Example of High Dimensional Contract

Transcription

1 Example of High Dimensional Contract An exotic high dimensional option is the ING-Coconote option (Conditional Coupon Note), whose lifetime is 8 years (24-212). The interest rate paid is flexible. In the first 2 years, a payment of 6 %, after that interest rate depends on asset prices. It depends on the value of the average value of the stocks in the Dow Jones Global Titan index: The starting price (2/24) was Euro 1.75, i.e., the value of 5 stocks of the Dow Jones Global Titans index. 15 / 2 / 212: Eur 1 will be paid guaranteed + the variable coupon Computational Finance (Summerschool) Hitotsubashi University August 29 1 / 38

2 Dow Jones Global Titans index, 5 stocks Abbott Laboratories New York Medical Supplies Altria Group Inc. New York Tobacco American International Group Inc. New York Full Line Insurance Astrazeneca PLC London Pharmaceuticals Bank of America Corp. New York Banks Barclays PLC London Banks BellSouth Corp. New York Fixed Line Telecommunications BP PLC London Integrated Oil & Gas Chevron Corp. New York Integrated Oil & Gas Cisco Systems Inc. NASDAQ Telecommunications Equipment Citigroup Inc. New York Banks Coca-Cola Co. New York Soft Drinks DaimlerChrysler AG NA XETRA Automobiles Dell Inc. NASDAQ Computer Hardware Eli Lilly & Co. New York Pharmaceuticals Computational Finance (Summerschool) Hitotsubashi University August 29 2 / 38

3 Dow Jones Global Titans index, 5 stocks ENI S.p.A. Milan Integrated Oil & Gas Exxon Mobil Corp. New York Integrated Oil & Gas General Electric Co. New York Diversified Industrials GlaxoSmithKline PLC London Pharmaceuticals HBOS PLC London Banks HSBC Holdings PLC (UK Reg) London Banks ING Groep N.V. Amsterdam Life Insurance Intel Corp. NASDAQ Semiconductors International Business Machines New York Computer Services Johnson & Johnson New York Pharmaceuticals JPMorgan Chase & Co. New York Banks Merck & Co. Inc. New York Pharmaceuticals Microsoft Corp. NASDAQ Software Morgan Stanley New York Investment Services Computational Finance (Summerschool) Hitotsubashi University August 29 3 / 38

4 Dow Jones Global Titans index, 5 stocks Nestle S.A. VIRTX Food Products Nokia Corp. Helsinki Telecommunications Equipment Novartis AG VIRTX Pharmaceuticals PepsiCo Inc. New York Soft Drinks Pfizer Inc. New York Pharmaceuticals Procter & Gamble Co. New York Nondurable Household Products Roche Holding AG Part. Cert. VIRTX Pharmaceuticals Royal Bank of Scotland Group PLC London Banks Royal Dutch Petroleum Co. Amsterdam Integrated Oil & Gas Samsung Electronics Co. Ltd. Korea Semiconductors SBC Communications Inc. New York Fixed Line Telecommunications Siemens AG XETRA Electronic Equipment Time Warner Inc. New York Broadcasting & Entertainment Computational Finance (Summerschool) Hitotsubashi University August 29 4 / 38

5 Dow Jones Global Titans index, 5 stocks Total S.A. Paris Integrated Oil & Gas Toyota Motor Corp. Tokyo Automobiles UBS AG VIRTX Banks Verizon Communications Inc. New York Fixed Line Telecommunications Vodafone Group PLC London Mobile Telecommunications Wal-Mart Stores Inc. New York Broadline Retailers Walt Disney Co. New York Broadcasting & Entertainment Wyeth New York Pharmaceuticals Computational Finance (Summerschool) Hitotsubashi University August 29 5 / 38

6 If an asset increases more than 8 %, a maximum of 8 % is considered. If it decreases more than 2 % a minimum -2 % is considered. Then, there is a maximum payment of 8 %, and a minimum of % each year. However, the interest rate cannot decrease from year to year. The required information to value a basket option is the volatility of each asset σ i and the correlation between each pair of assets ρ i,j. With Payoff: i=1 max {min { S (i) j+1 S (i) j, 8%}, 2%}, j = t j S (i) j Computational Finance (Summerschool) Hitotsubashi University August 29 6 / 38

