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OPTION PRICING WITH A
NON-ZERO LOWER BOUND
ON STOCK PRICE
MING DONG
Black, F. and Scholes, M. (1973) assume a geometric Brownian motion forstock prices and therefore a normal distribution for stock returns. In thisarticle a simple alternative model to Black and Scholes (1973) is presentedby assuming a non-zero lower bound on stock prices. The proposed stockprice dynamics simultaneously accommodate skewness and excess kurto-sis in stock returns. The feasibility of the proposed model is assessed bysimulation and maximum likelihood estimation of the return probabilitydensity. The proposed model is easily applicable to existing option pricingmodels and may provide improved precision in option pricing. © 2005Wiley Periodicals, Inc. Jrl Fut Mark 25:775–794, 2005
INTRODUCTION
The classic option pricing work of Black and Scholes (1973) assumes ageometric Brownian motion process of stock prices and consequently anormal distribution of stock returns. While the normality assumption on
I thank the editor Robert Webb, an anonymous referee, Zhiwu Chen, Craig Doidge, Al Goss, DavidHirshleifer, Andrew Karolyi, Moshe Milevsky, Christo Pirinsky, Gordon Roberts, Yisong Tian,Qinghai Wang, Ingrid Werner, and seminar participants at Ohio State University and FinancialManagement Association meetings in Seattle, for valuable comments and suggestions. All remainingerrors are my responsibility alone.For correspondence, Schulich School of Business, York University, 4700 Keele Street, Toronto,Ontario, M3J 1P3 Canada; e-mail: [email protected]
Received November 2002; Accepted October 2004
� Ming Dong is an Assistant Professor of Finance at Schulich School of Business of YorkUniversity in Toronto, Ontario, Canada.
The Journal of Futures Markets, Vol. 25, No. 8, 775–794 (2005) © 2005 Wiley Periodicals, Inc.Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/fut.20159
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asset returns is a fairly good approximation to the real world, it is at oddswith true stock return behavior. For example, Fama (1976) documentsthat U.S. stock returns possess excess kurtosis and are, in general, slightlypositively skewed with respect to the normal distribution. Black (1975),Cox and Ross (1976), Cox (1996), and Beckers (1980) document aninverse relationship between the variance of stock returns and the level ofstock prices. Lo and MacKinlay (1988) show that U.S. stock marketprices do not follow random walks.
A direct consequence of the normality assumption to option pric-ing is that the Black–Scholes type of formula suffers the “volatilitysmile” problem, which means that the Black–Scholes formula systemat-ically misprices deep-out-of-the-money and deep-in-the-money stockoptions. Bakshi, Cao, and Chen (1997) empirically test the pricing per-formance of S&P 500 index options and conclude that “as the smileevidence is indicative of negatively-skewed implicit return distributionswith excess kurtosis, a better model must be based on a distributionalassumption that allows for negative skewness and excess kurtosis.”(p. 2014) As such, numerous alternatives to the Black-Scholes analysishave emerged in the literature, including the stochastic volatility mod-els (SV) of Hull and White (1987), Scott (1987, 1997) and Wiggins(1987) and stochastic volatility with jumps models (SVJ) of Bates(1996) and Scott (1997). However, as Bakshi et al. (1997) show, allthese stochastic volatility models are significantly misspecified. Inparticular, “to rationalize the negative skewness and excess kurtosisimplicit in option prices, each model with stochastic volatility requiredhighly implausible levels of volatility-return correlation and volatilityvariation.” (p. 2043)
The purpose of this work is to propose a simple alternative to theBlack–Scholes (1973) stock price dynamics and therefore a new optionpricing formula. The new model differs from the Black–Scholes model inthat the underlying stock price follows a “displaced” geometric Brownianmotion with a non-zero lower bound (either positive or negative). Thebenefit of the new stock dynamics is that the resulting stock returns pos-sess both the non-zero skewness and excess kurtosis observed in reality.Although the natural lower bound of any stock price is zero because oflimited liability of shareholders, from the modeling perspective, if weassume a geometric Brownian motion for the stock price movement,then relaxing the zero lower bound constraint generates a return distri-bution that is more consistent with the real world. Consequently, abetter modeled return distribution should imply a higher precision of
Option Pricing With a Non-Zero Lower Bound 777
option pricing.1 The model proposed here has the lower bound on stockprice as the only new parameter, and includes the Black–Scholes modelas a special case when this lower bound is set to zero.
It turns out that if the lower bound on stock prices is positive(negative), then stock returns will present positive (negative) skewness. Theintuition for positive skewness in the presence of a positive lower bound isthat the expected stock return is relatively higher when the stock price ishigher, and lower when the stock price is lower. In the case of negative lowerbound, the expected stock return is higher (lower) when the price is lower(higher) and therefore returns present negative skewness. Furthermore,stock returns possess excess kurtosis when the lower bound sufficientlydeviates from zero, or when we allow the lower bound to change over time.
