Stochastic Receding Horizon Control for Dynamic Option Hedging

Transcription

1 Stochastic Receding Horizon Control for Dynamic Option Hedging Alberto Bemporad University of Siena Department of Information Engineering joint work with L. Bellucci and T. Gabbriellini, MPS Capital Services 1 /34

2 Talk outline Dynamic hedging problem for financial options Dynamic hedging as a linear stochastic control problem Stochastic receding horizon control (SRHC) Pricing engine + SRHC = dynamic option hedging tool Simulation results 2 /34

3 Dynamic hedging problem for financial options The financial institution sells a synthetic option to a customer and gets w() ( ) Such money is used to create a portfolio w(t) of underlying assets (e.g. stocks) whose prices at time t are x1(t), x2(t),..., xn(t) At the expiration date T, the option is worth the payoff p(t) = wealth ( ) to be returned to the customer option price portfolio wealth time (years) 2 wealth w(t) = payoff p(t)?... 1 How to adjust dynamically 4 the portfolio so that.. for any price realization xi(t)? 8 6 payoff p(t) wealth w(t)! asset price at expiration 3 /34

4 Dynamics of traded assets Trading instants: {, t, 2 t,..., t t,...,(t 1) t} Generic asset price dynamics: price of asset #i at time t internal variables (e.g. variance) x i (t + 1) = f i (x i (t),y i (t),z x i (t)) y i (t + 1) = g i (y i (t),z y i (t)) noise Example: xi(t)=stock price, log normal model (BS, Black Scholes) dx i =(µdt + σdz x i )x i x i (t + 1) = e (µ 1 2 σ2 ) t+σ tz x i (t) x i (t) geometric Brownian motion Ito s lemma Example: xi(t)=stock price, GARCH(1,1) model (Heston, Nandi, 2) Example: xi(t)=option price of European call, based on BS model y i (t), z y i (t) Note: numerical integration can be also used to express xi(t+1), yi(t+1) as a function of x i (t), y i (t). 4 /34

5 Portfolio dynamics Portfolio wealth at time t: w(t) =u (t)+ n i=1 x i (t)u i (t) money in bank account (risk-free asset) w(t + 1) = (1 + r)u (t)+ w(t + 1) = (1 + r)w(t)+ n i=1 number of assets #i price of asset #i Assets traded at discrete time intervals under the self balancing constraint: x i (t + 1)u i (t) n i= b i (t)u i (t) b i (t) x i (t + 1) (1 + r)x i (t) Assumption: transaction costs can be neglected (this can be relaxed...). Assumption: option pricing engine is available (more later in this talk ), only focus on dynamic hedging. In particular, w() is assigned. 5 /34

6 Payoff function x 1 (t) Let = vector of assets, and x(t) =. y(t) = x n (t) y 1 (t). y n (t) Option price p(t): p(t) =f(x(t),y(t)) p(t) =f(x(),x(1),..., x(t),y(t)) path dependent options Payoff p(t): p(t )=f(x(),x(1),..., x(t )) Examples (n=1): p(t ) = max{x(t ) K, } p(t ) = max {,C + min i {1,...,N fix } x(t i ) x(t i 1 ) x(t i 1 ) } European call Napoleon cliquet (t i = fixing dates) 6 /34

7 Talk outline Dynamic hedging problem for financial options Dynamic hedging as a linear stochastic control problem Stochastic receding horizon control (SRHC) Pricing engine + SRHC = dynamic option hedging tool Simulation results 7 /34

8 Option hedging = linear stochastic control Portfolio dynamics is linear, with multiplicative stochastic noise w(t + 1) = (1 + r)w(t)+ n i= b i (t)u i (t) input vector state variable B matrix affected by stochastic multiplicative noise Block diagram of stochastic control problem: z(t) asset dynamics x(t) option pricing engine p(t) hedging controller u(t) wealth dynamics w(t) Control objective: w(t) should be as close as possible to p(t), for any possible realization of the asset prices x(t) ( tracking w/ disturbance rejection ) 8 /34

