# Top PDF Two-State Option Pricing Model

### Two-State Option Pricing Model

Financial Economics Two-State Model of Option Pricing Two-State Model of Option Pricing Rendleman and Bartter [1] put forward a simple two-state model of option pricing. As in the Black-Scholes model, to buy the stock and to sell the call in the hedge ratio obtains a

### Option pricing with model-guided nonparametric methods

Option Pricing with Model-guided Nonparametric Methods Abstract Parametric option pricing models are largely used in Finance. These models capture several features of asset price dynamics. However, their pricing performance can be significantly en- hanced when they are combined with nonparametric learning approaches that learn and correct empirically the pricing errors. In this paper, we propose a new nonparametric method for pricing derivatives assets. Our method relies on the state price distribution instead of the state price density because the former is easier to estimate nonparametrically than the latter. A parametric model is used as an initial estimate of the state price distribution. Then the pricing errors induced by the parametric model are fitted nonparametrically. This model-guided method estimates the state price distribution nonparametrically and is called Automatic Correction of Errors (ACE). The method is easy to implement and can be combined with any model-based pricing formula to correct the systematic biases of pricing errors. We also develop a nonparametric test based on the generalized likelihood ratio to document the efficacy of the ACE method. Empirical studies based on S&P 500 index options show that our method outperforms several competing pricing models in terms of predictive and hedging abilities.

### Option Pricing for a General Stock Model in Discrete Time

In conclusion, we summarize our findings and indicate possible further topics of in- vestigation. We now know that we have a weak duality relationship between the pri- mal linear program and the dual linear program. By this relationship, the objective function of our dual linear program provides a lower bound for the objective function of the primal linear program, for feasible points of both. For the finite-dimensional linear program arising from the Trinomial Asset Pricing Model, strong duality of linear programming establishes that the optimal values of the two linear programs are equal. This strong result does not necessarily hold for infinite-dimensional linear programs.

### Model Risk Analysis for Risk Management and Option Pricing.

In Figures 4.4 and 4.5 we give the capital requirements based on the BIS capital requirements. All requirements are based on investments of \$100 in the market. Results can therefore be interpreted as percentages. We have used the BIS backtest procedure (see Basel Committee on Banking Supervision (1996b)) to backtest the Gaussian and the GARCH(1, 1) models and to determine the multiplication fac- tors. 16 The capital requirement can then be determined by multiplying the daily value-at-risks by the multiplication factor and √ 10. 17 The capital requirements are compared to the two-week returns. In addition to the BIS capital requirements we plot the capital requirements based on the model risk multiplication factors shown in Figure 4.3. In Figures 4.4 and 4.5 we see that the capital requirements for the GARCH(1, 1) model are much more variable than those of the Gaussian model. Fur- thermore, we see that in normal market conditions the model reserves based on the model risk measures cover the losses safely. The performance in terms of number of exceedances per daily returns, two week returns, and average regulatory capital, is more or less the same for both models as can be seen from Table 4.4. In Table 4.4 we see that the number of exceedances of the two-week VaR and ES’s is very small for all capital requirement schemes. Of course, the capital requirements set by the BIS are exceeded least, but they are also very large compared to the model risk multiplication factors. Eventually, the regulator needs to make a trade-oﬀ between the cost of exceedance of the capital requirements and the cost of impeding banks in their operations by charging high capital requirements.

### An empirical model of volatility of returns and option pricing

We now solve the inverse problem: given the exponential distribution (11) with (12) and (46), we use the F-P equation to determine the diffusion coefficient D(x,t) that generates the distribution dynamically. In order to simplify solving the inverse problem, we assume that D(x,t) is linear in ν (x- δ) for x> δ , and linear in γ(δ -x) for x< δ . The main question is whether the two pieces of D( δ ,t) are constants or depend on t. In answering this question we will face a nonuniqueness in determining the local volatility D(x,t) and the functions γ and ν. That nonuniqueness could only be resolved if the data would be accurate enough to measure the t-dependence of both the local and global volatility accurately at very long times, times where γ and ν are not necessarily large compared with unity. However, for the time scales of interest, both for describing the returns data and for pricing options, the time scales are short enough that the limit where γ,ν>>1 holds to good accuracy. In this limit, all three solutions to be presented below cannot be distinguished from each other empirically, and yield the same option pricing predictions.

### A Stochastic Volatility Model With Realized Measures for Option Pricing

The comparison between models in the SV class are reported in the on-line Appendix. SV-LHARG consistently shows the best option pricing performance among the models and, for this reason, hereafter, only SV-LHARG will be used in the comparison with the RV-LHARG and CGARCH models. Let us focus on the estimation period January 1, 1998 – December 31, 2007. The analogous analyses for the other two periods are available in the on-line Appendix. We proceed as follows: i) estimate the three models over the selected time interval and calibrate the variance risk-premium ν1; ii) price options over the estima- tion intervals (in-sample pricing); iii) keep the parameter values and risk premium fixed as at point i) and price options over the following year, from January 1, 2008 to December 31, 2008 (out-of-sample pricing). The overnight correction factor ϕ, the shape parameter δ for RV models, and the CGARCH parameter ω are fixed via targeting on the estimation period and kept unchanged out-of-sample.