The feasibility of the proposed stock price dynamics with non-zerolower bound is assessed by simulation and maximum likelihood estima-tion of the return probability density. First, it is shown through a largesample of simulated returns that the non-zero bound price dynamicscan yield significant skewness (positive or negative, depending on thesign of the lower bound) and excess kurtosis in the returns. Next, theprobability density function of stock returns is derived under the non-zero lower bound price dynamics. Maximum likelihood estimation(MLE) for the parameters of the density function is conducted on dailyreturns of the S&P 500 index. It is found that the MLE parameter esti-mates under the new price dynamics differ from those under theBlack–Scholes price dynamics, and the implied stock price lower boundis substantially different from zero.
An important advantage of the proposed price dynamics is that it isstraightforward to derive new option pricing formulas from existing mod-els. We only need to adjust for the underlying stock price and the optionstrike price to account for the non-zero lower bound of the stock price.Therefore, analytical solutions for option prices can be obtained immedi-ately from existing models, with one new parameter for the implied lowerbound of stock price. The option prices under the non-zero lower boundmodel can significantly deviate from the existing model prices. Option
1Economically, the motivation for a non-zero lower bound on stock price should, in principle, be con-sistent with the factors underlying the actual non-zero skewness and excess kurtosis of stock returns.For instance, in one period, returns can possess negative skewness due to managers’ reluctance torelease bad financial information (which causes large negative returns), while in another period,returns can be positively skewed due to unexpected technical innovations (which cause large positivereturns). However, for option pricing purposes, it is not necessary to identify the exact economicforces responsible for the non-zero lower bound, because the lower bound is a parameter to bebacked out from market price data. See also the subsection Non-Zero Lower Bound on Stock Price.
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prices are higher when the lower bound of stock prices is lower, ceterisparibus. The new parameter for the lower bound on stock price can beestimated together with the other ones from observed option prices andcan potentially reduce misspecifictions of the existing option pricingmodels. The empirical application of the new option pricing formula canbe utilized and tested along the line of Bakshi et al. (1997).
There have been quite a few authors that challenge the normalityassumption of asset returns in option pricing. For example, Jarrow andRudd (1982), Corrado and Su (1996, 1997) use serial expansion of theunderlying price or return distribution to price options. Savickas (2002)propose alternative return distribution to the normal distribution. Ait-Sahalia and Lo (1998) use nonparametric techniques to estimate state-price densities in asset pricing.
Several authors propose alternative price dynamics to theBlack–Scholes model. Diffusion models that extend Black–Scholesinclude Cox and Ross (1976), Cox (1996), Rubinstein (1983), and Kimand Yu (1996). In particular, Rubinstein (1983) decomposes the valueof a firm into riskless and risky components and generalizes theBlack–Scholes (1973) analysis. In fact, his analysis can be viewed as pro-viding an economic story for a positive (but not negative) lower bound ofthe stock price. In addition to the diffusion models, Heston (1993) andPopova and Ritchken (1998) propose jump price dynamics as an alterna-tive to the Brownian motion to accommodate skewness and kurtosis.However, it should be pointed out that none of the diffusion models inthe literature directly addresses the skewness and excess kurtosis instock returns without involving a jump process for stock price.
In the next section the alternative stock price dynamics are presentedand the return distribution and option pricing formula under the newdynamics are derived. A comparison of Rubinstein (1983) and this studyis also given in the following section. This is followed by a discussion ofthe feasibility of the proposed model through simulations and maximumlikelihood estimations on the stock returns. A conclusion is given in thelast section.
AN ALTERNATIVE TO THEBLACK–SCHOLES MODEL
In the original Black–Scholes model, the time-t stock price St is assumedto follow a geometric Brownian motion:
(1)dSt
St� a dt � s dvt
Option Pricing With a Non-Zero Lower Bound 779
where is the standard Brownian motion process and the instantaneousexpected return and instantaneous variance are constant. Under thisassumption, the stock price is lognormally distributed, with
(2)
The lower bound of the stock price under the assumed process is there-fore zero, which is consistent with limited liability of equity. The contin-uously compounded rate of return (per annum) r from time 0 to t isnormally distributed:
(3)
Non-Zero Lower Bound on Stock Price
To accommodate the skewness and excess kurtosis observed in the actualreturn data, consider the following transformation:
(4)
and
(5)
We will refer to as the displaced price. Figure 1 illustrates the relationbetween the stock price process and the displaced price process with apositive or negative lower bound on stock price. Under the above trans-formation, the displaced price , rather than the stock price , follows ageometric Brownian motion. In effect, this transformation moves thelower bound of the stock price to a new level b, which can be positive ornegative. Therefore, follows a shifted lognormal distribution (see, e.g.,Aitchison & Brown, 1966). As will be demonstrated below, assuming ashifting, non-zero lower bound of stock price can accommodate theskewness and excess kurtosis observed in stock returns.
The stock price dynamics can be rewritten as
(6)
Intuitively, if , the expected stock return is higher at higher stockprice level and vice versa. This is what is to be expected from a posi-tively skewed return distribution:2 There is a greater probability of higher
St
b � 0
dSt
St� a1 �
bStb (a dt � s dvt)
St
StXt
Xt
dXt
Xt� a dt � s dvt
Xt � St � b
r �N ca �12s2, s2�t d
St � S0e(a�1
2s2)t�svt
sa
vt
2Negative skewness means there is a substantial probability of a large negative return. Positive skew-ness means that there is a greater than normal probability of a large positive return.
