values I have simulated 1000 series 850 days long, and have used the last 750 observations of each series to estimate β. This is intended to reduce the impact of the fact that all share series begin in the same place, if indeed these initial conditions have an effect, and replicate the conditions under which β will be estimated for the real series. After 100 days, the share price will be a random variable whose properties have been discussed in Chapter 3. Recalling that the CEV model allows bankruptcy, it is possible that the 1000 share price series will result in fewer than 1000 estimates of β. In fact, given the initial conditions, the expected number of bankruptcies for some β is large, as is reflected in the observed numbers. This analysis is meant to estimate β values for series similar to those for the real data examined in the following section. Since these series are all approximately three years long, I have estimated β using only the complete simulated series. Estimating β from the shorter series, where bankruptcy is observed, should not improve the estimates, since fewer observations are available than in the full length series, nor should it affect them, since the bankrupted series contain no information not theoretically present in complete series. This same simulation procedure could be used to test any β estimation procedure.
They represent different schools of thoughts for the modelling of asset prices as random processes. In their seminal work in , Black and Scholes modelize the asset prices as a geometric Brownian motion i.e asset prices with continuous paths and a constant volatility. It leads to the famous Black-Scholes formula which gives a theoretical estimate of the price of European-style options. It is perhaps the world’s most well-known options pricingmodel and usually used as a benchmark model by the quantitative finance community.
The hats in the terms mean estimating the quantities using method-of-moments. The proposed pricingmodel complies with the put-call parity property of Stoll (1969). Using the current formula, it assumes that the historical mean and the standard deviation of data will persist in future price movements. Adjusting for desired mean and variance assumptions on the asset returns, µ ˇ t,T ,i and σ ˇ
with c a constant and H=O(1/2) after roughly Δ t>10-15 minutes in trading . With H ≠ 1/2 there would be a nonMarkovian stochastic process, fractional Brownian motion, with long time correlations |9] that could in principle be exploited for profit. The assumption that H ≈ 1/2 is equivalent to the assumption that it is very hard to beat the market , which is approximately true (economists call such a market ‘efficient’; such a market consists of pure noise plus hard to estimate drift, the expected return R). We assume a continuous time description for mathematical convenience, although this is also obviously a source of error. Levy distributions, with infinite variance, are not discussed here because the observed tail exponents for returns  are larger than the range required for Levy to be of interest.
This paper studies supershare and chooser options in a stochastic volatility economy. These two options are typical exotic options which suggest a broad range of usage and application in different fi nancial market conditions. Despite the popularity and longevity of the Black-Scholes model, the assumption of constant volatility in the Black-Scholes model contradicts to the existence of the non-fl at implied volatility surface observed in empirical studies. Although many studies are devoted to optionpricing under stochastic volatility model in recent years, to the best of our knowledge, research on exotic option such as supershare and chooser optionpricing have not been carried out in the stochastic volatility case. Supershare and chooser options are both important fi nancial instruments, research on these two exotic options in stochastic volatility model may give more insights on the pricing of supershare and chooser options. By extending the constant volatility in the Black-Scholes model, this paper studies the pricing problem of the supershare option and chooser options in a fast mean-reverting stochastic volatility scenario. Analytic approximation formulae for these two exotic options in fast mean-reverting stochastic volatility model are derived according to the method of asymptotic expansion which shows the approximation option price can be expressed as the combination of the zero-order and fi rst-order approximations. By incorporating the stochastic volatility effect, the numerical analysis in our model shows that stochastic volatility of underlying asset underprices the supershare options, while in the case of the chooser options its price in stochastic volatility model is higher than the price in the constant volatility model.
