IBEX-35 OPTION PRICING UNDER
ALTERNATIVE MODELS
Juan Manuel Prado Enrico
Tesina CEMFI No. 0505August 2005
CEMFI
Casado del Alisal 5; 28014 Madrid Tel. (34) 914 290 551. Fax (34) 914 291 056
Internet: www.cemfi.es
This paper is a revised version of the Master’s Thesis presented in partial fulfillment of the 2002-2004 Graduate Programme at the Centro de Estudios Monetarios y Financieros (CEMFI). I would like to thank Ángel León for his excellent guidance and helpful on this paper. I am also indebted to Manuel Arellano for his suggestions. I have also benefited from the comments and helps of Samuel Bentolila, Enrique Sentana, Rafael Repullo, Javier Mencía and my CEMFI classmates. And I would like to thank Mónica Gago for the option data set. I dedicate this paper to el Pá y la Má, for giving me the freedom; to Fede, Fla y Seba, my three big loves; and to my
Tesina CEMFI No. 0505 August 2005
IBEX-35 OPTION PRICING UNDER
ALTERNATIVE MODELS
Abstract
Several papers show that asset distributions are different to the Normal one. Using a statistic series expansion, Corrado and Su (1996) relaxed the lognormality assumption of the risk-neutral density of the terminal asset proposed by Black and Scholes (1973). Then, Jondeau and Rockinger found the values for skewness and kurtosis that guarantee the positivity of this function. An alternative approach to capture the deviations from normality is assuming that the risk-neutral density is a weighted combination between two lognormal distributions. I used IBEX-35 options from February 1996 to October 1998 to examine the in-sample fit, the predictability and the hedging performance of these three alternative models. And, I compared these with the results obtained using Black and Scholes (1973) model. I provide evidence that these alternative models improve the results and are easy to implement.
Juan Manuel Prado Enrico Indigo Value Analysis [email protected]
1
Introduction
Since the famous option pricing formula of Black and Scholes (1973) many re-searchers have been trying to find a model that could fit, in a better way, the behavior of the assets return. Although the Black and Scholes formula is, cer-tainly, one of the most utilized to price options, it presents some inconsistent patterns. The key of these inconsistencies is due directly to the assumption of the behavior of the underlying asset returns over the life of the option contract. The Black and Scholes model assumes that the log-price of the underlying asset follows a normal distribution. However, several empirical studies show that this assumption is not corroborated and that the model missprices eitherin-the-money
or out-of-the-money options. Thus, the implied volatilities are not constant as a function of the strike and maturity, and they depend on time.
There are a lot of papers that describe different methods in order to carry out a solution to these problems. A first type of solutions considers alternative un-derlying stochastic processes instead of the geometric Brownian process. Among these alternatives it can be mentioned: jump-diffusion process [see Merton (1976), Bates (1996)], stochastic volatilities [Hull and White (1987), Chesney and Scott (1989), Stein and Stein (1991), Heston (1993)], or jump-diffusion models with stochastic volatility [Bates (2000), Pan(2002)].
A second type of solutions considers implied binomial trees in order to in-troduce stochastic volatility for the option pricing model [see Rubinstein (1994, 1998), Derman and Kani (1994), Dupire (1994), Derman et al. (1996), Jackwerth (1997)].
It is important to mention that, although both solutions can yield skewed and leptokurtic risk-neutral density, they are not completely satisfactory. The most common critics on these kind of models are the lack of parsimony, leading sometimes to overfit the prices, and the presence of inadequate volatility term structure.
The aim of this paper is to price future options on the IBEX-35 index finding a model that combines simplicity and efficiency. Simplicity in terms of implemen-tation and application costs. Efficiency in terms of fit and performance results.
The models chosen in this paper are focused on the risk-neutral density func-tion that follows the terminal asset prices considering an alternative approach specification. On the one hand, I chose a semi-parametric model: the Corrado and Su (1997) model. This approach consists in approximating the risk-neutral density of the terminal stock prices by the Gram-Charlier series expansion. The series are truncated to a finite sum giving a tractable closed-formula and allow-ing to capture the risk-neutral skewness and kurtosis of the underlyallow-ing asset in a very natural way since the coefficients of statistical series expansion depend on the moments of the distribution. On the other hand, I used a fully parametric model that I called Mixture Normal model. In this approach, it is considered the risk-neutral density of the terminal asset price such as a weighted sum of two lognormal distributions. The underlying distribution should beflexible enough to capture some features of the asset price’s distribution such as fatness in the tails and either positive or negative skewness.
Besides, I implemented a third model in this paper, the Jondeau and Rockinger (2001) model. The Corrado and Su model has an issue. More specifically, some-times the risk-neutral density could be negative for some values of the skewness and excess kurtosis. Jondeau and Rockinger found the region where the values for skewness and excess kurtosis guarantee the positivity of the Gram-Charlier risk-neutral density implemented by Corrado and Su.
The results of this paper show that the mean relative pricing errors (in absolute values) obtained using the alternative models are more than 50 percent lower than Black-Scholes model for call options. And, for put options the results are still better because the mean relative pricing errors, using the alternative models, are more than 60 percent lower than those using Black-Scholes model.
The remainder of this paper is structured as follows. Section 2 describes the theoretical framework of the four models used here. Information about the dataset is outlined in section 3. Section 4 describes the estimation procedure and in-sample results. Section 5 presents the out-of-sample results and a regression analysis of the forecast pricing errors. Section 6 shows the volatility smiles and Section 7 shows a comparative static delta-hedge. Finally, Figures and Tables mentioned in the text are presented in the last section of this paper.
2
Option Pricing Model
2.1
Option contracts
In Finance, there are two kinds of option contracts, named call options and put options. A call option confers the right on its holder, without the obligation, to purchase the underlying asset at a certain date for a certain price. A put option confers the right on its holder, without the obligation, to sell the underlying asset at a certain date for a certain price. The underlying assets include stocks, stock indices, foreign currencies, debt instruments, commodities, and futures contracts. In this paper I used IBEX-35 index1 future contracts as the underlying assets. The price at which the underlying asset can be sold or bought is called the exercise price. The date when the option contract is exercised is known as the expiration date or maturity date. Furthermore, there are two basic types of call and put options, called European and American options. European options can only be exercised at the expiration date itself, while American options can be exercised at any time up to maturity. As options on the IBEX-35 future index are European-style, the option price models used in this study do not allow for early exercise.
Consider that you want to price an European call option at time t, with expiration date T, where ST is the price of the underlying asset at T,rtis the
free-risk interest rate of the market and K is the exercise price of the option contract. Then, under the assumptions of complete markets, no arbitrage opportunity and rt constant over time, the theoretical price (the fair price) of an European call
option is defined as the present value of the expected payoff at maturity. This price is given by the pricing kernel
Ct=e−rτEQ[ max(ST −K,0)]
where EQis the expectation under the risk-neutral probability measure.
Moreover, in terms of the risk-neutral density, the theoretical price of the call
1The IBEX-35 index is composed by 35 securities quoted on the Joint Stock Exchange System of the four Spanish Stock Exchanges.
option can be written as
Ct =e−rτ
Z +∞
ST=K
(ST −K)f(ST)dST. (1)
The option formula can be obtained once a certain distribution of the risk-neutral density for the terminal asset price is assumed. The Black-Scholes model assumes that the risk-neutral density for the terminal price of the asset is lognor-mal. The normal mixture model assumes that the risk-neutral density follows a mixture of two lognormal densities.
