The Empirical Performance of Option Based Densities of Foreign Exchange
Summary (2 min read)
1 Introduction
- It is well known that with complete markets, a sufficiently rich set of European options prices implies a state price density that one may interpret as a probability density over the price that underlies the derivative contract, if agents are risk neutral.
- Another possibility, explored in this paper, is to calculate the risk neutral densities from American option prices on the thickly traded market by using methods that are theoretically consistent with the early exercise option.
- The authors modify the tests based on the inverse probability functions to account for this correlation between their random variables that are uniform under the null.
- Less sophisticated models of the diffusion process, such as the simple log normal Black-Scholes model, do less well than more sophisticated models in forecasting the one-hundred-eighty day horizon.
2 The data
- The American options are exchange-traded, approach a fixed expiration date and can be exercised before maturity.
- This early exercise boundary is something that the authors take account of in calculating their risk neutral densities.
- In addition, because of the historical illiquidity in certain markets, other prices were excluded: options expiring within 10 days of the current trading date, options expiring more than 100 days from the current trading date, and options with strike prices that are greater than .05 in relative, time normalized moneyness.
- (In the two million data points this happened about 20 times).
3 Estimation of the Densities
- Following Dumas et.al. (1998), their procedure is to estimate the parameters of a diffusion process in order to approximate the risk neutral density for each day.
- Estimation of the daily diffusions σ̂t(X, τ, bβ) hinges on being able to calculate the price of a given option quickly and accurately, given an arbitrary function σ̂t(X, τ, bβ).
- Therefore the authors use higher order lattices that hold the intervals of discretization of the state space and time constant and have more branches.
4 Evaluating density forecasts
- Different methods of estimation lead to different forecasting densities, some of which necessarily must be wrong.
- This is because a ranking depends on the often unknown individual loss function of agents, that may include more arguments than the first two moments.
- To assess whether there is significant evidence whether the estimated densities coincide with the true densities at a first step the authors perform the probability integral transforms of the actual realizations.
- One possible way to jointly test the departures for each prn would be to sum up their squares, as was suggested by Karl Pearson (1905) very early in the history of specification tests.
- First the authors expand their discussion of the tests based on the stationary bootstrap.
4.1 The stationary bootstrap approach
- The stationary bootstrap approach (IFSB) of Politis and Romano (1994) uses a resampling procedure to calculate standard errors of estimators that account for weak data dependence in stationary observations.
- Bootstrapped 13 distribution functions, Fb(prb) are also formed and the CvMb statistic, CvMb ≡ 1Z 0 (F (prb)− Fb(prb))2d(prb) (9) is evaluated for each bootstrapped sample.
- The null hypothesis of correct forecasts corresponds to the dashed 45◦-line that connects the origin of the diagram (on the left side at the bottom) to the upper left corner.
- This issue arrises when the forecast horizon is longer than the sample frequency.
5 The results
- The results of the CvMb statistics are reported in table 1.
- *Bold numbers indicate, that the hypothesis of an accurate density can’t be rejected.
- These broad patterns were also supported by other tests based on the bootstrapped variance of the CvMb.
- Thus, the option forecast densities fail at the short horizon because they do not place enough mass at the extreme ends of the densities.
- Note also that the confidence bands are fairly tight with the one day horizon.
6 Concluding remarks
- The authors results fall into two groups, one the thirty to ninety day time horizon for which the forecasting densities seem to fit the data fairly well, and the very short and the very long horizons which are poor specifications for forecast densities (except for the cubic diffusion model which is not rejected for the long horizon.).
- More work can be done to specify a set of models that are sufficiently rich to match the option prices, either by increasing the dimension of the states, controlling the diffusion process or by incorporating time dependence into the process.
- In other research (Craig and Keller (2001)), the authors resoundingly reject densities on the thirty day horizon implied by other methods, such as a GARCH technique, or based on other options with lower liquidity, even though these tests are based only on less than three years of data.
- As shown in Csőrgő and Horváth (1993), the CvM test does not exploit much information that may be known about the null, such as behavior of the density in the tails.
Did you find this useful? Give us your feedback
Citations
38 citations
12 citations
Cites methods or result from "The Empirical Performance of Option..."
...For a graphical representation of the integral transformation see Craig and Keller (2002)....
[...]
...This idea goes back to Fischer (1930) and Rosenblatt (1952) and has been recently applied by Diebold et al. (1998), Clements and Smith (2001), Craig and Keller (2002), and de Raaij and Raunig (2002)....
[...]
...To derive a test statistic for d CvM in the presence of data dependency we use the stationary bootstrap approach, suggested by Politis and Romano (1994), that is based on a resampling procedure, which uses samples of blocks of random lengths (for details see Craig and Keller (2002). To assess whether the distance from the 45 degree line is significant we calculate the bootstrapped distribution functions, Fb(yn), for each bootstrapped sample....
[...]
...This is in contrast to Craig and Keller (2002) who report that these densities forecast exchange rates well....
[...]
...To derive a test statistic for dCvM in the presence of data dependency we use the stationary bootstrap approach, suggested by Politis and Romano (1994), that is based on a resampling procedure, which uses samples of blocks of random lengths (for details see Craig and Keller (2002)....
[...]
11 citations
Cites background from "The Empirical Performance of Option..."
