CS计算机代考程序代写 Excel Financial Engineering

Financial Engineering
IC302
8. Basic Numerical Procedures for Valuing Options (Monte Carlo Methods)
David Oakes
–
Autumn Term 2020/1

Monte Carlo Method
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Monte Carlo Method
• In risk-neutral valuation, we calculate the expected payoffs to the derivative under the risk-neutral probabilities and then discount them at the risk-free rate.
• Previously, we did this using binomial trees.
• Another way of calculating the expectation is to generate a sample of values for the derivative’s payoff and then estimate the expectation by averaging the sample values.
• This is the Monte Carlo method.

Modeling Asset Prices
• In this lecture, we explain how the Monte Carlo method works and how it can be used to value derivatives for which binomial methods are inappropriate or inefficient.
• For the Monte Carlo method to work, the sample must be generated from a population that has a distribution consistent with the relevant risk-neutral probabilities.
• We need to build a model for the asset price. We begin by showing how this can be done. Later, we will show how Monte Carlo methods are used to price derivatives.
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Modelling Randomness
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Brownian Motion
• Our starting point is Brownian motion.
• Brownian motion is a random process (i.e. a variable that changes randomly over time) that evolves continuously over time and that has the property that its change over any time period is normally distributed with mean zero and variance equal to the length of the time period.
• Brownian motion takes its name from the fact that it was initially used to describe the random movement of a microscopic particle suspended in a fluid, a physical phenomenon first studied by the botanist Robert Brown.

Brownian Motion
• Albert Einstein proposed a mathematical model for Brownian motion in 1905, which helped establish the validity of the molecular theory of matter.
• The first mathematically rigorous treatment of the model was given by Norbert Wiener in 1923; for this reason, the random process that we call Brownian motion is also known as the Wiener process.
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Randomness of Brownian Motion
• Brownian motion is very random. Suppose that we divide
atimeinterval 0,𝑇 asfollows:
0=𝑡! <𝑡" <⋯<𝑡# <⋯<𝑡$%" <𝑡$ =𝑇 • Let 𝐵 𝑡 be a Brownian motion, and calculate the sum of squared changes over the interval: $ ' • The limit of this sum of squared increments as the partition of the interval becomes increasingly fine is called the quadratic variation of the process. '𝐵𝑡# −𝐵𝑡#%" #&" 8 Randomness of Brownian Motion • For an ordinary function, the quadratic variation is zero, because the function is approximately linear at each point. • But for Brownian motion, the quadratic variation over the interval 0,𝑇 isequalto𝑇. • Similarly, the total variation of a process is the limit of the sum of the absolute value of changes over the interval. • For an ordinary function, this is finite. But for Brownian motion, it is infinite. No matter how closely we look at a path of Brownian motion, it is still random! 