Let the τ-period log-return of the underlying asset, log(ST/St), show a
condi-tional meanµτ and a standard deviationσ√τ, then, introducing the standardized variable
z = log(ST/St)−µτ
στ
where στ =σ√τ, the theoretical price of the call option becomes
Ct =e−rτ
Z +∞
z=log(ST /St)−µτ
στ
(Steµτ+zστ −K)f(z)dz. (2)
The expected price under the neutral probability measure should be equal to the current asset price compounded at the risk-free rate. To guarantee the risk-neutral measure, it must hold the well known martingale restriction2
EQ[ST] =erτSt (3)
and then, the risk-neutral density f(z) must satisfy (demonstration in appendix)
µτ =rτ −ln[
Z +∞ −∞
eστzf(z)dz]. (4)
2.2
Black-Scholes Model
The most common assumption in the option pricing literature is that the un-derlying asset price follows a geometric Brownian process, which implies that the risk-neutral density for the terminal asset prices is lognormal. This is the assump-tion behind Black and Scholes (1973) model for pricing European opassump-tions, as it was mentioned previously. This assumption leads to the following expression for the conditional mean of the equation (4)
µτ =rτ − 1
2σ
2
τ
usually named the “traditional” martingale restriction.
To price a call option, it is necessary to insert the risk-neutral density in equation (1) subject to the martingale restriction. Then, the call option price under Black-Scholes (1973) model (hereafter BS model) is defined by
CtBS =StΦ(d)−Ke−rτΦ(d−στ) where d= ln(St/K) +rτ + (σ 2 τ/2) στ
with Φ(.) as the standard cumulative Normal distribution function.
2.3
Corrado-Su Model
Under regularity conditions3, any continuous density function can be expressed in
terms of the expansion
f(z) =φ(z)
∞ X
i=0
biHi(z)
where Hi(z) is thei-thHermite polynomial andφ(.) is the Normal density function.
The Hermite polynomial of order iis defined by4
Hi(z) = (−1)i
∂iφ
∂zi
1
φ(z)√i!.
3See Kendall and Stuart, 1977.
The Hermite polynomials form the basis of a Hilbert space and can be used to get an expansion of the probability density function. Usually, this series is called Gram-Charlier. Corrado and Su (1996) truncated the series to afinite sum taken into consideration only thefirst few terms of this expansion. The resulting truncated series can be written as
f(z) =φ(z)[1 + √k3
3!H3(z) + k4 √
4!H4(z)] (5) where k3and k4 are the Fisher parameters for skewness and excess kurtosis5
k3 = µ3 µ32/2 and k4 = µ4 µ2 2 −3
with µi the corresponding centered moments of orderi, fori= [2,3,4].
Equation (5) represents the Normal probability density function multiplied by a polynomial that accounts for the effects of departure from normality allowing more flexibility. Plugging the density function (5) into equation (4), the condi-tional mean takes the following form
µτ =rτ −1 2σ 2 τ−ln(1 + k3 3!σ 3 τ+ k4 4!σ 4 τ) (6)
allowing for skewness and kurtosis.
As shown by Brown and Robinson (2002), the original Corrado and Su (1996) formula contains an error in the term which corresponds to the sensitivity of the option price to the skewness of the implied risk-neutral density. Jurczenko, Maillet and Negrea (2002) provide a closed-form for the call price that contains the correction of this typographic error and use the conditional mean shown in equation (6). The new formula for a call option price under the Corrado-Su model (hereafter CS model) becomes
CtCS =CtBS∗ +k3Q3 +k4Q4 H1(y) =y H2(y) = (y2−1)/ √ 2 Hk(y) = yHk−1(y)− √ k−1Hk−2(y) √ k
with Q3 = StστP3(d∗)φ(d∗) 3!(1 +w) Q4 = StστP4(d∗)φ(d∗) 4!(1 +w) and P3(y) = 2στ −y P4(y) =y2−3yστ+ 3σ2τ−1 and d∗ = [ln(St/K) +rτ + 1 2σ 2 τ −ln(1 +w)] στ =d− ln(1 +w) στ w= k3 3!σ 3 τ + k4 4!σ 4 τ where CBS∗
t is the Black-Scholes (1973) call price evaluated at a “correct”
stan-dardized moneyness level denoted d∗.
2.4
Jondeau-Rockinger Model
As I mentioned before, the Corrado and Su model has an analytical problem, some-times the risk-neutral density could be negative for some values of the skewness and excess kurtosis. Specifically, with the Corrado-Su model, f(z) from equation (5) can be negative, for some z and for some (k3, k4). Jondeau and Rockinger
(2001) characterized the regionD in the (k3, k4)-plane wheref(z) is positive
def-inite. This region defines a particular plot, like a seed, where k3 is between −1.05
and 1.05, andk4 is between 0 and 4. In figure 1 this region is presented.
Then, the fair price of a call option using Jondeau-Rockinger model (hereafter JR model) becomes CtJR=CBS ∗ t +k∗3Q3+k4∗Q4 with (k∗ 3, k∗4)∈D,∀z ∈R.
2.5
Normal Mixture Model
As it was suggested by Melick and Thomas (1997) the risk neutral density of the terminal price of an asset can be expressed like a mixture of two lognormal distri-butions which captures departures from normal returns. Under this assumption, the expression for the risk neutral density will be
f(ST) =αL(ST;σ1) + (1−α)L(ST;σ2) (7) with L(ST;σi) = 1 STσi √ 2πτe {−[ln(ST /St)−(r−σ 2 i /2)]2 2σ2 iτ } with i= 1,2
where the parameter 0 ≤ α ≤ 1 is the mixing parameter which weights the two lognormal distributions. Plugging equation (7) in equation (1) and solving the integral, it is possible to compute the theoretical price of a call option. Then, the expression for a call option price under Normal Mixture model (hereafter MIX model) becomes
CtM IX =αCtBS(σ1) + (1−α)CtBS(σ2)
where CBS
t (σi) is the option price of a call under Black-Scholes model for the
lognormal distribution i.
3
Data Description
The data set I used, consists of future options on the IBEX-35 index traded daily on the Mercado Oficial de Futuros y Opciones Financieros (MEFF)6. The
sample period extends from 1 February 1996 to 9 October 1998. All the option contracts written are European-style and expire the third Friday of the expiration month. Maturity lengths range from approximately six to forty days, because it was selected the most liquidity contracts. In addition to the option prices, each record in the dataset includes the future price of the index, Ft, the strike price,
6MEFF is the Spanish Official Exchange for Financial Futures and Options. It is fully regulated, controlled and supervised by the Spanish authorities.
K, and the time (in years) to maturity of the contract option, τ. The proxy for risk-free interest rate, r, is the repo daily annualized interest rate obtained from Bank of Spain.
I applied two filters to the data set. First, I included in the sample only observations for which the price of the call options exceeds the lower bounds
LBC = max{0, e−rτ(Ft−K)}
and the price of the put options exceeds the lower bounds LBP = max{0, e−rτ(K−Ft)}.