relevant parameters of the two models. In fact, the only parameter that we must care- fully examine is the scaling parameter of the L´evy-Stable process; we opt for one that can be related to the standard deviation used when the classical Black-Scholes model is used. One approach is to proceed as in [HPR99] and match a given percentile of the Normal and a symmetric L´evy-Stable distribution. For example, if we want to match the first and third quartile of a Brownian motion with standard deviation σ = 0.20 to a symmetric L´evy-Stable motion κdL α,0 with characteristic exponent α = 1.7, we
We extend the subordinated binomial optionpricingmodel with stochastic arrival intensity (Chang, Chang and Lu, 2010) to allow for untraded underlying assets by using matching futures prices to imply out the underlying asset values. We empirically apply the model to VIX optionpricing vis-à-vis the original model with constant arrival intensity (Chang, Chang and Tian, 2006) using a two-year set of daily VIX options and futures data to specifically examine the efficacy of adding stochastic arrival intensity and untraded underlying assets. We find that the extended version significantly outperforms the original model both in sample and out-of-sample in terms of the MSE, with pricing error reduction about 37% and 32%, respectively, and additionally the outperformance is robust to the selection of the constant arrival intensity level. We attribute the outperformance to the extended model’s incorporation of the stylized effects of mean-reversion and clustering in intensity arrivals as well as of the information content conveyed by the matching futures prices.
Given the estimator’s unbiasedness, the efﬁciency gain factor compared to full-dimensional (non-truncated) sam- pling is the ratio of expected work between the two es- timators. Tables 1 and 2 show results for the up-and-in call and the down-and-out call, respectively; we give 95% conﬁdence intervals on the expected work E[ M(d) ] and the estimated option prices, varying the barrier b and the problem dimension d . The ratio d/E[ M(d) ] may be viewed as a simple, albeit rough, measure of the efﬁciency gain. In all cases, we see that expected work grows very slowly with the dimension d; equivalently, efﬁciency increases rapidly.
Table 5 contains the summary of simulation results considering the duration, variance structure, skewness, and kurtosis, ignoring other simulation scenar- ios. Generally, the BS model gives higher valuations compared to the CMM, CMM.RN, QMM, QMM.RN, and CP. The RN models tend to have lower fre- quency of negative call option prices compared to the non-RN counterparts. For all proposed models except CP.RN, the occurrence of negative option prices tends to be spread to almost all cases but of differing frequency, so analysis will point out on which cases were higher-than-average frequency of price oc- currences are observed.
In addition, we propose several other models that can explain the leptokurtic features and the volatility smile in optionpricing. For example, Chen et al. (2016) proposed American optionpricing under generalized mixed fractional Brown motion (GMFBM), using numerical methods to solve the linear complementarity problem. However, the parameters of the model are difficult to estimate, and the SIR risk and credit risk cannot be included in the model. In addition to fractional Brownian motion, the Lévy process is one of the most popular topics among researchers. Kleinert and Korbel (2016) pointed out that the Lévy process can produce infinite jumps and infinite variance and, hence, is considered an ideal method to describe heteroscedasticity and rare events (jumps). Fajardo and Farias (2010) price options by applying the hyperbolic distribution function proposed by Barndorff‐Nielsen (1977), and assuming that the underlying asset return obeys the Lévy process generated by a multidimensional generalized hyperbolic distribution. Gong and Zhuang (2016) price options using the underlying asset price process that contains Lévy volatility and Lévy jump processes. However, the constrained nonlinear optimization method used by these models lacks stability in the estimated parameters.
merical computations for the location of free boundary are also carried out by many people (see, for examples, [14,25-28] and the references therein). More recently, some global property of the free boundary attracts some interest. The authors of [29,30] proved that the free boundary is convex if the volatility in the model is as- sumed to be a constant. However, this global property is not valid in the real financial market since the volatility depends on time and other economical factors. When the volatility depends on time and the security, the problem becomes much more challenging. In this paper we would like to study some global property of the free boundary. We want to find how the optimal exercising boundary changes when the volatility changes during the life-time of the option contract. This question is very important for structured products in the financial world.
The behaviour of the process at the zero-boundary can be also be inferred from Proposition 1.1.2 type of results and Feller classification. For a > ^, 0 is an exit boundary. For ol < \ , the origin is regular boundary point and we adjoin a killing boundary condition so th a t the process is an absorption boundary at 0 in any case. The process has no limit-distribution given th a t (1.78) has the non mean-reverting form. The density of the spot-equity is a straightforward transform of the corre sponding density given in Theorem 1.1.1. Analytics to compute vanilla options prices have been produced in the literature by integration against this density. Exotic op tions such as barrier and lookback options on this process have also been given some importance very recently thanks to the works of Boyle and Tian , Davydov and Linetsky  and Lo and al. . Yet, Asian types of derivative depending on the continuous average of the process have not received much consideration till now al though they constitute the next step to be taken to deepen our understanding of the derivatives market under this model. We will therefore try to derive here some properties concerning the distribution of the tem poral integral Yt = J q X tdt.
