How much do options usually cost?

How much do options usually cost?

Dr. Igor Halperin on Reinforcement Learning & IRL For Investing & The Dangers of Deep Learning

The illiquidity of the options markets results in temporary deviations of options  prices from their fundamental values. Being able to find arbitrage opportunities gives  investors a chance to exploit the money machine without actually taking any risks  theoretically. A successful options trading which is priced properly consists of a  holistic establishment of upper and lower bound, convexity conditions, put-call parity  conditions, and Black-Scholes model. In this paper, we will examine each factor  holistically, including any elements that might contribute to the fluctuation. The  following work will show how to use real-world data to come up with the results; then  other underlying elements that would help support the results will be discussed and  exploited.  

Literature Reviews  

In the paper Index Options Prices and Stock Market Momentum, Kaushik Amin,  Joshua D. Coval, and H. Nejat Seyhun innovatively checked the probability of  boundary violations and used the χ 2 test, suggesting that boundary condition  violations happen more likely when the stock prices change and are somewhat more  significant for the precast period by using the data of OEX options from 1983 to 1995  [1].

Researchers used the iterate model, where C(σ i) represents the price of the given potion with an implied volatility of σi computed from  the binomial model; C (σ) represents the observed options price; and Vega represents  the partial derivative of the binomial options price with respect to σi [1]. The result  suggested the total implied that declining stock prices would lead to increases in  volatility. Furthermore, after examining the influence of past stock returns on the  volatility smiles (the estimating function of the exercise price with U-shaped implied  volatility), they found the market momentum effect consistently declines with  increasing moneyness but cannot explain the volatility smiles. Hence, writers defined  a new measure of volatility smile and examined the relation in between the past stock  returns and the present stock returns.

After that, overall volatility spread became the  concern, and this differential response was precisely quantified. Result shows that  deep-in-the-money options and deep-out-of-money have the highest price pressure, while at-the-money options appear to have the lowest price pressure. To further  research the potential sources of price pressure, writers analyzed the kurtosis and  skewness, finding that past returns appears not to be a proxy variable for higher  moments of stock return, and the market momentum hypothesis is not rejected even  other factors are controlled and market momentum affects options valuations is not  produced by worries [1].  

In “volatility uncertainty and the cross-section of options returns,” published by  Jie Cao, Aurelio Vasquez, Xiao Xiao and Xintong Zhan, a holistic examination of the  relation between the future delta-hedged equity options returns and uncertainty of  volatility was finished [2]. Results showed that when the volatility is uncertain, the  delta-hedged options returns would decrease consistently. To test the hypothesis, the  authors constructed a delta-hedged options portfolio where volatility changes are the  main factor in the portfolio returns. 

To measure the volatility of the stock, three  different methods became used: Firstly, volatility becomes estimated from an EGARCH model in the timespan of 252 trading days. Secondly, implied volatility from 30 days to  maturity options. Thirdly, intraday realized volatility in the span of 5-minute stock.  The authors found that these three volatilities of volatility (VOV) measures could  predict returns from future options.

To further explain where VOV predictability  comes from, the authors explained that days around earning announcements do not  affect return spread [2]. Second, rather than by any systematic factors, most of the  predictability is driven by VOV, while it is more costly for market makers to hedge  options with the high volatility of idiosyncratic volatility. Third, the authors  decomposed VOV into VOV and VOV-, finding that options sellers prefer VOV+  to VOV-. In the end, the authors explain that when volatility is costly to forecast and  options sellers tend to charge a higher premium for equity options [2].

To further  illustrate the correlation between idiosyncratic volatility of options on stocks and  profit returns, Jie Cao and Bing Han assume that on average, options on low  idiosyncratic volatility stocks earn significantly higher returns than options on high  idiosyncratic volatility stocks [3]. To approve this hypothesis, writers examined a  cross-section of options on individual stocks for every month. It is interesting to see  that for a more meaningful result, before the examination, (the author did) several  variables were controlled which is unexpected. For instance, the author did control for  volatility-related options mispricing, finding that Goyal and Saretto (2009) applied wrong data to estimate stock price [4], but they figure out using cross-sectional distribution will limit the volatility of mispricing [3].

However, our paper is more  focused on the frequency of mispricing and how to apply strategy to future market, Cao, J, and Han, B (2013) didn’t address this question, instead they were trying to  figure out the how the change of idiosyncratic volatility of options affect profit return  [3]. Their study goal is more prone to analyze how the market will affect the volatility  of options on stocks, but this paper is more prone to analyze how individual investor’s  behavior causes the frequency of mispricing. Even though they stated useful  information for our study is that volatility risk premium or volatility-related options  mispricing are not necessary factors of negative cross-sectional relation between  delta-hedged options return and idiosyncratic volatility of the underlying stock [3].  