7 Simulation: Euler Discretization Suppose that we want to generate random paths from the process of the following form: ds t = µs t dt + σs t dw t first: we choose a time step t = T /N, and generate iteration of Algorithm S t+ t S t = µs t t + σs t (W t+ t W t ) Computational Finance (Summerschool) Hitotsubashi University August 29 7 / 38

8 S Simulation: Euler Discretization time Figure: Euler discretization for: S = 5, µ =.1, σ =.2, T = 1 and N = 1, LEFT: generated paths, RIGHT: histogram of the prices at maturity T Computational Finance (Summerschool) Hitotsubashi University August 29 8 / 38

9 Exercises Stochastic Integration: For a given deterministic function g(s) how to find: t g(s)dw s =? First we define partition: = t < t 1 < < t n = T then T n 1 g(s)dw s = g(t k ) (W (t k+1 ) W (t k )) k= Example: let us take g(t) = t 2 and T = 1. Theoretically we have: ( 1 E t 2 dw t ) ( 1 ) Var t 2 dw t = ( 1 )2 = E t 2 dw t = 1 t 4 dt =.2 Computational Finance (Summerschool) Hitotsubashi University August 29 9 / 38

10 Exercises Figure: LEFT: Histogram of simulated stochastic integral with T = 1, N = 1, g(t) = t 2. RIGHT: Matlab code ( 1 ) ( 1 ) E t 2 dw t.37, Var t 2 dw t.247 Computational Finance (Summerschool) Hitotsubashi University August 29 1 / 38

11 Exercises Stochastic Integration: For a given Brownian motion W t how to find: T W s dw s =? Analytically: ( ) T E W s dw s = In order to calculate the variance we define a function g = x 2, from Itô we have: dg = g x dx g x,x(dx) 2 = 2xdx (dx)2 so: dw 2 t = 2W t dw t + (dw t ) 2 = 2W t dw t + dt Computational Finance (Summerschool) Hitotsubashi University August / 38

12 Exercises Further we have: so finally: T T dwt 2 = 2 W 2 T W 2 = 2 T T T W t dw t + dt W t dw t + T W t dw t = 1 2 W 2 T 1 2 T. Example: Let us take T = 2. Theoretically we have: ( 2 ) ( 2 ) E W s dw s =, Var W s dw s = 2 Computational Finance (Summerschool) Hitotsubashi University August / 38

13 Exercises Figure: LEFT: Histogram of simulated stochastic integral with T = 2, N = 1. RIGHT: Matlab code ( 2 ) ( 2 ) E W t dw t.83, Var W t dw t Computational Finance (Summerschool) Hitotsubashi University August / 38

14 Analytic vs Monte-Carlo of BS model Exercise: We set: S = 5, σ =.3, r =.6, T = 1, M = 5 (= # time steps), and K = S. Exact solutions from BS formula are: Call t= =.7359, Put t= =.4447, with time.3[s] Table: Call and Put prices depending on number of Monte-Carlo Paths (Euler Approach) n (N = 2 n ) # paths Call price at t = Put price at t = Time [s] Computational Finance (Summerschool) Hitotsubashi University August / 38

15 Analytic vs Numerical Solution of BS model Algorithm Computational Finance (Summerschool) Hitotsubashi University August / 38

16 Numerical Solutions Monte Carlo simulation vs. BS formula for Call & Put Prices as a function of strikes. 7 Monte Carlo Exact Solution 6 Monte Carlo Exact Solution Value 4 3 Value strike K Strike K Figure: Call and Put prices as a function of Strikes,K, K = [.1 :.1 : 1], Steps=5 Computational Finance (Summerschool) Hitotsubashi University August / 38

17 Simulation Again X t denotes a stochastic process and solution of an SDE, dx t = µ(y t, t)dt + σ(x t, t)dw t for t T. where the driving process W t is a Wiener process. We already know the Euler discretization: { xi+1 = x i + µ(x i, t i ) t + σ(x i, t i ) W j, t j = j t W j = W ti+1 W ti = Z (1) t with Z N(, 1) where length t is assumed equidistant, i.e., t = T M. Definition (Absolute error) The absolute error at time T is defined as: ɛ(h) := E ( X T x h T ). Computational Finance (Summerschool) Hitotsubashi University August / 38