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�1 �0.8 �0.6 �0.4 �0.2 0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
A. Probability Density Functions. � = 0.17, � = 0.2, � = (b/S0) = 0.3, t = 1
New pdfNormal pdf
�1 �0.8 �0.6 �0.4 �0.2 0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
2.5
f(r)
f(r)
Continuously Compounded Return r
B. Probability Density Functions. � = 0.17, � = 0.2, � = (b/S0) = �0.4, t = 1
New pdfNormal pdf
FIGURE 1Probability density functions for the normal distribution and the new distribution. The figure
shows that the non-zero lower bound model of stock price can accommodate skewness instock returns. Returns are positively skewed (A) when the lower bound is positive ( )
and negatively skewed (B) when the lower bound is negative ( ).b � 0b � 0
returns at higher prices. Similarly, if , the expected return is lowerat a higher stock price level , which is characteristic of a negativelyskewed return distribution. This intuition is confirmed in the moreformal analysis below.
Proposition 1: Under assumptions (4) and (5), the probability densityfunction (p.d.f.) of continuously compounded return per annum r is
(7)
where .b � b�S0
f(r) �1
2 2ps2�t
ert
(ert � b) exp• �
c logaert � b
1 � bb � (a � 1
2s2)t d 2
2s2t¶
f(r)
St
b � 0
Option Pricing With a Non-Zero Lower Bound 781
Proof Under assumptions (4) and (5), the stock price at time t is
(8)
The annualized continuously compounded rate of return from time 0 tot is therefore
(9)
where y is normally distributed as Let .Then the probability density function for R is
(10)
The p.d.f. in Equation (7) can be obtained by noticing that and. ❑
This p.d.f. degenerates to that of a normal distribution when .Figure 1 presents the p.d.f. plots of and the standard normal density.3
It illustrates the property that there is a greater than normal probability oflarge negative returns when (characteristic of a negatively skeweddistribution), and there is a greater than normal probability of large posi-tive returns when (characteristic of positive skewness).
To see more directly how the non-zero lower bound price dynamicsaffect the skewness and kurtosis of stock returns, we plot the relation-ships between and (standardized) skewness and kurtosis in Figure 2.Since analytical expression for skewness and kurtosis is not possible, themoments in Figure 2 are computed numerically. The parameters are setat . Results for both the monthly andannual horizons are plotted.(t � 1)
(t � 1�12)a � 0.17, s � 0.2
b
b � 0
b � 0
f(r)b � 0
f(r) � tf(R)R � tr
�eR
(eR � bS0
)
1
2 2ps2t exp µ�
c log aeR � bS0
1� bS0
b � (a � 12s
2)t d 22s2t
∂�
eR
(eR � bS0
)
1
22ps2t exp e�
[y � (a � 12s
2)t]2
2s2tf
f(R) � f(y) ` dRdy` �1
R � r � tN[(a � 12s
2)t, s2t].
r �1t log a St
S0b �
1t log c a1 �
bS0bey �
bS0d
St � (S0 � b)e(a�12s
2)t�svt � b
3To make the new and the normal distributions have comparable (although not identical) levels ofexpected return and variance, adjustments have been made in Figure 1 based on Equation (6).Specifically, and of the new model are the parameter values for the normal case multiplied by afactor of .1�(1 � b)
sa
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�0.8 �0.6 �0.4 �0.2 0 0.2 0.4 0.6 0.8�0.5
0
0.5
Skn
ewne
ss o
f Ret
urn
A. Skewness of Return and the Lower Bound b. � = 0.17, � = 0.2
t = 1t = 1/12
�0.8 �0.6 �0.4 �0.2 0 0.2 0.4 0.6 0.82.8
3
3.2
3.4
3.6
3.8
� (= b/S)
Kur
tosi
s of
Ret
urn
B. Kurtosis of Return and the Lower Bound b. � = 0.17, � = 0.2
t = 1t = 1/12
FIGURE 2The impact of the lower bound on stock price on skewness and kurtosis of returns. The figure
shows how the skewness (A) and kurtosis (B) depend on in the non-zero lower boundmodel of stock price. Returns are positively (negatively) skewed when is positive (negative),
and possess excess kurtosis when deviates significantly from zero.b(�3)b
b
Panel A of Figure 2 shows that skewness of returns are positive(negative) when the lower bound is positive (negative), confirming theintuitive discussion below Equation (6). The effect of on skewnessincreases with the time horizon. Panel B shows that the effect of onkurtosis is rather limited when return is measured over the monthly hori-zon, but kurtosis becomes appreciably larger than that of the normaldistribution (3) when deviates substantially from zero, especially when
is in the negative territory. The effect of the new price dynamics onkurtosis when the lower bound is stochastic is discussed in theSimulation of Returns subsection.