However, will volatility be mispriced? Goyal and Saretto (2007) found an  economically important source of mispricing in the options implied volatilities [4]. It  is the trader’s conviction that lies behind a volatility trade, i.e., the market expectation  about future volatility that is implied by the options price is not entirely correct.  Because at least an estimate of the parameters that characterize the probability  distribution of future volatility is necessary for all the options pricing models, and  volatility mismeasurement is indeed the most obvious source of future options  mispricing. Underlying stocks are sorted according to the difference between  historical realized volatility and at-the-money implied volatility.

Then portfolios are  constructed using straddles and delta-hedged calls and puts.

The authors suggested  that a zero-cost trading strategy which involves a long position in the portfolio with a  huge positive difference in this historical realized volatility and implied volatility (and  vice versa), results in significant average monthly returns, both economically and  statistically [4]. The results hold regardless of market conditions, amount of risk  attached to the stock, industry groupings, or liquidity of options. Besides, linear factor  models do not explain the results. In this paper, Goyal and Saretto (2007) empirically  investigated this mispricing conjecture that comparing a measure of historical realized  volatility to the market volatility forecast is a possible way to identify whether an  option is mispriced or not, where the key aspects are implied by the options price by  researching the cross-section of equity options in U.S. market [4].  

In Coval and Shumway paper “Expected Options Returns”, authors mainly focused on examining options returns through different asset pricing theories and  backgrounds [5]. The first important thing to understand is the risk-return  characteristics of options returns. The authors considered two main ideas: leverage  effects and curvature of options payoffs.

In particular, using Black-Scholes Model,  they analyzed that call options should earn expected returns which exceed those of the  underlying security, while put options should earn expected returns below that of the  underlying security, and they carefully tested and rejected the hypothesis of “delta neutral options positions should earn the risk-free rate on average” [5].

They built the  test mainly to directly examine the leverage effect on various models. So that they can  use economic terms to check the violations of market efficiency. And therefore  consider the economic payoff for taking particular risks. Experiments indicate that  options return largely conform to most asset pricing implications. But due to  systematic risks, both call and put contracts earn exceedingly low returns. However,  they also figured out a strategy of buying zero-beta straddles that have an average  return of around -3 percent per week, showing that pricing assets may be significantly  caused by systematic stochastic volatility [5]. They also came up with a possible  extension of their results:

investigations of whether volatility risk is priced in the returns of other assets.  

Data and Methodology  

 In this section, we are going to introduce different methods of testing for  violations. This includes more details about principles, underlying assumptions, and  potential limitations. The notations we use: C(c) denotes the call price, P(p) denotes the put price, S denotes the stock price, X denotes the strike price, respectively.  

Put-Call Parity  

Before the checkout of the Black-Scholes model. Firstly we pay attention to the  upper bound. Lower bound. And Put-Call Parity conditions, and it is necessary to  introduce the principles of them. In the Put-Call Parity, the following conclusion has  become obtained. Considering the American options, we will not discuss the present  value of exercise prices.

Once the equation is impossible, we can do arbitrage. In reality, it is possible to  find the inequity of the two sides. Under this condition, we can buy low and sell  high. Here we just introduce and demonstrate the violation of the upper bound. For  instance, for call options, if c>S, we can do arbitrage by selling the calls and buying  the stocks, and not loss. (Also, if c=S, we can do the same operation), then the “upper  bound” on call price displays, i.e., it cannot exceed stock price S.  

Nevertheless, American Put-Call Parity is different from European one. More  specifically, we write down the equation of American Put-Call Parity without dividend.

where S0, X, C, P, r, T denote current stock price, exercise price, call price, put price,  risk-free rate and maturity, respectively. Usage of natural constant e means  compounding continuously. Similarly, if the inequalities are violated, we can arbitrage.  

Upper Bound and Lower bound  

Based on the data from CBOE market ranging between half of a month to a full  month in January, 2017, this section will discuss below how to find the risk-free  arbitrage opportunity and the method used for calculating the percentage of a risk-free  arbitrage opportunity. 

Firstly, for the lower bounds on call prices, in general, call options  must sell above their call intrinsic values at all time C>S-X (C stand for call price, S  stand for the stock price and X stand for exercise price), if the call intrinsic values are  higher than call price it is a money machine. After minus stock prices and exercise  prices of call options we compare it to call prices to see which one is higher. 