9 Control objective Define hedging error e(t) w(t) p(t) we want e(t) small! π(e(t)) π(e 2 (T)) π(e(t)) e(t) Minimize expected hedging error: e 2 (T) min E [e(t )] 2 Very risky, variance of e(t) may be large! e(t) Minimize expected squared error: min E [ e(t ) 2] variance If minimum is then both mean and variance of hedging error are In fact: E [ e(t ) 2] = E [ (e(t ) E[e(T )]) 2] + αe [e(t )] 2 for α =1 mean Minimize variance of hedging error: min E [ (e(t ) E[e(T )]) 2] How about E[e(T)]? 9 /34

10 Minimum variance control Let us minimize the variance of hedging error: min E [ (e(t ) E[e(T )]) 2] Theorem Let w() = p(). Under non-arbitrage conditions, if a strategy exists such that Var[w(T ) p(t )] = (=perfect hedging) then w(t) p(t) =, t =, 1,..., T and in particular E[w(T ) p(t )] =, that is the final hedging error is deterministically. (cf. Black Scholes theory) Proof: By induction. Note: Var[e(T)]= means e(t)=w(t)-p(t) is deterministic e(t)> would imply w(t)>p(t) always gain wealth, which is impossible if no arbitrage exists, as w() = (1 + r) T E[p(T )] 1 /34

11 Existing approaches to dynamic option hedging Analytical approaches: choose u(t) to reject exactly the effect of stochastic noise z(t) PROS: lots of insight! CONS: limited to simple stock models & payoff functions (Black and Scholes, 1973) (Merton, 1973) Multi stage stochastic programming: choose u(t) by solving a (large scale) optimization problem. PROS: pricing & hedging in one shot, check arbitrage conditions CONS: rough discretization of probability space & trading dates (Edirisinghe et al., 1993) (Gondzio et al., 23) (Klaassen, 1998) (Zhao and Ziemba, 1998) Stochastic dynamic programming: PROS: rather versatile CONS: rough discretization of probability space & trading dates (Bertsimas et al., 21) Stochastic model predictive control: solve recursively and on line an optimization problem over a short prediction horizon PROS: very versatile CONS: requires on line optimization (Primbs, 29) (this talk) 11 /34

12 Talk outline Dynamic hedging problem for financial options Dynamic hedging as a linear stochastic control problem Stochastic receding horizon control (SRHC) Pricing engine + SRHC = dynamic option hedging tool Simulation results 12 /34

13 Model Predictive Control (MPC) model based optimizer reference p(t) process input output u(t) w(t) measurements A model of the process is used to predict the future evolution of the process in order to optimize the control signal A. Bemporad Stochastic RHC for Dynamic Option Hedging Siena, September 7, /34

14 Receding horizon philosophy At time t: solve an optimal control problem over a finite future horizon of N steps: min z N 1 k= w t+k p(t) 2 + ρ u t+k 2 s.t. w t+k+1 = f(w t+k,u t+k ) w t+k p(t) Predicted outputs Manipulated Inputs u t+k t t+1 t+n u min u t+k u max w min w t+k w max w t = w(t), k =,..., N 1 t+1 t+2 t+n+1 Only apply the first optimal move u*(t) At time t+1: Get new measurement w(t+1), repeat the optimization. And so on Stochastic MPC: minimize expected value of cost function 14 /34

15 Stochastic linear MPC Assume stochastic model of the process w(t + 1) = A(z 1 (t))w(t)+b(z 1 (t))u(t)+ez 2 (t) w R n z 1 (t), z 2 (t) = stochastic noise Optimize stochastic performance under (chance) constraints min E w N Pw N + N 1 k= w k Qw k + u k Ru k Ensure mean square convergence E[w (t)w(t)] = A few SMPC approaches exist in the control literature (Schwarme & Nikolaou, 1999) (Wendt & Wozny, 2) (Batina, Stoorvogel, Weiland, 22) (van Hessem & Bosgra 22) (Munoz de la Pena, Bemporad, Alamo, 25) (Couchman, Cannon, Kouvaritakis, 26) (Primbs, 27) (Oldewurtel, Jones, Morari, 28) (Ono, Williams, 28) (Bernardini & Bemporad, 29) 15 /34

16 Talk outline Dynamic hedging problem for financial options Dynamic hedging as a linear stochastic control problem Stochastic receding horizon control (SRHC) Pricing engine + SRHC = dynamic option hedging tool Simulation results 16 /34