18 Approximation Error Example Suppose we study a linear SDE of the form: dx t = µx t dt + σx t dw t, We know that the solution is of the following form: X T = X exp ((µ 12 ) σ2 )T + σw T Now, we perform an experiment in order to check the Euler method s convergence. In the experiment we use a discrete measure of the absolute error: ɛ(h) = 1 N X T,k XT h N,k k=1 Computational Finance (Summerschool) Hitotsubashi University August / 38

19 Approximation Error- Euler Example (Euler Scheme) We set X = 5, µ =.6, σ =.3, T = 1 and find ɛ(h) = 1 N X T,k XT h N,k k=1 Table: Table of the absolute error ɛ(h), wrt time and h. error:ɛ M=1 M=1 M=2 M=3 M=5 seed seed seed time [s] Computational Finance (Summerschool) Hitotsubashi University August / 38

20 Approximation Error Example The numerical results for obtained estimates ɛ are assumed to be valid for ɛ. We postulate: ɛ(h) C h 1 2 = O(h 1 2 ) error.6.5 value h time Figure: LEFT: error against value of step size h for Euler discretization. RIGHT: Generated paths for M = 1, red stars indicate value of exact solution. Computational Finance (Summerschool) Hitotsubashi University August 29 2 / 38

21 Approximation Error Algorithm Computational Finance (Summerschool) Hitotsubashi University August / 38

22 Strong and weak convergence Definition (Strong Convergence) x h T converges strongly to X T with order γ > if ɛ(h) = E ( X T x h T ) = O(h γ ), x T converges strongly, if lim h E ( X T x h T ) =. The Euler method converges strongly with order 1/2. Definition (Weak Convergence) x h T converges weakly to X T with respect to g with order β >, if E(g(X t )) E(g(x h T )) = O(h β ) The Euler scheme is weakly O(h 1 ) convergent wrt polynomials g. Computational Finance (Summerschool) Hitotsubashi University August / 38

23 Milstein Algorithm As before X t denotes a stochastic process and solution of an SDE, dx t = µ(y t, t)dt + σ(x t, t)dw t for t T. where the driving process W t is a Wiener process. Now we introduce another discretization method: The Milstein Scheme. { xi+1 = x i + µ t + σ W j σσ (( W ) 2 t ) W j = W ti+1 W ti = Z t with Z N(, 1) (2) where length t is assumed equidistant, i.e., t = T M, and where σ = σ(x i, t i ) x i Computational Finance (Summerschool) Hitotsubashi University August / 38

24 Milstein Algorithm In the case of GBM we obtain: { xi+1 = x i + µx i t + x i σ W j σ2 x i (( W ) 2 t ) W i = W ti+1 W ti = Z t with Z N(, 1) (3) The additional correction term in the Milstein scheme improves the speed of convergence compared to Euler method. The improved method is convergent with order one. Although the Milstein Scheme is definitely manageable in the one-dimensional case, its general multidimensional extension may be very difficult. Computational Finance (Summerschool) Hitotsubashi University August / 38

25 Approximation Error- Milstein Example (Milstein Scheme) We set X = 5, µ =.6, σ =.3, T = 1 and find ɛ(h) = 1 N X T XT h N,k i=1 Table: Table of the absolute error ɛ(h), wrt time and h. error:ɛ M=1 M=1 M=2 M=3 M=5 seed 1.5 7E-4 4E-4 2E-4 2E-4 seed 2.7 6E-4 4E-4 3E-4 2E-4 seed 3.5 8E-4 4E-4 3E-4 2E-4 time [s] Computational Finance (Summerschool) Hitotsubashi University August / 38

26 Approximation Error- Milstein Computational Finance (Summerschool) Hitotsubashi University August / 38

27 Euler vs Milstein- Trajectories 6 6 Euler Discretization Euler Discretization 58 Milstein Discretization 58 Milstein Discretization value value time time Figure: LEFT: Trajectories generated for σ =.1, RIGHT: Trajectories generated for σ =.3 Computational Finance (Summerschool) Hitotsubashi University August / 38

