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Hedging Barriers Hedging Barriers Liuren Wu Zicklin School of Business, Baruch College (http://faculty.baruch.cuny.edu/lwu/) — Based on joint work with Peter Carr (Bloomberg) Modeling and Hedging Using FX Options, March 13, 2008 Liuren Wu Hedging Barriers Risk, March 13, 2008 1 / 27 Hedging barriers: Overview Barrier options are actively traded in the OTC currency market. Hedging barriers faces two major challenges: Compared to delta hedging of vanilla options, delta hedging of barriers is subject to larger errors in practice. —The errors are larger when the delta varies more over time. Reliable barrier option quotes are hardly available, making performance comparison across different models/hedging strategies difficult. The traditional approach: Simulation based on an assumed environment. Liuren Wu Hedging Barriers Risk, March 13, 2008 2 / 27 Comparing vanilla call to down-out call Option values and delta (under Black-Scholes) 0.9 20 Call Call Down−and−out Call Down−and−out Call 18 0.8 16 0.7 14 0.6 12 0.5 10 Delta 0.4 Option value 8 0.3 6 0.2 4 2 0.1 0 0 70 80 90 100 110 120 70 80 90 100 110 120 S S Down-and-out: The option has zero value once the lower barrier is crossed. Delta drops to zero once the barrier is crossed. Model parameters: r = q = 0, σ = 20%, K = 100, t = 6/12, L=90. Liuren Wu Hedging Barriers Risk, March 13, 2008 3 / 27 Comparing vanilla put to down-out put Option values and delta (under Black-Scholes) 30 Put 0 Down−and−out Put −0.1 25 −0.2 20 −0.3 −0.4 15 Delta −0.5 Option value −0.6 10 −0.7 5 −0.8 −0.9 Put Down−and−out Put 0 70 80 90 100 110 120 70 80 90 100 110 120 S S The terminal payoff (conditional on survival) increases with lowering spot, but the down-out feature works against it. Model parameters: r = q = 0, σ = 20%, K = 100, t = 6/12, L=90. Liuren Wu Hedging Barriers Risk, March 13, 2008 4 / 27 Hedging barriers: Objective I propose an approach, with which we can systematically examine the barrier hedging effectiveness of several strategies empirically based on historical paths on the FX and vanilla FX option prices, but without resorting to barrier option quotes. Two objectives What is the best hedging strategy really? How many dimensions to hedge. Which models to use. How many dimensions of risk do we need to model? Liuren Wu Hedging Barriers Risk, March 13, 2008 5 / 27 Tracking the P&L of a hedging strategy Consider N hedging strategies or models. At each date t, sell the target instrument for an unknown price Pt , Put on hedging positions on the underlying currency and/or vanilla currency options (straddles, risk reversals, butterflies). n N Let {Ht }n=1 denotes the hedging cost of the N strategies, observable based on the market quotes on currency prices and vanilla option prices. n N Track the aggregate hedging cost to maturity, {Ht+τ }n=1. The terminal cost of selling the barrier, Pt+τ , is observable based on the underlying currency path, even though the barrier price is not before expiry. Terminal P&L of hedging a barrier sold at time t with strategy n is: n rτ n PLt = Pt e − Pt+τ − Ht+τ n Compare the time-series variation of PLt for different strategies. n Issue: How to calculate PL without observing the barrier selling price Pt ? Liuren Wu Hedging Barriers Risk, March 13, 2008 6 / 27 Hedging performance comparison without observing the target instrument price We consider two solutions, each with different underlying assumptions: Compute the P&L excluding the barrier sales revenue: Replace n n rτ n n PLt = Pt e − Pt+τ − Ht+τ with PLft = −Pt+τ − Ht+τ . n n If Pt is constant over time, Var(PLt ) = Var(PLft ). m n More generally, for hedging strategy m, n, Var(PLt ) > Var(PLt ) if m n Var(PLft ) > Var(PLft ) and the covariance terms do not dominate: m rτ n rτ m n Cov(PLt , Pt e ) − Cov(PLt , Pt e ) < Var(PLt ) − Var(PLt ). Choose a barrier contract whose value does not vary much over time. Compute the P&L using the model fair value as the selling price. If there exists a model that generates the perfect hedging strategy, then the P&L computed based on the fair value of the model should always be zero. To the point that the P&L is not zero and varies over time, it reflects both the pricing error and the hedging ineffectiveness. Good news: Both solutions generate the same ranking in terms of hedging performance. Liuren Wu Hedging Barriers Risk, March 13, 2008 7 / 27 Hedging exercise design Target: One-touch paid at expiry with a lower