Skip to main content. You're using an out-of-date version of Internet Explorer. By using our site, you agree to our collection of information through the use of cookies. To learn more, view our Privacy Policy. Log In Sign Up. Christiaan Oosthuizen. Modeling is neither science nor mathematics; it is the craft that builds bridges between the two.

Progress in modeling dynamics has always been closely associated with advances in computing. It is used to scrutinise the impact of risk and uncertainty in financial and other forecasting models. It is very useful when complex financial instruments need to be priced. Most listed exotic options are marked-to-model and the JSE needs accurate values at the end of every day.

Monte Carlo methods in a local volatility framework are implemented. This paper discusses how Monte Carlo MC simulation is implemented when exotic options like Barriers are valued. We further summarise the historical development in modern computing and the development of the Monte Carlo method.

Introduction Why is simulation or modeling an important component of analysis for scientists? Simulation is the imitation of a real-world process or system. Simulation games are fun too and one gains valuable experience at the same time. Experience and insight are gained by simulating the valuation of financial products, constructing portfolios and testing trading rules McLeish, Through simulation work is transferred to a computer.

Models can be handled that involve greater com- plexity and fewer assumptions, and a more faithful representation of the real world is possible. Morrison states that modeling is neither science nor mathematics; it is the craft that builds bridges between the two. Modeling is important because scientists investigate the world around us by build- ing models that simulate real-world problems. Our insight in a physical system, com- bined with numerical mathematics, gives us the rules for setting up an algorithm the model , or a set of rules for solving a particular problem Steinhauser, These models usually take the form of differential equations that have to be solved to obtain physical answers.

Researchers usually start with a very simplistic model and try to solve it analytically or algebraically. Such models are mostly easier to analyse and scrutinise.

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This inevitably means they have to make a lot of simplifying assumptions. This means that modeling practitioners need to be familiar with a wide variety of mathematical specialities, computer science and one or more disciplines which provide data. This is exactly the route the evolution of the Black-Scholes option pricing model took. Black, Scholes and Merton made some simplifying assumptions that enabled them to devise a backward parabolic partial differential equation PDE. They solved this PDE analytically using Fourier series methods. It is now well-known that this model is far from the real world and stock prices behave in a much more complex manner.

Since most of the original simplifying assumptions have been relaxed. When some of these assump- tions are relaxed, one finds that the model cannot be solved analytically anymore. Most problems where there is significant uncertainty, can be solved using Monte Carlo techniques. Monte Carlo methods are techniques utilising random numbers and probability to solve problems. The analysis is based on artificially recreating a chance process, running it many times and directly observing the results.

Monte Carlo methods are attractive in evaluating integrals in high dimensions Glassermann, What does this have to do with financial engineering? The foundation of the theory of derivative pricing is the random walk of asset prices. According to the Feynman-Kac theorem, the solution to this PDE can be represented by an expected value — valuing derivatives is reduced to computing expectations.

Monte Carlo simulation is widely used in statistics in calculating an expected value of a particular function. Boyle showed that all financial options are always the expected value of certain functions. If we were to write the relevant expectation as an integral, we would find that its dimension is large or infinite. This is precisely the setting in which Monte Carlo methods become attractive Glassermann, Weber stated that the Monte Carlo method is widely used in the financial markets as a valuation tool.

It is used with path-dependent options and in models with more than one state variable. It is sometimes preferred to finite difference or tree methods, even in situations where these methods could work well — simply because its generality and its robustness in contexts where a portfolio of options is being valued. In this paper, we consider the Monte Carlo approach to value exotic options. However, it is also known that these formulas do not lead to market related and realistic prices and hedge ratios. This is due to the assumption of a fixed volatility. However, such option are path-dependent meaning that the actual path the stock takes to get to the expiry value on the expiry date does matter.

To price them correctly one should either use stochastic volatility models or local volatility models. The choice here is to use either finite difference techniques or Monte Carlo simulation. This note will focus on Monte Carlo techniques. Section 3 gives a brief overview of exotic options.

In section 4 we bring local volatility into the Black-Scholes framework and we discretise the Black-Scholes stochastic differential equation. Section 5 is crucial where we show how to use Monte Carlo simulation when pricing options. We conclude in section 8. Note that there are three Appendices where we elaborate on some of the theory described in this paper. Appendix A shows why Monte Carlo simulation can be used when pricing options and we show how to discretise the Black-Scholes stochastic differential equation.

