# The Empirical Performance of Option Based Densities of Foreign Exchange

## Summary (3 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.
- As pointed out by 1 Breeden and Litzenberger (1978), differentiation of this expression twice with respect to the strike price, K, gives the risk neutral density, πT (X) times a discount factor, e−ρ(T−t).
- 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.
- Thus, for a small distance around the early exercise boundary, in the neighborhood where e−ρ∆tPVXt and Fp(X, t−∆t)−K are nearly equal, the authors use the value function, φ(e−ρ∆tPVXt) + (1 − φ)(Fp(X, t − ∆t) − K) where φ is a many times differentiable kernel between 0 and 1, with the property that φ→ 1 for values of e−ρ∆tPVXt that are ‘much’ greater than Fp(X, t−∆t)−K and φ→ 0 for e−ρ∆tPVXt that are ‘much’ smaller than Fp(X, t−∆t)−K.

### 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 sequence of actual {z1, ...zt, ...zN} are on the left side at the top and the estimatedbF (prn) are plotted below.
- 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.
- Stat. 23, pp. 470-472. Shimko, D.[1993]: ”Bounds of Probability,” Risk, 6, 4, pp. 33-37.

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### Cites methods or result from "The Empirical Performance of Option..."

...For a graphical representation of the integral transformation see Craig and Keller (2002)....

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...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)....

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...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....

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...This is in contrast to Craig and Keller (2002) who report that these densities forecast exchange rates well....

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...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)....

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### 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)....

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##### References

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### Additional excerpts

...The basic univariate integral transformation theorem is due to Fischer (1930) and has been generalized for the multivariate case by Rosenblatt (1952)....

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### "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…...

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