...…as the markets quantification of the uncertainty about the future course of the underlying asset prices (techniques for estimating risk neutral densities may for example be found in Craig and Keller, 2002; Anagnou et al., 2002; Soderlind and Svenson, 1997; and Aparicio and Hodges, 1998)....
[...]
3 citations
2 citations
References
630 citations
361 citations
"The Empirical Performance of Option..." refers background in this paper
...However, as shown in Csőrgő and Horváth (1993), the CvM test does not exploit much information that may be known about the null, such as behavior of the density in the tails....
[...]
...However, as shown in Csőrgő and Horváth (1993), the CvM test does not exploit much information that may be known about the null, such as behavior of the density in the tails. Further, in the space of probability distribution functions, the CvM is only optimal for deviations in the L2-direction cos(σx) as shown by Gregory (1980). The theory of statistical distribution specification testing is still fruitful, offering major new advances each year....
[...]
...However, as shown in Csőrgő and Horváth (1993), the CvM test does not exploit much information that may be known about the null, such as behavior of the density in the tails....
[...]
269 citations
"The Empirical Performance of Option..." refers methods in this paper
...We report results from a distance statistic, the so-called Cramer-von Mises statistic [von Mises (1931)]....
[...]
257 citations
244 citations
"The Empirical Performance of Option..." refers background in this paper
...The subsequent literature (e.g. Shimko (1993), Malz (1997), Jackwerth and Rubinstein (1996) and Stutzer (1996)) has concentrated on estimation of the density from noisy or, in the Malz case, extrapolated data on prices by using parametric distributions, mixtures of parametric distributions, or…...
[...]
Frequently Asked Questions (16)
Q2. What future works have the authors mentioned in the paper "The empirical performance of option- based densities of foreign exchange" ?
Thus, the authors did not use all price data of options which expired within ten days of the trading day, options that are perhaps best designed to forecast the future exchange rate at one and seven days ahead. Amore powerful test might have much to say about which of the densities represent the market ’ s true assessment of possibilities. Malz, A. M. [ 1997 ]: ” Estimating the Probability Distribution of the Future Exchange Rate from Option Prices, ” The Journal of Derivatives, 4, pp. 18-36. These densities can be computed daily, and thus form a useful policy tool, as well as providing an important set of data with which to test deeper theories of foreign exchange determination.
Q3. Why is the sample distribution function dCvM calculated?
Because the sample distribution functiondCvM and all bootstrapped sample distribution functions CvMb are step functions, the integral expression in CvMb is calculated directly.
Q4. How long does prob suboptimally affect the bootstrap?
as long as prob → 0 and Nprob → ∞ fundamental consistency properties of the bootstrap are unaffected by choosing prob suboptimaly.
Q5. How do the authors handle the end effects of a block?
End effects (in case of a block going beyond the last observation) are handled by ordering the observations in a circle, so that the series ”restarts” after the last observation.
Q6. What is the method used to calculate the risk neutral density of the underlying futures contract?
The method adopted in this paper to calculate the risk neutral density in this case is to first estimate the underlying process of the underlying futures contract for foreign exchange, based on the traded price of the American puts and calls reported for the end of the trading day.
Q7. How did the value function of an option quickly converge to the theoretical value?
for a wide range of ω, calculation of the value of an option quickly converged to the theoretical true value where these were known.
Q8. Why is the CvM test of lower power than other more specific tests?
The tests of the densities that are explored in this paper are of lower power than other more specific tests in part because of their all encompassing character.
Q9. What is the definition of a risk neutral density?
It is well known that with complete markets, a sufficiently rich set of European options prices implies a state price density that one may interpret as a probability density over the price that underlies the derivative contract, if agents are risk neutral.
Q10. What is the way to test the departures for each prn?
One possible way to jointly test the departures for each prn would be to sum up their squares, as was suggested by Karl Pearson (1905) very early in the history of specification tests.
Q11. What is the simulated density of the diffusion function t(X,,?
The simulated density bΠt of the diffusion function σ̂t(X, τ, bβ) is on the right side at the top and bπt, the corresponding first empirical derivative of bΠt with respect to K, is situated at the bottom.
Q12. How is the inverse probability of the realized thirty day ahead spot correlated with the same number?
The inverse probability of the realized thirty day ahead spot at time, t, is correlated with the same corresponding number at time t− 1, because the spot shares twenty-nine days of history.
Q13. What is the probability of drawing a block of length L?
The procedure requires a sample of random blocks of random lengths out of the original time series, where the length L of each block is drawn from a geometric distribution, so that the probability of drawing a block of length L is (1−prob)L−1prob for L = 1, 2, ....
Q14. What are the tests that the authors use to evaluate implied densities?
Next the authors describe the tests that the authors use to evaluate their implied densities, especially those that take into account the time series nature of the overlapping windows of the forecasts.
Q15. What is the second group of conclusions?
The second group of conclusions concern the thirty to ninety day horizons where their tests clearly do not reject any of the specifications of the diffusion process as forecasting densities.
Q16. How do the authors circumvent the problem of ”too small” values of t?
In another version of this paper the authors circumvent the problem of ”too small” values of σ̂t(X, τ, bβ)2, and therefore of values of α below 1/6, by augmenting, if necessary, the state space increment ∆h , so that the critical value of α is only reached by smaller values of σ̂t(X, τ, bβ)2.