Two remarks are in order. First, the non-zero lower bound stockprice dynamics in Equation (6) hold strictly when the lower bound onprice b is positive. When , there is a theoretical probability,b � 0
b
b
b
b
Option Pricing With a Non-Zero Lower Bound 783
4Intuitively, the fact that returns have negative skewness is not because that stock prices may turnnegative. Rather, as discussed above, negative skewness arises from the fact that the expected returnis lower when stock price is high when .b � 0
however small, that stock price will turn negative, which would contra-dict the principle of limited liability of equity. Therefore, the model onlyapplies in the absence of negative stock prices, i.e., when the probabilityof bankruptcy is negligible.
Second, when , returns are negatively skewed. However, it isthe change in probability distribution of returns from the normal distri-bution, rather than the possibility of negative stock prices per se, thataccommodates negative skewness. For example, if we set the modelparameters at , then accord-ing to the p.d.f. given by Equation (7), the probability of negative stockprice is merely . This probability is even smaller when is setcloser to zero. This means that it is not the part of the probability whereprices are negative that generates the negative skewness of returns;rather, it is the entire new return distribution that is responsible for theskewness and kurtosis.4 This also shows that ignoring negative stockprices only has a small impact on the proposed price dynamics for a typ-ical firm. In what follows, we only consider cases when the firm’s bank-ruptcy risk can be ignored.
Option Pricing
Since the non-zero lower bound stock price dynamics change the proba-bility distribution of stock returns, the option pricing formula willchange accordingly. However, it is straightforward to derive the newoption pricing formula from existing models under the non-zero lowerbound dynamics of the underlying asset.
Proposition 2: Under the non-zero lower bound dynamics for theunderlying asset with lower bound b and initial asset price , the formulafor European call option retains the same form as those existing in the lit-erature, except that the strike price K is replaced by and the initialasset price replaced by .
Proof The price of a European call option with strike price K and time-to-maturity T is:
(11) � C(S0 � b, K � b)
� E�[e��T0 ru�
du(XT � (K � b))�]
C(S0, K) � E�[e��T0 ru�
du(ST � K)�]
S0 � bS0
K � b
S0
b8.7 � 10�5
a � 0.17, s � 0.2, t � 1, b � b�S0 � �0.4
b � 0
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5In principle, it is possible to model the lower bound b as a stochastic process and treat it as a statevariable. However, the lower bound is not observable and not a feasible candidate for a state variable.
where is the expectation operator and is the (generally stochastic)risk-free rate, both under the risk-neutral probability measure. Since theonly relevant probability for a call option is the probability of , thecall option formula remains the same as in Equation (11). The formulafor a European put can be obtained simply by making use of put-callparity. ❑
Proposition 2 is a general statement and applies to theBlack–Scholes model as well as models of stochastic interest rates, sto-chastic volatilities and jumps, and so on. In the case of stochastic volatil-ity models, the volatility state variable needs to be adjusted to reflect thenon-zero lower bound, i.e., the volatility of stock returns needs to be con-verted into the volatility of the returns of the displaced prices.5
In the case of the Black–Scholes model, the option pricing formulacorresponding to the original C(S, K, T, r0, s) now becomes
More specifically,
(12)
with
(13)
where is the cumulative standard normal distribution function.To see how option prices change with the lower bound parameter
Figure 3 plots how the European call option price given by Equations (12)and (13) is affected by according to moneyness of the call, holding allother parameters constant. The graphs show that call prices are higherwhen the lower bound is negative, regardless of the moneyness of the call.Call price is linearly related to only if the call is at-the-money. In othercases the slope is steeper when is negative. Intuitively, when the lowerbound of stock price is negative, returns are negatively skewed and theprobability of large negative returns is greater, raising the put optionprice. By put-call parity, the call price is also higher. The reasoning holdsfor all strike prices.
The magnitude of the sensitivity of option price to is also remark-able. For the out-of-the-money case, fixing the parameters at S � 100,
b
b
b
b
b,£
z �log(S � b
K � b) � (r0 � s2�2)T
s2T
C(S, K, T, r0, s) � (S � b)£(z) � e�r0T(K � b)£(z � s2T)
C(S � b, K � b, T, r0, s)
St � K
r~uE�
Option Pricing With a Non-Zero Lower Bound 785
�0.8 �0.6 �0.4 � 0.2 0 0.2 0.4 0.6 0.80
5
10
15C
all P
rice
A. In-the-money Call Price and The Lower Bound b. S0 = 100, K = 80, T = 0.5, � = 0.2, r0 = 0.06
�0.8 �0.6 �0.4 �0.2 0 0.2 0.4 0.6 0.80
5
10
15
Cal
l Pric
e
B. At-the-money Call Price and The Lower Bound b. S0 = 100, K = 100, T = 0.5, � = 0.2, r0 = 0.06
�0.8 �0.6 �0.4 �0.2 0 0.2 0.4 0.6 0.80
2
4
6
Cal
l Pric
e
C. Out-of-the-money Call Price and The Lower Bound b. S0 = 100, K = 120, T = 0.5, � = 0.2, r0 = 0.06
� (= b/S0)
FIGURE 3The relation between the call option price and the lower bound on stock price.
