We believe that there are a lot of exciting avenues open for further research. Here we suggest four possible topics.
9.2.1
Improvements of the Model
Though we focused our calibration on the Chaotic Approach, it is exciting work to model the state price density paying particular attention to those volatility drifts of the underlying dynamics for improving fitting ability into the volatility surface. Our final goal is to establish a model which enables to fit well into a volatility surface across the maturity and the strike. We here list shortcomings of the One-variable Chaos Models so that possible improvement may be discussed.
• Option premiums can be computed only one at a time.
• An explicit implied volatility form is not available.
• Analytical forward LIBOR rate correlation form is not available.
The first shortcoming is discussed in Section 8.1. We proposed to take an exponential form for the state price density. In particular, we suggested modelling the LIBOR rate and swap rate volatility by the application of the short rate models. In other words, we suggested to incorporate the Short Rate Model, the Market Model and the Potential Approach. Since the Vasicek Model in 1977 there have been many interest rate models developed by various researchers. However, as is suggested in Chapter 8, it is possi- ble to combine all the previous described techniques of interest rate theory, in order to make the best advantage of existing work. We observed that the Affine Term Structure Model gives an analytical stochastic differential equations of the underlying assets. As examples, we showed that the Vasicek Model belongs to the Shifted-lognormal Market Model, and the Squared Gaussian Model belongs to the SVM. Here, it would be opti- mistic to generate the market by only one factor, we would need multiple factors. For example the SABR Model applies two correlated factors. Therefore we should consider two-factor Affine Term Structure Model so that we indirectly model the distribution of the volatility drift terms in the underlying assets. It is also advantageous to have an intuitive meaning for each parameter. A question now is about the market price of risk, which is not present in the volatility drift terms. However, the state price density is expressed by the short rate and the market price of risk. Hence, it is important to model the market price of risk for the propose of pricing options. To model the state price density we should also incorporate the discussions in Economics, see for example [25] and [31]. Some other ideas are also proposed in Chapter 8, but we leave the remaining questions open.
Finally, having explicit implied volatility and forward rate correlation forms is a de- sirable feature in interest rate modelling. Though it is straightforward to compute
premium from implied volatility, the other way around is not simple and is often ap- proximated. We understand that not many practitioners apply the lognormal distribu- tion for underlying assets any more, but they still apply the Black formula to measure volatilities, using implied volatility as a benchmark. We observe that the SABR Model outperforms the Wu-Zhang Model ([93]) in this sense. However, Chaos Models do not have that capacity either. Moreover, as is stated in [62] we should extract the forward LIBOR rate correlation information from the market in the calibration work, not only the volatility information. Therefore, we should also derive the correlation form in the Potential Approach for future work.
9.2.2
Improvements of the calibration
Though we understand that the market does not apply historical data but estimates volatilities only from the current data, the model assumption claims that the parame- ters are time independent. For example, Rogers ([80]) suggests time series calibration methodologies. Kalman Filtering, General Method of Moments, or Maximum Like- lihood method may be applied where the bid-ask spreads or liquidity would work to estimate volatility for the Maximum Likelihood function. Moreover we should also cali- brate the models proposed in Chapter 8 and the other popular SVM such as Wu-Zhang Model and Piterbarg Model ([70]), not only the SABR Model.
Calibration is implemented for pricing and hedging purpose. Here we take exotic options into account. Particularly, the chooser flexible cap and the Bermudan Swaption are liquidly traded in the market. However, as is the case for the SABR Model, it is often not easy to price those options, we must rely on either Monte Carlo Simulation or the trinomial tree algorithm, (see, for example [87]). One of the most appealing parts of the Potential Approach is its tractability to price options, because we are modelling the stochastic discount factor itself. We could investigate pricing method for those exotic options and also check pricing errors, using calibrated parameters by European
options.
In addition to checking the pricing error, model performance may be evaluated by its hedging performance. Although we observe some literature about the hedging the delta and vega risks under the SABR Model, (see for example [3] and [41]), we do not find it under the Potential Approach. As stated in Rebonato’s book ([76]), we may not say a model is perfectly good unless hedging ability is checked. A desirable model has to have a stable and non-erratic feature of prediction in the future time. For example, as stated in the book [76], the Local Volatility Market Models do not have great ability in this sense, since there the dynamics move the other way around, even though it satisfies fitting ability to the volatility smiles. We notice that nobody has investigated evolution in time of the term structure of volatility in the Potential Approach. Here again the SABR Model would work as a benchmark of the performance.
9.2.3
Further investigations in Mathematics
In this thesis we proposed the One-variable Wiener-Chaos expansion. Although we did not find loss of generality under the One-variable Chaos Models, it is still an open topic to compare the convergence speed between the usual Wiener-Chaos expansion and the One-variable Wiener-Chaos expansion. As an alternative direction, may we suggest applying the Winker-Askey Polynomial Chaos Expansion (or Generalized Polynomial Wiener-Chaos expansion, sometimes written as GPCE), which has been used recently in Physics and Engineering fields to estimate a square integrable random variable as an alternative to the Monte Carlo Method, see for example [94]. This method is appropriate to estimate a non-Gaussian variable.
9.2.4
Application to other products
The expressions of the stock price process and FX system are derived in [48]. Hence, it is straightforward work to price stock options under the Potential Approach in particular