Thesefilters ensure us that there are no static arbitrage opportunities in the data set. Second, I put into the sample only days with more than three prices, calls or puts. This restriction is necessary for system identification because the Corrado-Su model has three parameters to be solved. Based on these criteria I eliminated 16 observations, representing 0.04 percent of the original sample, approximately. The resulting data set contains 37998 observations divided into two groups: 22303 call options and 15701 put options in 597 days. The mean is 64 contracts per day. Besides, I divided the data set into several categories according to moneyness with the purpose of a richer analysis. Call options for whichK/F are between 0.985 and 1.015 are categorized as at-the-money (ATM), those for which K/F≥1.015 as out-of-the-money (OTM), and those for which K/F≤0.985, as in-the-money
(ITM). Put options for which K/F range from 0.985 to 1.015 are categorized as
at-the-money (ATM), those for which K/F≥1.015 as in-the-money (ITM), and those for whichK/F≤0.985, as out-of-the-money (OTM). So, the dataset is then divided into six groups: 11608 ITM call options, 9247 ATM call options, 1444 OTM call options, 790 ITM put options, 5518 ATM put options, and 9391 OTM put options.
4
In-Sample Results
In order to do the empirical analysis I used the four models described in section 2. The aim is to show which of them fits better the sample used here. That is why, I will explain in this section the estimation procedure and describe the results in terms of statistical errors. I selected five measurements to do this, defined as follow:
• The mean error (ME) is the average deviation of the market option prices from the values of the theoretical model.
• The mean absolute error (MAE) is the average absolute deviation of the market option prices from the values of the theoretical model.
• The mean absolute relative error (MARE) is the average absolute relative error defined as the market option price minus the model’s theoretical value, divided by the market option price.
• The root mean squared error (RMSE) is the square root of the average square deviation of the reported option prices from the theoretical values of the model.
• The Akaike Information Criterion (AIC) is the root mean squared error multiplied by a coefficient that penalizes the goodness-of-fit as more degree of freedom are added to the model.
4.1
Estimation procedure
I implemented a non-linear least squares method7 to estimate risk neutral
param-eters for the four models. At each date t, the non-linear least squared estimator
7Another method was implemented. It consists of minimizing the square relative errors to obtain the implicit parameters. The results are similars to the method described in this section and that is way they are omitted.
bθ is obtained by minimizing the square distance between the observed and the theoretical option prices
b θ = arg min nt X i=1 ³ PiM −PiT ³ b θ´´2 (8)
where bθ is a vector of parameters that will be estimated and depends on the model8, PM
i is the i-th option price at date t and PiT
³
bθ´ is the i-th theoretical option price different for each model.
The dataset contains prices of future options. With a simple transformation in all the formulae, replacing St by e−rτFt it is possible to calculate the theoretical
future option price. In theory, European options on futures and on spots with the same strike price and time to maturity are equivalent. This is because the underlying variable is equal at maturity (ST =FT). Consequently, the risk-neutral
density of the underlying asset is also equal at maturity.
I implemented the minimization problem in equation (8) for the four models using GAUSS9 software. The MIX model has some weaknesses and sometimes,
it may produce a density function which is characterized by a sharp spike. The reason is that one of the two lognormal distributions is estimated with a very small volatility. It also happens that the optimization procedure fails in finding any solution for a particular day. To overcome these problems, following Andersoon and Lomaka (2001), the restriction
0.25< σ1
σ2
<4
was added to the σ values.
I used all the option contracts, calls and puts, for each day t to solve the minimization problem in equation (8). I obtained the formulae for puts using the Put-Call Parity, where
Pt=Ct−St+Ke−rτ
8In Black-Scholesbθ=σ, in Corrado-Su and Jondeau-Rockingerbθ= (σ, k
3, k4),and in Normal Mixturebθ= (α,σ1,σ2).
with Pt as the put option price in t, Ct as the call option price in t, St as the
underlying asset price in t, and Ke−rτ the present value of the strike price.
Figures 2 and 3 present the implicit parameters estimated by the models. Fig-ure 2 shows the implied volatilities for BS, CS and JR models for the whole sample. Figure 3 compares the implied parameters of skewness and kurtosis estimated by CS and JR models, and shows the region D where the density function f(z) is positive.
4.2
In-Sample Performance
The analysis of in-sample pricingfit is distributed as follows: Table 2 resumes the
five statistic errors for total calls and puts, respectively. Table 3 and 4 present the
first four statistic errors divided by contract moneyness for call and put options, respectively.
The ME statistic informs about the bias of the fit. Table 2 shows how all the models negatively bias call option prices and how BS does it worth. For put options the bias is positive, and again, pricing under BS model display the biggest bias. The MIX model makes the bestfit either for call or put options in terms of the MAE, RMSE, AIC and MARE. The CS and JR models have a good fit near to MIX model in terms of RMSE. In three of the four statistics10 JR model does
better than CS model. One can conclude that the restriction which guarantees the positivity of the density function does not make worsed the results, and sometimes it can improve them.
When I calculated the statistic errors categorized by moneyness the results were similar. The best fit was for MIX model in all cases. The typical fit’s problems of BS model are reflected in tables 3 and 4. BS model did the worsefit for at-the-money call options and for in-the-money put options. For both at-the-money options, calls and puts, the pricing is good but not enough to improve the other models.
CS model has a positive bias in out-of-the-money call options but near zero (0.04). We believe the correction was so hard that the bias sign changed. This is
an important issue, and a further analysis11 in this study will show that the mon-eyness is statistically significant to explain the pricing error, when it is regressed with moneyness, interest rate and time to maturity.
5
Out-of-Sample Results
It is showed that the in-samplefit is better for the CS, JR and MIX models than for BS model. As one may argue, this better fit might simply be a consequence of having a big number of structural parameters. In order to reduce the impact of this argument, I used the AIC statistic, which penalizes for the number of parameters, to compare the estimation results. Another alternative is to examine the out-of-sample pricing performance for each model. For out-of-sample pricing, the presence of more parameters may actually cause overfitting and penalize the model if the extra parameters do not improve its structural fitting.
5.1
Out-of-Sample Performance
I constructed five statistics from the errors12 series to analyze the out-of-sample
fit of the four models. To set the forecast price of date t for a given model, I computed the price of each option using the previous day’s implied parameters as inputs. Table 5 summarizes the results of these five statistic errors for total calls and puts, respectively. In tables 6 and 7 the results are classified by option contract moneyness. Table 6 has call options and Table 7 put options.
Table 5 shows a negative bias in out-of-sample pricing for the four models, but with a huge difference in bias magnitude between BS model and the remainder ones, because in the last models the bias is near zero. In terms of put options, Table 5 shows an opposite bias. The difference in the results is that only MIX model improves all the bias with a ME of 0.41. While analyzing the other statistic errors, I found all of them rank the JR model first. And these results are similar
11Regression pricing errors of subsection 5.2. 12These statistics are described in section 4.
to the ones obtained in table 6. But, for the second place, the results are different. For call options the MIX model makes it better than CS and BS models, but for put options CS model makes it better than the other ones.
In terms of the moneyness the results are quite different. For call options JR model makes the bestfit in the three categories but the differences with the other models decrease forat-the-money and in-the-money call options. In terms of the MARE statistic there are not differences in the out-of-samplefit forin-the-money
call options, and forat-the-money call options the differences are very small. The key is that the CS, JR and MIX models improve the results for out-of-money call options because their formulae allow to capture departures from normal returns. For put options, all models have a similar fit for at-the-money and in-the-money
options, but results are not too good forin-the-money options. The values of the RMSE statistics are between 26 and 27 for in-the-money options. In the case of
out-of-the-money put options, the fit is better for the three alternative models to BS model, and JR model is the best. These results are similar to call options because the three models improve the results in this category of moneyness.