The dotted lines represent the original Black–Scholes prices; the solid lines representthe option prices under the non-zero lower bound price dynamics. The figure shows
that the call option price is negatively related to the lower bond of stock price, regardless of the moneyness of the call. The relationship is linear for at-the-money call (B),
but is convex for both the in-the-money (A) and out-of-the-money (C) calls.
and , a of 0.1 causes a 30%decrease in call price from the the Black–Scholes price.6
The lower bound parameter b (or, equivalently, ) is not observableand needs to be estimated. In principle, b needs to be lower than thestrike price K to ensure that the option pricing formula is well defined
b
br0 � 0.06K � 120, T � 0.5, s � 0.2,
6Figure 3 shows the effect of on call option prices when all other parameters are fixed. As dis-cussed above, when the lower bound on stock price is non-zero, the volatility corresponds to thereturns of the displaced prices and therefore should change from the Black–Scholes case.
s
b
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[see Equation (13)]. However, this condition is almost surely satisfied inpractice because there are no traded options with strike price K very farfrom the stock price S from below. The interest of options quickly dimin-ishes as the strike price gets close to zero (or even half of the stockprice), because such options become more like the underlying stock andthere is no point trading options any more.7
It is well known that the Black–Scholes model underprices in-the-money calls and overprices out-of-the-money calls, resulting in the volatil-ity smile. Since such biases can be attributed to negative skewness in thereturn distribution (see, e.g., Hull, 2002), introducing negative skewness(i.e., with a negative ) should mitigate the volatility smile problem.
Comparison With Rubinstein (1983)
Rubinstein (1983) presents alternative stock price dynamics by decom-posing the firm’s assets into risky and riskless parts. It turns out the stockprice dynamics under his asset decomposition are related to the non-zerolower bound stock price dynamics presented above.
In Rubinstein (1983), the firm’s initial assets includes debt and equity , with a debt–equity ratio of . At time 0, pro-portion of the total value is invested in the risky assets and in risk-less assets. The value of the risky assets follows a lognormal diffusionprocess with instantaneous volatility , while the value of the risklessassets grows at a discrete compounded annual rate of . At time t,the value of the firm is
(14)
where y is normally distributed with standard deviation . Since thedebt is assumed to be riskless, the time-t value of the debt becomes
(15)
and the value of the stock is
(16)� aRS0ey � bRS0
� a(1 � d)S0ey � (1 � a � ad)rtS0
St � Vt � Dt
Dt � D0rt � dS0r
t
sR2t
Vt � [aey � (1 � a)rt]V0
r � 1sR
1 � a
0 a 1dS0
D0V0
b
7Take the S&P 500 call and put options for example. In the first 2 months of 2003, the moneynessratio, , ranges from a minimum of 0.47 to a maximum of 2.08, with a mean of 1.13. There isvirtually no trading for options with below 0.5 (less than 0.036% of all the trading volume of 5.5 million contracts).
K�SK�S
where and . Comparing Equa-tion (16) with Equation (8), we can see the similarities betweenRubinstein (1983) and the alternative price dynamics. In fact, in the spe-cial case of an all-equity firm with , if we ignore the riskless rate ofgrowth (i.e., ), the two specifications are the same, except thatRubinstein (1983) only allows a non-negative lower bound of the stockprice .8
Rubinstein (1983) therefore provides one plausible economic storyfor a positive lower bound of stock price—the lower bound may be viewedas the riskless part of total assets. However, this story does not admit thecase of a negative lower bound, which is actually necessary to accommo-date the empirically observed negative skewness of stock returns fromtime to time. Also, according to this story, the higher the lower bound ofstock prices, the safer the firm’s assets should be. Evidence providedhere does not support such a relationship in reality.9 In fact, the new pricedynamics proposed here are more flexible in the sense that the “displace-ment” of the lower bound is not limited to the riskless assets explana-tion—it can potentially reflect any economic forces underlying the positiveor negative skewness of returns observed in reality (see footnote 1 forexamples). Finally, the option pricing model of Rubinstein (1983) hastwo additional parameters (the riskless part of the assets and thedebt–equity ratio ) compared to the Black–Scholes model, whereas inthe absence of default risk, the model proposed here is more parsimo-nious with only one new parameter (the lower bound of stock price b).
FEASIBILITY OF THE NON-ZERO LOWERBOUND PRICE DYNAMICS
It is shown in the subsection Non-Zero Lower Bound on Stock Price thatthe non-zero-lower-bound price dynamics yield a skewed probabilitydensity function of the returns and accommodates skewness in observedreturns. When the lower bound on stock price substantially deviatesfrom zero, returns also possess excess kurtosis with respect to the normaldistribution. It is also claimed that if we allow for a changing lowerbound of stock prices, then stock returns will present even higher excesskurtosis. In this section, the magnitude of the skewness and kurtosis bysimulation is investigated. A large sample of returns is simulated and the
d
a
(1 � a)S0
r � 1d � 0
bR � (1 � a � ad)rtaR � a(1 � d)
Option Pricing With a Non-Zero Lower Bound 787
8More generally, with positive riskless rate ( ), the lower bound on stock price is time-varying andequals , which represents the growing value of the riskless portion of the firm’s assets.9The implied lower bound appears to switch from positive to negative randomly for the same stock/index.See the subsection Maximum Likelihood Estimation for the MLE estimation of the lower bound.
