The Results

We now present the results achieved using our algorithm. We performed simulations on 90 day options setting an initial volatility of 25% which is a figure comparable to the historical data. We performed simulations for $\alpha = 0,\, \frac{1}{2},\, \frac{3}{4}$ and 1 with correlations ranging from -0.5 to 0.5 in steps of 0.1. We set $\lambda = \mu = 0$ for most of the simulations. Some simulations with mean-reverting volatility processes were performed for the purposes of investigating the effect of mean-reversion on option prices.

We present the results in the form of implied volatility curves. These curves are plots of the implied volatility against the strike price of the option. Implied volatility is defined as that volatility which when put into the Black-Scholes equation gives the calculated price of the option. Since the price increases with implied volatility, we can consider the implied volatility to be a sensitive measure of the price of the option. If the volatility is constant, the implied volatility will always be equal to the constant value for the volatility. For a deterministic volatility process dependent only on the time, the implied volatility will be the average volatility over the time period of the option (due to Merton's theorem). In either case, there is no variation of the implied volatility with the strike price and the curve is just a straight line parallel to the x-axis.

However, empirical implied volatility curves vary strongly with the strike price (we will have a look at one of them later). This phenomenon is referred to as the ``volatility smile'' (this terminology arises from the fact that the first few empirical studies gave implied volatility curves that looked like smiles). Hence, if stochastic volatility is to be useful, it should also give rise to implied volatility which varies with the strike price. We will see that this is indeed true.

The following parameters have the following values unless otherwise stated : $S_0 = 100,\, r = 5\%,\, q = 0\% \text{ (the average annualized dividend rate)},\, t = 90 \text{ days},\, \lambda = 0$ and $\mu = 0$. Most of the simulations have been performed using 128 time steps and 500,000 Monte Carlo configurations. The exceptions have been for $\alpha = \frac{1}{2}$ where we have used 512 time steps and the simulations we have performed for the other values of $\alpha$ to check that the number of time steps is sufficient. The error bars in all the graphs refer to the standard error obtained from the Monte Carlo simulation.

The Effect of $\rho$ on Option Prices

Figure 6.1: Implied volatility curves showing the effect of $\rho$ on option prices when $\alpha = 0$. We can see that positive $\rho$ leads to an increase in the option price when the strike price is high and a decrease when the strike price is low while negative $\rho$ has the opposite effect.

Figure 6.2: Implied volatility curves showing the effect of $\rho$ on option prices when $\alpha = \frac{1}{2}$. We can see that positive $\rho$ leads to an increase in the option price when the strike price is high and a decrease when the strike price is low while negative $\rho$ has the opposite effect.

The correlation $\rho$ has a very large impact on the implied volatility curve irrespective of the values of the other parameters. When $\rho = 0$, we see that the implied volatility is in the form of a smile with the minimum near the present value of the underlying security ($S_0$). As $\rho$ increases, we find that the implied volatility increases for large strike prices and decreases for small strike prices. This can be easily seen in the graphs in figures 6.1, 6.2, 6.3 and 6.4. We can also see that the deviation of the implied volatility from the initial volatility is much higher when the correlation is non-zero. These results are consistent with those reported in Hull and White[18], Heston[20], Johnson and Shanno[31] and Scott[16].

This can be explained in terms of the propagator obtained by the Monte Carlo simulation. From figure 6.5, we see that the propagator for positive correlation is greater for very large $x'- x$ (where $x = \ln S_0$ and $x' = \ln S$) as compared to the propagator for zero correlation. This essentially means that the probability for the underlying security price reaching very large values is higher when the correlation is positive. Hence, the price of options with a very large strike price is larger when the correlation between the processes for the underlying security price and the volatility process is positive. For negative correlation, we see that the propagator for small, positive $x'-x$ is greater than that for zero or positive correlation. Hence, for options whose strike price is smaller, there are two competing factors, the propagator for $x'$ slightly larger than $x$ and the propagator when $x'>>x$. For relatively large strike prices, the latter is more important and the implied volatility is higher when the correlation is positive while for relatively small strike prices, the former is more important and the price when the correlation is positive is lesser than for zero or negative correlation. The same analysis can be performed for the case of negative correlation and is consistent with the simulated results.

We can also intuitively examine why the propagator has this form when the two processes are correlated. If the two processes are positively correlated, we have two posibilities. If the price initially increases, so will the volatility and hence, large price increases become more likely while small price increases become less likely. On the other hand, if the price initially decreases, so will the volatility and the price is more likely to remain at that value. Hence, for positive correlation, the propagator must be higher for prices slightly lower and much greater than the initial price and lower for prices slightly larger than the initial price. The reverse holds true for negative correlation. This naive reasoning is fully borne out in figure 6.5.

We were unable to simulate frowns or any sort of kinks in the implied volatility curve for any values of the parameters. This seems to suggest that stochastic volatility even with arbitrary correlation places some constraints on the shape of the implied volatility curve. We can use this to check whether the hypothesis that the volatility is stochastic is reasonable or otherwise. We note that the empirical implied volatility curve used for our calibration does satisfy this criterion.