5.2
Regression Analysis of Pricing Errors
I regression analysis was implemented to study the association between the out-of-sample pricing errors and factors that are either contract-specific or market condition-dependent. This analysis tries to understand the structure of remaining pricing errors. I ran two regressions for the whole sample, one for call options and other for put options as follows
εi(t) =α0 +α1
Ki
F(t) +α2τi+α3r(t) +ςi(t)
where εi(t) is the out-of-sample relative error between the market price and the
theoretical model’s value of option i at date t, Ki
F(t) is the moneyness of the i-th
option,τi is the time to expiration of optioni, andr(t) is the free-risk interest rate
at date t. The idea is that moneyness and expiration date are contract-specific factors, and that free-risk interest rate is a factor of the market conditions. In
some sense, the contract-specific variables help us to detect the existence of cross-sectional pricing biases, whereas the free-risk interest rate serves to indicate if the pricing errors over time are related to the dynamically changing market conditions. Table 8 reports the regression results for call and put options. The standard errors for the estimated coefficients are robust standard deviations according to the White (1980) heteroskedasticity-consistent estimator. To obtain these stan-dard errors, I ran the regression with Stata software13 using a cluster to identify
the options that belonged to the same day. Table 8 presents the p-values in parenthesis.
For call options the coefficient of moneyness is significant only for two models, BS and CS models. For BS model the results are not a surprise because all previous analysis in this paper show a bias in the estimation related to the option moneyness,e.g. the results related in table 8. But for CS model, the significance of the estimated coefficient for moneyness and its positive sign (1.08) show that the model corrects in excess the overfitting problem of BS model. The R-square goes in the same way, because it is very small (0.02) meaning that the pricing errors have little structure in terms of moneyness. These results imply that the flexibility of the Gram-Charlier in adjusting the real density function of the underlying performs the results. The problem is that the CS percentage pricing errors will on average be 1.08 points higher when the moneyness increases by one point. It holds that both JR and MIX percentage pricing errors will not be affected for these increments of the moneyness. In contrast BS percentage pricing errors will on average be 3.27 points lower. The interest rate and the expiration time estimated coefficients are not significant for all models, meaning that the pricing errors for call options do not present an expiration-day or a risk-free interest bias. For put options the results are quite different. First, for BS model all the coefficients are statistically significant atfive percent and the explanatory power of the variables is quite high with a R-square of 0.62. Which shows us the misspricing of BS model for Ibex-35 put options from 1996 to 1998. What is important to stand out here is that the coefficient of moneyness is -4.69, bigger in absolute value than the one for call options, -3.28.
The results are a surprise for MIX percentage pricing errors. All their coeffi -cients estimated are significative although the R-square is small. This means that all the variables explain only two percent of the pricing errors,.e.g. if the free-risk interest rate rises one point, the MIX percentage pricing errors will on average be 0.97 points higher. The results for CS model are similar to those of call options. The JR percentage pricing errors are the ones that do not have structure in terms of this variables. These results coincide with the results of section 4 where JR model did the best out-of-sample fit.
6
Volatility Smile
A good diagnosis of the relative model misspecification is to compare the implied-volatility patterns of each model across moneyness or both moneyness and matu-rity. In this analysis, I used a subsample data from 1 April 1996 to 30 September 199614. On the one hand, I included in the BS formula the values of the future
index, the interest rate, the time to maturity and the strike price for each option contract of the subsample, to obtain the BS implied-volatility. Then, I obtained the implied-volatility by equating the observed option price with the pricing for-mula. On the other hand, to compute both CS and JR implied-volatilities I included the same values of contract characteristics in each formula and it was also needed to include the implicit parameters of skewness and excess kurtosis which had been calculated with all the options for each day. So, the volatility is the only parameter to be determined. Then, for each given call option, I obtained the volatility of each model equating the observed price of the option contract with the pricing formula. Finally, after repeating these steps for all options in the subsample and for each model, I obtained an average implied-volatility value for each moneyness-maturity category. These estimates are shown in Figure 4 for BS, CS and JR models15, respectively. After that, I divided the subsample
ac-14Due to changes of volatility over time, I chose the average implied-volatility for a relatively short period of time where the volatility varies in a small range, rather than for the entire sample. Results choosing another subperiod are similar.
15I did not found a representative estimation of the implied-volatility for the MIX model to compare with other models.
cording to the days-of-expiration of the option contracts, classifying in short-term (less than 21 days to maturity) and long-term (more than 20 days to maturity). Figure 5 and 6 display the results for short-term and long-term, respectively.
In figure 4, the implied-volatility of BS model presents a similar form to a smile named commonly “smirk”, indicating a misspecification of the model. In contrast, CS and JR models correct this bias and they obtain implied-volatilities values showing nearly a flat line along s subset of moneyness values.
In terms of the expiration days (see figure 5), the short-term implied-volatility values show an U-shaped moneyness-related bias stronger than long-term values, for all the models.
Summing up, it is important to mention that CS and JR patterns are quite closed, on a maturity-by-maturity basis. And, in more detail, JR model makes a better fit than CS model when the moneyness is bigger than 1.01516. Another
thing to say is that when the call options arein-the-money or the put options are
out-of-the-money BS model estimates the biggest volatilities, and at-the-money
options the lowest.
7
Static Hedging Performance
Table 9 and 10 show the IBEX-35 option contracts needed to delta-hedge one million euro stock portfolio with a beta of one. This illustrates how a hedging strategy based on alternative models might differ. I presented two cases. Thefirst one corresponds to a specific day, 8 August 1996. The number of contracts needed was obtained with the implied parameters for this day for the four models, and are presented in Table 9. The second one corresponds to the whole sample. In this case, the number of contracts needed was obtained with an average of the implied parameters of the entire sample for each model, and are showed in Table 10. The results are calculated for a contract with the following characteristics: a future index level of F = 6500 euros, a free-risk interest rate of r = 5%, at three weeks
16Remember that K/F>1.015 implies out-of-the-money call options and in-the-money put options.
to expiration and with an exercise price varying between 6000 to 6900 euros in increments of 100. This range of exercise prices allows to analyze the needed delta-hedge for in-the-money asout-of-the-money options. For all cases, the number of contracts required are computed as follows
N = P
∆
where N is the number of contracts, P is the value of the stock portfolio and ∆
is the option delta for the different models. The option delta for the four models are calculated as follows17.
Under BS model ∆BSC =e−rτN(d) where d= ln(Ft/K) + (σ 2 τ/2) στ
with N(.) as the standard cumulative normal distribution function. Under CS and JR models
∆CSC =e−rτ{N(d∗) + φ(d∗) (1 +w)[ w στ +k3 1 3!P6(d ∗) +k 4 1 4!P7(d ∗)]} with P6(y) =y2−3yστ+ 2σ2τ−1 P7(y) =−y3+ 4y2στ + 3y(1−2σ2τ) + 3σ 3 τ −4στ and d∗ = [ln(Ft/K) + 1 2σ 2 τ −ln(1 +w)] στ =d−ln(1 +w) στ w= k3 3!σ 3 τ+ k4 4!σ 4 τ.
Under MIX model
∆M IXC =e−rτ{α∆BSC (σ1) + (1−α)∆BSC (σ2)}
17It is presented the formulae for delta call options. The deltas for put options can be obtained by applying the Put-Call Parity. The results of table 9 and 10 are for call options.
where the parameter 0≤α ≤1 is the mixing parameter for weighting and∆BSC (σi)
is the delta call option price under Black-Scholes model for the lognormal distri-bution i.