(1 � a)S0rt
r � 1
skewness and kurtosis computed, with various specifications of the lowerbound. In addition, the feasibility of the non-zero lower bound of stockprice is assessed by fitting the actual return data with the proposedmodel. The maximum likelihood method is used to back out the parame-ters from the S&P 500 index data.
Simulation of Returns
Under the non-zero lower bound model, stock prices follow Equation (6).For initial stock price , the time-t price is given by Equation (8).Therefore, we can simulate a time-series of stock prices according toEquation (8) and then compute a time-series of (annualized) continu-ously compounded returns (CCR) using Equation (9).
Both the monthly ( ) and annual ( ) stock prices aresimulated according to Equation (6). The parameters are set at and , and the initial stock price is set at $50. The returns aregenerated under various specifications of to investigate the effect ofthe lower bound on the return distribution. The first is set at zero asthe starting point of the normal distribution case. Two positive s are setat 0.2 and 0.5, respectively. Two negative s are set at �0.2 and �0.5.To investigate the effect of a random lower bound on kurtosis of thereturns, is allowed to vary uniformly between certain values in fourspecifications. These specifications are (symmetric andcentered around zero), (with positive expected ),
(with negative expected ), and (centeredaround zero with more variation). Here, means the random vari-able is uniformly distributed between values a and b.
For each specification, 1,000 CCRs are generated, and fourmoments (mean, variance, skewness, and kurtosis) of the time-series ofCCRs are calculated. This is one realization of the moments. Thisprocess is repeated 100 times so that the average and standard deviationof each of the moments can be calculated.
Panels A and B of Table I report monthly and annual simulationresults, respectively. The statistical significance of skewness (withrespect to zero) and kurtosis (with respect to 3) is assessed using thet-statistic. For example, the t-statistic of kurtosis is the mean kurtosisminus 3, divided by the standard deviation of kurtosis, multiplied by thesquare root of the number of simulated kurtosis.
All results confirm the discussion in the section An Alternative to theBlack–Scholes Model but illustrate the magnitude of the effect of thelower bound parameter . First, the sign of skewness always agrees withb
b
U[a, b]U[�0.5, 0.5]bU[�0.5, 0.1]
bU[�0.1, 0.5]U[�0.3, 0.3]
b
b
b
b
b
s � 0.2a � 0.17
t � 1t � 1�12
S0
788 Dong
TA
BL
E I
Sim
ulat
ion
of R
etur
ns U
nder
the
Non
-Zer
o L
ower
Bou
nd P
rice
Dyn
amic
s
00.
20.
5�
0.2
�0.
5[�
0.3,
0.3
][�
0.1,
0.5
][�
0.5,
0.1
][�
0.5,
0.5
]
Pane
l A. S
tati
stic
s of
ann
uali
zed
cont
inuo
usly
com
poun
ded
retu
rn (
CC
R)
impl
ied
by m
onth
ly r
etur
ns
Mea
n0.
1486
0.12
140.
0798
0.17
260.
2092
0.14
920.
1226
0.17
600.
1456
SD
Mea
n0.
0223
0.01
780.
0107
0.02
260.
0338
0.02
430.
0185
0.02
320.
0228
Var
ianc
e0.
4779
0.31
060.
1224
0.69
301.
0757
0.49
410.
3198
0.70
180.
5223
SD
varia
nce
0.02
220.
0147
0.00
630.
0282
0.04
660.
0249
0.01
500.
0384
0.02
92
Mea
n sk
ewne
ss0.
0071
0.03
610.
0885
�0.
0380
�0.
0816
0.00
410.
0674
�0.
0317
0.03
83S
Dsk
ewne
ss0.
0743
0.07
620.
0901
0.08
080.
0751
0.08
550.
1029
0.09
850.
1169
t-S
tatis
tica
0.95
4.74
9.82
�4.
70�
10.8
60.
486.
54�
3.22
3.28
Mea
n ku
rtos
is2.
9781
2.98
812.
9861
3.01
373.
0073
3.32
983.
4744
3.23
623.
8199
SD
kurt
osis
0.16
190.
1628
0.17
250.
1480
0.17
010.
2972
0.24
880.
2151
0.32
21t-
Sta
tistic
b�
1.35
�0.
73�
0.80
0.93
0.43
11.1
019
.07
10.9
825
.45
Pane
l B. S
tati
stic
s of
ann
uali
zed
cont
inuo
usly
com
poun
ded
retu
rn (
CC
R)
impl
ied
by a
nnua
l ret
urns
Mea
n0.
1488
0.12
470.
0833
0.17
440.
2057
0.15
040.
1240
0.17
290.
1488
SD
Mea
n0.
0070
0.00
540.
0034
0.00
770.
0091
0.00
640.
0054
0.00
720.
0060
Var
ianc
e0.
0399
0.02
700.
0116
0.05
520.
0818
0.04
150.
0281
0.05
620.
0434
SD
varia
nce
0.00
190.
0015
0.00
050.
0027
0.00
380.
0021
0.00
130.
0022
0.00
23
Mea
n sk
ewne
ss�
0.00
760.