### The Constant Elasticity of Variance Option Pricing Model

The final Figure 4.9 shows the three graphs for the BOR time series. BOR has ˆ β = 6.6570 which indicates that periods of high volatility should correspond to high share price level, and that periods of low volatility should correspond to periods of low share price level. The share series in the first graph does not show the same oscillation as the other two series, but has a period of oscillation to begin with, and then a period of growth in the latter part of the series. Yet again, the daily returns have a mean that appears to oscillate about zero, and a changing standard deviation. The standard deviation of the daily returns is highest at the end of the series, consistent with the size of ˆ β, since this is where the share price is largest. Lower volatility is seen at other times. The third graph indicates that there is a positive relationship between share price level and the standard deviation of the daily returns. This and the relationship seen for the other two series, is probably not linear though, as the specification of the CEV model would suggest.

### Quanto option pricing in the parsimonious Heston model

We apply this model to the pricing of Quanto options. These are options whose payoff is in another currency as the underlying is traded. The term Quanto is short for quantity adjusted. The (fixed) exchange rate incorporated in these options is called Quanto rate and is usually set to 1. An investior can use Quanto options when he wants to participate in gains in the underlying, but without carrying risks from the foreign exchange (FX) rate. The Quanto feature can also be applied to other derivatives like futures. In section 3 we explain how to apply the model to Quanto options. We then price two different options on two different dates with the model and compare the obtained prices with market and Black–Scholes prices in section 4.

### The Cox-Ross-Rubinstein Option Pricing Model

The model in these notes makes the assumption that the underlying asset, hereafter referred to as a stock, takes on one of only two possible values each period. While this may seem unrealistic, the assumption leads to a formula that can accurately price options. This “binomial” option pricing technique is often applied by Wall Street practitioners to numerically compute the prices of complex options. Here, we start by considering the pricing of a simple European option written on a non-dividend-paying stock.

### A GARCH option pricing model with filtered historical simulation

The GARCH literature provides several specifications for the conditional volatility and we could adopt a different GARCH model in this study. We favor the asymmetric GJR GARCH model in Equation (3) mainly for two reasons: (1) the flexibility in capturing the leverage effect and (2) the ability in fitting daily S&P 500 Index returns (used in our empirical application). Engle and Ng (1993) document that the GJR model provides an adequate modeling of the news impact curve, outperforming the EGARCH model of Nelson (1991). Rosenberg and Engle (2002) fit a number of GARCH models to daily S&P 500 returns and find that the GJR model in Equation (3) describes the data best. In Section 2, we undertake an extensive empirical analysis using several years of S&P 500 returns and in some instances there might be other GARCH models that outperform the asymmetric GARCH model in Equation (3). For comparison purposes and to simplify implementation, we always maintain the GJR model in our study. If on specific occasions we use other GARCH models that fit the data better, we could obtain more accurate pricing results. Hence, our findings can be interpreted in a conservative way.

### The Model of Lines for Option Pricing with Jumps

models are capable of explaining the skew of implied volatilities of medium and long dated options; however, intrinsic model limitations are encountered with short dated options. The observed steepness of implied volatility skews cannot be justified by a model where paths are continuous and volatility driven by a stationary process. Models with state and time dependent volatility, as in Rubinstein et al. [18],[25], Derman and Kani [11], [12] and Stutzer [28], are more effective with short dated options. Unfortunately, this approach requires the introduction of a highly non-stationary process that requires frequent readjustments. On the other hand, jump models reproduce the skew of short and medium dated options, while the predicted smiles for longer dated claims are flatter than observed. For a comparison among these models the reader is referred to the empirical studies [2], [26]. The present authors believe that a model which combines both jumps and stochastic volatility, possibly of the GARCH type, will perform considerably better than either models separately. In a forthcoming paper, we demonstrate how such a synthesis can be implemented by combining the model of lines with a two-level stochastic volatility process that gives rise to a recombining stochastic volatility tree. It suffices to say that the model of lines is not limited to pure jump models, but rather, is an essential element of a more elaborate pricing framework.

### On the Internal Consistency of the Black Scholes Option Pricing Model

In order to value derivatives on an asset by arbitrage, we cannot allow both to be risky simultaneously. This leaves two possibilities. We can treat the dividends as a diffusion and fix the discount rate. This would seem somewhat unrealistic for most asset types such as stocks whose dividends tend to be smooth and paid only inter- mittently. Alternatively, we can assume dividends are purely deterministic and let the discount rate be a diffu- sion.