Columns 2 and 7 in both tables list the number of contract options needed to make a delta-hedge under BS model. Columns 3 and 8 report the number of contracts needed to do a delta-hedge based on the CS model, columns 4 and 9 list the number of contracts needed under JR model and the columns 5 and 10 report the number of contracts needed to do a delta-hedge based on the MIX model.
Table 9 shows that for in-the-money call options are necessary more contracts under BS model than under MIX model, while if the options are out-of-money
this relation changes. An inverse relation is found between BS model and CS or JR models. It is needed more contracts to delta-hedge the portfolio for out-of-the-money call options under CS or JR models than BS model. On the contrary, it is needed few contracts for in-the-money call options. These results are a particular case of the sample.
For a more general analysis Table 10 is presented. There, it can be seen that in the case of call options, for any moneyness, it is needed more contracts for delta-hedge a portfolio under BS model that under CS or JR models. The results obtained by Corrado & Su (1996) or by Jurczenkoet al. (2002) between BS model and CS model are similar to the relation obtained in this table between BS model and MIX model.
In conclusion, to make a delta-hedge for one million euro portfolio with beta one, of the future index of the IBEX-35, it is needed a lower number of contracts under JR model than under BS or MIX models.
8
Conclusion
The in-sample pricing fit, for the three alternative models selected above, per-forms the results obtained with Black-Scholes model. The relative pricing errors average (in absolute values) for call options of the alternative models are more than a 50 percent lower than Black-Scholes model. For put options the results are still better, because the relative pricing errors average (in absolute values) of the alternative models are more than a 60 percent lower than those for Black-Scholes model ones.
Pricing options under Corrado-Su model with the restriction for skewness and excess kurtosis values that guarantee the positivity of the risk-neutral density function (the Jondeau-Rockinger model) does not worsen the quality of the fit. Moreover, sometimes the results are better.
The Mixture Log-normal model is a good alternative to Corrado-Su model. Due to itsflexibility, the model can capture the particular conditions of the market structures. The results obtained with this model are as good as the Corrado-Su model. But sometimes the results are better.
These models have two important sections that deserves to be highlighted. On the one hand, they solve the overfitting problems of the Black-Scholes model in out-of-the-money options. On the other hand, they maintain the quality of the BS model good results.
References
[1] Andersson, M. and M. Lomakka (2001), “Evaluating implied RNDs by some new confidence interval estimation techniques”, Stockholm School of Eco-nomics, Working paper.
[2] Bates, D. (1996), “Jumps and Stochastic Volatility: Exchange Rate Process Implicit in Deutsche Mark Options”, Review of Financial Studies 9 (1), 69-107.
[3] Bates, D. (2000), “Post-’87 Crash Fears in the S&P 500 Futures Option Market”, Journal of Econometrics 94, 181-238.
[4] Black, F. and M. Scholes (1973), “The Pricing of Options and Corporate Liabilities”, Journal of Political Economy 81, 637-659.
[5] Brown, C. and D. Robinson (2002), “Skewness and Kurtosis Implied by Op-tion Prices: A CorrecOp-tion”, Journal of Financial Research, forthcoming, 9 pages.
[6] Chesney, M. and L. O. Scott (1989), “Pricing European Currency Options: A Comparison of the Modified Black-Scholes Model and a Random Variance Model”, Journal of Financial and Quantitative Analysis 24, 267-284.
[7] Corrado, C. and T. Su (1996), “Skewness and Kurtosis in S&P 500 Index Returns Implied by Option Prices”, Journal of Financial Research 19 (2), 175-192.
[8] Derman, E. and I. Kani (1994), “Riding on a Smile”,Risk 7 (2), 32-39. [9] Derman, E., I. Kani and N. Chriss (1996), “Implied Trinomial Trees of the
Volatility Smile”, Journal of Derivatives 3 (4), 7-22.
[10] Dupire, B. (1994), “Pricing with a Smile”, Risk 7 (1), 18-20.
[11] Heston, S. (1993), “A Closed-form Solution for Option with Stochastic Volatility with Applications to Bond and Currency Options”, Review of
Fi-[12] Hull, J. and A. White (1987), “The Pricing of Option with Stochastic Volatil-ities”, Journal of Finance 42, 281-300.
[13] Jackwerth, J. (1997) “Generalized Binomial Trees”,Journal of Derivatives 5 (2), 7-17.
[14] Jondeau, E. and M. Rockinger (2001), “Gram-Charlier Densities”, Journal of Economics Dynamics and Control 25, 1457-1483.
[15] Jurczenko, E., B. Maillet and B. Negrea (2002),“Skewness and Kurtosis Im-plied by Option Prices: A Second Comment”, Universit´e de Paris I Panth´eon-Sobornne, Working Paper, 32 pages.
[16] Jurczenko, E., B. Maillet and B. Negrea (2002), “Multi-moment Approximate Option Pricing Models: A General Comparison (Part 1)”, Universit´e de Paris I Panth´eon-Sobornne, Working Paper, 57 pages.
[17] Kendall, M. and A. Stuart (1977), “The Advanced Theory of Statistics”, Fourth Edition, Vol. 1, Macmillan Publishing Company, New York, 472 pages.
[18] Le´on, A., J. Menc´ia and E. Sentana (2004), “Parametric Properties of Semi-nonparametric Distributions, with Applications to Option Valuation”, Work-ing Paper.
[19] Longstaff, F. (1995), “Option Pricing and the Martingale Restriction”, Re-view of Financial Studies 8 (4), 1091-1124.
[20] Melick, W. and C. Thomas (1997), “Recovering an Asset’s Implied PDF from Options Prices: An Application to Crude Oil during the Gulf Crisis”,Journal of Financial and Quantitative Analysis 32 (1), 91-115.
[21] Merton, R. (1976), “Option Pricing when the Underlying Stock Returns are Discontinuous”, Journal of Financial Economics 4, 125-144.
[22] Pan, J. (2002), “The Jump-risk Premia Implicit in Options: Evidence from an Integrated Time-series Study”, Journal of Financial Economics 63, 3-50.
[23] Rubinstein, M. (1994), “Implied Binomial Trees”, Journal of Finance 49, 771-818.
[24] Rubinstein, M. (1998), “Edgeworth Binomial Trees”, Journal of Derivatives
5 (3), 20-27.
[25] Stein, E. and J. Stein (1991), “Stock Price Distributions with Stochastic Volatility”, Review of Financial Studies 4, 727-752.
[26] White, H. (1980), “A Heteroscedasticity-Consistent Covariance Matrix Esti-mator and a Direct Test for Heteroskedasticity”, Econometrica, 48, 817-838.
A
Appendix
Proof of the formula in equation (4): the expectation in equation (4) can be written as
EQ[ST] =
Z ∞ 0
STf(ST)dST. (A1)
Then (A1) can be rewritten as EQ[ST] =
Z +∞ −∞
STfz(z)dz (A2)
where it is applied the following transformation
P(ST ≤sT) =P(Stexp(µτ +στz)≤sT) =P(z ≤ ln(ST/sT)−µτ στ ) then fST(ST) = ∂P(ST ≤sT) ∂ST = ∂P(Z ≤z) ∂Z ∂Z ∂ST =fz(z) 1 STστ and dST =STστdz.