1048
0.27
76�
0.11
29�
0.26
160.
0573
0.20
08�
0.08
460.
1277
SD
skew
ness
0.08
060.
0806
0.08
000.
0876
0.08
250.
0853
0.09
130.
0817
0.12
53t-
Sta
tistic
a�
0.94
13.0
134
.72
�12
.89
�31
.70
6.72
22.0
0�
10.3
610
.19
Mea
n ku
rtos
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0083
2.99
953.
1017
3.05
603.
2194
3.25
453.
3778
3.26
343.
6244
SD
kurt
osis
0.17
650.
1545
0.19
560.
1593
0.24
840.
2883
0.21
100.
2270
0.30
36t-
Sta
tistic
b0.
47�
0.03
5.20
3.52
8.83
8.83
17.9
111
.61
20.5
7
Not
e.F
irst,
mon
thly
(P
anel
A)
or a
nnua
l (P
anel
B)
stoc
k pr
ices
are
sim
ulat
ed u
nder
the
non
-zer
o lo
wer
bou
nd p
rice
dyna
mic
s of
Equ
atio
n (8
). T
hen,
the
ann
ualiz
ed c
ontin
uous
lyco
mpo
unde
d re
turn
s (C
CR
) ar
e ca
lcul
ated
fro
m t
he p
rices
. F
or e
ach
spec
ifica
tion
of t
he p
aram
eter
b(w
hich
is e
qual
to
), t
he d
aily
or
mon
thly
pric
es a
re s
imul
ated
1,0
00 t
imes
,an
dth
e m
ean,
var
ianc
e, s
kew
ness
and
kur
tosi
s of
the
CC
R a
re c
ompu
ted.
Thi
s pr
oces
s is
the
n re
peat
ed 1
00 t
imes
so
that
the
mea
n an
d st
anda
rd d
evia
tion
of t
he f
our
mom
ents
are
com
pute
d. T
his
tabl
e re
port
s th
e m
ean
and
stan
dard
dev
iatio
n (S
D)
of th
e fo
ur m
omen
ts. F
or th
e b
spec
ifica
tion,
[a, b
] mea
ns u
nifo
rmly
dis
trib
uted
ran
dom
var
iabl
e be
twee
n th
e va
lues
aan
d b.
a T
he t-
stat
istic
for
skew
ness
is fo
r te
stin
g w
heth
er th
e m
ean
skew
ness
is d
iffer
ent f
rom
0.
b The
t-st
atis
tic fo
r ku
rtos
is is
for
test
ing
whe
ther
the
mea
n ku
rtos
is is
diff
eren
t fro
m th
e ku
rtos
is o
f the
nor
mal
dis
trib
utio
n (3
).b�S
0
b
790 Dong
the sign of . When , the skewness is not statistically differentfrom zero for both monthly and annual returns. The skewness is signifi-cantly positive (negative) when is positive (negative), with the level ofsignificance increasing with the absolute value of (or expected in thecase of random bounds).
Second, shifting the lower bound on stock price indeed results inexcess kurtosis of stock returns (greater than 3), with the kurtosisincreasing with the level of variation. In the monthly horizon, for differ-ent constant values of , the kurtosis is all not significantly differentfrom the normal value of 3. As takes random values, the kurtosisincreases to approximately 3.3 for a range of variation of 0.6, and 3.8for a range of 1.0. Kurtosis in annual returns behaves similarly, exceptthat it is higher than 3 even when is below �0.2 or above 0.5.
Third, also affects the expected return and variance of the returns,in accordance with Equation (6). Specifically, the expected return andvariance decreases with the value (or expected value) of .
Fourth, the variance of CCR implied from the monthly returns ismuch larger than from the annual returns, regardless of specification.This is a result of continuous compounding of the return under consid-eration. The variance of the CCR is of the order of (seeEquation (3)). In summary, the proposed non-zero-lower-bound stockprice dynamics are capable of generating skewness and excess kurtosisthat are observed in actual stock returns.
Maximum Likelihood Estimation
It is shown in the subsection Option Pricing that the lower boundparameter can significantly affect option prices. In this subsection themagnitude of is estimated from the actual return data to assess theimpact of on option pricing. With negligible probability of bankruptcy,the p.d.f. is given by Equation (7), and it is easy to back out the parame-ters of the p.d.f. ( , and ) using the maximum likelihood estimation(MLE) method.