9 Simulating Brownian Motion • We can simulate Brownian motion using a discrete-time approximation. • Divide the time interval 0, 𝑇 to be covered by the simulation into 𝑛 intervals of length Δ𝑡 = 𝑇⁄𝑛 . • Define the value of the Brownian motion at the end of each interval Δ𝑡 as its value at the beginning of the interval plus a normally distributed variable with mean 0 and variance Δ𝑡. 10 Simulating Brownian Motion • The increment in the simulated Brownian motion over the interval is: Δ𝐵=𝜖 Δ𝑡 where 𝜖 is a random variable with a standard normal distribution (i.e. a normal distribution with mean 0 and standard deviation 1). • Over each interval Δ𝑡, the increment Δ𝐵 will have expected value zero and variance Δ𝑡, just as we expect for a Brownian motion. 11 Simulating Brownian Motion • We can model this discrete-time approximation in Excel, using the function =NORM.S.INV(RAND()) to generate a standard normal random number 𝜖 at each time step. • Brownian motion has the kind of randomness that we might associate with stock prices. But we know that different stocks have different volatilities (i.e. standard deviations of return) and levels of expected return. • The same is true of other risky assets. How can we build these properties into our model of stock prices? 12 Itô Process • An Itô process is a variable 𝑋 that evolves over time according to: 𝑑𝑋𝑡 =𝜇𝑋,𝑡𝑑𝑡+𝜎𝑋,𝑡𝑑𝐵𝑡 • Here, 𝑑𝑋 𝑡 is to be understood as the change in 𝑋 (denoted Δ𝑋 ) evaluated in the limit, as the time increment Δ𝑡 goes to zero. • The notation is similar to that used in ordinary calculus when we talk about the differential of a variable. • 𝐵 𝑡 is a Brownian motion. 13 Drift and Diffusion • The function 𝜇 𝑋, 𝑡 is the drift of the process. We can think of 𝜇 𝑋, 𝑡 𝑑𝑡 as the expected change in 𝑋 in the next ‘instant’. • The function 𝜎 𝑋, 𝑡 is the diffusion of the process. It scales the randomness of Brownian motion so that its instantaneous variance is 𝜎2 𝑋, 𝑡 . • We want a stock price process that has non-zero drift and a diffusion that is scaled to the stock’s volatility, so we model stock prices (and the prices of other risky assets) as Itô processes. 14 Mathematics of Itô Processes • An Itô process is a function of Brownian motion. • Unfortunately, the mathematics of functions of Brownian motion are complicated by the fact that Brownian motion has non-zero quadratic variation. • Consider a twice-differentiable function 𝐺 𝑋, 𝑡 . the rules of ordinary calculus, we would write the differential 𝑑𝐺 of this function as: 𝑑𝐺 = 𝜕𝐺 𝑑𝑋 + 𝜕𝐺 𝑑𝑡 𝜕𝑋 𝜕𝑡 Using 15 Mathematics of Itô Processes • In our case, however, 𝑋 is an Itô process, with stochastic (i.e. random) differential: 𝑑𝑋𝑡 =𝜇𝑋,𝑡𝑑𝑡+𝜎𝑋,𝑡𝑑𝐵𝑡 • This means that the rules of ordinary calculus do not apply. To see why, consider a Taylor series expansion of Δ𝐺, an increment in 𝐺: Δ𝐺=𝜕𝐺Δ𝑋+𝜕𝐺Δ𝑡+1𝜕'𝐺 Δ𝑋 '+ 𝜕'𝐺 Δ𝑋Δ𝑡+1𝜕'𝐺 Δ𝑡 '+⋯ 𝜕𝑋 𝜕𝑡 2𝜕𝑋' 𝜕𝑋𝜕𝑡 2 𝜕𝑡' • If 𝑋 is an ordinary variable, then we can ignore the ‘second-order’ terms in Δ𝑋 2, Δ𝑋 Δ𝑡, and Δ𝑡 2, since they will go to zero in the limit, as Δ𝑋 and Δ𝑡 go to zero. 16 Mathematics of Itô Processes • This leads to the differential expression from ordinary calculus that we saw earlier. • But if 𝑋 is an Itô process, we cannot ignore the term in Δ𝑋 2, since it will include a term in Δ𝐵 2. • As we saw earlier when discussing the properties of Brownian motion, squared increments of Brownian motion do not ‘disappear’ in the limit, as Δ𝑡 goes to zero. • Instead, the limit of the sum of these squared increments (i.e.quadraticvariation)goesto𝑇overanyinterval 0,𝑇. 17 Mathematics of Itô Processes • It follows that these terms will be of the same order as Δ𝑡. • Taking limits in the Taylor series expansion as Δ𝑋 and Δ𝑡 go to zero, this leads us to write the differential of 𝐺 as: 𝑑𝐺=𝜕𝐺𝑑𝑋+𝜕𝐺𝑑𝑡+1𝜕'𝐺𝜎' 𝑋,𝑡 𝑑𝑡 𝜕𝑋 𝜕𝑡 2𝜕𝑋' where the extra term in 𝑑𝑡 is a result of the non-zero quadratic variation of Brownian motion. 