CASE STUDY III EVALUATING MODEL RISK WITHIN THE BLACKSCHOLES - PowerPoint PPT Presentation

CASE STUDY III EVALUATING MODEL RISK WITHIN THE BLACK–SCHOLES FRAMEWORK Limiting Model Risk by Short-Selling Constraints

Outline for case study III • Samuelson (Black–Scholes) model • Exotic options, unlimited short positions • Mitigation of model risk by short-selling constraints • Resulting market incompleteness, upper hedging price • Incorporation of constraint into option price • Option price as stochastic control problem • Explicit valuation for several examples

References for case study III • U. Schmock, S. E. Shreve, U. Wystup: Valuation of Exotic Options under Shortselling Constraints Finance and Stochastics, Vol. 6 (2002) 143–172. • U. Schmock, S. E. Shreve, U. Wystup: Dealing with Dangerous Digitals In: J. Hakala and U. Wystup (eds.): Foreign Exchange Risk: Models, Instruments and Strategies Risk Books, Risk Waters Group (2002) 327–348.

Samuelson model Geometric Brownian motion for the exchange rate: dS t = ( r d − r f ) S t dt + σS t dW t , S 0 > 0 S t price of one unit of foreign currency in domestic currency at time t ∈ [0 , T ] r d ∈ R risk-free domestic interest rate r f ∈ R risk-free foreign interest rate σ > 0 volatility ( W t ) t ∈ [0 ,T ] Brownian motion (Wiener process) under the risk-neutral measure P r � r d − r f mean rate of return of the exchange rate

Samuelson model (cont.) Equity model S t stock price at time t r d ∈ R risk-free domestic interest rate f ∈ R continuously paid dividend rate r Solution of the SDE � � rt + σW t − σ 2 t ∈ [0 , T ] S t = S 0 exp 2 t , Canonical probability space Ω = C ([0 , T ] , R ) ∋ ω �→ W t ( ω ) = ω ( t ) σ -algebra F t is the P -completion of σ ( W s ; s ∈ [0 , t ]), i. e., {F t } t ∈ [0 ,T ] is a Brownian filtration.

Hedging a European call option with strike K • Pay-off at maturity T is ( S T − K ) + where S T is the price of the underlying at time T and K > 0 is the strike price. • To hedge the call, buy a fraction of the underlying (delta-hedging). In the Samuelson model: Calculation of the fraction ∈ (0 , 1) at time t by differentiation of the Black–Scholes formula w. r. t. S t � log( S t /K ) + ( r + σ 2 / 2)( T − t ) � √ N σ T − t

Option value 12 10 8 50 days 6 4 10 days 2 1 day Stock price S t 90 95 100 105 110 Price of a European call option for three different maturities, interest rate r = 5%, volatility σ = 30%, strike price K = 100

1 Fraction of stock in hedging portfolio 0.8 0.6 0.4 50 days 1 day 0.2 10 days Stock price S t 90 95 100 105 110 Hedge for a European call option for three different maturities, interest rate r = 5%, volatility σ = 30%, strike price K = 100

Reverse up-and-out call European call option with strike K > 0 and knock- out barrier B > K . Pay-off at maturity T g ( S ) � ( S T − K ) + 1 { max t ∈ [0 ,T ] S t <B } No-arbitrage Black–Scholes price at time t ∈ [0 , T ] v ( t, x ) = E t,x � � e − r d ( T − t ) ( S ( T ) − K ) + 1 { max u ∈ [ t,T ] S u <B } if S t = x > 0 and no knock-out occurred before t . Delta hedging: – If S t is well below B : Buy a fraction of the underlying. – If S t is just below B : Go short in the underlying.

Price of the reverse up-and-out call f τ � � v ( t, x ) = xe − r N ( b − θ + ) − N ( k − θ + ) f τ +2 bθ + � � + xe − r N ( b + θ + ) − N (2 b − k + θ + ) − Ke − r d τ � � N ( b − θ − ) − N ( k − θ − ) − Ke − r d τ +2 bθ − � � N ( b + θ − ) − N (2 b − k + θ − ) where N is the standard normal distribution function, � r � √ τ σ ± σ τ � T − t, θ ± � 2 and σ √ τ log B 1 σ √ τ log K 1 b � k � x , x .

50 Option value option pay-off 40 1 day 30 10 days 20 50 days 10 Stock price S t 90 100 110 120 130 140 150 Price of a European call option with knock-out barrier B = 150 for three different maturities together with the option pay-off, interest rate r = 5%, volatility σ = 30%, strike price K = 100

Number of stocks in hedging portfolio 1 0.5 50 days 10 days Stock price S t 130 0 100 110 120 140 150 − 0 . 5 1 day − 1 − 1 . 5 Hedge of a European call option with knock-out barrier B = 150 for three different maturities, interest rate r = 5%, volatility σ = 30%, strike price K = 100

Fraction π t of capital in the stock 135 140 145 150 0 Stock price S t 10 days 50 days − 20 1 day − 40 − 60 − 80 − 100 Fraction π t of capital invested in the stock to replicate a European call option with knock-out barrier B = 150 for three different maturities, interest rate r = 5%, volatility σ = 30%, strike price K = 100

Problems with large FX positions • Large exposure for one sold barrier option • Liquidity risk and transaction costs • High model risk! Possible solutions • Pay a rebate at maturity or at the first hitting time of the barrier, when the option knocks out. • Modify the knock-out regulation (soft barrier option, step option, Parisian option). • Impose constraint for the hedge portfolio. → incomplete market, superhedge the option

