In today’s market economy it is crucial that households and firms are able to select a suitable level of risk in their transactions. This occurs on the financial markets, which restructures risks towards those agents who are willing and able to assume them Markets for options and other derivatives are important as the agents who foresee future revenues or payments can ensure a profit above a certain level or insure themselves against a loss above a certain level. Due to their design, options allow for hedging against one-sided risk, options give the right, but not the obligation, to buy or sell a certain security in the future at a pre-specified price.

The challenge to value derivatives has a long history. As far back as 1900, the French mathematician Louis Bachelier reported one of the earliest attempts in his doctoral dissertation, although the formula he derived was flawed in several ways. Subsequent researchers have handled the movements of stock prices and interest rates more successfully. But all of these attempts suffered from the same fundamental shortcoming, risk premium was not dealt with in a accurate method.

The value of an option to buy or sell a share depends on the uncertain development of the market price to the date of maturity. It is therefore obvious to suppose, the same way as earlier researchers, that the valuation of an option requires taking a stance on which risk premium to used. This risk is determined when calculating the present values and the future physical investment with uncertain returns. Conveying a risk premium is difficult, however, in that the correct risk premium depends on the investor’s attitude towards risk. Whereas the attitude towards risk can be defined in theory, it is difficult to observe in reality.

Barrier options are path-dependent options, which come in many flavours and forms, but their key characteristic is that these types of options are either initiated or exterminated upon reaching a certain barrier level; that is, they are either knocked in or knocked out. Having mentioned that barriers come in all shapes and sizes, we consider the most basic type of barrier option- the single barrier. This option comes in 8 flavours, each with its own characteristics (www.global-derivatives.com).

These are classified as (1) Up & In (2) Up & Out (3) Down & In (4) Down & Out. Each type can take the form of a call or a put, giving a total of 8 single barrier types. An "In" barrier means that a barrier becomes active once crossing a particular barrier level; for example, an Up & In barrier becomes active when the underlying price hits a barrier from below (www.global-derivatives.com).

A barrier event occurs when the underlying event crosses the barrier level. While it seems straightforward to define a barrier event as "underlying trades at or above a given level," in reality it’s not so simple. Qurstions arise such as What if the underlying only trades at the level for a single trade? How big would that trade have to be? Would it have to be on an exchange or could it be between private parties? When barrier options were first introduced to options markets, many banks and institutions founf themselves in legal hot water resulting from a mismatched understanding with their counterparties regarding exactly what constituted a barrier event (Watson, D & Head A 2004).

Standard options permit the owner the right to buy (or sell) some asset in the future for a fixed price; this is known as the strike price. Call options allow the right to buy the asset, while put options confer the right to sell the asset. Path-dependent options represent expansion of this concept. One example is a look back call option allows the right to buy an asset at its least price over a certain time period. A barrier option is similar to a standard option , except that the payoff also depends on whether the asset price crosses a certain barrier level during the option’s life. Look back options and barrier options are two of the most popular types of path-dependent options (Davydov, D & Linetsky, V 2001).

However, the evidence point towards that this hypothesis is not enough. Davydov, & Linetsky (2001) discussed that a down-and-out call is identical to a European call with the additional provision that the contract is cancelled (knocked out) if the underlying asset price hits a pre-specified lower barrier level. The contract may also specify a cash rebate to be received by the option holder if cancellation occurs The rebate is received when the knockout barrier is first reached (Davydov, D & Linetsky, V 2001:952).

An up-and-out call is the same, except the contract is invalid when the underlying asset price first reaches a pre-specified upper barrier level. A cash rebate may be received when the barrier is first reached. Unlike down-and-out calls, which are cancelled out-of-the-money (the lower barrier is placed below the strike price), up-and-out calls are cancelled in-the-money (the upper barrier is placed above the strike). Generally, contracts that are cancelled in the money are called reverse knockout options (Davydov, D & Linetsky, V 2001).

