The Empirical Performance of Option Based Densities of Foreign Exchange

TL;DR: In this paper, the authors used higher order lattices combined with smoothing the value function of the American option at the boundaries in order to mitigate the non-differentiability of both the payoff boundary at expiration and the early exercise boundary.

Abstract: Risk neutral densities (RND) can be used to forecast the price of the underlying basis for the option, or it may be used to price other derivates based on the same sequence. The method adopted in this paper to calculate the RND is to firts estimate daily the diffusion process of the underlying futures contract for foreign exchange, based on the price of the American puts and calls reported on the Chicago Mercentile Exchange for the end of the day. This process implies a risk neutral density for each point of time in the future on each day. I order to estimate the diffusion process we need methods of calculating the prices of American options that are fast and accurate. The numercial problems posed by American options are tough. We solve the pricing of American options by using higher order lattices combined with smoothing the value function of the American Option at the boundaries in order to mitigate the non-differentiability of both the payoff boundary at expiration and the early exercise boundary. By calculating the price of an American option quickly, we can estimate the diffusion process by minimizing the squared distance between the calculated prices and the observed prices in the data. Risikoneutrale Dichten (RND) konnen dazu genutzt werden, den Preis des der Option unterliegenden Basiswertes vorherzusagen, oder aber nur um den Wert anderer Derivative, die sich auf den gleichen Basiswert beziehen, zu berechnen. Die in diesem Aufsatz vorgestellte Methode zur Berechnung der RND besteht darin, in einem ersten Schritt taglich den Diffusionsprozess der Futures Kontrakte des Devisenmarktes zu schatzen, indem auf die Tagesendnotierungen von amerikanischen Kauf- und Verkaufsoptionen zuruckgegriffen wird, die an der Chicagoer Produktenborse gehandelt werden. Dieser taglich geschatzte Prozess impliziert dann fur beliebige Zeitpunkte in der Zukunft risikoneutrale Dichten. Allerdings erfordert die Schatzung des Diffusionsprozesses schnelle und genaue Berechnungsmethoden fur die Preise amerikanischer Option. Wir losen das Problem der Bewertung amerikanischer Optionen dadurch, dass wir an nicht differenzierbaren Stellen glatten, also zum Falligkeitszeizpunkt der Option bzw. bei Wertentwicklung des Basiswertes, an dem sich moglicherweise das Ausuben der Option vor Falligkeit lohnt. Wir schatzen den Diffusionsprozess, indem wir den Kleinstquadratabstand zwischen den vom Diffusionsprozess implizierten Preisen und den beobachteten Notierungen amerikanischer Optionen bestimmen; ein Verfahren, das nur bei schneller Berechung der Optionspreise durchfuhrbar ist.

Summary

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.

Figures

Figure 3: The deviations of the empirical density from individual quantiles

Figure 2: The integral transformation

Figure 1:

Full text

working
paper
FEDERAL RESERVE BANK OF CLEVELAND
03 13
The Empirical Performance of Option-
Based Densities of Foreign Exchange
by Ben R. Craig and Joachim G. Keller

Working papers of the Federal Reserve Bank of Cleveland are preliminary materials
circulated to stimulate discussion and critical comment on research in progress. They may not
have been subject to the formal editorial review accorded official Federal Reserve Bank of
Cleveland publications. The views stated herein are those of the authors and are not necessarily
those of the Federal Reserve Bank of Cleveland or of the Board of Governors of the Federal
Reserve System, the Bundesbank, or Oestereichische Nationalbank.
Working papers are now available electronically through the Cleveland Fed’s site on the World
Wide Web: www.clevelandfed.org/research.

Ben R. Craig is at the Federal Reserve Bank of Cleveland and may be reached at ben.craig@clev.frb.org
or (216) 579-2061. Joachim G. Keller is at the Bundesbank and may be reached at
joachim.keller@bundesbank.de
. This working paper was previously presented at the regular joint research
workshop of the Deutsche Bundesbank and the Oesterreichische Nationalbank. It is also available as a
Deutsche Bundesbank Working Paper and as Oesterreichische Nationalbank Working Paper No. 60.
The authors thank Christoph Fischer, Heinz Herrmann and Christian Upper as well as seminar
participants at the Deutsche Bundesbank, at the joint research workshop of Deutsche Bundesbank and the
Oesterreichische Nationalbank and at the “Joint Lunchtime Seminar” organized by the Center for
Financial Studies, ECB, and Deutsche Bundesbank.
Working Paper 03-13
November 2003
The Empirical Performance of Option-Based Densities of Foreign Exchange
By Ben R. Craig and Joachim G. Keller
In this paper, we calculate risk-neutral densities (RND) by estimating the daily diffusion process of the
underlying futures contract for foreign exchange, based on the price of the American puts and calls
reported on the Chicago Mercantile Exchange for the end of the day. Our quick and accurate method of
calculating the prices of the American options uses higher-order lattices and smoothing of the option’s
value function at the boundaries to mitigate the nondifferentiability of the payoff boundary at expiration
and the early exercise boundary. We estimate the diffusion process by minimizing the squared distance
between the calculated prices and the observed prices in the data. We also test whether the densities
provided from American options provide a good forecasting tool. We use a nonparametric test of the
densities that depends on inverse probabilities. We modify the test to compensate for an inherent problem
that arises from the time-series nature of the transformed variables when the forecasting windows overlap.
We find that the densities based on the American option prices for foreign exchange do considerably well
for the longer time horizons.
Key words: risk-neutral density, option prices, diffusion process
JEL codes: G13, G15

