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

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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.

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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.

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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.

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

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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**.

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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.

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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.

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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

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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.

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Abstract
This thesis investigates a methodology for quantification of **model** risk in **option** **pricing**. A set of different **pricing** models is specified and each **model** is assigned a probability weight based on the Akaike Information Criteria. It is then possible to obtain a price distribution of an exotic derivative from these probability weights. **Two** measures of **model** risk inspired by the regulatory standards on prudent valuation are proposed based on this methodology. The **model** risk measures are studied for different equity options which are priced using a set of stochastic volatility models, with and without jumps. The models are calibrated to vanilla call options from the S&P 500 index, as well as to synthetic **option** prices based on market data simulated using the Bates **model**. For comparable options, the **model** risk is higher for up-and-out barrier options compared to vanilla, digital and Asian options. Moreover, the **model** risk measure, in relative terms of **option** price, increases quickly with strike level for call options far out of the money, while the **model** risk in absolute terms is lowest when the **option** is deep out of the money. The **model** risk for up-and-out barrier options tends to be higher when the barrier is closer to the spot price, although the increase in risk does not have to be monotonic with decreasing barrier level.

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Thus, by suitable specifications of parameters (u, v and y) the **two**-**state** Markov chain finan- cial market **model** can be constructed to reflect a possible feature of a real market, having bullish or bearish trends. It could create forgetful markets and markets with long memory, markets with different sorts of dependencies between assets returns. So, the **two**-**state** Markov chain market **model** has the bull and bear features of underlying asset price fluctuations, and it gives better results with the evaluation of **option** price of companies from DJIA.

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