Using the martingale restriction (3) the equation A2 can be expressed as
Z +∞ −∞ Stexp(µτ+στz)fz(z)dz =Stexp(rτ) then exp(µτ) Z +∞ −∞ exp(στz)fz(z)dz = exp(rτ) and finally µτ + ln[ Z +∞ −∞ exp(στz)fz(z)dz] =rτ. Q.E.D.
B
Tables and Figures
Table 1
Summary Statistics of the Variables from the Whole Sample
All Moneyness (K/F) Contracts <0.985 0.985-1.015 >1.015 Call Mean 101.02 67.92 119.28 250.18 St. Dev. 97.86 60.88 90.48 180.25 Minimum 1 1 9 74 Maximum 1415 540 490 1415 Number 22299 11608 9247 1444 Put Mean 103.44 340.04 121.76 72.76 St. Dev. 116.89 271.57 98.08 71.73 Minimum 1 75 12 1 Maximum 2700 2700 515 410 Number 15699 790 5518 9391 Strike Price Mean 6,574
St. Dev. 2303 Minimum 3400 Maximum 12500 Underlying Mean 6,568 St. Dev. 2272 Minimum 3620 Maximum 10977 Skewness -1.11 Kurtosis 3.39
The sample period is 1 February 1996 to 10 October 1998. K is the strike price of the option contract. F is the future price of the option contract.
Table 2
In-Sample Pricing Errors
Options Statistics BS CS JR MIX Call ME -4.26 -0.27 -0.70 -0.13 (10.41) (7.48) (7.46) (6.91) MAE 7.11 3.94 3.86 3.52 (8.72) (6.37) (6.42) (5.95) MARE 0.13 0.06 0.06 0.05 (0.20) (0.13) (0.12) (0.12) RMSE 11.25 7.49 7.49 6.91 AIC 11.87 8.79 8.80 8.12 Put ME 7.50 0.31 0.50 0.07 (12.53) (7.16) (7.06) (6.81) MAE 9.56 4.04 3.85 3.67 (11.04) (5.92) (5.93) (5.73) MARE 0.18 0.07 0.06 0.06 (0.22) (0.12) (0.10) (0.11) RMSE 14.60 7.16 7.07 6.81 AIC 15.76 9.00 8.89 8.55
BS is Black-Scholes model, CS is Corrado-Su model, JR is Jondeau Rockinger model and MIX is Normal Mixture model. ME is the mean error that computes the average deviation of the reported call prices from the model’s theoretical values. MAE is the mean absolute error that computes the average absolute deviation of the reported option prices from the model’s theoretical values . MARE is the mean absolute relative error that computes the average absolute relative error defined as the reported option price minus the model’s theoretical values, divided by the reported option price. RMSE is the root mean squared error that computes the square root of the average square deviation of the reported option prices from the model’s theoretical values. AIC, the Akaike Information Criterion, is RMSE multiplied by exp(2*k*N/T), where k is the number of parameters for each model, N is the number of days in the sample, and t is the number of options in the sample. The sample period is 1 February 1996 to 9 October 1998, with a total of 22297 call option prices and 15678 put option prices. In parenthesis, the standard deviations.
Table 3
In-Sample Pricing Errors Call Options by Moneyness
BS CS JR MIX OTM ME -8.17 0.04 -1.09 -0.24 (10.51) (7.65) (7.85) (7.29) MAE 9.15 3.84 4.01 3.59 (9.67) (6.61) (6.84) (6.35) MARE 0.20 0.09 0.08 0.08 (0.25) (0.17) (0.15) (0.17) RMSE 13.31 7.65 7.93 7.29 ATM ME -1.03 -0.51 -0.21 -0.06 (7.25) (6.89) (6.52) (6.06) MAE 4.43 4.00 3.62 3.39 (5.83) (5.63) (5.42) (5.03) MARE 0.04 0.04 0.03 0.03 (0.05) (0.05) (0.04) (0.04) RMSE 7.33 6.91 6.52 6.06 ITM ME 6.49 -1.24 -0.57 0.30 (11.98) (9.42) (9.47) (8.65) MAE 7.89 4.42 4.21 3.75 (11.12) (8.41) (8.5) (7.8) MARE 0.03 0.02 0.02 0.02 (0.03) (0.03) (0.03) (0.02) RMSE 13.63 9.50 9.48 8.65
BS is Black-Scholes model, CS is Corrado-Su model, JR is Jondeau Rockinger model and MIX is Normal Mixture model. OTM are out-of-the-money call options for which K/F≤0.985, ATM are at-the-money call options for which K/F are between 0.985 and 1.015, ITM are in-the-money call options for which K/F ≥1.015. ME is the mean error that computes the average deviation of the reported call prices from the model’s theoretical values. MAE is the mean absolute error that computes the average absolute deviation of the reported call prices from the model’s theoretical values . MARE is the mean absolute relative error that computes the average absolute relative error defined as the reported call price minus the model’s theoretical values, divided by the reported call price. RMSE is the root mean squared error that computes the square root of the average square deviation of the reported call prices from the model’s theoretical values. The sample period is 1 February 1996 to 8 October 1998, with a total of 22297 call option prices, where 11608 are ITM, 9245 are ATM and are 1444 OTM. The standard deviations are in parenthesis.
Table 4
In-Sample Pricing Errors Put Options by Moneyness
BS CS JR MIX ITM ME -7.84 -1.56 -1.86 -1.31 (18.04) (15.36) (15.42) (13.55) MAE 11.09 7.62 7.51 6.77 (16.25) (13.42) (13.59) (11.81) MARE 0.03 0.03 0.02 0.02 (0.05) (0.04) (0.04) (0.04) RMSE 19.66 15.43 15.52 13.60 ATM ME 1.65 0.68 1.05 0.05 (7.02) (6.58) (6.16) (5.95) MAE 4.40 4.00 3.63 3.41 (5.72) (5.26) (5.09) (4.87) MARE 0.04 0.04 0.03 0.03 (0.04) (0.05) (0.04) (0.04) RMSE 7.21 6.61 6.25 5.95 OTM ME 12.25 0.26 0.37 0.20 (11.98) (6.32) (6.34) (6.4) MAE 12.47 3.75 3.67 3.56 (11.75) (5.09) (5.19) (5.32) MARE 0.27 0.09 0.08 0.08 (0.24) (0.15) (0.12) (0.13) RMSE 17.13 6.33 6.35 6.41
BS is Black-Scholes model, CS is Corrado-Su model, JR is Jondeau Rockinger model and MIX is Normal Mixture model. OTM are out-of-the-money put options for which K/F≥1.015, ATM are at-the-money put options for which K/F are between 0.985 and 1.015, ITM are in-the-money put options for which K/F ≤0.985. ME is the mean error that computes the average deviation of the reported put prices from the model’s theoretical values. MAE is the mean absolute error that computes the average absolute deviation of the reported put prices from the model’s theoretical values . MARE is the mean absolute relative error that computes the average absolute relative error defined as the reported put price minus the model’s theoretical values, divided by the reported put price. RMSE is the root mean squared error that computes the square root of the average square deviation of the reported put prices from the model’s theoretical values. The sample period is 1 February 1996 to 8 October 1998, with a total of 15636 put option prices, where 790 are ITM, 5504 are ATM and 9342 are OTM. The standard deviations are in parenthesis.