The daily returns of S&P 500 index in the bull market year of 1997and the bear market year of 2001 are chosen as the data sample forestimation. The S&P 500 index returns do not include dividends and aretaken from the Center for Research in Security Prices (CRSP). For eachparticular month, the parameters are estimated using the daily returns inthat month. The parameters corresponding to the normal distribution case( ) are estimated first for comparison. Those estimates are used as thestarting values for the estimation of the non-zero lower bound model.b � 0
ba, s
b
b
b
s2�t
b
b
b
b
b
b
b
bb
b
b � 0b
Table II presents the result of the MLE estimates. The first thing tonotice is that the MLE estimates of are substantially different fromzero for all months. This means that if we want to impose geometricBrownian motion on the process of stock prices, then it is more reason-able to assume a non-zero lower bound on stock price. Second, the signof always agrees with the skewness of the actual returns, as expected.Furthermore, the value changes vastly from month to month and themagnitude of this parameter is sufficient to make a substantial differ-ence in option prices, given the result described in the Option Pricingsubsection. Finally, the MLE estimates of and also substantiallysa
b
b
b
Option Pricing With a Non-Zero Lower Bound 791
TABLE II
Maximum Likelihood Estimation of Probability Density Function Parameters
Normal distribution Non-zero lower boundmodel distribution
Date Skewness a s a s b
A. The Bull Market of 1997
1997–01 �0.118 0.696 0.120 0.604 0.100 �0.1701997–02 0.074 0.098 0.144 0.175 0.234 0.4011997–03 �0.496 �0.524 0.159 �0.374 0.109 �0.4091997–04 �0.342 0.684 0.185 0.558 0.145 –0.2501997–05 0.088 0.707 0.153 0.782 0.163 0.0831997–06 �0.225 0.534 0.154 0.409 0.114 �0.3191997–07 �0.234 0.884 0.145 0.736 0.115 �0.2331997–08 �0.141 �0.680 0.170 �0.739 0.180 0.0761997–09 1.160 0.652 0.177 1.018 0.257 0.3291997–10 �0.838 �0.279 0.327 �0.187 0.227 �0.4101997–11 0.029 0.614 0.189 0.740 0.219 0.1611997–12 0.374 0.205 0.164 0.347 0.264 0.391
B. The Bear Market of 2001
2001–01 0.890 0.469 0.248 0.649 0.348 0.3102001–02 0.469 �1.253 0.169 �1.371 0.179 0.0812001–03 �0.463 �0.677 0.289 �0.490 0.199 �0.4072001–04 0.267 1.026 0.307 1.257 0.357 0.1632001–05 0.638 0.088 0.174 0.199 0.254 0.3252001–06 0.093 �0.286 0.137 �0.426 0.197 0.3212001–07 �0.076 �0.094 0.191 �0.073 0.141 �0.3232001–08 0.168 �0.702 0.153 �0.760 0.163 0.0812001–09 0.043 �1.311 0.348 �1.424 0.368 0.0812001–10 �0.073 0.233 0.193 0.175 0.143 �0.3222001–11 0.035 0.895 0.156 0.987 0.166 0.0812001–12 0.148 0.118 0.154 0.263 0.254 0.405
Note. The probability density function (p.d.f.) of returns under the non-zero lower bound price dynamics isgiven by Equation (7) in the text. The parameters a, s, and b of the p.d.f. are estimated on the S&P 500 indexfor each month of 1997 and 2001 using the maximum likelihood method. The skewness and parameters for aparticular month are estimated using the daily returns in that month. The normal distribution case correspondsto .b � 0
(b � 0)
differ from the normal distribution parameters. All these results apply toboth the bull and bear markets, although the mean estimate of is posi-tive for the bull market and negative for the bear market.
The MLE estimation here uses only the underlying index data toback out the model parameters. For option pricing purposes, it may bepreferable to back out the parameters from the option prices, along theline of Bakshi et al. (1997). The parameters backed out from the optionsdata would better reflect the valuation standards of the options market.Of course, it is ultimately an empirical question which approach of esti-mation is best for option pricing.
CONCLUSION
Most asset pricing models are based on the assumption that stockreturns are normally distributed. In this article, an alternative stock pricemodel with a non-zero lower bound, and therefore a new option pricingformula is proposed. This model of stock price accommodates the skew-ness and excess kurtosis observed in actual stock returns. By relaxing thezero-bound restriction imposed by the assumption of geometricBrownian motion of stock prices, stock returns present positive (nega-tive) skewness with positive (negative) values of the lower bound, and byallowing the lower bound to shift over time, the returns can presentexcess kurtosis as compared to the normal distribution.
Since the non-zero lower bound changes the probability distributionof stock returns, the option prices change accordingly. It is shown thateven moderate values of the lower bound can substantially affect optionprices. The new parameter for the lower bound can be estimated simul-taneously with the other ones from observed option prices and canpotentially reduce misspecifictions of the existing option pricing models.
The ultimate test of the non-zero lower bound model would be theempirical performance of the model in combination with existing optionpricing models. It would be interesting to compare the performance ofoption pricing models with and without the non-zero-lower-boundassumption. Tests can also be conducted to see whether the impliedparameter values of the stochastic volatility and stochastic interest ratemodels become more reasonable and whether the volatility smile disap-pears or mitigates, once a non-zero lower bound of the underlying assetis allowed. In addition, it would be interesting to compare the effective-ness of different approaches that accommodate skewness and kurtosis ofreturns: the serial expansion approach (as in Jarrow & Rudd, 1982 andCorrado & Su, 1996), the alternative return distribution approach
a
792 Dong
(e.g., Savickas, 2002), the jump price dynamics (Heston, 1993; Popova &Ritchken, 1998), and the non-zero lower bound diffusion modeldescribed in this article. Finally, option pricing with bankruptcy risk con-siderations under the proposed price dynamics is worth exploration.
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Option Pricing With a Non-Zero Lower Bound 793
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