Secondly,  upper bounds on call prices must be that stock prices are always higher than call prices  C<S, if not there is a risk-free arbitrage opportunity. Also, call prices and stock prices  have been compared. Third, lower bound on put prices must always sell above its  intrinsic value P>X-S, if its intrinsic value is higher than put prices, it creates a money  machine. Then the intrinsic value of the put options becomess calculated, and compared to put options prices to find out which prices are higher. Lastly, upper bounds on put prices must always become exercised higher than put prices, i.e., P<X, if not, it creates a risk-free  arbitrage opportunity. Hence, all the put prices become selected and compared to the  exercise price to find out which prices are higher.  

Convexity condition  

To be convex, succeeding equal increases in exercise prices must decrease the  call prices by less and less. And a $5 increase in exercise price decreases the call prices  by less than $5, but in decreasing increments. Succeeding equal increases in exercise  prices must increase the put prices by more and more. And a $5 increase in exercise  price increases the put prices by less than $5 but in increasing increments. If the  convex condition becomes violated, you can do a butterfly or vertical spread to arbitrage,  which means buy low and high and sell 2 middles.  

Black-Scholes Model and Delta  

The pricing of options depends on the model (i.e., the prices obtained by different  models are different). So the arbitrage strategy cannot become implemented without the  framework of the model, only if the market meets the framework of this model can the arbitrage strategy become implemented? The basic assumptions of the Black – Scholes  model is the following:  

(1) During the life of the options, there is no dividend payment and other  distributions of the underlying asset of the options (to simplify, the asset here denotes  stock);  

(2) When dealing the options or the stocks, transaction cost can become neglected, the short-term risk-free interest rate is known and remains unchanged over the  lifecycle;  

(4) Any purchaser of securities can borrow any amount of money at a short-term  risk-free rate;  

(5) Short selling is allowed, and the short seller will receive the money for the price  of the stock sold short on that day immediately;  

(6) The options are European;(The American options in the data are temporarily  treated as European options) 

(7) All securities trading occurs continuously, and short-term stock price random  walks.  

(8) Stock prices are subject to a lognormal distribution.  

The Black-Scholes formula applied to value a European option written on a stock. 

Where d1 and d2 are preliminary calculations. N(x) represents the cumulative  probability function for a standardized normal variable. Which is also the probability  that a variable with a standard normal distribution will be less than x. N(d1) and N(d2) are to assess the probability that the stock price will exceed the strike price. So that the  call options will end up being exercised at maturity. Specifically, N(d1) measures the  probability that the present value of future stock price will exceed the current stock  price; N(d2) measures the risk-adjusted probability that the call options will become  exercised.  

Delta is defined as the rate of change of the options price concerning the price  of the underlying asset. The diagram below shows the relationship between a call price  and the underlying stock price: 

Figure 1. Relationship between call price and underlying stock price 

The Black-Scholes Options Pricing Model can be used to calculate Delta. The  Delta of a long European call option on a non-dividend-paying stock is N(d1). The  Delta of a long European put option on a non-dividend-paying stock is N(d1) – 1.  

For a call option, if C > BS is expensive. Ignoring the transaction fee. You need  to sell the actual call and buy a synthetic call or buy N(d1) shares. If C < BS, it is cheap,  you need to buy the actual call and sell synthetic call or sell N(d1) shares.  

For a put option, if P > BS is expensive. You need to sell the actual put and sell  N(d1) -1 of the stock. If P < BS it is cheap, you need to buy the actual put and buy  N(d1) -1 of the stock.  


We then implemented our testing methods into our data and came out with our  arbitrage strategies and conclusions.  

Upper Bound Violations  

First, the actual data becomes checked. Before calculating. We control the maturity to  ensure the consistency. And group each call/put price and stock price in the CBOE  market (from 1.1 to 1.31) in 2017. Getting 1048575 groups for each side of calls and  puts. 3057 groups of call options and 2758 of put options offer arbitrage  opportunities. Based on the analysis of the upper bound, and the proportions are 0.29%  and 0.26%. The average scale of profit in upper bound arbitrage is about $1.31 per  share of put and $1.08 per share of call.

To arbitrage, take call options for an example. We can sell the calls. And buy the corresponding stocks, then it will be a money machine.  

More interesting finding is not about the data which violate the upper bound. But the factors that affect the violated conditions. Because the maturities have become adjusted to become the same, we will start with the stock prices. More precisely. We will  figure out how much will the profits become influenced. When stock prices change under  the violated condition. And the result is, there is a significant negative relationship between them.  

Here is the figure:  

Figure 2. Relationship between stock price and profit  

To qualify the effect, we add the trendline, considering that the linear fitting is  too inexact. When the stock price is large enough (the profit will be too negative). We use a logarithmic function to model the relation. Indeed, the result is better than linear  fitting, and it is as follows: 

Figure 3. The adjusted fitting-figures (based on Logarithmic fitting) 

By constructing a graph of a function, the effect brought by stock price displays.  We can conclude when the stock price is low (or, lower than $5), the negative effect  of stock prices is strong because of the large slope. When stock prices get higher and  higher, the strength of the effect decreases, but is still negative.  