1 Introduction
It is well kno w n t hat w it h complete markets, a sufficiently rich set of European
optionspricesimpliesastatepricedensitythatonemayinterpretasaprobability
density over the price that under lies the derivative contract, if agents are risk neutral.
In this case the state price density is called a risk neutral densit y. European options
ha ve been used to recover the risk neutral densities for a variet y of prices and
indices, including oil and the Standard and Poor’s 500 index. The ric hest market
for foreign exchan ge options presen t a difficulty in app lying this theory, ho wever.
The most liquid foreign exc han ge options, sold on the Chicago Mercan tile Exchange
are Am erican options based o n a futures p rice. As is w ell known, this type of option
ha ve an early exercise feature that d estroys the logic behind com p utin g the risk
neutral densities from European options. To see this, th e European option price,
c
t
(K, X, T − t), (in this case of a call op tion ) at time, t, with a strik e price, K,
expiring at time T , in a one state model can be expressed as
c
t
(K, X, T − t)=e
−ρ(T −t)
∞
Z
K
(X
T
− K)π
T
(X)dX (1)
where ρ is the discount rate, (here assumed constan t) and π
T
(X) is the risk neutral
densit y o ver the state space of X at the expiration date T . As pointed out b y
1

Breeden and Litzen berger (1978 ), differentiation of this expression tw ice with respect
to the strike price, K, gives the risk neutral density, π
T
(X) times a discoun t factor,
e
−ρ(T −t)
. The subsequent literature (e.g. Shimko (1993), M alz (1997 ), Jackwerth and
Rubinstein (1996) and Stutzer (1996)) has concen tr ated on estimation of the density
from noisy or, in the Malz case, extrapolated data on prices by using parametric
distribution s, mixtures of para m etric distributions, or non-p ar am e tric smoothers
to fit the second derivative of the option price function w ith respect to the strik e
price. Others, like Neuhaus (1995) do not rely on smoothing equations and calculate
probabilities at and betw een strik e prices. Once the risk neutral densit y is calculated,
then it can be used to forecast the price of the underlying basis for the option , or it
maybeusedtopriceotherderivativesbasedonthesamesequence.
With an American option based on a future price, the relationsh ip in equation (1)
breaks down. The expectation operator m ust take into accoun t the early exercise
boun d ar y, which will differforeachoptionbasedonadifferent strik e price, and differ
by time to expiration for the same option . Und er this regime, equation (1) is no
longer true, and arguments which generate equation (1) from the theory of option
pricing, suc h as application of Feynmann -K ac to the pa rtial differen tial equation
system defin ing the evolution of the option price no longer mak e sense. T h is leaves
a researcher with tw o cho ices. O ne can use a thinner market, suc h as the European
options offered b y the Philadelphia exc ha nge or use the European options prices
where they are quoted b y a single bank. Another possibility, exp lored in this paper,
is to calculate the risk neutral den sities from America n option prices on the thickly
traded market by using methods that are theoretically consisten t with the early
exercise option.
The method adopted in this paper to calculate the risk neutral densit y in this
case is to first estimate the underlying process of the underlying futures contract
for foreign exc hange, based on the traded price of the American puts and calls
reported for the end of the trading da y. This estimated process implies a risk
neutral density for each point of time in the future. In order to estim ate the diffusion
process w e need methods of calculating the prices of American options that are fast
and accurate. The n um erical problems posed b y Am erican options are tough. We
2

Frequently asked questions

Q1. What contributions have the authors mentioned in the paper "The empirical performance of option- based densities of foreign exchange" ?

In this paper, Fischer et al. proposed a method to estimate the risk neutral density of a call option based on the early-expiration of a foreign exchange option. 

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.