But if this was an European digital option, we can see from the pic that the underlier level at maturity is way below , so the holder of the option would have received nothing at maturity. Here we are simulating many different price paths for the underlier by breaking the overall time period of the option into tiny time steps. As the no. We have stored all these different price paths in a matrix and then with the help of simple if,else, for statements we check if the binary condition has been met or not.

If the binary condition is met for a sample price path we fix the payoff at maturity for that price path and if the binary condition is never met for a sample price path we fix the final payoff as 0. Then the price of the option is just the average of the various payoffs expected value of the payoff multiplied by the discount factor. And that gives us the price of the option.

Monte Carlo Option Pricing with Excel

In the above results we notice a very interesting phenomenon. When the strike is very close to the initial level and the volatility of the underlier is very low, the price of such an option is higher than when the volatility is very high. Time to delve into more exotic derivatives. We are going to look at pricing binary options with the help of MC simulations. I am going to do this in 2 parts. In the 1st part we will be discussing about a binary option where the observation for the binary event happens only on 1 single day which we will call the valuation day, and this day will be the last day of the maturity period for the Option.

In part 2 we will look at binary options with continuous observations but with payoff at maturity. Please go through it, and we will look at how the code works. The above code is for pricing a binary call, if the underlier is at or above a certain level at maturity, it will have a fixed payoff. We already know the meaning of the terms, [ s , k , r , t , vol ] from my previous post.

For ex:- If we say this amount is , it means if the underlier is at or above a certain pre-defined level at maturity, the party who is long the option will get We can easily tweak the code a little bit for pricing Binary PUT options. Maybe the readers can try that out for practice. Lets start with something easy and simple. We will also assume that the stock is a non-dividend paying stock just makes our life a little easier, though if assume that the stock pays dividends, the formula just changes a little bit. Please find below the function that can be used to price an European Call option using MC simulation.

This is based on the R platform Here we are doing simulations of the Final Stock Price in order to price the call option. Before we proceed, we must detail the Monte Carlo approach to pricing options in general. In recent years, the complexity of simulation methods has increased tremendously, in a continuous search for accuracy and speed.

The Monte Carlo method in particular can be applied for a variety of purposes: valuation of securities, estimation of their sensitivities, assessment of the hedging performance, risk analysis, stress testing, etc. The literature on simulations is voluminous, starting with the seminal paper by Boyle until the recent papers on quasi-monte Carlo. The expectation is taken with respect to a risk neutral probability measure. The Monte Carlo approach is an efficient application of this theory as summarized by the followings: simulate a path of the underlying asset under the risk neutrality condition, over the desired time horizon; discount the payoff corresponding to the path at the risk-free interest rate; repeat the procedure for a high number of simulated sample paths; average the above discounted cash flows over the number of paths to obtain the option s value.

The Law of Large Numbers guarantees the convergence of these averages to the actual price of the option and the Central Limit Theorem insures that the standard error of the estimate tends to 0 with a rate of convergence of 1 N where N is the number of simulations.

Monte Carlo Simulation: European and Asian Options Pricing

This convergence rate is based on the assumption that the random variables are generated with the use of pseudo-random numbers. It is possible to achieve an even higher rate of convergence provided that quasi-random numbers are used. Overall, the method proves to be flexible and easy to implement or modify. It can deal with extremely complicated or high-dimensional problems. As shown, the rate of convergence does not depend on the dimension of the problem. Another advantage of the simulation approach is the confidence interval that it provides for the estimate.

This interval shows how accurate the estimate really is and if more time and effort are needed for additional precision. Last, the current advances in technology have reduced the computation time and have made the method more attractive.

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There are, also, several disadvantages to this methodology: very complicated problems may require a very high number of simulations for an acceptable degree of accuracy and this may be rather time-consuming and expensive. However, many variance reduction methods have been proposed such as: antithetic variables, control variables,. Another improvement is to use deterministic sequences of numbers instead of pseudo-random ones.

This procedure is particularly useful for high-dimension integrals. Given that Monte Carlo can be easily extended for complex derivatives and complex stochastic processes, we have decided to implement it in order to price exotic options. We shall first describe the main characteristics of the most important classes of exotic options: path-dependent options, correlation options and other popular types that cannot be integrated in one of the previous categories. In each class, we have chosen the most traded and used exotic options for which analytical formulas have already been found.

The majority of these formulas, under Black-Scholes assumptions, can be found in Zhang The prices obtained by simulations will be compared with the ones given by these formulas in order to assess the performance of our numerical approach. The basis of all the pricing procedures is represented by the construction of price paths for the underlying asset. Path dependent options Path dependent options As their name shows, path-dependent options are options whose payoffs at exercise or expiry depend on the past history of the underlying asset price as well as on the spot price at that moment.