17 SMPC for dynamic option hedging Stochastic finite horizon optimal control problem: min {u(k,z)} s.t. Var z [w(t, z) p(t, z)] w(k +1,z) = (1 + r)w(k, z)+ w(t, z) =w(t) n i= b i (k, z)u i (k, z), k = t,..., T 1 z = {z(t + 1),..., z(t )} Z t w(t+t-1,z) w(t,z) p(t,z) w(t+n,z) p(t+n,z) w(t,z) p(t,z) w(t+1,z) w(t) w(t) Optimize up to time t+n Perfect hedging assumption from time t+n to T 17 /34

18 SMPC for dynamic option hedging Stochastic finite horizon optimal control problem (fixed horizon N): min {u(k,z)} s.t. Var z [w(t + N, z) p(t + N, z)] w(k +1,z) = (1 + r)w(k, z)+ w(t, z) =w(t) n i= b i (k, z)u i (k, z), k = t,..., t + N w(1+n,z) p(1+n,z) w(3+n,z) p(3+n,z) w(t,z) p(t,z) w(t,z) p(t,z) w(n,z) p(n,z) w(2+n,z) p(2+n,z) w(t,z) p(t,z) w(t,z) p(t,z) w() w(1) w(2) w(3) 18 /34

19 SMPC for dynamic option hedging Drawback: the longer the horizon N, the largest the number of scenarios! Special case: use N=1! minimum variance control! min u(t) s.t. Var z [w(t +1,z) p(t +1,z)] w(t +1,z) = (1 + r)w(t)+ n i= b i (t, z)u i (t) w(t+1,z) p(t+1,z) w(t,z) p(t,z) Only one vector u(t) to optimize (no dependence of u on z!) No further branching, so we can generate a lot of scenarios for z Need to compute target wealth p(t+1,z) for all z w(t) Perfect hedging assumption from time t+1 to T Optimize up to time t+1 19 /34

20 SMPC hedging algorithm Let t=current hedging date, w(t)=wealth of portfolio, x(t) R n = asset prices Use Monte Carlo simulation to generate M scenarios of future asset prices x 1 (t + 1), x 2 (t + 1),..., x M (t + 1) y 1 (t + 1), y 2 (t + 1),..., y M (t + 1) Use a pricing engine to generate the corresponding future option prices p 1 (t + 1), p 2 (t + 1),..., p M (t + 1) Optimize sample variance, get new asset quantities u(t) R n, rebalance portfolio: min u(t) M j=1 w j (t + 1) p j (t + 1) 1 M M i=1 w j (t + 1) p j (t + 1) 2 This is a very simple least squares problem with n variables! (n = number of traded assets) With transaction costs, the problem can be rewritten as mixed integer quadratic program using a mixed logical dynamical (MLD) reformulation of the portfolio dynamics (Bemporad, Morari, 1999) 2/34

21 Talk outline Dynamic hedging problem for financial options Dynamic hedging as a linear stochastic control problem Stochastic receding horizon control (SRHC) Pricing engine + SRHC = dynamic option hedging tool Simulation results Is SMPC good for option hedging? 21 /34

22 Example: BS model, European call Portfolio wealth vs. payoff at expiration Black Scholes model (=log normal) volatility=.2, risk free=.4 T =24 weeks (Δt=1 week) 5 simulations M =1 scenarios Pricing method: Monte Carlo sim. SMPC 5 4 Portfolio wealth, option price option price p(t) Stock price at expiration 3 portfolio wealth w(t) 2 CPU time = 7.52 ms per SMPC step (Matlab R29 on this mac) /34

23 Example: BS model, European call Portfolio wealth vs. payoff at expiration Black Scholes model (=log normal) volatility=.2, risk free=.4 T =24 weeks (hedging every week) 5 simulations M =1 scenarios Delta hedging Hedging error e(t)=w(t) p(t) MPC hedging delta hedging Stock price at expiration 1.5 CPU time =.2 ms per SMPC step (Matlab R29 on this mac).5 1 SMPC and delta hedging are almost indistinguishable Simulation test 23 /34