Table 5
Out-Sample Pricing Errors
Options Statistics BS CS JR MIX Call ME -4.42 -0.25 -0.83 -0.79 (14.84) (14.12) (13.4) (13.65) MAE 9.82 8.37 7.90 8.17 (11.97) (11.38) (10.85) (10.96) MARE 0.16 0.12 0.11 0.11 (0.23) (0.21) (0.17) (0.16) RMSE 15.48 14.13 13.42 13.67 AIC 16.33 16.59 15.76 16.05 Put ME 9.42 2.06 2.01 0.41 (17.2) (15.33) (14.2) (15.48) MAE 12.31 8.58 8.09 8.88 (15.27) (12.88) (11.84) (12.68) MARE 0.20 0.12 0.10 0.13 (0.22) (0.16) (0.14) (0.21) RMSE 19.61 15.47 14.34 15.48 AIC 21.16 19.44 18.02 19.45
BS, CS, JR and MIX stand for Black-Scholes, Corrado-Su, Jondeau Rockinger and Normal Mixture models. For a given model, it is computed the price of each option using the previous day’s implied parameters. ME is the mean error that computes the average deviation of the reported option prices from the model’s theoretical values. MAE is the mean absolute error that computes the average absolute deviation of the reported option prices from the model’s theoretical values . MARE is the mean absolute relative error that computes the average absolute relative error defined as the reported option price minus the model’s theoretical values, divided by the reported option price. RMSE is the root mean squared error that computes the square root of the average square deviation of the reported option prices from the model’s theoretical values. AIC, the Akaike Information Criterion, is RMSE multiplied by exp(2*k*N/T), where k is the number of parameters for each model, N the number of days in the sample, and t the number of option (call or put) in the sample. The out-of-sample period is 2 February 1996 to 9 October 1998, with a total of 22259 call option prices and 15678 put option prices. In parenthesis, the standard deviations.
Table 6
Out-Sample Pricing Errors Call Options by Moneyness
BS CS JR MIX OTM ME -7.82 0.79 -0.61 -0.15 (15.69) (15.18) (14.37) (14.44) MAE 11.88 9.09 8.69 8.73 (12.88) (12.18) (11.46) (11.5) MARE 0.24 0.17 0.16 0.16 (0.28) (0.27) (0.22) (0.20) RMSE 17.53 15.20 14.38 14.44 ATM ME -1.76 -1.25 -1.00 -1.58 (12.25) (12.58) (12.02) (12.34) MAE 7.41 7.57 7.02 7.48 (9.91) (10.13) (9.81) (9.95) MARE 0.07 0.07 0.06 0.06 (0.07) (0.07) (0.06) (0.07) RMSE 12.37 12.64 12.06 12.44 ITM ME 5.90 -2.10 -1.61 -0.90 (14.71) (14.05) (13.62) (14.74) MAE 8.69 7.80 7.11 8.09 (13.25) (11.87) (11.73) (12.35) MARE 0.03 0.03 0.03 0.03 (0.04) (0.04) (0.04) (0.04) RMSE 15.84 14.20 13.71 14.76
BS is Black-Scholes model, CS is Corrado-Su model, JR is Jondeau Rockinger model and MIX is Normal Mixture model. OTM are out-of-the-money call options for which K/F ≤ 0.985, ATM are at-the-money call options for which K/F are between 0.985 and 1.015, ITM are in-the-money call options for which K/F ≥ 1.015. For a given model, it is computed the price of each option using the previous day’s implied parameters. ME is the mean error that computes the average deviation of the reported call prices from the model’s theoretical values. MAE is the mean absolute error that computes the average absolute deviation of the reported call prices from the model’s theoretical values . MARE is the mean absolute relative error that computes the average absolute relative error defined as the reported call price minus the model’s theoretical values, divided by the reported call price. RMSE is the root mean squared error that computes the square root of the average square deviation of the reported call prices from the model’s theoretical values. The sample period is 1 February 1996 to 8 October 1998, with a total of 22297 call option prices, where 11601 are ITM, 9224 are ATM and 1434 are OTM. The standard deviations are in parenthesis.
Table 7
Out-Sample Pricing Errors Put Options by Moneyness
BS CS JR MIX ITM ME 0.14 6.58 6.00 4.09 (25.69) (26.21) (25.9) (26.35) MAE 14.40 15.27 14.77 15.50 (21.27) (22.29) (22.1) (21.7) MARE 0.04 0.05 0.04 0.05 (0.05) (0.06) (0.06) (0.05) RMSE 25.67 27.00 26.57 26.65 ATM ME 3.44 2.45 2.79 0.42 (14.56) (14.65) (14.19) (14.94) MAE 8.16 8.27 7.81 8.18 (12.54) (12.34) (12.17) (12.51) MARE 0.06 0.07 0.06 0.07 (0.07) (0.07) (0.06) (0.07) RMSE 14.96 14.85 14.46 14.95 OTM ME 13.71 1.46 1.21 0.09 (16.36) (14.39) (12.66) (14.48) MAE 14.56 8.19 7.69 8.73 (15.61) (11.92) (10.13) (11.55) MARE 0.29 0.15 0.14 0.17 (0.24) (0.2) (0.16) (0.26) RMSE 21.35 14.46 12.71 14.48
BS is Black-Scholes model, CS is Corrado-Su model, JR is Jondeau Rockinger model and MIX is Normal Mixture model. OTM are out-of-the-money put options for which K/F ≥ 1.015, ATM are at-the-money put options for which K/F are between 0.985 and 1.015, ITM are in-the-money put options for which K/F ≤ 0.985. For a given model, it is computed the price of each option using the previous day’s implied parameters. ME is the mean error that computes the average deviation of the reported put prices from the model’s theoretical values. MAE is the mean absolute error that computes the average absolute deviation of the reported put prices from the model’s theoretical values . MARE is the mean absolute relative error that computes the average absolute relative error defined as the reported put price minus the model’s theoretical values, divided by the reported put price. RMSE is the root mean squared error that computes the square root of the average square deviation of the reported put prices from the model’s theoretical values. The sample period is 1 February 1996 to 8 October 1998, with a total of 15636 put option prices, where 790 are ITM, 5507 are ATM and 9381 are OTM. The standard deviations are in parenthesis.
Table 8
Regression Analysis of Pricing Errors
Options Statistics BS CS JR MIX Call Constant 3.21 -1.11 0.17 -0.22 (0.00) (0.001) (0.59) (0.48) K/F -3.27 1.08 -0.19 0.19 (0.00) (0.001) (0.53) (0.97) t 0.00 0.00 0.00 0.00 (0.85) (0.76) (0.6) (0.84) r 0.44 0.16 0.31 0.11 (0.44) (0.79) (0.56) (0.43) R-square 0.19 0.02 0.00 0.00 Put Constant 4.75 0.59 0.15 0.67 (0.00) (0.02) (0.49) (0.01) K/F -4.69 -0.61 -0.16 -0.80 (0.00) (0.01) (0.44) (0.00) t 0.00 0.00 0.00 0.00 (0.00) (0.58) (0.71) (0.01) r 0.74 0.15 0.40 0.97 (0.04) (0.79) (0.3) (0.03) R-square 0.63 0.02 0.00 0.03
The above regression results are based on the equation
εi(t) =α0+α1KFi((tt)) +α2τi +α3r(t) +ζi(t) whereεi is the percentage pricing error
of the i-th option date-t, K/F and τ respectively represent the moneyness and the term to-expiration of the option contract; and r is the free-interest-rate on date-t. The p-value, reported in parenthesis are white’s (1980) heteroskedasticity consistent estimator. The percentage pricing errors are obtained using the parameters implied by all of the previous day’s options. The sample period is February 1996-October 1998 for a total of 22259 call options and 15678 put options. BS, CS, JR and MIX, respectively stands for the Black-Scholes model, the Corrado-Su model, the Jondeau Rockinger model and the Normal Mixture model.