barrier (OTPE). Maturity: 30 calendar days. Barrier (L) set to 25 Balck-Scholes put delta. Considering barrier in terms of delta instead of percentage (98%) spot generates more stable barrier values. Spot normalized to $100. Notional set at $100. Data and sample period: Currencies: dollar-yen, dollar-pound, both with dollar as domestic. Sample period: 1996/1/24–2004/1/28, 2927 days. Instruments: currencies, delta-neutral straddle, 10- and 25-delta risk reversal and butterfly spread at 1 week, and 1,2,3,6,9, 12 months. Liuren Wu Hedging Barriers Risk, March 13, 2008 8 / 27 Hedging exercise design Procedure: Sell a one touch at each date (2897 days) and monitor hedging (rebalancing if needed) over the next 30 days. Set the selling price to (i) zero and (ii) the model fair value. Store the two types of terminal P&L from the hedging exercise. Report/compare the summary statistics on the P&L over the 2897 exercises for each currency pair across different hedging strategies. Also report the summary statistics of no hedging. Liuren Wu Hedging Barriers Risk, March 13, 2008 9 / 27 Hedging strategies Dynamic Strategies (Rebalancing based on partial derivatives): Hedging dimensions (sources of risks hedged): Delta (spot sensitivity) using the underlying currency. Vega (volatility/variance sensitivity) using delta-neutral straddle. Vanna (spot/volatility cross sensitivity) using risk reversal. Volga using butterfly spread. Hedging models (used to compute the sensitivities): Black-Scholes (Garman-Kohlhagen). Heston stochastic volatility (daily parameter calibration). Carr-Wu stochastic skew model (daily parameter calibration). Dupire local volatility model Rebalancing frequencies: Daily. Weekly. Once (vanna-volga approach?). Static strategy (No rebalancing until barrier crossing or expiry) Liuren Wu Hedging Barriers Risk, March 13, 2008 10 / 27 The Black-Scholes model Black-Scholes model: dS/S = (r − q)dt + σdW Definitions of greeks: ∂P ∂P ∂2P ∂2P Delta ≡ , Vega ≡ , Vanna ≡ , Volga ≡ , ∂St ∂σ ∂S∂σ ∂σ∂σ σ input in calculating the greeks is the IV (L, τ), obtained via linear interpolation across strikes. When L is outside the available delta range, using the IV at the boundary (no extrapolation). At maturities shorter than one month (as time goes by), use one month quotes on implied volatility and interest rates. Delta, vega are analytical. Vanna and volga are numerical. Liuren Wu Hedging Barriers Risk, March 13, 2008 11 / 27 Pricing and hedging under the Black-Scholes model Vanilla and barrier pricing: −q(T −t) −r(T −t) Ct (K, T ) = St e N(d1) − Ke N(d2), 2p −r(T −t) St OTPEt (L, T ) = e N(−d2) + L N(−d3) , with ln(S /K)+(r−q±σ2/2)(T −t) ln(S /L)−(r−q−σ2/2)(T −t) p ≡ 1 − r−q , d = t √ , d = t √ . 2 σ2 1,2 σ T −t 3 σ T −t Hedging ratios on vanillas and barriers: −q(T −t) −q(T −t)√ Delta(Ct (K, T )) = e N(d1), Vega(Ct (K, T )) = St e T − tn(d1), −q(T −t) Vanna(Ct (K, T )) = −e n(d1)d2/σ. „ « −r(T −t) 0 2p “ St ”2p−1 “ St ”2p 0 Delta(OTPEt (L, T )) = −e n(d2)d2(St ) − L L N(−d3) + L n(d3)d3(St ) , „ « −r(T −t) “ St ”2p “ St ”2p S 4(r−q) Vega(OTPEt (L, T )) = e n(d )d + n(d )d + N(−d ) ln( ) , 2 4 L 3 5 L 3 L σ3 −x2/√ 0 0 √ with n(x) ≡ e 2π denoting a normal density function and d2(S) = d3(S) = 1/(Sσ T − t), and 2 2 ln(St /L) + (r − q + σ /2)(T − t) ln(St /L) − (r − q + σ /2)(T − t) d4 = √ , d5 = √ . σ2 T − t σ2 T − t Liuren Wu Hedging Barriers Risk, March 13, 2008 12 / 27 The Heston stochastic volatility model Heston model: √ √ dS/S = (r −q)dt+ vt dW , dvt = κ(θ−vt )dt+σv vt dZ, ρdt = E[dWdZ] Definitions of greeks: ∂P ∂P ∂2P Delta ≡ , Vega ≡ , Vanna ≡ , ∂St ∂vt ∂S∂vt Model parameters and vt are obtained from daily calibration to vanilla options (1m to 12m). κ is fixed to one. Use finite-difference method to determine values and greeks for the barrier. Use FFT method to determine greeks for vanilla options. Also consider adjusted delta: dOTPE (S ) ∂OTPE (S ) ∂OTPE (S ) σ ρ Adj. Delta ≡ t t = t t + t t v . dSt ∂St ∂vt St Liuren Wu Hedging Barriers Risk, March 13, 2008 13 / 27 Pricing and hedging under Heston Under the Heston model, we can derive the conditional generalized Fourier transform of the log currency return st,τ = ln St+τ /St in closed form: h i iu ln St+τ /St iu(r−q)τ−a(τ)−b(τ)vt φs (u) ≡ Et e = e , u ∈ D ⊆ C, (1) where h i κ η−κe −ητ a(τ) = σ2 2 ln 1 − 2η (1 − e ) + (η − κe)τ , v (2) 2ψ(1−e−ητ ) b(τ) = −ητ , 2η−(η−κe)(1−e ) and q 1 η = (κ)2 + 2σ2ψ, κ = κ − iuρσσ , ψ = σ2(iu + u2).
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