Volatility Trading Strategies: Dynamic Hedging via A Simulation Approach Antai Collage of Economics and Management Shanghai Jiao Tong University Advisor: Professor Hai Lan June 6, 2017
Outline 1 The volatility trading strategy Pricing and hedging under incomplete information Derivation of the PnL and its properties 2 Hedging & Simulations Discrete replication interval Transaction cost Model misspecification 3 An application: volatility arbitrage on Chinese 50 ETF options
Three Volatility Values: σ, Σ, σ h Incomplete Information makes volatility arbitrage possible: the market participants do not know the true data generating process (DGP) of risky asset price We formalize this notion with three different volatilities Definition (Parametrized Pricing Formula) The parametrized function F (τ, x; b) solves the Black-Scholes PDE: { F τ F + (r q)x x + 1 2 b2 x 2 2 F rf = 0, x 2 0 t < T, x R F (T, x; b) = (x K) +, x R Where b is the volatility input And we denote F [b] t := F (t, S t ; b) σ: The true scale of the underlying diffusion process: ds t = µs t dt + σs t dw t Σ: The volatility implied by market maker s quotes of options, F t,quote = F (t, S t ; Σ) σ h : The volatility parameter used by trader to calculate hedging position: π [σ h],1 t = [σ h] t = F x (t, S t; σ h )
The PnL Process of Delta Hedged Option Proposition (Cumulated PnL of Delta Hedged Option) If we long call at F [Σ] t, and delta hedge with [σ h] t shares; then the cumulative profit & loss from time t to expiration T is: PnL(t, T ) = 1 T 2 (σ2 σh 2 ) e r(t τ) Γ [σ h] τ Sτ 2 dτ t }{{} path-dependent integral ( ) +e r(t t) F [σ h] t F [Σ] t }{{} deterministic value If we hedge with real volatility, ie σ h = σ, then the final PnL is deterministic: PnL(t, T ) = e r(t t) (F [σ] t F [Σ] t ) If we hedge with implied volatility, ie σ h = Σ, then the final PnL is the integral: 1 2 (σ2 Σ 2 ) T t e r(t τ) Γ [Σ] τ Sτ 2 dτ driven by the difference between Gamma profit and Theta decay Otherwise, for an arbitrary σ h, the final PnL is a composite of these two terms
Simulated PnL Paths Figure: Monte-Carlo Simulation of 5 PnL Paths of Delta Hedged 1YR-ATM Call, σ h = σ = 04, Σ = 02, µ = 0, S 0 = 100, r = 002, q = 0; 10000 Steps Bounds of the PnL at time s, t s T (hedge with σ h = σ): F [σ] t F [Σ] t PnL(t, s) (F [σ] t F [Σ] t ) Ke r(t t) 2[N( σ Σ 2 T s) 1]
The Distribution of PnL The distribution of PnL(t, T ) is dependent on hedging volatility (σ h ) and drift parameter of stock return (µ) If σ h > σ, the mean PnL increases with µ ; otherwise decreases σ h = σ, µ = 0 is a saddle point on the mean PnL surface The standard deviation of PnL increases with σ h σ And the PnL is deterministic when σ = σ h Figure: Mean (left) and Standard Deviation (right) of PnL Distributions; σ = 04, Σ = 02, 5000 samples for each (σ h, µ) combination
Outline 1 The volatility trading strategy Pricing and hedging under incomplete information Derivation of the PnL and its distribution 2 Hedging & Simulations Discrete replication interval Transaction cost Model misspecification 3 An application: volatility arbitrage on Chinese 50 ETF options
Replication Error from Discrete Hedging Proposition (Replication Error with Discrete Hedging) Given time grid t j = t 0 + jδt, δt = (T t)/n, the total hedging error is n 1 n 1 [ ] 1 HE(t 0, T ) := δν j+1 = 2 Γ jσ 2 Sj 2 (Zj+1 2 1)δt + O(δt 3 2 ) j=0 j=0 Where {Z j } are iid standard normal variables Moreover, the conditional expectation E[δν j+1 F j ] = 0; E[δν 2 j+1 F j] = 1 2 (Γ js 2 j )2 (σ 2 δt) 2 In a small time interval, the hedging error is of order O(δt) If σ h = σ, HE(t 0, T ) is in fact the PnL The standard deviation of it is proportional to 1/ n, n is the hedging frequency In the above case, the standard deviation of PnL is approximately π 4n σv 0, V 0 = F b (t 0, S 0 ; b) If σ h is of other value, increasing the hedging frequency cannot reduce the variance of PnL
Replication Error from Discrete Hedging: Numerical Result Figure: Distribution of PnL with Increasing Hedging Frequency; σ h = σ = 04 (left), and σ h = Σ = 02 (right); 5000 samples for each
The Impact of Transaction Cost Proposition (Transaction Cost) If the transaction cost is a fixed proportion of trading volume, ietc j = κ j j 1 S j from t j 1 to t j, κ is constant, then n 1 n 1 [ TC(t, T ) := TC j+1 = κσsj 2 Γ j Z j+1 ] δt + O(δt) j=0 j=0 Where {Z j } are iid standard normal variables Moreover, the conditional expectation E[TC j+1 F j ] = κσsj 2 Γ 2δ j π ; E[TC2 j+1 F j] = κ 2 (Γ j Sj 2)2 σ 2 δt In a small time interval, the transaction cost is of order O(δt 1 2 ) Therefore, the transaction cost dominates the PnL asymptotically, for the latter only has order O(δt) With increasing hedging frequency, the PnL will be ultimately wiped out (its mean decreases), and its standard deviation will first decrease then increase
The Impact of Transaction Cost: Numerical Result Figure: Distribution of PnL with Transaction Cost and Increasing Hedging Frequency; σ h = σ = 04, Σ = 02; 5000 samples for each
The Impact of Model Misspecification We have discussed two types of model misspecifications: 1 Jump-Diffusion Process, where N(t) Pois(λt), Y i N(µ J, σ J ): ds t = µs t dt + σs t dw t + dj t ; dj t = S t Σ N(t) i=1 Y i The jumps add more variation to the stock returns Increases the mean PnL for long position, decreases the mean for short position Increases the standard deviation of PnL 2 Stochastic Volatility with Markov Switching (to lower volatility): ds t = µs t dt + σ(m t )S t dw t ; Reduces the variation of stock returns P (M t = M j M t δt = M i ) = P ij Decreases the mean PnL for long position, Increases the mean for short position Increases the standard deviation of PnL
Outline 1 The volatility trading strategy Pricing and hedging under incomplete information Derivation of the PnL and its distribution 2 Hedging & Simulations Discrete replication interval Transaction cost Model misspecification 3 An application: volatility arbitrage on Chinese 50 ETF options
Configuration of the Trading Strategy Idea: The market tends to overprice short-term options On Jan 3 2017, the volatility impled by 50ETF option expiring in March at 230 is higher than rolling estimate of real volatility Portfolio: We construct a delta hedged short Straddle as follow Short 1 unit (10000 shares) 50ETF-Call-March-2300 (SH10000787) Short 1 unit (10000 shares) 50ETF-Put-March-2300 (SH10000792) Long 996 ( ) shares of underlyting asset Figure: Normalized Equity Curve / Drawdown from Jan-3-2017 to Mar-22-2017
Post-Trade Analysis & Identification of the Model Risk Table: Post-Trade Analytics Name Cum Return Tot Volatility Sharpe Ratio Max Drawdown Value 6318% 4935% 126 2615% Figure: Drawdown (blue) VS Rolling Standard Deviation (red) of Stock Return
Q & A Session Thanks for your attention