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# 数学代写|期权定价理论代写Option Pricing Theory代考|FINS3635 Empirical Results

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## 数学代写|期权定价理论代写Option Pricing Theory代考|Description of the data

Summary statistics of both interest rates and stock returns are reported in Table $6.1$, a time-series plot and salient features of both data sets can be found in Figures $6.1$ and 6.2. The interest rates used in this paper as a proxy of the riskless rates are daily U.S. 3-month Treasury bill rates and the underlying stock considered in this paper is $3 \mathrm{Com}$ Corporation which is listed in NASDAQ. Both the stock and its options are actively traded. The stock claims no dividend and thus theoretically all options on the stock can be valued as European type options. The data covers the period from March 12 , 1986 to August 18, 1997 providing 2,860 observations. From Table 6.1, we can see that both the first difference of logarithmic interest rates and that of logarithmic stock prices (i.e. the daily stock returns) are skewed to the left and have positive excess kurtosis $(>>3)$ suggesting skewed and fat-tailed distributions. Similarly, the filtered interest rates $Y_{r_t}$ as well as the filtered stock returns $Y 1_{s_t}$ (with systematic effect) and $Y 2_{s_t}$ (without systematic effect) are also skewed to the left and have positive excess kurtosis. However, the logarithmic squared filtered series, as proxy of the logarithmic conditional volatility, all have negative excess kurtosis and appear to justify the Gaussian noise specified in the volatility process. As far as dynamic properties, the filtered interest rates and stock returns as well as logarithmic squared filtered series are all temporally correlated. For the logarithmic squared filtered series, the first order autocorrelations are in general low, but higher order autocorrelations are of similar magnitudes as the first order autocorrelations. This would suggest that all series are roughly $\operatorname{ARMA}(1,1)$ or equivalently $\operatorname{AR}(1)$ with measurement error, which is consistent with the first order autoregressive SV model specification. Estimates of trend parameters in the general model are reported in Table 6.2. For stock returns, interest rate has significant explanatory power, suggesting the presence of systematic effect or certain predictability of stock returns. For logarithmic interest rates, there is an insignificant linear mean-reversion, which is consistent with many findings in the literature.

## 数学代写|期权定价理论代写Option Pricing Theory代考|Structural models and Estimation Results

The general model: the model specified in Section $2.1$ assumes stochastic volatility for both the stock returns and interest rate dynamics as well as systematic effect on stock returns. This model nests the Amin and $\mathrm{Ng}$ (1993) model as a special case when $\lambda_2=0$. Following are four alternative model specifications:

• Submodel 1: No systematic effect, i.e. $\phi_s=0$ and $\alpha=0$, i.e. a bi-variate stochastic volatility model;
• Submodel 2: No stochastic interest rates, i.e. interest rate is constant, $r_t=r$, which is the Hull-White model and the Bailey and Stulz (1989) model;
• Submodel 3: Constant stock return volatility but stochastic interest rate, $\sigma_{s t}=$ $\sigma$, which is the Merton (1973), Turnbull and Milne (1991) and Amin and Jarrow (1992) models;
• Submodel 4: Constant stock return volatility and constant interest rate, $\sigma_{s t}=$ $\sigma, r_t=r$, which is the Black-Scholes model.

## 数学代写|期权定价理论代写Option Pricing Theory代考|Structural models and Estimation Results

• 子模型1: 无系统效应，即 $\phi_s=0$ 和 $\alpha=0$ ，即双变量随机波动率模型；
• 子模型2: 无随机利率，即利率是恒定的， $r_t=r$ ，即 Hull-White 模型和 Bailey 和 Stulz (1989) 模型;
• 子模型 3: 股票收益波动恒定但利率随机， $\sigma_{s t}=\sigma$ ，即 Merton (1973)、Turnbull 和 Milne (1991) 以及 Amin 和 Jarrow (1992) 模型;
• 子模型 4: 恒定的股票收益波动率和晅定的利率， $\sigma_{s t}=\sigma, r_t=r$ ，这是布莱克-斯科尔斯模型。

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