Down-and-out and up-and-out puts are similar modifications of European put options. Knock-in options are complementary to the knockout options: They pay off at expiration if and only if the underlying asset price does reach the pre-specified barrier. The combination of otherwise identical in and out options is equivalent to the corresponding standard European option. Rubinstein and Reiner (1991) derive closed-form pricing formulae for all eight types of single-barrier options under the lognormal assumption (Rubinstein and Reiner (1991) cited in Davydov, D & Linetsky, V 2001:958).

Amongst the major models that have been developed to help the financial markets is the Black-Scholes model in 1973. Although financial historians have traced the origins of this model to the seminal work of Bachelier (1900) and to the extensions of Sprenkle (1962), Boness (1964), and Samuelson (1967), it is clearly the insights provided by Black and Scholes as well as Merton (1973) that revolutionised the financial merkets (Chance, D 1999).

The key factor is the notion that an option can be perfectly hedged with a unit of the underlying asset. This forms a risk-free position with a return over an instant time interval equal to the risk-free rate. If this was not the case then investors could buy and hedge under priced options or sell and hedge overpriced options, thereby eliminating market risk. This would earn a risk free profit in excess of the risk-free rate. Such transactions are known as arbitrage. This combined buying and selling leads to the situation that if we know all necessary parameter values can solve the option price (Chance, D 1999).

The Black-Scholes model is more and more being viewed as the model of choice, but it is questioned whether it merits should earn this status. Outside the investment industry, the model isn’t well understood, and vital considerations such as the impact of vesting on option values remain unresolved. For example, the FASB’s current position on recognising option vesting restrictions is flawed. It recognises the impact on the value of stock-option forfeitures created by vesting restrictions, but not the loss of the right to exercise the option for the vesting term. The original model’s inputs are the volatility of the underlying stock’s returns, the option strike price, and the stock’s market price, a risk-free interest rate and the option’s term (Young C 1993).

Other versions of the model allow the use of less-restrictive assumptions in valuing stock options. These include adjustments for dividends, as well as variable-interest-rate and non-constant-volatility assumptions. Although many users employ modified versions of the model that permit relaxing the underlying assumptions, generally the resulting values do not differ radically from the original model adjusted for dividends. When valuing executive stock options, this version seems to appropriately balance the trade-offs between ease of use and theoretical accuracy (Young C 1993).

The Black-Scholes model in its fundamental form applies to European options on non-dividend-paying stock. In actual options, those restrictions are not upheld. Black (1975) discussed that European options on dividend-paying stocks can be easily priced if it can be assumed that the dividends over the life of the option are known Then one can simply subtracts the present value of the dividends over the option’s life from the stock price and inserts this adjusted option price into the Black-Scholes formula (Black, F 1975:38).

Merton (1976) demonstrates that if dividends can be expressed as a continuously compounded yield, one can first discount the stock price at the yield rate, and then insert the discounted stock price into the Black-Scholes formula The Merton continuous-yield case provides a framework within which one can price options on foreign currency. The continuous yield can be replaced with the foreign interest rate; the underlying asset price is the exchange rate; and the volatility represents the volatility of the exchange rate. The ensuing formula correctly prices a European option on a foreign currency (Merton, R 1976:128).

Since its formation the Black-Scholes model has received a lot of consideration from academics and practitioners. Investors’ attitudes toward risk and risk aversion can sustain the Black-Scholes model. With such a utility function, which displays constant proportional risk aversion (CPRA), the percentage invested in risky assets is unchanged as the wealth of the investors’ increases. Rubinstein (1976) remarked that the Black-Scholes model can be obtained in an equilibrium economy, when agents have power utility functions characterised by CPRA and aggregate wealth and the stock price are jointly distributed (Rubinstein (1976) cited in Camara A 2005:1686).