After checking the upper bound, we turn to Put-Call Parity. Simply put, we will  check if the inequalities of American options make sense. (We use LIBOR to measure  the risk-free rate). Surprisingly, there are too many violated data. Before discussing  the reasons why these data violate the Put-Call Parity, we firstly show the data in detail.  

After checking the left side of the inequation, we find 488404 violated data in all of  1048575 data. And the violated proportion reaches 46.58%, a very high scale.  Meanwhile, the average scale of profit is about $0.88 per share. Data which violate  the right side of the inequation show a number of 473827. A proportion of 45.19% and  an average profit scale of $0.82 per share.  

Apparently, the violated data of Put-Call Parity are far more than the ones of  upper bound and lower bound.

In theory, it is impossible to violate the Put-Call Parity  so many times, so the strange phenomenon needs an explanation. For the assignment  of each variable, there is a flaw in that we use the LIBOR whose maturity is between 2  weeks to 1 month to denote the fixed risk-free rate, while in reality the maturity of  each option can be different.

However, this impact just has an influence on the violation of the right side of the inequation, making the result less precise. For the left  side, the assignment of risk-free rate cannot bring any impact, so there must be another  explanation. Besides the options premium, fees may be a reason causing the huge  proportion of violated data. Because of the fees, investors have to give up the arbitrage  opportunities, or we can say there are no arbitrage opportunities when considering the  fees. Hence, to get the “real” violated data, we should consider the payment of fees.  

Lower Bound Violations  

To exploit profit opportunity, we also need to take a close look at any violations  of lower bound or strict lower bound. For call options, in general, C must be greater  than S-X, and for the strict lower bound, C must be greater than S-PV(X). Based on  1048575 data we collected, we found 469, 0.04% in total, put options and 713, 0.068% 

In total, call options violate the lower bound rule, which provides investors an  opportunity to make arbitrages. The average scale of profit in lower bound arbitrage  is about $0.31 per share of put and $0.082 per share of call.  

If we assume such an event is not happening under systematic conditions; in  other words, there will be a money machine: risk-free arbitrage opportunity. For call  options, we can buy low, sell high, and make money for sure. To best illustrate this,  we take a violation on lower bound for example: if we assume C=11, S=$40, X=$40,  Rf=10%, T=1 year; then we would buy the call for $11, short the stock for $50, and  lend the present value of the exercise price, $36.3 (36.36=40/(1+R)). We could get an  immediate $2.64 (50-36.36-11) in cash. For put options, if the put is selling below  intrinsic value; then it will create a money machine.

We assume the lower bound becomes violated, and P <PV(X)-S, and X=40, S=30, P=4, T=1, r=10%. We could buy the put  at $4, and buy the stock at $30 and borrow the present value of the exercise price at  $36.36; therefore, we could get immediate cash of $2.36. Then we could wait till  maturity, if the put is out of money, then we could make an extra future profit; in other  words, it will also be a money machine.  

 The average transaction fee is about $0.65 per contract fee; thus, we could  conclude that the majority of the arbitrage will not exist in the real world since  transaction cost > money gained from arbitrage.  


This paper examines upper and lower bound violations, convexity violations,  and put-call parity violations in the market. Black-Scholes model and delta hedging  strategy become implemented to exploit any arbitrage profit caused by mispricing.  Generally, there are still quite a several mispricing and arbitrage opportunities.  

The data shows that the average scale of profit in upper bound arbitrage is about  $1.31 per share of put and $1.08 per share of call. The average scale of profit in lower  bound arbitrage is about $0.31 per share of put and $0.082 per share of call. So, we  can sell the calls and buy the corresponding stocks to make a money machine. 



[1] Amin, K., Coval, J. D., & Seyhun, H. N. (2004). Index options prices and stock  market momentum. Journal of Business, 77(4), 835–873.  

[2] Cao, J., Vasquez, A., Xiao, X., & Zhan, X. (2018). Volatility Uncertainty and the  Cross-Section of Options Returns. SSRN Electronic Journal.  

[3] Cao, J., & Han, B. (2013). Cross-section of options returns and idiosyncratic  stock volatility. Journal of Financial Economics, 108 (1), 231–249.  [4] Goyal, A., & Saretto, A. (2011). Options Returns and Volatility Mispricing.  SSRN Electronic Journal, (404).  [5] Coval, J. D., & Shumway, T. (2001). Expected options returns. Journal of  Finance, 56 (3), 983–1009.

How much do options usually cost?