Question

g. Assume Investor Casey is optimistic that a vaccine for COVID-19 will be found soon, thus has estimated the following probabilities pertaining to the S&P performance: 16% that S&P falls below the low strike price, X1, so Casey pays Sam; 71% that the S&P remains between the strike prices, so Casey makes no payment; and 13% that the S&P will move at and above the higher strike price, X2, so Sam pays Casey. Graph the lognormal probability distribution, with the S&P index movements represented on the x-axis and the positive investment returns on the y-axis. (NOTE: Probability distributions and profit- loss diagrams are different, but they arguably convey similar information.)

Answer 1

The dynamics of the hidden Markov process is characterized by the matrix of transition prob-
abilities between the states. For the four states assumed in our model of the S&P returns, we
obtain
The elements of this matrix are averages over 200 initializations. For each run, the states are sorted
by decreasing noise levels, oil in order to make the averaging meaningful. Note that the expert
with the largest noise level has the smallest self-transition probability, all = 0.904: on average,
the system stays in this state for only ten days. Looking back to Fig. 6, we can see that this expert
takes responsibility for some of the large returns in the training set, as well as for the region of
high volatility in late 1982.
Table 2 lists the noise levels of the experts for both HME and GE. For each run, the experts
were ordered in terms of decreasing noise levels, and means and standard deviations of the square
roots of the variances of the Gaussians are shown.
Table 2: The average noise levels oi of the individual experts for HME and GE for the S&P500
density predictions. In each run, i.e., for each set of initial conditions, the expert with the largest
variance is assigned the label "Expert l", etc. The table gives the means of the square roots of
the variances of the Gaussians. The standard deviations are indicated in parentheses. High-noise
experts have more relative variation in the noise levels than the low-noise experts than in those of
high-noise experts. Furthermore, GEs are more sensitive to initial conditions than HMEs.
5.2.3 Evaluation of the S&P500 Density Predictions
The function optimized in training is significantly different for HME and GE-we have emphasize.

This paper started out by discussing different tasks for prediction, and proceeded by presenting
hidden Markov experts (HME) in detail. The main focus is the prediction of the full conditional
density distribution. This is in contrast to the literature on Markov switching models that focuses
on point predictions and segmentation, and on the literature on stochastic volatility and GARCH
models that focuses on conditional variances. The density predictions we obtained as mixture
models were evaluated in comparison to these standard approaches using several methods, including
Diebold et al. (1998).
The approach was illustrated with two time series. Section 4 showed the results of a computer
generated example where the true regimes are known. This helped us obtain intuitions for model
misspecification, e.g., by revealing the signature of misapplying GE to data generated by HME.
When the right model class is used (HME), the parameters are estimated correctly and the density
is predicted well.
Section 5 applies the approach to the density of daily S&P500. On the test set, about 98 percent
of the HMEs estimated (they differed by their initial conditions) outperformed a GARCH(1,l)
model. While HME found a solution rather reliably, GE showed a large dispersion for two reasons:

(i) in any task with very high noise levels it is very difficult for the gate to learn a mapping from
some exogenous variables to the expected probabilities of the experts, and (ii) in the specific case
of financial returns, volatility is often estimated better recursively (as in GARCH and stochastic
volatility models) than with a feedforward architecture without memory, such as GE, see Timmer
and Weigend (1997).
This paper focused on introducing hidden Markov experts. The examples were chosen to
communicate some intuitions and illustrate several methods to evaluate the performance of density
predictions. An identical set of inputs, consisting of lags of the time series, was used to facilitate the
comparisons between the methods. When using this architecture in trading, we find that carefully
selected exogenous inputs lead to better predictions than autoregressive models. In addition to
trading applications, we have also used HME in risk management in combination with Independent
Component Analysis (Back and Weigend 1997) to capture non-Gaussian tails and compute Value-
at-Risk, as discussed in Chin and Weigend (1998).

So finally Investor Casey is optimistic that a vaccine for COVID-19 will be found soon, thus has estimated the following probabilities pertaining to the S&P performance: 16% that S&P falls below the low strike price, X1, so Casey pays Sam; 71% that the S&P remains between the strike prices, so Casey makes no payment; and 13% that the S&P will move at and above the higher strike price, X2, so Sam pays Casey. Graph the lognormal probability distribution, with the S&P index movements represented.

Thank you!!!