In their paper, Black and Scholes (1973) assume that "the stock price follows a random walk in continuous time … and thus that the distribution of possible stock prices is lognormal (Black and Scholes (1973) cited in Camara A 2005:1686). It is well documented that the Black-Scholes model miss-prices options in the marketplace and there have been many competitive explanations for these pricing anomalies. Rubinstein (1994) suggested, preferences could be important causes of miss pricing in the options market (Rubinstein (1994) cited in Camara A 2005:1687).

The availability of a accurate estimate of an option’s theoretical price contributed to the explosion of trading in options. Other option pricing models have since been developed for different markets and situations using similar arguments, assumptions, and tools, including the Black model for options on futures, Monte Carlo methods and Binomial options models (Watson, D & Head A 2004).

Therefore in theory traders could buy cheap options and sell expensive options (relative to their theoretical prices), in quantities such that the overall is zero, and expect to make a profit. Nevertheless, implementing this in practice may be difficult because of "stale" stock prices, large bid spreads, market closures and other symptoms of stock market illiquidy. If stock market prices do not follow a random walk (due, for example, to insider trading) this delta neutral strategy or other model-based strategies may encounter further difficulties (Sloman, J. 2003)

Closely linked to the Black-Scholes model was a parallel development of the binomial model. The binomial model’s origins are not as well documented, although it is traced to a textbook by to William Sharpe in 1978. The binomial model demonstrates, using an asset with only two possible future prices, how an option is combined with the asset to form a risk-free hedge, leading to a process for obtaining the option price. It is considered a discrete time model, allowing trading only at finite time intervals. This model was further developed by Cox, Ross, and Rubinstein (1979) and Rendleman and Bartter (1979). In their papers the life of an option is split into an increasing number of smaller and smaller time intervals, the option price obtained from the binomial procedure joins to the Black-Scholes price (Chance, D 1999).

The binomial and Black-Scholes models offer insight on the derivatives, which can be priced using a risk neutral approach. Risk neutrality, is the detestation to modern portfolio theory, it is the conception that investors do not care about risk and are content to price a risky asset as the accepted payoff of the asset discounted at the risk-free rate. Regrettably, much of the research on derivatives has left the notion that the use of risk neutrality is equivalent to the assumption of risk neutrality (Chance, D 1999).

Although it is argued that theorists do not assume risk neutrality. They accept that (1) the prices of underlying assets are determined in a fair market and are observable, along with other necessary parameters such as volatility or interest rates, and (2) arbitrage opportunities do not exist. The work of Harrison and Kreps (1979) though full of complex mathematical results, builds on those assumptions and correctly demonstrates that the prices of all assets, suitably discounted at the risk-free interest rate, follow a martingale, which is a random variable whose value is not expected to change (Chance, D 1999).

Barrier options are a well-accepted type of path-dependent options traded over-the-counter on stocks, stock indexes, currencies, commodities, and interest rates. There are several reasons to use barrier options rather than standard options. First, barrier options strongly match investor beliefs about the future behaviour of the asset. Second, barrier option premiums are generally lower than those of standard options because an additional condition has to be met for the option holder to receive the payoff. The premium can be reduced considerably, in particular when the volatility is high (Davydov, D & Linetsky, V (2001).

Another form of path-dependent options are look back options. Their payoff depends on the maximum or minimum underlying asset price attained during the option’s life. A standard look back call gives the option holder the right to buy at the lowest price recorded during the option’s life, and the right to sell at the highest price recorded during the option’s life. A call on maximum pays off the difference between the realised maximum price and some pre-specified strike or zero, whichever is greater. These options are called fixed-strike look backs. In contrast, the standard look back options are also called floating-strike look backs, because the floating terminal underlying price serves as the strike price in standard look back options Davydov, D & Linetsky, V (2001).

