − t = 0 . 25 years). � 2001 (RiskLab) c 14

Hedge Simulation – Tracking Error Computation (1) In order to check the robustness of our model and to compare different hedging strategies we carry out some hedge simulation. First, we use the stochastic differential equation (SDE) (1) satisfied by the stock price process under feedback: = v ( t, S t ) S t dW t + b ( t, S t ) S t dt. dS t Then we use Euler-Maruyama scheme to solve it numerically. We discretisize the time interval [0 , T ] with a fixed step-size (∆ t = T n ) and for k = 0 , . . . , n − 1,  ˜ S 0 = S 0 ,  � � S i ( k +1)∆ t = S i k ∆ t + v ( k ∆ t , S i W i ( k +1)∆ t − W i + b ( k ∆ t , S i k ∆ t ) k ∆ t )∆ t ,  k ∆ t where � � W ( k +1)∆ t − W k ∆ t (0 ≤ k ≤ n − 1) denote independent N (0 , ∆ t )-distributed Gaussian random variables. � 2001 (RiskLab) c 15

Hedge Simulation – Tracking Error Computation (2) Then, for each simulated path i, 1 ≤ i ≤ N, we approximate the tracking error (1) as follows:   n − 1 � � � � � e i T ≈ h ( S i k ∆ t , S i S i ( k +1)∆ t − S i  , T ) −  V 0 + Φ k ∆ t k ∆ t k =0 where T ) is the payoff of the derivative at maturity ( h ( S ) = ( S − K ) + ) , h ( S i V 0 is the initial value of the hedge-portfolio, Φ( k ∆ t , S i k ∆ t ) is the hedging strategy value. We define the tracking error average by N e T = 1 � e i T . N i =1 � 2001 (RiskLab) c 16

Tracking Error Density 1.4 1.4 1.2 1.2 rho=0 Black-Scholes rho=0.01 Nonlinear rho=0.05 1.0 1.0 rho=0.1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0.0 0.0 -3 -2 -1 0 1 2 -2 0 2 4 Tracking error density in an illiquid market using Tracking error density in an illiquid market the nonlinear strategy for various values of ρ using various strategies ( N = 5000 , n = 240 , T = 0 . 5 years). ( ρ = 0 . 02 , N = 5000 , n = 240 , T = 0 . 5 years). � 2001 (RiskLab) c 17

Properties of the Tracking Error Distribution (1) 0 0.01 0.02 0.05 ρ e M – 0.08 – 0.08 – 0.08 – 0.07 T � � e M 0.67 0.7 0.73 0.83 V aR 0 . 99 T � � e M 0.84 0.89 0.93 1.07 ES 0 . 99 T Properties of the tracking error distribution for the nonlinear hedging strategy used to replicate an European call option for different values of ρ ( T = 0 . 5 , K = 100, S 0 = 100, 5000 simulations with n = 240 (number of trades)). � 2001 (RiskLab) c 18