24 Example: BS model, European call TIME x(t) w(t) p(t) u(t) x(t)*u1(t) x(t)*dp/dx(t) t=.: S=1., P= 6.196, O= 6.196, P(B)= , P(S)= (BS delta= ) t=.185: S=11.367, P= 6.955, O= 6.865, P(B)=-56.91, P(S)= (BS delta= ) t=.37: S= , P= 4.134, O= 4.261, P(B)= , P(S)= (BS delta= 46.37) t=.556: S= , P= 2.985, O= 3.18, P(B)=-35.8, P(S)= (BS delta= 37.67) t=.741: S= 93.57, P= 2.345, O= 2.415, P(B)= , P(S)= (BS delta= ) t=.926: S= , P= 2.431, O= 2.395, P(B)=-3.2, P(S)= (BS delta= ) t=.1111: S= , P= 2.732, O= 2.591, P(B)= , P(S)= (BS delta= 34.76) t=.1296: S= , P=.426, O=.859, P(B)= , P(S)= (BS delta= 15.53) t=.1481: S= 9.411, P=.83, O= 1.199, P(B)= , P(S)= (BS delta= 2.147) t=.1667: S= , P=.373, O=.754, P(B)= , P(S)= (BS delta= ) t=.1852: S= , P=.214, O=.544, P(B)= , P(S)= (BS delta= ) t=.237: S= 9.998, P=.641, O= 1., P(B)= , P(S)= (BS delta= 18.91) t=.2222: S= , P= 1.425, O= 1.867, P(B)=-3.734, P(S)= (BS delta= ) t=.247: S= 99.89, P= 3.149, O= 3.945, P(B)=-52.32, P(S)= (BS delta= ) t=.2593: S=12.72, P= 4.682, O= 5.466, P(B)= , P(S)= (BS delta= ) t=.2778: S= , P= 2.69, O= 3.439, P(B)= , P(S)= (BS delta= ) t=.2963: S= , P= 2.499, O= 3.147, P(B)=-5.513, P(S)= (BS delta= ) t=.3148: S= , P= 1.79, O= 2.233, P(B)=-42.46, P(S)= (BS delta= ) t=.3333: S=1.471, P= 2.79, O= 3.142, P(B)= , P(S)= (BS delta= 57.34) t=.3519: S=12.84, P= 4.12, O= 4.363, P(B)= , P(S)= (BS delta= ) t=.374: S= , P=.144, O= 1.22, P(B)= , P(S)= (BS delta= ) t=.3889: S= , P=.238, O= 1.3, P(B)= , P(S)= (BS delta= ) t=.474: S= , P=.62, O= 1.89, P(B)= , P(S)= (BS delta= 4.275) t=.4259: S= , P=.71, O=.38, P(B)=-22.85, P(S)= (BS delta= 2.3) t=.4444: S= 96.2, P= -.344, O=., P(B)= -.344, P(S)=. (BS delta=.) 24/34

25 Example: Heston model, European call Portfolio wealth vs. payoff at expiration Heston s model T =24 weeks (hedging every week) 5 simulations M =1 scenarios risk free=.4 Pricing method: Monte Carlo sim. SMPC Stock price at expiration CPU time = 85.5 ms per SMPC step (Matlab R29 on this mac) Heston's model dx i (τ) = (µ x i dτ + y i (τ)dz x i )x i(τ) dy i (τ) = θ i (k i y i (τ))dτ + ω i y i (τ)dz y i 25 /34

26 Example: Heston model, European call Portfolio wealth vs. payoff at expiration Heston s model T =24 weeks (hedging every week) 5 simulations M =1 scenarios risk free=.4 Delta hedging 2 6 Hedging error e(t)=w(t) p(t) MPC hedging delta hedging Stock price at expiration 2 CPU time = 1.85 ms per SMPC step (Matlab R29 on this mac) Simulation test 26/34

27 Example: BS model, Napoleon cliquet Final wealth in portfolio Portfolio wealth vs. payoff at expiration Black Scholes model (=log normal) volatility=.2 T =24 weeks (hedging every week) 5 simulations M =1 scenarios risk free=.4 Pricing method: Monte Carlo sim. SMPC: only trade underlying stock Payoff p(t ) = max {,C + min i {1,...,N fix } x(t i ) x(t i 1 ) x(t i 1 ) } CPU time = 14 ms per SMPC step (Matlab R29 on this mac) ti =,8,16,24 weeks 27 /34

