Adverbios de tiempo

function is Lipschitz continuous and we apply the discretisation scheme in Definition 7.2.2. Fix α = 0

and β = 1/2 as the scheme parameters, and choose k = 1 (the upper bound, since 2κβ ≤ 1), and k′ =10 (arbitrary choice).

Figure 7.2: Example 7.2.2: Mean absolute error and MSE for process Y, using different step sizes.

In Figure 7.2, we show the rates of convergence for the mean absolute error (MAE) and the MSE, using a reference solution with 216 _{steps. The rates obtained are 0.45 and 0.90.}

7.3

Discontinuous drift function

We consider the stochastic differential equation

dYt =µ f(Yt)dt+σdWt, Y0 ∈R, (7.3.1)

where µ and σ are strictly positive parameters and f(x) = 2(1/2−1_x≥0). Clearly, we have a discontinuity at x =0 for the drift function; more so, the function is one-sided Lipschitz.

neighbourhood of the discontinuity at the origin. We define this approximation by fε(x) =        1 if x_{≤ −}ε, x3 2ε3 −3x_2ε if −ε<x <ε, −1 if x≥ε. Note that f′

ε(x) ≤Cε(1+ |x|2). For this example the locally Lipschitz parameters are α =2 and

β = 0. Fix the explicit Euler scheme parameters such that 0 < _k′ _≤1/4, and k arbitrarily. For

ε = n−k, the map pn,x will project points in E0n to Dn+ or D−n. We consider the SDE in (7.3.1) with parameters (µ, σ, Y0, T) = (2, 2, 1, 1). The MAE and MSE rates for this example are 0.47 and 0.96.

8. Monte Carlo Acceleration

We present applications using the strong rate of convergence for the modified Euler scheme introduced in Chapter 6. We consider the multilevel Monte Carlo (MLMC) technique which requires this strong rate of convergence, and an accelerating scheme for a stochastic volatility model.

8.1

MLMC

We combine the modified Euler scheme and the multilevel Monte Carlo approach introduced by Giles [Gil08b, GS12]. The original paper focused on approximating the expected value of Lipschitz continuous payoffs. The MLMC method has also been justified for digitals, lookback and barrier options [GHM09]. Multischeme MLMC techniques use different discretisation schemes in order to further improve the computational efficiency [Abe11]. The use of MLMC techniques has also been applied to compute Greeks [BG12].

We target a root mean squared error (RMSE) of O(ε) for the option price. Using an Euler- Maruyama scheme, the MSE of an option price is C1/N+C2h2, where N is the number of Monte Carlo paths, and h is the step size of the discretisation. By choosing N := O(ε−2), and h:_{= O(}ε), the total cost isO(ε−3).

The idea behind MLMC is to use different time steps, at different levels of the simulation. We increase the number of time steps at each level by a factor M, where level l uses Ml _{steps of} size h_l := T/Ml. We define Pl to be the numerical approximation of the payoff at level l, for l =0, . . . , L, where L is the maximum number of levels. By linearity of the expectation operator we note that E_{[PL] =} E_{[P0] +} L

∑

where the difference in the payoff approximation on levels l and l ₋1 is estimated using the same Brownian path, for both levels. The variance of the payoff difference, V_l :=

V_{(Pl −Pl−1)}, decreases quickly with increasing levels, and it has been shown that for

186 8.1 MLMC

the strong convergence rate of the scheme. At each level l, we simulate N_l paths and estimate E[Pl−Pl−1]. The multilevel estimator has variance 1/Nl∑l=0L Vl, and Nl := C√Vlhl minimises the computational cost [Gil08b], to achieve a RMSE ofO(ε). The strong convergence rate is required for the MLMC techniques, and the complexity theorem provides a general result for the computational cost of the MLMC method [Gil08b]. MLMC methods have been shown to improve the computational efficiency using an Euler-Maruyama discretisation to

O�ε−2_{(log ε)}2�

, andO(ε−2)for a Milstein scheme [Gil08b, Gil08a].

8.1.1 CIR model ZCB with MLMC

We consider the Cox-Ingersoll-Ross model (6.4.1) for the process(vt)t≥0[CIR85]; the price of a zero-coupon bond (ZCB) with maturity T, at time t, reads

B(t, T) =E � exp � − � T t vsds � � � � �Ft � ,

which admits a closed-form solution [CIR85, BM07]. This solution at time zero is B(0, T) =

Aexp(−Cv0), where Λ := � κ2+2ξ2_and A:= � 2Λ exp[(κ+Λ_)T/2] 2Λ+ (κ+Λ_{)(exp TΛ−1)} �2κθ/ξ2 , C := 2(exp(TΛ) −1) 2Λ+ (κ+Λ_{)(exp(TΛ) −1)}. We consider a CIR model with parameters (κ, θ, ξ, v0, T) = (2, 1, 0.5, 1, 1), (N, M, L) =

(2000000, 4, 5), and RMSE thresholds(0.001, 0.0005, 0.0002, 0.0001, 0.00005).

In Figure 8.1, we compute the standard Monte Carlo, and MLMC approximations for the ZCB. The first plot demonstrates the average variance for the approximations P_l and the differences P_l−P_l−1. Observe that the variance of the differences decreased roughly twice as fast as the rate of weak convergence of an Euler scheme. Also, the variance of P_l is asymptotically a constant. The second plot shows the mean of P_l and the mean of P_l−P_l−1. The third plot shows how decreasing the target ε, require more steps N_l and the number of levels increasing from 3 to 5. The fourth plot shows the ratio of savings between the standard Monte Carlo approach for approximating the bond price (Std MC), and the MLMC counterpart. The ratio of savings is a factor of 27 for ε = 0.00005 between the standard Monte Carlo and the MLMC approach. We adapt code freely available from [Gil08b].

0 1 2 3 4 5 −12 −10 −8 −6 −4 −2 0 l log M variance Variance of P_l and P_l − P_l−1 P_l P_l− P_l−1 0 1 2 3 4 5 −12 −10 −8 −6 −4 −2 0 l log M |mean| Mean of P_l and P_l − P_l−1 P_l P_l− P_l−1 0 1 2 3 4 5 6 104 106 108 1010 l Nl

Runs N_l per level for different ε

ε=0.00005 ε=0.0001 ε=0.0002 ε=0.0005 ε=0.001 10−4 10−3 100 101 102 103 ε ε 2 Cost

Saving ratio of cost vs target error Std MC MLMC

Figure 8.1: CIR model, and ZCB pricing using MLMC.