MSc.MF Theses and Dissertations (2022)
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- ItemEstimating Heston stochastic volatility model parameters(Strathmore University, 2022) Musya, MartinThe Black Scholes model is widely favoured for pricing derivatives such as the European put and call options. This model while having the benefit of ease of application has some restrictive assumptions. First there is the assumption that volatility of asset returns is constant. This assumption is easily violated in the volatility smile that is widely documented in literature as well as observed in option market data, another assumption is that asset returns are normally distributed. The assumption of normal distribution is reasonable for long term horizons but not for shorter horizons. Market are rarely if ever complete. There always exists informational asymmetry where some investors know more about the market than others. It is also well known that a single asset is insufficient to hedge away risk. The Heston Model improves on previous assumption of constant variance (homoskedasticity) by allowing correlation between volatility and the price of the underlying. This research sought to estimate the Heston model parameters over various periods using maximum likelihood method. This was to compare the performance of both the Heston model and the Black Scholes model in periods of market uncertainty and in relatively stable periods where markets are performing well. The three periods, also called epochs, in this study are during the 2008 financial crisis, the Covid-19 pandemic and the relatively stable years between the two periods. The data for this thesis was obtained from the S&P 500 volatility index (VIX). Both the option data and the underlying data was available. A¨ıt-Sahalia (2002) constructed the sequence of approximations to the transition probability for a diffusion process. The first step involved standardizing the diffusion function using the Lamperti transform in order to remove some state dependent term in order to get rid of boundary conditions and correlation structures resulting in simple diffusion terms. This allowed for parameter estimation for the differentiable unit diffusion function. After this, the pseudo normalized” increment for random variable of the diffusion function was obtained and maximum likelihood estimation was done on this transformed variable to get rid of the issue of transition probabilities getting peaked when the step size gets small. The state variables for the Heston diffusion function were those of the underlying process and the diffusion process. The option price and that of the underlying in the data were first organized into a matrix. The determinant of that matrix which was the Jacobian term was then used to estimate the Heston and Black-Scholes parameters. A hypothesis test on the output for the three periods showed that at the 95% significance level the Heston model parameters are significant unlike the Black-Scholes parameters for all periods under study. The Heston model performed better even in times of financial instability such as during the 2008 crisis and the Covid-19 pandemic.
- ItemMathematical modeling of house price dynamics and their impact on the cost of No Negative Equity Guarantee: evidence from Kenya(Strathmore University, 2022) Mau, Erick OmondiEquity Release Mortgages (ERMs) are significantly required in an aging population with high homeownership levels. The capacity to identify the risks associated with the cost of the No Negative Equity Guarantee is an essential aspect of a risk management tool for most annuity and pension providers. Therefore, the main objective of this research is to; institute a stochastic modeling framework for the No Negative Equity Guarantee (NNEG) in an Equity Release Mortgage (ERMs) loan, and the payoff structure of the NNEG, and finally price the Equity Release Mortgages. An ARMA-EGARCH model that can capture auto-correlation and volatility clustering characteristics is proposed based on the model fittings. To analyze the regional and the national effect, we evaluate different models using the bench-marked loan data obtained from the nationwide building society database in the United Kingdom, for the period between 1991-2020 and had details such as the amount borrowed, age, marital status, and sex among others during the period. House Price Index (HPI) data was used to calibrate the loan data. Four baseline scenarios were used to simulate the NNEG valuation: the loan-to-value ratio, the roll-up rate, the risk-free rate, and house price volatility. The model forecasting power was evaluated using: root means squared errors, mean average error, the Diebold-Mariano forest accuracy test, and Occam's razor method. However, due to fluctuations in the house price data-generating process and goodness of fit, the Diebold-Mariano forest accuracy test was used as the metric to evaluate the model's performance in providing superior forecasting power. The study adopts the suggestion of (Hosty et al., 2008) to investigate the model risk on the cost of NNEGs and further develops a risk-neutral valuation methodology using the Conditional Esscher Transform Technique as proposed by (Buhlmann et al., 1996). The findings indicate that ARMA (4, 3)-EGARCH (1, 1) outperformed both the Black (1976) and the Geometric Brownian Motion-risk-neutral (GBM-rn) with a score of 0.2637. The simulation results further established that the cost of NNEG is critically sensitive and robust to; the Roll-up rate, Loan-to-Value (LTV) ratio, the volatility of the house prices, the risk-free rate, and the rental yield. Also, under current market settings, the Geometric Brownian Motion (GBM)-rn and Black' 76 may suddenly increase the NNEGs values via higher than obligatory volatilities at longer time horizons.