In Appendix B we discuss the generation of random numbers and in Appendix C we elaborate on convergence issues when simulating stock price paths and option values. A Lesson in History Progress in modeling dynamics2 has always been associated with advances in computing Morrison, As such dynamics has reached maturity with the devel- opment of digital computers, both as concept and technological product. But, why is reading the history of science important? Donald Knuth motivates, why, as a computer scientist he reads the history of science. First, reading history helped him to understand the process of discovery.

Sixth, history teaches how human experience has changed over time. As humans we should care about that Haigh, A bit of Computing History The modern history of computing is quite short Copeland, In Wil- helm Schickard , constructed a machine for his mathematician friend Jo- hannes Kepler which was able to perform addition, subtraction, multi- plication and division Steinhauser, Charles Babbage is generally credited with originating the concept of a programmable computer Dasgupta, Babbage built a small working model in but he never completed a full-scale machine.

The punch card was developed by Hermann Hollerith during It was developed for a population census. Dynamical is what concerns change. This computer was not programmable but a full-scale machine was built in Turing , at Cambridge University, invented the principle of the modern computer.

He described an abstract digital computing machine consisting of a limit- less memory and a scanner that moves back and forth through the memory, symbol by symbol, reading what it finds and writing further symbols. He further stated that the actions of the scanner are dictated by a program of instructions that is stored in the memory in the form of symbols. Alan Turing designed the Bombe. It was built in and was an electro-mechanical special purpose computing device. The second world war brought much needed progress though.

Both the British and Americans developed electronic computing machines. The first fully function- ing electronic digital computer was Colossus, used by the Bletchley Park and British cryptanalysts from February The Colossus computer was built on the theoretical framework set by Turing Colossus had two problems: First, it had no internally stored programs.

Second, Colossus was not a general-purpose ma- chine, being designed for a specific cryptanalytic task involving counting and Boolean operations. The primary function for which ENIAC was designed was the calculation of tables used in aiming artillery. These earliest large-scale electronic digital computers, the Colossus and the ENIAC, did not store programs in memory. These machines were massive which led the IBM chairman, Thomas Watson, to state in that there might be a world market for five computers and no more Steinhauser, The main character was Stanilaw Ulam Ulam and Edward Teller developed the first thermonuclear weapon also known as the hydrogen bomb or H-bomb.

Ulam was intensely interested in random processes. He relaxed by playing solitaire and poker. His extensive mathematical background made him aware that statistical sampling techniques had fallen into desuetude because of the length and tediousness of the calculations. Due to the computational issues, this method did not really take off. This triggered the spark that led to the Monte Carlo method. One of the first problems solved on the ENIAC in was a computational model of a thermonuclear reaction6. Los Alamos got its own computer early in A signif- icant advance in the use of the Monte Carlo method came out of the collaboration between Nicholas Metropolis and Edward Teller.

Together they introduced the idea of what is today known as importance sampling, also referred to as the Metropolis algorithm7. Monte Carlo Methods and the Pricing of Options Boyle was the first to relate the pricing of options to the simulation of random asset paths. This reaction is responsible for the energy produced in the sun. The future important role of Monte Carlo simulations in finance was assured. Boyle et al. We can narrow this definition down slightly, by stating that exotic options are options for which payoffs at maturity cannot be replicated by a set of standard options de Weert, Further to this, a structured derivative product is a bespoke instrument that enables an investor to pursue strategies tailored to his or her market view Tan, Such a product allows an investor more control over the yield-risk tradeoff in his investment.

Exotic options and structured notes have traditionally been traded over-the-counter OTC. The JSE was the first exchange in the world to list such products. Since , the types of exotic listed on the JSE have grown tremendously. Most exotic options are European in nature — this means they can only be exercised on the expiry date.

If an instrument is liquid, a full mark-to-market MtM process can be run be- cause on-screen traded prices or bid-ask spreads are available. However, all exotic instruments are very illiquid, and a mark-to-model process is used. This means mod- els are used in estimating the end of day levels.

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In this note, we will describe how these exotic instruments can be evaluated using Monte Carlo simulation. Deterministic Local Volatility One of the original assumptions made by Black, Scholes and Merton was that volatility is constant. In Equation 4. The local volatility is the instantaneous volatility for each point in space and time i. In this case, local volatility is a scaler field Boas, ; Reif, These concepts come from mean field theory MFT where the Ising model is a standard many-body system discussed in solid state physics textbooks Harras, ; McCauley, ; Sornette, It is clear that W , and consequently its infinitesimal increment dWt , still represents the only source of uncertainty in the price history of the security.

Black, Scholes and Merton made some assumptions in order to facilitate a better understanding of the dynamics of the security price St. One of the main assumptions is that of risk neutrality. In its simplest form, this infers that all risk-free portfolios can be assumed to earn the same risk-free rate.

