Option Pricing When Changes of the Underlying Asset Prices Are Restricted

Exchanges often impose daily limits for asset price changes. These restrictions have a direct impact on the prices of options traded on these assets. In this paper, we derive closed-form solution of option pricing formula when there are restrictions on changes in underlying asset prices. Using numerical examples, we illustrate that very often the impact of such restrictions on option prices is substantial.


Introduction
Conventional option pricing models assume that there are no restrictions on changes of underlying asset prices.For example, [1,2] specify that stock prices follow a geometric Brownian motion and stock returns follow a normal distribution.Based on a portfolio replication strategy or equivalently the risk-neutral method, option prices are derived as expected payoff of the contract under the risk-neutral distribution, discounted by the riskfree rate.
In practice, however, asset price changes may subject to restrictions imposed by exchanges.For example, CBOT (Chicago Board of Trade) and CME (Chicago Mercantile Exchange) both have daily price limits for futures contracts except currency futures.Daily price limit serves as a precautionary measure to prevent abnormal market movement.The price limits, quoted in terms of the previous or prior settlement price plus or minus the specific trading limit, are set based on particular product specifications.For example, the current limit of daily price changes on short term corn futures is 4.5%.In 1996, Chinese stock market also introduces restrictions on daily stock price changes.Specifically, except the first trading day of newly issued stocks, the limit of stock price change in a trading day relative to previous day's close price is 10%, and for stocks that begin with S, ST, S*ST letters, the limit is 5%.It is clear that such restrictions reduce the value of options since extreme returns on a daily level are ruled out.Nevertheless, how to evaluate the prices of option contracts when changes on underlying asset prices are restricted?How much is the exact impact of such daily price limits on option prices?These questions are yet to be examined in the extant literature.
In this paper, we first derive the option pricing formula when there are restrictions on daily changes of underlying asset prices.We perform the analysis under the Black-Scholes-Merton model framework.Then, we provide numerical comparisons between option prices with and without restrictions on underlying asset price changes.

Option Pricing with Restrictions on
Underlying Asset Price Changes

Risk-Neutral Valuation under the Black and Scholes [1] and Merton [2] Framework
In this section, we first review the risk-neutral approach of option pricing under Black-Scholes-Merton framework.The same approach will be used in the next subsection to derive option pricing formula when there are restrictions on underlying asset price changes.Black and Scholes [1] and Merton [2] assume that stock price follows a geometric Brownian motion: where  and  are expected return and volatility, t is a standard Brownian motion.It is also assumed that the continuously compounded risk-free interest rate, W denoted by , is constant.The key feature in Black-Scholes-Merton framework is that asset return volatility is constant and market is complete.As such, in a riskneutral world, expected return of the underlying stock is equal to risk-free interest rate.That is, where is a standard Brownian motion under the risk-neutral probability measure .
Q t W Q Consider a European call option with strike price K and maturity measured in the number of trading days.The price of such option can be computed as where , and is the time interval of a trading day.As shown in many derivatives textbooks, for example [3], the option pricing formula is given as: where is the cumulative distribution function (cdf) of standard normal distribution, and This is the famous Black-Scholes-Merton option pricing formula.By constructing a riskless portfolio with option and underlying stock and based on no arbitrage argument, [1] and [2] derive the above option pricing formula as a solution to a partial differential equation (PDE).

Closed-Form Option Pricing Formula with Restrictions on Underlying Asset Price Changes
As mentioned in the introduction, many exchanges impose restrictions on daily price changes of traded assets.As a result, the range of asset return (in logarithmic form) is no longer , but truncated from both below and above.The restriction is particularly important in option pricing since the tail behavior of asset returns has a significant effect on the payoff of options contracts.In addition, and are two positive constants.Truncating the left tail of the normal density b a below the mean, and the right tail by b above the mean, the truncated normal distribution is illustrated in Figure 1.
Normalizing the truncated density function to make sure the total probability is equal to , we obtain the pdf of a truncated normal random variable where ( ) f x is the normal density function and where     is the cdf of standard normal distribution.In practice, price restrictions are often imposed in terms of daily simple returns.For example, daily simple returns in absolute value are restricted to be less than  , then for log returns, these restrictions are ln(1 a ) For the purpose of option pricing, it is also convenient to obtain the characteristic function (CF) of stock returns.As derived in the Appendix, the characteristic function of the truncated normal variable X is given by: trc is the CF of a normal random variable and Since limits are typically imposed on daily price changes and option maturity can be more than one day, we need to derive the distribution of returns over multiple days. .The log price at the end of day is T Similarly, as derived in the Appendix, the CF of T X is given by 0 ( ) ( ) e ( ) In the following, we follow the same risk-neutral approach as outlined in the previous subsection to price options when there are restrictions on underlying asset price changes.As seen in the Black-Scholes-Merton framework, when we move from real world into riskneutral world, volatility remains the same, but expected return is equal to risk-free interest rate.Option prices are then calculated as expected payoff under the risk-neutral measure, further discounted by risk-free interest rate.In the following, we first derive the risk-neutral distribution of asset returns when daily returns follow truncated normal distributions, and then derive a closed-form formula for European call options.
Lemma Let 2 ~iid ( , ) and the time interval between observations is , we have 1, , t   T i) Under the risk-neutral measure where the expected return of the asset is given by the risk-free rate , we have ii) The price of a European call option with strike price K and maturity T is given by trc 0 1 2 ( ln ) e ( ln The probabilities and can be computed numerically as where ( )     is the CF corresponding to .P Proof: i) Let be the time interval of each trading day, and the fact that expected return is equal to risk free rate leads to: From the moment generating function of a truncated normal distribution as derived in the Appendix, we have From the above two equations, we have the expression for  .
ii) According to risk-neutral pricing method, the price of a European call option with strike price K and maturity is T where T X ( ) f x is the pdf of log price T X at time .In addition, under the risk-neutral measure,  g x is a pdf.Therefore, we can write the European call option price as 0 1 2 ( ln ) e ( ln à End of proof. As shown in [4] that the probabilities in (8) can be computed numerically by their corresponding CFs as follows.From the Fourier inversion, we have To compute the CF corresponding to ( ) g x , denoted as 1 ( )     , by definition, we have e ( e ( )d where the numerator is given by 0 , and the denominator is given by 0 (1) e ( )d (1) e (1; , ) ( 1 ; , )

Numerical Illustrations
In this section, we illustrate numerically the d of option prices with and without restrictions on under- ), the relative difference in ption price with restriction and without restriction is more than 30% when the price limit is set as 5%.

Conclusions
In practice, ex c ect since impact on prices of th ule ou matic c es in prices p fo we de closed-f solutio underly option g asset c es. Usi umeric ample illustr very often the impact of such restrictions on option prices can be substantial.

Acknowledgements
The research of Lei Sh (111 ppendix the appendix, we first derive the moment generating nction (mgf), and characteristic function (CF) of a al random variable.Recall that the pdf of truncated normal random variable ) where ( ) f x is the normal density function and is the mgf of a normal random variable, and Similarly, the characteristic function (CF) of a truncated normal random variable ) where c is given by (10), and is the CF of a normal random variable, ( ) Next, we derive the CF of the log price with truncated distribution.Denote where

Table 1
reports the differences in European call option pr mit as ices under different scenarios.The Black-Scholes-Merton price, denoted by CallBSM, is the call option price without price restriction and is computed from formula (3), the call option price with price restriction, denoted by CallTrc, is computed from formula (8) derived in the previous subsection.