Evolution of the hedge capital X t π t fraction of X t in foreign currency (adapted) 1 − π t fraction of X t in domestic currency C t capital consumed in [0 , t ] dX t = π t X t dS t + r f π t X t dt + r d (1 − π t ) X t dt − dC t S t = r d X t dt + σπ t X t dW t − dC t Option pay-off Lower semi-continuous function g : C + [0 , T ] → [0 , ∞ ) Short-selling constraint for foreign currency π t ≥ − α for all t ∈ [0 , T ] with α ≥ 0

Upper hedging price v (0 , S 0 ; α ) � inf { X 0 | ∃ ( π, C ) with X T ≥ g ( S ) and π t ≥ − α ∀ t ∈ [0 , T ] } Dual maximization problem Theorem: ( Cvitani´ c & Karatzas 1993, El Karoui & Quenez 1995 ) � � e − r d T − αλ T g ( S ) v (0 , S 0 ; α ) = sup E λ λ ∈L L contains all adapted, non-decreasing λ with λ (0) = 0, which are Lipschitz-continuous in t , uniformly in ω . � � � T � T d P λ − 1 1 t ) 2 dt λ ′ ( λ ′ t dW t − d P = exp 2 σ 2 σ 0 0

Simplification for dependence on final value Theorem: ( Broadie, Cvitani´ c & Soner 1998 ) � � e − r d T � If g ( S ) = ϕ ( S T ), then v (0 , S 0 ; α ) = E ϕ α ( S T ) with face-lift e − αλ ϕ ( xe − λ ) , ϕ α ( x ) � sup x ≥ 0 . � λ ≥ 0 Aim of our work • Generalization to path-dependent options by conversion of the dual maximization problem to a stochastic control problem. • Explicit computation of the upper hedging price for several examples.

Idea behind Broadie–Cvitani´ c–Soner theorem ( t, x ) �→ v ( t, x ; α ) is the smallest function which • satisfies the Black–Scholes PDE v t + rxv x + 1 2 σ 2 x 2 v xx − r d v = 0 , • dominates the final pay-off, i.e. v ( T, x ; α ) ≥ ϕ ( x ), • satisfies the constraint αv ( t, x ; α )+ xv x ( t, x, α ) ≥ 0. ϕ α is the smallest function dominating the pay-off � ϕ ′ α ( x ) ≥ 0. and satisfying the constraint α � ϕ α ( x ) + x � → Solve Black–Scholes PDE with pay-off � ϕ α . Pleasant surprise: Solution satisfies constraint!

Extension to path-dependent up-and-out call Observation: If v ( t, x ; α ) solves the Black–Scholes PDE, then w � αv + xv x solves the PDE, too. Strategy: • Boundary conditions for v give boundary condi- tions for w . • Require w = 0 at the boundary where the uncon- strained value function violates the constraint. • Solve Black–Scholes PDE for w . • Solve w � αv + xv x for v .

Formulation of the dual problem as singular stochastic control problem Theorem (Schmock/Shreve/Wystup): � � e − r d T − αλ T g ( Se − λ ) v (0 , S 0 ; α ) = sup E λ ∈C where C � { λ | λ adapted, non-decreasing, continuous process, λ (0) = 0 } . Remarks: • Maximization w. r. t. processes is easier. • Maximizing process can be found in many examples. • Maximizing processes can be singularly continuous. • Since g is lower instead of upper semi-continuous, maximizing processes need not exist.

Application to a European call option with strike K and knock-out barrier B > K Obligation at maturity T : g ( S ) � ( S T − K ) + 1 { max t ∈ [0 ,T ] S t <B } Maximization problem: � e − r d T − αλ T � � + 1 { max t ∈ [0 ,T ] S t e − λt <B } � S T e − λ T − K sup E λ ∈C Supremum unchanged for < → ≤ . Maximizing process: S t e − λ t ≤ B ⇐ ⇒ λ t ≥ log S t − log B ⇒ λ t ≥ λ ∗ u ∈ [0 ,t ] (log S u − log B ) + t � max =

Upper hedging price f τ � v ∗ ( t, x, α ) = xe − r N ( b − θ + ) − N ( k − θ + ) �� � 2 s ( s − 2 θ + ) × 1 e sb N ( − b + θ + − s ) − e sk N ( − k + θ + − s ) + e � f τ +2 bθ + + sxe − r 1 2 s ( s − 2 θ + ) N ( b + θ + ) − N ( ℓ + θ + ) + e s − 2 θ + �� � e ( s − 2 θ + ) b N ( − b + θ + − s ) − e ( s − 2 θ + ) ℓ N ( − ℓ + θ + s ) × − Ke − r d τ � N ( b − θ − ) − N ( k − θ − ) �� s − 2 θ − ) � 1 2 ˜ s (˜ e ˜ s ) − e ˜ sb N ( − b + θ − − ˜ sk N ( − k + θ − − ˜ + e s ) � sKe − r d τ +2 bθ − − ˜ 1 2 ˜ s (˜ s − 2 θ − ) N ( b + θ − ) − N ( ℓ + θ − ) + e s − 2 θ − ˜ �� � e (˜ s − 2 θ − ) b N ( − b + θ − − ˜ s ) − e (˜ s − 2 θ − ) ℓ N ( − ℓ + θ − − ˜ × s )