**Executive Summary:**

In this dissertation we look at the ability of non-linear Kalman filters and forward calibration using the Fast Fourier Transform (FFT) applied to the Heston model to predict volatility for a short time frame, using Realised Volatility as a proxy.

Parameters are found using one bounded and one unrestricted calibration. Results from the calibration demonstrate that commonly employed bounds most often do not contain the global minima for the loss functions used. The results equally indicate that the Heston model is not able to capture characteristics in the stock price process where extreme values for the reversion rate and volatility of volatility tries to mimic jumps.

Testing the similarity between the calibrated instantaneous variance and the Realised Volatility shows significance for the non-restricted FFT, but none of the other models. This demonstrates that the market has the ability to predict the variance for a short period of time, but the calibration needs to be allowed to fluctuate freely to match characteristics optimally.

Comparing the realised volatility with the simulated realised volatility found from the parameter values show no significance of being similar, giving a first indication that the calibrations do not provide a good distributional fit. This becomes obvious when comparing the distribution of the true realised volatility to the ones found by the calibrated values, where we find no significance that the calibration results can reproduce the distribution of the variance or the location for the individual assets. This demonstrates that the state space models are inadequate at matching the current distributional properties of the variance.

For the results from the forward calibration a suitable risk adjustment might be necessary to provide a good fit due to the market price of volatility risk. Theoretical adjustments using results from the Kalman filtering provides generally poor results with no significant fits between the real and simulated distributions, but some significance with respect to location. This is attributed to how the filters remove noise to match the Heston model and therefore cannot approximate jumps, and that the models cannot accurately capture the market price of volatility risk due to its time-heterogeneity.

Several authors make the argument for a constant risk compensation for volatility, however we find no evidence of a constant market price of volatility risk or a price linear in variance as suggested by Heston. The best method seems to be one consisting of both a constant and a linear component, which provides significance in most cases. Equally, the linear and the constant and linear risk compensation provide results that match well with expected values found from previous research by other authors.

The level of the variance can therefore be estimated using an unrestricted forward calibration. Furthermore, no ex-ante prediction of the distribution of variance can be made based on the techniques presented here, since the market price of volatility risk is essential in matching a forward calibrated distribution to its empirical counterpart. Nonetheless, evidence clearly suggests that a stock price process incorporating jumps may be necessary to provide a perfect fit.