This equation is a backward parabolic partial differential equation also known as the back- ward Kolmogorov equation Rebonato, ; Duffie, Discretising the Black-Scholes Equation Fourier solved his simplistic heat conduction equation analytically by introducing Fourier transforms. The extended version is not solved that easily. However, we will understand the SDE in Equation 4. See Appendix A for the derivation. Equation 4. This, together with equations 5. Equations 4. So the question is how do we sample from the continuous distribution for the variable ST? In order to start the simulation we need a starting asset value S t0.

Such a price path is shown in Figure 1 where we have 25 time steps. In that case we can take larger time steps and get by with a smaller number of time steps N.

The general solution to the Black-Scholes backward parabolic partial differential equation in Equation 4. Note that the expectation is taken under the risk-neutral probability measure Q where the stochastic term in Equation 4. Using the mathematical law of expectation, the expectation for a call option in Equation 5. We thus need to integrate over all possible S-values that is larger than the strike K at expiry.

We can use either Equation 4. Note: N is the number of time steps and M the number of simulations. By scrutinising equations 5. It is evident that such an analysis is based on artificially recreating a chance process, running it many times and directly observing the results. Figure 2 shows 5 price paths generated with Equation 4. If we have a call option with a strike price of , Equation 5. This is shown in Table 1. However, crucial to obtaining the correct terminal values ST is that the volatilities we use in equations 4. This shows we first of all need the stock price at each time step, i.

We have given some examples in Table 1. But, further to this, we also need the instantaneous volatility at each time step for each stock price. We can obtain all of this from a three dimensional local volatility surface. We will discuss this in section 6 below. Whereas stochastic volatility and jump-diffusion models introduce new risks into the modeling process, local volatility models stay close to the Black-Scholes theoretical framework and only introduce more flexibility to the volatility.

This is one of the main reasons of fierce criticism of local volatility models Ayache et al. Thus, it is a mistake to interpret local volatility as a complete representation of the true stochastic process driving the underlying asset price. Local volatility is merely a simplification that is practically useful for describing a price process with non-constant volatility.

A local volatility model is a special case of the more general stochastic volatility models. So there is still just one source of randomness, ensuring that the completeness of the Black-Scholes model is preserved. Completeness is important, because it guarantees unique prices, thus arbitrage pricing and hedging Dupire, Note that Equation 6. The main problem is that the implied or traded volatilities are only known at discrete strikes K and expiries T.

This is why the parameterisation chosen for the implied volatility surface is very important. If implied volatilities are used directly from the market, the derivatives in Equation 6. This can still lead to an unsta- ble local volatility surface. Furthermore we will have to interpolate and extrapolate the given data points unto a surface. Since obtaining the local volatility from the data involves taking derivatives, the extrapolated implied volatility surface cannot be too uneven. If it is, this unevenness will be exacerbated in the local volatility surface showing that it is not arbitrage free in these areas.

This function is quadratic across strike and exponential across time. This three dimensional function is fitted to traded data. They further showed that all derivatives in Equation 6. There are no functional forms available and all derivatives in Equation 6. The local volatility surfaces are used when exotic options are evaluated. However, we also see from the local volatility surface that it has more curvature. This shows that the local volatility skew is twice that of the implied volatility skew.

Here we also show the local volatility surface with steeper sides. Continuing with our example: in section 5 and Table 1 we tabulated some price paths. We now want to calculate the Dupire local volatility for each stock price at each time step. This is the local volatility that should then be used in equations 4. On a practical note: in equations 4.

## Serie: The Wiley Finance Series

To apply Equation 6. In our example, the price paths were shown for a one year time period. We thus cannot generate the same price paths under a local volatility regime. So we run this experiment and generate 5 price paths under the ALSI local volatility surface. The newly generated price paths are shown in Figure 5. The actual numbers are listed in Table 2 and the corresponding local volatilities are listed in Table 3. Comparing graphs 5 and 2 and Tables 2 and 1 reveal that the stock prices are not that much different.

This is the way it should be because the local volatility does not differ that much from the implied volatility. However, even these slight differences, can lead to vastly different exotic option prices and especially, Greeks. We price this option using Monte Carlo simulation under a local volatility surface and using the closed-form solutions.

To explain the differences between the MC and closed-form solutions, we look at an example of a one month down-and-out put. Monte Carlo simulation is implemented by using Equation 5. We can further use either Equation 4. Having calculated the stock price at each time step ti , makes it quite easy to implement the boundary conditions. At each time step one needs to check if the stock price S ti is above or below the barrier H.

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