### A Nonparametric Option Pricing Model Using Higher Moments

zero or symmetric, and positive or skewed to the right. The value of skewness is based on the S&P 500 data and would be different for every contract duration considered. Table 2 shows the value of skewness per case and duration, based on the skewness of the S&P 500 data. With respect to kurtosis, two cases are considered: the mesokurtic case, where kurtosis is equal to 3, and the leptokurtic or heavy-tails case, where the kurtosis is based on the S&P 500 data and returns based on contract duration. Table 2 contains the values of kurtosis that will be used for the simulation studies. Since nonnormal features will be part of the cases of the simulations, when nonzero skewness or nonnormal kurtosis is the case, then the Johnson family of distributions Johnson (1949) is used to generate the simulated returns data. The Johnson family of distributions has the following cumulative distribution formula F (x) = Φ { η + ζ × g [(x − ξ) /λ] } The function g( • ) is a function that determines the type of distribution used for generating returns. If g(z) = z, then the Johnson distribution type is the normal or Gaussian distribution, denoted as SN. If g(z) = ln(z), then the Johnson distribution type is the lognormal distribution, marked as SL. The Johnson SU or unbounded distribution type is generated by letting g(z) = sinh − 1

### PRICING EXOTIC OPTION UNDER STOCHASTIC VOLATILITY MODEL

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 option pricing under stochastic volatility model in recent years, to the best of our knowledge, research on exotic option such as supershare and chooser option pricing 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.

### Option Pricing Model Based on Telegraph Processes with Jumps

where IE ∗ denotes the expectation with respect to the equivalent martingale measure P ∗ defined in section 1. We decompose F into two parts, i. e. F = F + + F − , where F + and F − respect to the even and odd number of turns at time T − t of the telegraph particle.

### A GARCH Option Pricing Model with Filtered Historical Simulation

This paper presents two main contributions: a new GARCH pricing model and the analysis of aggregate intertemporal marginal rate of substitutions. An in-depth empirical study underlies the previous contributions. Our GARCH pricing model relies on the Glosten, Jagannathan, and Runkle (1993) asymmetric volatility model driven by empirical GARCH innovations. The nonparametric distribution of innovations captures excess skewness, kurtosis and other nonstandard features of return data. We undertake an extensive empirical analysis using European options on the S&P 500 Index from 1/2002 to 12/2004. We compare the pricing performances of our approach, the GARCH pricing models of Heston and Nandi (2000) and Christoffersen, Heston, and Jacobs (2006) and the benchmark model of Dumas, Fleming, and Whaley (1998). Interestingly, our GARCH pricing model outperforms all the other pricing methods in almost all model comparisons. We show that the flexible change of measure, the asymmetric GARCH volatility and the nonparametric innovation distribution induce the accurate pricing performance of our model. To economically validate our approach, we estimate the state price densities per unit probability (or aggregate intertemporal marginal rate of substitutions) for all the available maturities in our sample. Compared to previous studies, such as Jackwerth (2000) and Rosenberg and Engle (2002), we undertake a larger empirical analysis. More importantly, our estimates of the state price densities per unit probability tend to display the expected level and shape, as predicted by economic theory.

### Heterogeneity and option pricing

The intuitive discussion above can be alternatively considered as an illustration of Proposition 2, that an economy with heterogeneous agents with constant relative risk aversion will price assets as if it consisted of a single investor with declining relative risk aversion. Another way to present our case is to use Ross’s (1976) idea that options can be considered as completing the market structure, in the absence of trade in state contingent commodities. In this case, if there exists a stock market, and options can be traded for any strike price, both investors in our two-agent economy will hold a long position in the stock. In addition, the less risk averse agent will purchase call options with high strike prices, written by the more risk averse agent. Complementing these transactions, the more risk averse investor will purchase the put options with low strike prices that the less risk averse investor issues. In effect, the two agent will thus be able to obtain a Pareto-efficient

### Strategic option pricing

The study formulates and solves an extension of the well-known binomial approach to option pricing for the single-period case (see, e.g., Black & Scholes, 1973; Cox, Ross, & Rubinstein, 1979). Hence a two-date, two-asset, two-state economy as the starting point is taken. There is one risky asset and one risk-free asset whose initial price and rate, respectively, is exogenously given. The state of the economy at the second date is chosen according to a binomial distribu- tion the parameter of which is also given exogenously. The final or end-of-pe- riod price of the risky asset is state-dependent. It is customary to express this price in terms of the initial price. These end-of-period prices are considered as a date of the economy. They are not affected by actions of market participants. The classical option pricing question in this framework is: What is the time- zero price of an option on the risky asset? The traditional answer is obtained with the help of a replicating portfolio by ruling out arbitrage.