Diagnóstico PESTE

In 1993, Heston [27] proposed a model for option pricing that aimed at improving the -according to Heston - 'somewhat dubious' assumption of constant variance in the Black-Scholes model. His model assumes the stock price to follow the SDE

dS_t= µSdt +p

V_tSdW_t⁽¹⁾, (7.12)

where the variance process is modeled as a Cox-Ingersoll-Ross (CIR) process,⁴⁰ dVt= κ(θ − Vt)dt + p

VtdW_t⁽²⁾. (7.13)

The Wiener processes Wt⁽¹⁾ and Wt⁽²⁾ are correlated with instantaneous correlation ρ.

A CIR process has desirable features when modeling a volatility, e.g. positivity and reversion to a long-term mean.

The additional assumption of stochastic volatility introduces another dimension into the pricing problem. Since volatility cannot be hedged directly, the risk-neutral portfolio is created using a term called the price of volatility risk. We do not cover the theory of the Heston model here, but rather concentrate on applying the MLMC method to models that are readily available in Theta Suite. For this, we simulate the stock price paths under the risk-neutral measureQe as

dSt= rSdt +p

VtSdW_t⁽¹⁾. (7.14)

Diering from the Black-Scholes model, an analytic solution of the stock price process is not available in the Heston model and it is necessary to approximate the solution to (7.14) numerically. In his rst paper, Giles published results on European options in the Heston model with a Euler discretization. As Alfonsi [3] points out, the Euler scheme and the Milstein scheme admit negative values in the CIR process and consequently have to be modied, for example by only taking the positive part of the variance. Nevertheless, this introduces a bias.

Instead of the Milstein or Euler scheme, we employ an implementation of the scheme by Andersen [4] that is readily available in Theta Suite. In extensive numerical experiments, Andersen shows that his scheme outperforms the Euler scheme tremendously. Apart from the strong performance and simplicity of the scheme, the author states that the method is robust under changes in the model parameters, and in particular that it performs well for realistic parameters. Theoretical results on the convergence rate are not given.

If we recall the derivation of the Multilevel estimator in 5.3, we created the telescopic sum by adding and subtracting the expected payos for a timegrid and rearranging them into the level estimators:

To avoid introducing a bias, we have to make sure that

E⁽¹⁾_l−1= E⁽²⁾_l−1. (7.16)

In the absence of a type 1 discretization error, which is the setting we created in 7.1, the payo evaluation on the same timegrid alone satises this requirement, so we did not

40Following the notation in Theta Suite

10⁶ 10⁷ 10⁸ 10⁻³

10⁻² 10⁻¹

Heston Asian call standard deviation

# random points

Heston Asian call mean and 95% CI

# random points

value

std MC MLMC

Figure 7.10.: Asian option in the Heston model: Empirical standard deviation of MC estimator compared to MLMC estimator on the left, mean of all samples and 95 % condence region on the right. Both plots in dependence of the number of generated random variables in one replication

put any further attention to it.⁴¹ In the Heston model, where we approximate the path of the underlying (i.e. with a type 1 discretization error), we have to make sure that the path is also simulated on the same timegrid.

Hence, we have to simulate every path of a level estimator twice: Once for the ne timegrid, and once for the coarse timegrid. In practice, this can be done by summing up the Brownian increments on the ne path and using them subsequently to simulate the same path on a coarse timegrid. The size of the groups corresponds to the renement factor M.

7.2.1. Asian options

We priced an Asian call option with the parameters S0= 100, K = 100, T = 1, r = 0.05, V0 = 0.04, θ = 0.0375, κ = 0.9933,  = 0.1 and ρ = −0.7178. The option was calculated on a timegrid with 125 equidistant timesteps and the timestep levels in the MLMC method were dened as hl = 5^−l, l = 0, . . . , 3.

In gure 7.10 we can observe that the standard deviation in the MC and MLMC esti-mators declines at approximately the usual Monte Carlo rate of O(n^−1/2). Multilevel simulation reduces the standard deviation by a constant factor between 3 and 4 for the same number of generated random variables. Thus, the condence bounds for the Multi-level method are much tighter than for the standard Monte Carlo method. As the means

41Note that with an exact solution of the SDE, the ne and coarse paths coincide at any simulation time of the coarse path. Hence, sampling two paths is obsolete.

10⁰ 10¹ 10² 10⁻³

10⁻² 10⁻¹

Heston Asian call standard deviation

cputime (s)

Heston Asian call mean and 95% CI

cputime (s)

value

std MC MLMC

Figure 7.11.: Asian option in the Heston model: Empirical standard deviation of MC estimator compared to MLMC estimator on the left, mean of all samples and 95 % condence region on the right. Both plots in dependence of the actual computation time for one estimator

of both methods match almost perfectly, the MLMC condence bounds lie perfectly within the standard MC bounds.

A consideration that is more important in this setting, however, is the time requirement.

Since we have to approximate the same path both on a ne and on a coarse timegrid instead of calculating its exact course once on a ne timegrid, there is additional com-putational eort involved in the MLMC method. While this was not an issue in the simulations in section 7.1, where the time requirement was roughly equivalent, it is here.

Hence, we also present the same simulation results in dependence on the simulation time per replication in gure 7.11. Either way, the MLMC estimator is more ecient than the standard MC estimator.

7.2.2. American options

In the American option example, we again used the parameters S0 = 100, K = 100, T = 1, r = 0.05, V0 = 0.04, θ = 0.0375, κ = 0.9933,  = 0.1 and ρ = −0.7178. The option was an American put option with only 12 equidistant exercise opportunities, and the timestep levels in the MLMC method were dened as h0= 1, h1 = 1/4, h2 = 1/12. Unfortunately, the ThetaML model from the Asian option in the Heston model could not be used for the American option, because of dependencies in the regression that led to a substantial upward bias in the results. Instead, we assumed the approximation by Andersen to be exact and accepted the small approximation error that is introduced by this false assumption. As in the Black-Scholes experiments, the payos for the level

10⁶ 10⁷

American option value empirical standard deviation

MC Least squares

American option value and 95% CI

MC Least squares ML Least squares

Figure 7.12.: American option in Heston Model: Empirical standard deviation of stan-dard MC estimator compared to MLMC estimator on the left, mean of all samples and 95 % condence region on the right

estimators are calculated on two timegrids of the same actual path.⁴²

In gure 7.12 we see that the standard deviation in the Multilevel estimator exceeds the standard deviation from the Monte Carlo estimator. The mean of the Multilevel estimates is slightly higher, but since the condence intervals intersect we can not assume a bias.

It is hard to draw a clear conclusion from these observations. On the one hand, the applicability of the method can not be excluded, since the dierence in the simulation results is not signicant. Further work in adapting the model that was used for the Asian option in the Heston model, such that the payos for the coarse set of paths and the payos for the ne set of paths can be calculated independently, could remove any remaining bias in the MLMC estimator. On the other hand, the increased variance makes it questionable whether or not the MLMC estimator - should it prove to be unbiased - can actually return a lower variance in the result. As we mentioned earlier, our method for the initial variance estimation is not correct in the American option case, because the payos are not independent. Since the optimal distribution of the eort to the timegrid levels is determined by these variance estimates, a better distribution might be possible. Hence, a more sophisticated method of variance estimation could lead to variance reduction in the Multilevel estimator.

42In our terminology from 5.3.2, we assume that only a type 2 discretization error is involved.