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2015 | OriginalPaper | Chapter

7. Nonparametric Bounds for European Option Prices

Authors : Hsuan-Chu Lin, Ren-Raw Chen, Oded Palmon

Published in: Handbook of Financial Econometrics and Statistics

Publisher: Springer New York

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Abstract

There is much research whose efforts have been devoted to discovering the distributional defects in the Black-Scholes model, which are known to cause severe biases. However, with a free specification for the distribution, one can only find upper and lower bounds for option prices. In this paper, we derive a new nonparametric lower bound and provide an alternative interpretation of Ritchken’s (1985) upper bound to the price of the European option. In a series of numerical examples, our new lower bound is substantially tighter than previous lower bounds. This is prevalent especially for out-of-the-money (OTM) options where the previous lower bounds perform badly. Moreover, we present that our bounds can be derived from histograms which are completely nonparametric in an empirical study. We first construct histograms from realizations of S&P 500 index returns following Chen, Lin, and Palmon (2006); calculate the dollar beta of the option and expected payoffs of the index and the option; and eventually obtain our bounds. We discover violations in our lower bound and show that those violations present arbitrage profits. In particular, our empirical results show that out-of-the-money calls are substantially overpriced (violate the lower bound).

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The only assumption is that both option and its underlying stock are traded securities.

To further explore the research work of Ritchken and Kuo (1989) under the decreasing absolute risk aversion dominance rule, Basso and Pianca (1997) obtain efficient lower and upper option pricing bounds by solving nonlinear optimization problem. Unfortunately, neither model provides enough information of their numerical examples for us to compare our model with. The Ritchken-Kuo model provides no Black-Scholes comparison, and the Basso-Pianca model provides only some partial information on the Black-Scholes model (we find the Black-Scholes model under 0.2 volatility to be 13.2670 and under the 0.4 volatility to be 20.3185, which are different from what are reported in their paper (12.993 and 20.098, respectively)) which is insufficient for us to provide any comparison.

Inspired by Lo (1987), Grundy (1991) derives semi-parametric upper bounds on the moments of the true, other than risk-neutral, distribution of underlying assets and obtains lower bounds by using observed option prices.

Since our paper only provides a nonparametric method on examining European option bounds, our literature review is much limited. For a more complete review and comparison on prior studies of option bounds, please see Chuang et al. (2011).

Christoffersen et al. (2010) provide results for the valuation of European-style contingent claims for a large class of specifications of the underlying asset returns.

Given that our upper bound turns out to be identical to Ritchken’s (1985), we do not compare with those upper bound models that dominate Ritchken (e.g., Huang (2004), Zhang (1994) and De La Pena et al. (2004)). Also, we do not compare our model with those models that require further assumptions to carry out exact results (e.g., Huang (2004) and Frey and Sin (1999)), since it is technically difficult to do.

For the related empirical studies of S&P 500 index options, see Constantinides et al. (2009, 2011).

Without loss of generality and for the ease of exposition, we take non-stochastic interest rates and proceed with the risk-neutral measure $ \widehat{\boldsymbol{\mathbb{P}}} $ for the rest of the paper.

In the Appendix, ε > 0.

Perrakis and Ryan (1984) and Ritchken (1985) obtain the identical upper bound.

This is same as Proposition 3-i (Eq. 7.26) in Ritchken (1985).

By the definition of measure change, we have E _t[C _T S _T] = E _t[C _T]E _t ^(C) [S _T] which implies E _t ^(C) [S _T]/E[S _T] = E _t[C _T S _T]/{E _t[C _T]E _t[S _T]} > 1.

We also compare with the upper bound by Zhang (1994), which is an improved upper bound by Lo (1987), and show overwhelming dominance of our upper bound. The results (comparison to Tables 7.1, 7.2, and 7.3 in Zhang) are available upon request.

Table 7.2

Comparison of upper and lower bounds with the Gotoh and Konno (2002) model

Stk	Lower bound		Blk-Sch	Upper bound
Stk	Our	GK	Blk-Sch	Our	GK
S = 40; rate = 6 %; vol = 0.2; t = 1 week
30	10.0346	10.0346	10.0346	10.1152	10.0349
35	5.0404	5.0404	5.0404	5.1344	5.0428
40	0.4628	0.3425	0.4658	0.5225	0.5771
45	0.0000	0.0000	0.0000	0.0000	0.0027
50	0.0000	0.0000	0.0000	0.0000	0.0003
S = 40; rate = 6 %; vol = 0.8; t = 1 week
30	10.0400	10.0346	10.0401	10.1202	10.1028
35	5.2644	5.0404	5.2663	5.3483	5.4127
40	1.7876	1.2810	1.7916	1.8428	2.2268
45	0.3533	0.0015	0.3548	0.3717	0.5566
50	0.0412	0.0000	0.0419	0.0444	0.1021
S = 40; rate = 6 %; vol = 0.8; t = 12 week
30	11.9661	10.4125	12.0278	12.7229	12.8578
35	8.7345	6.2980	8.8246	9.4774	9.7658
40	6.2141	3.8290	6.3321	6.8984	7.5165
45	4.3432	2.5271	4.4689	4.9421	6.8726
50	2.9948	1.5722	3.1168	3.4990	4.5786