Figure 6.3: Implied volatility curves showing the effect of $\rho$ on option prices for $\alpha = 1$. We can see that positive $\rho$ leads to an increase in the option price when the strike price is high and a decrease when the strike price is low while negative $\rho$ has the opposite effect.

Figure 6.4: Implied volatility curves showing the effect of $\rho$ on option prices for $\alpha = 1$. The curves for the different values of $\rho$ are in ascending order according to the slope (in other words, the slope increases monotonically with $\rho$). Hence, we can see that positive $\rho$ leads to an increase in the option price when the strike price is high and a decrease when the strike price is low while negative $\rho$ has the opposite effect.

Figure 6.5: Propagators for different $\rho$ when $\alpha = 0,\, 0.5 \text{ and }1$.

The Effect of Mean Reversion on the Option Price

Several authors including Heston[20] and Hull and White[18] have considered mean-reverting processes as there is some empirical evidence that the volatility follows a mean-reverting process. We note that our process also includes mean reversion since $\lambda$ and $\mu$ can be adjusted so that the volatility performes a mean-reverted process. We find that the effect of mean reversion is straightforward in that it only seems to change the implied volatility curve so that it moves closer to the mean value.

Figure 6.6: Comparison of a mean-reverting process and a non mean-reverting process

For example, consider figure 6.6. In this case, we used an initial volatility of $V_0 = 0.0625$ (so that $\sigma_0 = 0.25$). We set $\lambda = 0.125 = 2 \times 0.0625$ and $\mu = -2$ so that the volatility is performing a mean-reverting process with the mean the same as the initial value. We see that we indeed obtain the expected behaviour as compared to the non mean-reverting process. The mean reverting process just produces an implied volatility curve of a similar shape which is closer to the mean value.

Figure 6.7: The implied volatility curve for European call options on the S&P 500 Index maturing on Feb. 21, 1998 on Jan. 5, 1998 at 3 p.m.

Table 6.1: The prices of European call options on the S&P 500 Index whose maturity was on 21 Feb, 1998 on Jan 5, 1998 at 3:00 p.m. The prices were taken to be those of the closest trade if there was a trade within half an hour and the average of the bid and ask prices otherwise.

Strike Price	Option Price
920	68.0
925	64.125
930	60.25
940	52.75
950	45.5 (trade at 3:20 p.m.)
960	39.0
970	31.5 (trade at 3:00 p.m.)
975	29.75
980	27.0
990	21.0 (trade at 3:00 p.m.)
1010	13.0 (trade at 3:20 p.m.)

Calibration with Market Data

We compare our model with market data to see how well it works. Since volatility information is available only as 10 day or 50 day averages and the options we were comparing the market data to had 47 days to expiration, we used the initial volatility as a free parameter. The market data used were the prices of European call option on the Standard and Poor's 500 Index at 3 p.m. on Jan 5, 1998 with the maturity date given as Feb 21, 1998. The prices taken were either the trade nearest to 3 p.m. if it was within half an hour and the average of the bid and ask prices closest to 3 p.m. otherwise. The data are presented in table 6.1. We note that the number of option prices we have is much larger than the number of free parameters in the model. The value of the Standard and Poor's 500 Index at the same time was given as 965.61 (since the S&P 500 is traded several times every minute on the average, obtaining data for it presented no problem). The risk-free interest rate $r$ was 5.131% and the annualized dividend yield was 1.617% at the same time. The implied volatility curve for the market data is shown in figure 6.7.

Looking at the market data, we immediately see that the implied volatility is almost monotonically declining. Hence, according to our model, the correlation is very probably negative. We also see that the implied volatilities vary within quite a wide range implying that $\xi$ must be quite high as the volatility must vary widely for the implied volatility to do so. Further, the value of the initial volatility can be seen to be about 25% (according to our model, not the actual initial volatility which we could not determine with reasonable accuracy).

Since there is no simple functional form for the option price, the calibration was performed manually. The reasoning above enabled us to start with fairly accurate values. Thus, while we cannot guarantee that the result is the best fit curve in any precise sense, we can see from figure 6.8 that the fit is very good. While the in the money options seem to not fit so well, this might be because these options are thinly traded.

The sceptical reader might comment that the number of free parameters (4) is quite large and that a fairly wide range of empirical curves might be fitted. However, we note that our model predicts only three possibilities for the implied volatility curve, namely a ``smile'' (low correlation), monotonically increasing (positive correlation) or monotonically decreasing(negative correlation). The existence of a ``frown'' or kinks in the implied volatility curve would be disastrous for the model. (There does appear to be a small kink in the empirical curve but the scale of the kink is very small and occurs for only one value.)

Figure 6.8: The above graph shows the implied volatility curve produced by the fitted values together with the market data.