### 3.1 The Monte Carlo Model overview

The Monte-Carlo approach resolves to be much simple compared to the Binomial model and the Black Scholes approach when the variance reduction routine is applied. the Monte-Carlo approach generates precise approximations drastically close to the exact value of the option, which at the same time, sanctions greater flexibility when valuing options. This flexibility is significantly helpful when valuing options and originates from the fact that the distributions of terminal stock prices is determined by the future stock price process mostly simulated a computer. Once the terminal prices are gathered, the value of the option can be determined as well as the standard deviation of the estimation in order to determine the accuracy of the results. The Monte-Carlo approach enables one to value options even when the underlying stochastic process is not continuous as it is required by the Black-Scholes approach or when it does not pursue a discrete binomial process as required by the binomial model. A number of methods such as the jump processes, or a combination of continuous with jump processes, even processes like the empirical distribution can be illustrated using the Monte-Carlo approach. (3)

(The established approach of utilising simulation to propagate the price of the option is straightforward:

1. Exerting the risk-free method, simulate sample paths of the underlying state variables (e.g. underlying asset prices and interest rates) over the relevant time perpective.
2. Investigate the discounted cash flows of a security on each sample path, as determined by the structure of the security in question, making sure to discount the payoff corresponding to the path at the risk-free interest rate; Hence
3. Average the discounted cash flows over the sample paths in order to obtain the options value.(4)

( Practically, this approach generates an estimate of a multi-dimensional integral-the expected value of the discounted payouts over the space of sample paths. The upsurge in the complexity of derivative securities induces a need to evaluate high-dimensional integrals. The Monte-Carlo simulation is commendable to use relative to other analytical techniques because it is flexible, simple to implement and modify, and the error convergence rate is independent of the dimension of the problem. (2))

### Background

The Monte-Carlo simulation approach is implemented to propagate paths for the underlying asset price, and then to obtain estimates for the payoff of a European call. The average of the estimated payoffs calculated then updated to a present date value utilising the risk-free interest rate as the discount rate. In this course, the price of the simulation together with each path were generated in steps, defined by the number of steps used. As in the Black and Scholes approach, we deduced that the underlying asset path of prices pursue a Brownian geometric motion, defined by the differential stochastic equation: (1)

(The criteria of large numbers certify the convergence of these averages to the actual price of the option whereas the 'central limit theorem' assures that the standard error of the estimate disposes to 0 with a rate of convergence of 1/√N where N is the number of simulations. This convergence rate is established on the assumption that the random variables are generated with the application of pseudo-random numbers. It is attainable to achieve an even higher rate of convergence provided that quasi-random numbers are utilised. Overall, the Monte-Carlo approach proves to be flexible and simple to implement or modify. It can confer with radically complicated or high-dimensional problems. As illustrated, the rate of convergence does not depend on the dimension of the problem.4)

### 2.1 Introduction

Over the years the magnitude of the exotic options market has developed widely. At present a paramount variety of such instruments are available to investors within the market which can be utilised for many purposes. Numerous factors contribute the current success of these instruments. One out of many resposible factors is their possibility of an almost unlimited flexibility in which they can be prepared to the specific needs of a given investor. Hence the names 'special-purpose options and 'customer-tailored' options are sometimes associated to exotic options. The second factor is its significant approach to the hedging role and thus, they meet then hedgers' needs in cost effective ways. Corporates have diversed away from buying some form of general protection and are creating strategies to meet specific exposures at a given point in time. These strategies are based upon exotic options that are essentially less expensive with greater efficiency than standard instruments. Thirdly, exotic options are utilised as desirable investments and trading opportunities. As a result, views on the spot evolution, various preferences on time horizons and premium contingency are all accustomed by exotic patterns. Furthermore it is also possible to develop a prevailing position that would be beyond reach in the spot or standard options market. The crucial types of exotic options are priced either numerically or analytically. Hence the method adopted for pricing this project is based on Monte-Carlo simulations.

### 2.2 Barrier Options

(A barrier option is one of the most primitive exotic options. The barrier option payoff is similar to the payoff of the standard option if the option still exists at maturity and 0 otherwise. Meaning that the underlying asset price has to remain in a predefined area or region for the exercised option. There are two main types of barrier options that depend on the defined region or area. There is the 'in' or 'knock-in' barrier options and the 'out' or 'knock-out' ones. The former has a payoff similar to a standard call if only the price of the underlying asset reaches the barrier, whereas the latter has the payoff if the barrier is not touched during the option's life. Furthermore it is important how the barrier is hit; if the price is initially under the barrier, the barrier will be hit from below, an 'up knock-in' and 'up knock-out' options will be attained. Conversely, when the barrier is hit from above, a 'down knock-in' and 'down knock-out' options will be attained. Also, the final classification is in a call or a put option, which ultimately when given the possibilities should result in a total number of eight plain vanilla barrier options. Apart from vanilla type variations there are many different types that are more or less complicated: Dual-barriers, Asian barriers, time-dependent barriers, forward-start barriers window barrier options and so on.. Barrier options are used for many purposes and are more economic than standard options, from hedging to speculation. One of the few important contributions in pricing these instruments belong to: Merton (1973), Goldman, Sosin & Shepp (1979) or Rubinstein & Reiner (1991). So far it is understood that analytical solutions have been proposed for the plain vanilla barrier options under log-normality and risk neutrality, similar ones may not exist for more complicate payoffs. Therefore the Monte-Carlo approach appears to be a good candidate for pricing these instruments. 4)

(A barrier option is the most important exotic option for structured products. Barrier options consist of unique characteristics that distinguish them from vanilla options. The most popular standard barrier options are `knock-out' and `knock-in' options. These options are expired or exercisable automatically when the stock price hits the specified barrier level. Both out- and in-options are divided into down and up options by the level of current stock price compared to the barrier level, so there are eight kinds of basic standard barrier options. At times a `knock-out' option may be called a `live-out' option if it is significantly in-the-money when it knocks out. It is also possible that it has some payoff (called a `rebate') when the barrier level is hit; capped call however for this study a standard knock out option was considered. (7)

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