The fundamental supposition behind all finance models is avarice, all other things being equal; more wealth (or more consumption) is better than less. This opinion is universally believed, it is an important statement about human psychology. Investor behaviour is the hypothesis that investors are, by and large, risk averse. Risk aversion may not apply at all times and places, but it is generally believed to be a enveloping aspect of human behaviour. The supposition of risk aversion is supported by a long history of evidence drawn from many diverse situations. Finance models often become quite specific about risk aversion; they may take for granted, for example, that (all else being equal) as an investor’s wealth increases, the investor is willing to invest more money in risky assets (*Rubinstein, M 2001)*.

Financial economists have the practice of modelling rational choice in terms of time-additive utility functions. These utility functions imply that how much one consumes today has no effect on the utility of consumption tomorrow. An ornately staged behavioural experiment is not requires to demonstrate that this assumption cannot be right. Financial economists need to allow for practice formation; doing so may be analytically difficult, but it is completely consistent with rationality. One of the most distinguishing features of prospect theory involves reference points, similar to the older idea of habit formation in economics. For many results in finance, habit formation is of no importance, but for many so called abnormal observations, it is assets (*Rubinstein, M 2001)*.

The Black-Scholes model gives theoretical values for European put and call options on non-dividend paying stocks. The vital argument is that traders could without risk hedge a long option, with a short position in the stock and incessantly adjusts the hedge ratio as required. Assuming that the stock price pursues a random walk, a price for the option can be considered where there is no arbitrage profit. This price depends on five factors: the current stock price, the exercise price, the risk-free interest rate, the time until expiration, and the volatility of the stock price (Watson, D & Head A (2004).

Barrier options are path-dependent options that are comparable in some ways to ordinary options. There are put and call, as well as the European and American range. But they become activated, or null and void if the underling value reaches a predetermined barrier. "In" options start their lives valueless and only become active in the event a predetermined knock-in barrier price is breached. "Out" options start their lives active and become null and void in the event a certain knockout barrier price is breached.

With either case, if the option terminates inactive, then there may be a cash rebate paid out. This could be zero, in which case the option ends up valueless, or it could be some fraction of the premium. Up-and-out (Down-and-Out): spot price starts below (above) the barrier level and has to move up (down) for the option to be knocked out (become null and void). Up-and-In (Down-and-In): spot price starts below (above) the barrier level and has to move up (down) for the option to become active.

The evaluation of barrier options can be complicated because not like other simpler options they are path-dependent, the value of the option at any time depends not just on the underlying at that point, but also on the path taken by the underlying (since, if it has crossed the barrier, a barrier event has occurred). Although the classical Black-Scholes approach does not directly apply, several more complex methods can be used, one of these is the Monte Carlo option model. Therefore whichever model is applied, barrier options are path-dependent options, and these barriers are initiated or exterminated when they reach a certain barrier level.

Sloman, J. (2003) (5^{th} Edition) Economics

Prentice Hall, London

Watson, D & Head A (2004) (3^{rd} Edition) Corporate Finance

Prentice Hall, London

Black, F (1975) Fact and Fantasy in the Use of Options Financial Analysts Journal, 31 (July-August 1975)

Camara A (2005) Option prices sustained by risk-preferences The Journal of Business, Sept 2005 v78 i5

Chance, D (1999) Research Trends in Derivatives and Risk Management Since Black-Scholes (Special Theme: Derivatives & Risk Management) Journal of Portfolio Management, May 1999

Davydov, D & Linetsky, V (2001) Pricing and Hedging Path-Dependent Options Under The CEV Process (constant elasticity of variance) Management Science, July 2001 v47 i7

Merton, R (1976) Option Pricing When Underlying Stock Returns are Discontinuous Journal of Financial Economics

*Rubinstein, M (2001)* Rational markets: Yes or no? The affirmative case Financial Analysts Journal Charlottesville: May/Jun 2001.Vol.57, Iss. 3

Young. C (1993) What’s the right Black-Scholes value? (Valuation of executive stock options) (Corporate Reporting) Financial Executive, Sept-Oct 1993 v9

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