28 Example: BS model, Napoleon cliquet Final wealth in portfolio Portfolio wealth vs. payoff at expiration Black Scholes model (=log normal) volatility=.2 T =24 weeks (hedging every week) 5 simulations M =1 scenarios risk free=.4 Delta hedging, only trade underlying stock Payoff p(t ) = max {,C + min i {1,...,N fix } x(t i ) x(t i 1 ) x(t i 1 ) } CPU time = 2.41 ms per SMPC step (Matlab R29 on this mac) ti =,8,16,24 weeks 28/34

29 Example: BS model, Napoleon cliquet Final wealth in portfolio Portfolio wealth vs. payoff at expiration Black Scholes model (=log normal) volatility=.2 T =24 weeks (hedging every week) 5 simulations M =1 scenarios risk free=.4 Pricing method: Monte Carlo sim. SMPC: Trade underlying stock & European call with maturity t+t Payoff p(t ) = max {,C + min i {1,...,N fix } x(t i ) x(t i 1 ) x(t i 1 ) } CPU time = 1625 ms per SMPC step (Matlab R29 on this mac) ti =,8,16,24 weeks 29/34

30 Approximate option pricing Bottleneck of the approach for exotic options: price M future option values p 1 (t + 1), p 2 (t + 1),..., p M (t + 1) Monte Carlo pricing can be time consuming: say L scenarios to evaluate a single option value need to simulate ML paths to build optimization problem (e.g.: M=1, L=1, ML=1 6 ) Use off line function approximation techniques to estimate p(t) as a function of current asset parameters and other option related parameters Example: Napoleon cliquet, Heston model p(t) =f(x(t), σ(t),x(t 1 ),..., x(t Nfix ) Most suitable method for estimating pricing function f: least squares Monte Carlo approach based on polynomial approximations (Longstaff, Schwartz, 21) 3/34

31 Example: BS model, Napoleon cliquet Final wealth in portfolio Portfolio wealth vs. payoff at expiration Black Scholes model (=log normal) volatility=.2, risk free=.4 T =24 weeks (hedging every week) 5 simulations M =1 scenarios Pricing method: LS approximation SMPC: only trade underlying stock Payoff CPU time = 14 ms per SMPC step (Matlab R29 on this mac) CPU time = 5.5 ms per SMPC step (Matlab R29 on this mac) CPU time = 76.7 s to compute LS approximation (off line) Hedging quality very similar! 31 /34

32 Example: BS model, Napoleon cliquet Final wealth in portfolio Portfolio wealth vs. payoff at expiration Black Scholes model (=log normal) volatility=.2, risk free=.4 T =24 weeks (hedging every week) 5 simulations M =1 scenarios Pricing method: LS approximation SMPC: Trade underlying stock & European call with maturity t+t Payoff CPU time = 1625 ms per SMPC step (Matlab R29 on this mac) CPU time = 59.2 ms per SMPC step (Matlab R29 on this mac) CPU time = 76.7 s to compute LS approximation (off line) Hedging quality very similar! 32 /34

33 Example: Heston model, Napoleon cliquet.2 Portfolio wealth vs. payoff at expiration Final wealth in portfolio Portfolio wealth vs. payoff at expiration Payoff SMPC: only trade underlying stock CPU time = 22 ms per SMPC step Final wealth in portfolio Payoff SMPC: Trade underlying stock & European call with maturity t+t CPU time = 277 ms per SMPC step Final wealth in portfolio Portfolio wealth vs. payoff at expiration Payoff Delta hedging only trade underlying stock CPU time = 156 ms per SMPC step CPU time = 156 s to compute LS approximation (off line) 33 /34

34 Conclusions Dynamic option hedging = stochastic control problem Propose Stochastic MPC as a very versatile tool for dynamic option hedging: it s based on Monte Carlo simulation can use arbitrary stock models can use any payoff function for which a numerically efficient pricing engine is available Computational demand mostly due to pricing future option values On line use of SMPC: suggest trading moves to traders Off line use of SMPC: run extensive simulations to quantify the average hedging error for a given market model and option type On going work: test SMPC s robustness w.r.t. market model mismatch 34/34