Table 9
Number of Option Contracts Needed to Delta Hedge a one million euros Stock Portfolio at August 3, 1996
In-the-Money Options Out-of-the-Money Options
Strike BS CS JR MIX Strike BS CS JR MIX 6000 192 174 187 198 6500 296 281 283 294 6100 204 182 197 210 6600 335 338 323 328 6200 220 195 211 226 6700 385 419 375 369 6300 240 214 229 244 6800 448 538 442 420 6400 265 242 253 267 6900 528 714 531 482
Number of contracts of the future option on IBEX-35 needed to delta-hedge one million euro stock portfolio in 3 August 1996. It is assumed the following: future index F=6500 euros, free-risk interest rate r=5%, and three weeks to maturity. Black-Scholes volatility is 20%, Corrado-Su volatility is 17%, skewness sk=-0.58, kurtosis k=5.09; Jondeau-Rockinger volatility is 21%, skewness sk=-0.4, kurtosis k=4; and Mixture Lognormal volatilities are sigma1=0.23, sigma2=0.32 and a weight of 0.99.
Table 10
Number of Option Contracts Needed to Delta Hedge a one million euros Stock Portfolio for the whole period
In-the-Money Options Out-of-the-Money Options
Strike BS CS JR MIX Strike BS CS JR MIX 6000 201 193 192 198 6500 294 272 273 294 6100 213 202 201 210 6600 325 301 303 328 6200 228 215 214 226 6700 363 337 341 369 6300 246 230 229 244 6800 409 382 388 420 6400 268 248 249 267 6900 464 439 448 482
Number of contracts of the future option on IBEX-35 needed to delta-hedge one million euro stock portfolio in 3 August 1996. It is assumed the following: future index F=6500 euros, free-risk interest rate r=5%, and three weeks to maturity. Black-Scholes volatility is 24%, Corrado-Su volatility is 26%, skewness sk=-0.69, kurtosis k=3.33; Jondeau-Rockinger volatility is 26%, skewness sk=-0.67, kurtosis k=4.22; and Mixture Lognormal volatilities are sigma1=0.22, sigma2=0.21 and a weight of 0.55.
-1.5 -1 -0.5 0 0.5 1 1.5 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 Kurtosis SK ewness
Figure 1: Region where f(z) is positive definite for skewness and kurtosis
f(z) > 0 10% 20% 30% 40% 50% 60% 70%
feb-96 abr-96 jun-96 ago-96 oct-96 dic-96 feb-97 abr-97 jun-97 ago-97 oct-97 dic-97 feb-98 abr-98 jun-98 ago-98 oct-98
BS CS JR
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 Kurtosis Skewness
Corrado & Su Model Positive definite region Jondeau & Rockinger Model
Figure 3: Daily Implied Skewness and Kurtosis of the Future Option on the Ibex-35 (1996-1998)
Figure 4: Implied Volatility (April,1 1996 - September, 30 1996)
0.135 0.14 0.145 0.15 0.155 0.16 0.165 0.17 0.175 0.18 0.185 0.97 1.00 1.03 Moneyness Implied volat ilit y BS JR CS
Figure 5: Days-to-expiration < 21 (April,1 1996 - September, 30 1996) 0.135 0.14 0.145 0.15 0.155 0.16 0.165 0.17 0.175 0.18 0.185 0.97 1.00 1.03 Moneyness Implied volat ilit y BS JR CS
Figure 6: Days-to-expiration > 20 (April,1 1996 - September, 30 1996)
0.135 0.14 0.145 0.15 0.155 0.16 0.165 0.17 0.175 0.18 0.185 0.97 1.00 1.03 Moneyness Implied volat ilit y BS JR CS
CEMFI MASTER’S THESES
0101 Tatiana Alonso and Pedro L. Marín: “Cooperación en I+D: el papel de los centros públicos de investigación”.
0102 Pedro Rey: “Research incentives in competing markets: A model of the development of new vaccines”.
0103 Daniel Santabárbara: “Discriminación de precios y entrada en el mercado de las telecomunicaciones.”
0104 Fernando Navarrete: “Supply and demand shocks and the new Phillips curve”. 0201 Olga Pascual: “Riesgo soberano: ¿Existe?”.
0202 Mercedes Morris: “Conditional skewness in multifactor asset pricing models: An application to the Spanish stock market”.
0203 Jordi Soy: “Diversificación sectorial y geográfica”.
0204 Esther Moral: “Modelos dinámicos para datos de panel censurados: movilidad salarial en España en los años 80”.
0205 María Oroz: “Márgenes y ciclo en la industria manufacturera española”. 0206 Valentín Bote: “Sorting, job contacts and inequality”.
0207 Pedro del Río: “Institutions and the direction of technological progress: an analysis for OECD countries”.
0208 José Antonio Caracena: “Un procedimiento completo para la detección de estacionalidad en series económicas”.
0301 Gabriel Jiménez Zambrano: “Modified maximum likelihood estimation of Tobit models with fixed effects: Theory and application to earnings equation”.
0302 Gema Zamarro: “Evaluación de políticas educativas: un análisis microeconométrico”.
0303 Abel Elizalde: “Credit default swap valuation: Application to Spanish firms”. 0304 Ramón Adalid: “Can we trust wealth to predict stock returns?”.
0305 Eva del Barrio: “Factores explicativos de los tipos de interés de mercado: análisis del caso español”.
0306 Carmen Martínez Carrascal: “Análisis del sector aéreo europeo: Modelización de la demanda y análisis del contacto multimercado entre compañías”.
0307 Sergio Gavilá Alcalá: “Technology shocks, nominal rigidities and real wages”. 0308 Esther Espeja: “Valoración de contratos forward de la electricidad”.
0309 Carlos Thomas: “Budget deficits in a fiscal federation: The role of central government commitment”.
0401 Alfredo Martín: “Supervivencia de las cajas de ahorros: ¿Territorialidad o eficiencia?”.
0402 Raquel Lago: “Estimating business cycle effects on default probabilities and ratings migrations”.
0403 Jacinto Marabel: “A microeconometric analysis of investment decision with financial constraints”.
0404 Ester Eusamio: “El diferencial de tasas de paro de hombres y mujeres en España (1994-1998)”.
0406 Meritxell Soler: “Banking plus currency crises: Are twins longer than singles?”. 0407 Elena López Dehesa: “Estimation of dynamic models for incomplete
micropanels. An application to production functions”.
0408 Laura Hospido Quintana: “Movilidad laboral y salarios de los jóvenes en España. Diferencias por sexos”.
0501 Andreu Castro Ortega: “Externalidades geográficas: Un análisis empírico a partir de citas de patentes”.
0502 Marta Piñol: “Testing for speculative bubbles in the Spanish housing market”. 0503 Pilar Castrillo: “La externalización como instrumento de gestión pública.
Aplicación a los servicio generales de los hospitales españoles”.
0504 Iban Hidalgo: “Gasto de las familias en educación básica y elección entre colegio público y privado: Un análisis empírico”.