Note: S is the stock price; Stk is the strike price; vol is the volatility; rate is the risk-free rate; Blk-Sch is the Black-Scholes (1973) solution; GK is the Gotoh and Konno (2002) model; Our is our model; $error is error in dollar; %error is error in percentage

Table 7.3

Comparison of upper and lower bounds with the Rodriguez (2003) model

S	Lower bound		Blk-Sch	Upper bound
S	Our	Rodriguez	Blk-Sch	Our	Rodriguez
30	0.0221	0	0.0538	0.1001	0.1806
32	0.0725	0.0000	0.1284	0.2244	0.3793
34	0.1828	0.0171	0.2692	0.4451	0.7090
36	0.3878	0.1158	0.5072	0.7973	1.2044
38	0.7224	0.3598	0.8735	1.3100	1.8900
40	1.2177	0.7965	1.3950	2.0044	2.7767
42	1.8982	1.4521	2.0902	2.8927	3.8619
44	2.7711	2.3329	2.9676	3.9697	5.1315
46	3.8319	3.4286	4.0255	5.2211	6.5640
48	5.0709	4.7177	5.2535	6.6302	8.1337
50	6.4703	6.1724	6.6348	8.1753	9.8149
52	8.0072	7.7635	8.1494	9.8327	11.5835
54	9.6574	9.4631	9.7758	11.5794	13.4187
56	11.3974	11.2462	11.4933	13.3950	15.3032
58	13.2067	13.0922	13.2832	15.2621	17.2235
60	15.0683	14.9845	15.1292	17.1674	19.1693
62	16.9716	16.9101	17.0179	19.1024	21.1328
64	18.9035	18.8593	18.9384	21.0573	23.1086
66	20.8559	20.8250	20.8822	23.0262	25.0926
68	22.8239	22.8020	22.8429	25.0056	27.0822
70	24.8018	24.7867	24.8157	26.9915	29.0754

Note: S is the stock price; Blk-Sch is the Black-Scholes (1973) solution; Rodriguez is the Rodriguez (2003) model; Our is our model; $error is error in dollar; %error is error in percentage

The upper bounds by the Gotoh and Konno model perform well in only in-the-money, short maturity, and low volatility scenarios, and these scenarios are where the option prices are close to their intrinsic values, and hence the percentage errors are small.

The term “dollar beta” is originally from Page 173 of Black (1976). Here we mean β _c and β _ρ.

This is so because the initial volatility is 0.2.

This Black-Scholes case is from the highlighted row in the first panel of Table 7.1.

Table 7.4

Lower bound under the random volatility and random interest rate model

https://static-content.springer.com/image/chp%3A10.1007%2F978-1-4614-7750-1_7/MediaObjects/313729_1_En_7_Fig5_HTML.gif

Note: S is the stock price; BCC/Scott are Bakshi et al. (1997) and Scott (1997) models; Our is our model; $error is error in dollar; %error is error in percentage

The data are used in Bakshi et al. (1997).

The (ex-dividend) S&P 500 index we use is the index that serves as an underlying asset for the option. For option evaluation, realized returns of this index need not be adjusted for dividends unless the timing of the evaluated option contract is correlated with lumpy dividends. Because we use monthly observations, we think that such correlation is not a problem. Furthermore, in any case, this should not affect the comparison of the volatility smile between our model and the Black-Scholes model.

We use three alternative time windows, 2-year, 10-year, and 30-year, to check the robustness of our procedure and results.

The conversion is needed because we use trading-day intervals to identify the appropriate return histograms and calendar-day intervals to calculate the appropriate discount factor.

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Title: Nonparametric Bounds for European Option Prices
Authors: Hsuan-Chu Lin
Ren-Raw Chen
Oded Palmon
Publisher: Springer New York
Book: Handbook of Financial Econometrics and Statistics
Print ISBN: 978-1-4614-7749-5

Electronic ISBN: 978-1-4614-7750-1

Copyright Year: 2015
DOI: https://doi.org/10.1007/978-1-4614-7750-1_7