In literatures many contributors have studied the maximization of the utility of terminal value or minimizing the probability of ruin for the insurer of which Brown  made the contribution of giving an explicit solution to the problem of a firm that maximized the exponential utility of terminal wealth and minimized the probability of ruin with its surplus process given by Lundberg risk model.
Stochastic volatility models play an important role in ﬁnance modeling. Under a mixed fractional Brownian motion environment, we study the continuity and estimates of a solution to a kind of stochastic diﬀerential equations with double volatility terms. Besides, we propose to price the vulnerable option with the discretization method and present the results using a Monte Carlo simulation.
This paper intends to find the optimal investment strategy of the optimal portfolio for the financial market, in which interest rate of the risk-free asset is a linear function of time and the risky asset assumed to follow constantelasticity of variance (CVE) model and look into the variation that may occur when transaction costs and taxes are charged on the risky asset only and total investment.
To implement the first approach, Carr, Madan, Geman and Yor deter- mine the risk neutral distribution for the stock price at each future time as a the exponential of the VGSV process and of the GCMYSV process, normalized to reflect the initial term structure of forward prices. Note that here the exponential is just the ordinary exponential and not a stochastic exponential as it will be in the second approach. By doing this the model is spot-forward arbitrage free; furthermore arbitrage of calendar spreads of option is also not allowed since it is based only on the stock price. Variance gamma and CGMY models modifications realized with this approach are de- fined VGSA and CGMYSA, where the letters “SA” remembers that they are free only from stock price based arbitrage. To write the formal expressions of the characteristic functions, let’s define S(t) the stock price at time t, r the constant continuously compounded interest rate and q the constant continu- ously compounded dividend yield. Consider the class of stochastic volatility L´evy processes, Z(t) from equation (3.13), then we have and we can define the stock price at time t as
Recently Lo  proposed a new simple approach to tackle the long-standing problem: “Given two correlated log- normal stochastic variables, what is the stochastic dy- namics of the sum or difference of the two variables?”; or equivalently, “What is the probability distribution of the sum or difference of two correlated lognormal stochastic variables?” The solution to this problem has wide appli- cations in many fields including financial modelling, actu- arial sciences, telecommunications, biosciences and phys- ics [2-15]. By means of the Lie-Trotter operator splitting method , Lo showed that both the sum and difference of two correlated lognormal stochastic processes could be approximated by a shifted lognormal stochastic proc- ess, and approximate probability distributions were de- termined in closed form. Unlike previous studies which treat the sum and difference in a separate manner [2-5, 8,13,15,17-27], Lo’s method provides a new unified ap- proach to model the dynamics of both the sum and dif- ference of the two stochastic variables. In addition, in terms of the approximate solutions, Lo presented an ana- lytical series expansion of the exact solutions, which can allow us to improve the approximation systematically.
In this paper, we extended the work of  by studying the optimal portfolio investment strategy for a commercial bank with exponential utility under constantelasticity of variance (CEV) model. In doing this, we considered a portfolio consisting of one risk free asset (treasury security) and two risky assets (marketable security and a loan) such that the risky assets are modelled by CEV model. By using power transformation, change of Variable approach, we obtain explicit solutions of the optimal investment portfolio strategies and the Value function. Furthermore, we observed that our result generalizes the result in .
The constantelasticity of variance (CEV) model with stochastic volatility is a natural extension of geometric Brownian motion and can explain the empirical bias ex- hibited by the GBM model, such as volatility smile. The CEV model allows the volatility to change with the un- derlying price and was first proposed by Cox and Ross . In comparison with other stochastic volatility mod- els, the CEV model is easier to deal with analytically and the GBM model can be seen as its special case. The CEV model was usually applied for optionpricing and sensi- tivity analysis of options in most literatures, see [16-19] for example. Recently, the CEV model has been applied in the research of optimal investment, as was done by Xiao, Zhai and Qin . Gao [21,22] investigated the utility maximization problem for a participant in a de- fined-contribution pension plan under the CEV model. Gu, Yang, Li and Zhang  used the CEV model for studying the optimal investment and reinsurance pro- blems.