18 Itô’s Lemma • Substituting for 𝑑𝑋 in this expression gives: 𝑑𝐺= 𝜕𝐺𝜇 𝑋,𝑡 +𝜕𝐺+1𝜕'𝐺𝜎' 𝑋,𝑡 𝑑𝑡+𝜕𝐺𝜎 𝑋,𝑡 𝑑𝐵 𝑡 𝜕𝑋 𝜕𝑡 2𝜕𝑋' 𝜕𝑋 • This result is known as Itô’s lemma. • Itô’s lemma is a rule for finding the stochastic differential of a function of an Itô process. 19 Using Itô’s Lemma • In derivatives markets, we model asset prices as Itô processes. Itô’s lemma tells us how to express the random behaviour of the derivative in terms of the random behaviour of the underlying asset. • This is of enormous practical value, since we need to understand how a derivative will behave in order to work out how to hedge and price it. • Ito’s lemma therefore plays a key role in deriving pricing results such as those in the Black-Scholes-Merton model. 20 Simulating Asset Prices 21 Stock Price Process • We can use Itô’s lemma to help us construct and simulate a model for the price of a stock or some other risky asset. • Suppose that the stock price 𝑆 𝑡 at some future date 𝑡 is given by: 𝑆 𝑡 = 𝑆 0 e x p 𝜇 − 12 𝜎 ' 𝑡 + 𝜎 𝐵 𝑡 where 𝜇 and 𝜎 are constants, 𝑆 0 is the current stock price, and 𝐵 𝑡 is a Brownian motion. Geometric Brownian Motion • Applying Itô’s lemma, the stochastic differential corresponding to this stochastic integral is: 𝑑𝑆𝑡 =𝜇𝑆𝑡𝑑𝑡+𝜎𝑆𝑡𝑑𝐵𝑡 • This is an Itô process, with drift 𝜇𝑆 𝑡 and diffusion 𝜎𝑆 𝑡 . • We interpret 𝜇 as the expected return on the stock and 𝜎 as the volatility of the stock price. • The stock price process that we have just described is called a geometric Brownian motion. It is widely used as a model of asset price behaviour in derivatives modelling. 23 Discrete-Time Approximation • Following the same principle that we applied earlier when simulating Brownian motion, we can write a discrete-time approximation to the stock price process as: 𝑆𝑡+Δ𝑡 −𝑆𝑡 =𝜇𝑆𝑡Δ𝑡+𝜎𝑆𝑡𝜖 Δ𝑡 where 𝜖 is a standard normal random variable. • In practice, it turns out to be more accurate to work with the natural logarithm ln 𝑆 rather than with 𝑆. The natural logarithm of the integral form of 𝑆 given earlier is: l n 𝑆 𝑡 = l n 𝑆 0 + 𝜇 𝑡 − 12 𝜎 ' 𝑡 + 𝜎 𝐵 𝑡 24 Discrete-Time Approximation • Applying Itô’s lemma, we find the stochastic differential: 𝑑 l n 𝑆 𝑡 = 𝜇 − 12 𝜎 ' 𝑑 𝑡 + 𝜎 𝑑 𝐵 𝑡 • The discrete-time approximation to this equation is: ln𝑆 𝑡+Δ𝑡 −ln𝑆 𝑡 = 𝜇−12𝜎' Δ𝑡+𝜎𝜖 Δ𝑡 25 Discrete-Time Approximation • By applying the exponential function to both sides, we see that this can also be written as: 𝑆 𝑡 + Δ 𝑡 = 𝑆 𝑡 e x p 𝜇 − 12 𝜎 ' Δ 𝑡 + 𝜎 𝜖 Δ 𝑡 • We can use this equation to simulate a path for 𝑆. • Notice that, at each time step along the path, we must generate a new standard normal random number 𝜖. 26 Girsanov’s Theorem • We saw in Lecture 5 that, when we move from the real world to the risk-neutral world, the expected return on the stock becomes equal to the risk-free interest rate. • It turns out, however, that this change does not affect the volatility of the stock. It remains the same in the risk- neutral world as in the real world. • This is a consequence of a more general result known as Girsanov’s theorem, which we also used in Lecture 6. 