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The aim of this paper is to present some of the stylized features of financial data which have received a lot of attention both from practitioners and those with more theoretical backgrounds

# Realistic Statistical Modelling Of Financial Data

INTERNATIONAL STATISTICAL REVIEW, no. 3 (2000): 233-258

The aim of this paper is to present some of the stylized features of financial data which have received a lot of attention both From practitioners and those with more theoretical backgrounds. Some of the models resulting from these efforts are reviewed and discussed. To facilitate the discussion two data sets are used: one of these contai...更多

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• The evaluation of risk and the pricing of derivative assets are based on statistical models describing underlying asset prices.
• Whereas all the ARCH type models are build on statistical knowledge of the financial market the binomial model is rather stylized and more of a toy model, since stock prices change randomly at random time points rather than move to one of two levels at deterministic time points.
• Both the discrete and the continuous models share the common feature that the variance of the distribution of the log returns is unobserved and specified by some stochastic process.

• The evaluation of risk and the pricing of derivative assets are based on statistical models describing underlying asset prices
• The quality of the risk measures and the validity of prices are strongly dependent on how well the statistical model captures the behavior of the underlying asset
• From a mathematical finance point of view, where the focus is on pricing derivatives, they have serious drawbacks
• The financial market described by the discounted price process X which is a solution to a SDE of the above form, is complete (see Hansen (1995)) if: The self financing trading strategies4 are well defined and the contingent claim H, H ∈ FT, H > 0, such that EP [NT H] < ∞, is attainable at the fair price given by π(H) = EP [NT H]
• Just to recapture the main reasons: mispricing of derivatives: hedge strategies concerned with holding a certain portfolio which is based on a misspecified model will not uncover the all the risk involved: measures and other economic questions could be given the wrong answers

• The attempt to incorporate statistical features in the models are limited, one of the reasons is perhaps that the goal is to price derivative assets and this becomes rather complicated as soon as the authors leave the Black & Scholes framework.
• The model incorporates this fact by letting the volatility be a function of the stock price in the following way γ dSt = μStdt + σSt2 dWt. Again W is a Wiener process and γ < 2.
• Some of the later diffusion type models have, in spite of the problems of pricing, tried to incorporate some of the statistical aspects of financial data.
• In order to use the multinomial model it is necessary to assume that the price movements can only exist on a fixed number of states in this case it is assumed that they live on −2, −1, 0, 1, 2 which is the case for 98.6% of all the trades in the data set considered in Russell and Engle (1998).
• The approach in Rydberg and Shephard (1998a) is somewhat different in that they use a decomposition technique which enables them to model the activity, the direction and the size of a given trade.
• One of the goals in modelling the intra–daily data is to gain insight into how different types of trades affect the long term price levels.
• More generally fair prices of a derivative assets such as a European or American call option, may be calculated as the mean value of the discounted payoff function of the derivative, under the unique equivalent martingale measure provided that this measure does exist.

• The financial market described by the discounted price process X which is a solution to a SDE of the above form, is complete (see Hansen (1995)) if: The self financing trading strategies4 are well defined and the contingent claim H, H ∈ FT , H > 0, such that EP [NT H] < ∞, is attainable at the fair price given by π(H) = EP [NT H] .
• Option pricing based on ARCH-type models is complicated by their discrete nature, which causes them to be incomplete.
• Being able to hedge and asses risk, is of interest to a small elite of financial speculators: it is of interest indirectly to almost everyone via pension schemes

• Table1: Estimated parameters for IBM log returns in the period January 2, 1973 to January

• Funding via EU grant on “Econometric inference using simulation techniques.” is gratefully acknowledged. The paper was specially prepared for the 51th Session of the International Statistical Institute held in Istanbul August ’97

observations: 462
However, several other models have been proposed, which also build on independent increments of the returns. Praetz (1972) presents a scaled t–distribution and this distribution is shown to give a good fit to weekly observations from the Sydney Stock Exchange for the period 1958–1966, a total of 462 observations. This type of model can also be viewed as a subordinate normal model, where the conditional distribution y|σ2 is a normal distribution and σ2 is an inverted gamma type of distribution 0 