Furthermore, this path-dependence can be either strong, as for Asian options or weak as for Barrier options. Strong path dependence means that we must keep track of an additional variable besides the asset level at every observation and time. For example, in the case of Asian options, this variable is the average to date of the asset values Asian options Asian options have payoffs that depend on the average value of the underlying asset over some period of time before expiry. This average can be defined in multiple ways: it can be either arithmetic or geometric, weighted or unweighted, calculated with the help of continuous or discrete observations.

The payoff is the difference between this average and a pre-defined strike price. Path dependent options 7 It is possible to have the so-called average strike Asian options whose payoffs are represented by the difference between the last asset price and one of the above averages. We should also add the flexible Asian options that allocate different weights to price observations and which are now being extensively used. The majority, however, are European style options based on unweighted arithmetic averages of the underlying asset.

Asian options are mostly used in commodity and currency markets because they are cheaper hedging alternatives to a string of standard options. The main difference between arithmetic averages and geometric ones in the case of options is that the latter are lognormally distributed, while the former are not. It is the reason why geometric Asian options can be relatively easy to price in a Black-Scholes environment.

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However, the prices of arithmetic average options can only be approximated using the formulas for the corresponding geometric averages. Monte Carlo simulations have been used quite often to price arithmetic Asian options and the geometric average based option is used as a control variate. We have implemented two procedures in Gauss, one for geometric Asian options and one for arithmetic Asian options, both for call and put. We have used antithetic variables in both programs in order to reduce the variance.

The simulation of stock price paths is the one presented in 1. Moreover, this numerical approach can be extremely useful when the averaging period is more complicatedly defined, i. The payoff of a barrier option is identical to the one of a standard option if the option still exists at maturity and 0 or a rebate otherwise. It simply means that the underlying asset price must stay in some predefined region for the option to be exercised. Depending on how this region is defined, there are two main types of barrier options: the in or knock-in barrier options and the out or knock-out ones.

The former have a payoff identical to a standard call if and only if the price of the underlying asset hits the barrier, while the latter have this payoff if the barrier is not touched during the option s life. It is also relevant how the barrier is hit; if, initially, the price is under the barrier, so the barrier will be hit from below, we have up knock-in and up knock-out options. Conversely, if the barrier is hit from above, we have down knock-in and down knock-out options.

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Of course, one last classification is in call or put options, so that finally, given all these possibilities, we shall have a total number of eight plain vanilla barrier options. Besides vanilla types, there are many other variations, more or less complicated: time-dependent barriers, Asian barriers, dual-barriers, forward-start barriers window barrier options, etc. Barrier options are cheaper than standard options and they can be used for various purposes, from hedging to speculation.

While 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. So, Monte Carlo becomes a good candidate for pricing these instruments. Path dependent options 9 We have constructed two programs for pricing a down knock-in and an up knock-out barrier option. For the down-in option, the parameters are the same as above, except that we have a barrier level of 95, a rebate of 1. The value given by simulations for a down-in call with a barrier of 95 and a strike of 98 is , the interval of confidence, 2.

For example, a up-out put with a strike of has a value, by simulations, of , a confidence interval of 4. There are several types of such options: floating strike lookback options, fixed strike lookback options, American lookback options, partial lookback options, etc. Path dependent options 10 where S T : stock price at expiration M T t : maximum price observed during the option s life The payoff of a fixed strike lookback option is similar to the one of a standard option except that the terminal price is replaced, for a call, with the maximum of the asset s price and for a put, with the minimum.

Nevertheless, these advantages are counterbalanced by the high premiums charged for such instruments.

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Regarding their pricing, the above authors were the first to study the European-style floating strike lookback options. The framework is the Black-Scholes one and the prices are derived by discounting the expected payoffs at the risk-free rate. The two Gauss programs we have developed deal with both floating and fixed strike lookback options. The parameters are maintained, but we must mention the current. For the floating strike lookback options we obtained the following results: the call price by simulations was and the confidence interval , ; the put price was , with a confidence interval of ,.

The corresponding Black-Scholes type prices were: for call and for put. The Black-Scholes prices were and , respectively. The pricing of lookback or barrier options can be further developed under more complicated specifications for the asset s variance. In particular, it can be assumed that the variance follows a CEV constant elasticity of variance process or a mean-reverting process and simulations can be performed for the variance process as well.

Pricing options under stochastic volatility will be treated in detail in the next chapter. These assets can be extremely different or they can belong to the same asset class. It is easy to infer that correlation among these assets will have a major role in the pricing and hedging of these instruments. The problems raised by correlation can be significant since it is even more unstable than the variance Exchange options Exchange options give their owner the right to exchange one risky asset for another. Practically, at maturity, the value of one asset is paid while the value of the other asset is received.