28 Antithetic sampling It is well known that if a random variable Z N(, 1), then Z N(, 1). We can use this property to drastically reduce the number of paths needed in the Monte-Carlo simulation. Suppose that ˆV is the approximation obtained from MC, and Ṽ is the one obtained using Z. By taking average V = 1 ) (Ṽ + ˆV 2 we obtain a new approximation. Since ˆV and V are both random variables we aim at: Var(V ) < Var( ˆV ). We have: Var(V ) = 1 4 Var(Ṽ + ˆV ) = 1 4 Var(Ṽ ) Var( ˆV ) Cov(Ṽ, ˆV ). So it is clear that: Var(V ) 1 2 (Var( ˆV ) + Var(Ṽ ) ). Computational Finance (Summerschool) Hitotsubashi University August / 38

29 Antithetic sampling paths path1 path2 paths time time Figure: Usual and Antithetic paths. LEFT: one path, RIGHT: five paths Computational Finance (Summerschool) Hitotsubashi University August / 38

30 Antithetic sampling Now, we check different implementations vs. number of paths, accuracy and time. For standard GBM model we set: T = 1, µ =.4, σ =.3, =.1, S = 5. Standard N = 1 1 N = 1 3 N = 1 4 N = 1 5 ɛ Time Antithetic N = 1 1 N = 1 3 N = 1 4 N = 1 5 ɛ Time The antithetic approach converges much faster, without any extra time needed for calculation. Computational Finance (Summerschool) Hitotsubashi University August 29 3 / 38

31 Antithetic sampling Figure: Matlab implementation- semi-efficient way of programming. Computational Finance (Summerschool) Hitotsubashi University August / 38

32 Antithetic sampling Efficient Implementation No.Samp. N = 1 1 N = 1 3 N = 1 4 N = 1 5 ɛ Time Figure: Matlab implementation-efficient way of programming. Computational Finance (Summerschool) Hitotsubashi University August / 38

33 Simulating Jumps ds t = dn t Figure: Poisson Process realizations with = 1, LEFT: λ =.1, RIGHT: λ = 1. Computational Finance (Summerschool) Hitotsubashi University August / 38

34 Simulating Jumps Suppose we have given a stock variable S t which jump at time τ j. We denote τ + the moment after one particular jump and τ the moment before. The absolute size of the jump is: S = S τ + S τ, which we model as a proportional jump, S τ + = qs τ with q >, so S = qs τ S τ = (q 1)S τ. The jump sizes equal q 1 times the current asset price. Assuming that for given set of i.i.d. q τ1, q τ2,... r.v.the process ds t = (q t 1)S t dj t, is called Compound Poisson Process. Computational Finance (Summerschool) Hitotsubashi University August / 38

35 Simulating Jumps- Jump Diffusion Process If we combine geometric Brownian motion and jump process we obtain: ds t = µs t dt + σs t dw t + (q t 1)S t dj t Figure: Geometric Brownian motion with jumps: =.1, µ =.4, σ =.2, λ =.5, LEFT: q = 1.4,RIGHT q =.6. Computational Finance (Summerschool) Hitotsubashi University August / 38

36 Simulating Jumps 5 8 x Figure: Geometric Brownian motion with jumps: =.1, µ =.4, σ =.2, q = 6, LEFT: λ =.4,RIGHT λ =.5. Computational Finance (Summerschool) Hitotsubashi University August / 38

37 Market and Jumps- Algorithm Computational Finance (Summerschool) Hitotsubashi University August / 38

38 Simulating Jumps- Jump Diffusion Process An analytical solution of the equation ds t = µs t dt + σs t dw t + (q t 1)S t dj t, can be calculated on each of the jump-free subintervals τ j < t < τ j+1 where the SDE is just a GBM. When at time τ 1 a jump of size: ( S) = (q τ1 1)S τ, 1 occurs, and thereafter the solution is given by: ) S t = S exp ((µ σ2 2 )t + σw t + (q τ1 1) S τ 1 In general we obtain: ) S t = S exp ((µ σ2 2 )t + σw t J t + j=1 S τ j ( qτj 1 ). Computational Finance (Summerschool) Hitotsubashi University August / 38