27 Girsanov’s Theorem • In practice, Girsanov’s theorem means that in order to simulate stock prices (or the prices of other risky assets) in the risk-neutral world, we set the drift 𝜇 equal to the risk- free interest rate 𝑟, but the diffusion 𝜎 remains the same. • The randomness in the stock price continues to be generated by a Brownian motion under the risk-neutral probabilities. 28 Risk-Neutral Stock Price Process • The discrete-time approximations that we use to simulate the stock price and the logarithm of the stock price in the risk-neutral world are therefore the same as those defined earlier, but with the drift set equal to the risk-free rate: 𝑆𝑡+Δ𝑡 −𝑆𝑡 =𝑟𝑆𝑡Δ𝑡+𝜎𝑆𝑡𝜖 Δ𝑡 ln𝑆 𝑡+Δ𝑡 −ln𝑆 𝑡 = 𝑟−12𝜎' Δ𝑡+𝜎𝜖 Δ𝑡 𝑆 𝑡 + Δ 𝑡 = 𝑆 𝑡 e x p 𝑟 − 12 𝜎 ' Δ 𝑡 + 𝜎 𝜖 Δ 𝑡 29 Monte Carlo Simulation • We will use the last of these approximations to generate stock prices under the risk-neutral probabilities so as to generate a sample of payoffs to a derivative. • This is a Monte Carlo simulation. • The average of the discounted payoffs in the sample is an estimate of the derivative’s price. 30 Monte Carlo Valuation 31 Monte Carlo Valuation • To price a derivative by the Monte Carlo method, we first generate a very large number of simulated paths for the stock price under the risk-neutral probability measure. • We then calculate the payoff to the derivative along each path and discount this payoff to the present at the risk- free rate. • The average of the discounted payoffs over all the paths is a Monte Carlo estimate of the derivative’s value. Valuation of European Call • We can illustrate the principles underlying Monte Carlo valuation by pricing the same option to which we applied the multi-period binomial model in Lecture 5. • The option is a one-year European call with parameters: Parameter Value 𝑆! 100 𝐾 100 𝑟 0.06 𝑇 1 year 𝜎 0.20 (i.e. 20% per year) • 𝑆> is the current stock price, 𝐾 is the exercise price, 𝑟 is the risk-free interest rate, 𝑇 is the time to expiry, and 𝜎 is the volatility of the stock price.
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Monte Carlo Simulation Trials
• In order to price the option, we will generate 𝑀 simulated paths for the stock price.
• Each of these paths is one trial in the simulation.
• Since the option is European, our task is made a bit easier. A European option can only be exercised at expiry, so we only need to simulate the terminal value of the stock price (i.e. its price on the option expiry date) in each trial.
• In the next section we will look at a more complicated example in which we must simulate the whole price path.
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Stock Price Simulation
• We saw in the last section that the geometric Brownian motion that we are using to model the stock price in the risk-neutral world has the stochastic integral form:
𝑆 𝑡 = 𝑆 0 e x p 𝑟 − 12 𝜎 ‘ 𝑡 + 𝜎 𝐵 𝑡
• We can therefore simulate the value of the stock price at the option expiry date 𝑇 by:
𝑆 𝑇 = 𝑆 0 e x p 𝑟 − 12 𝜎 ‘ 𝑇 + 𝜎 𝜖 T
where 𝜖 is a standard normal random variable. We can generate a standard normal random number in an Excel spreadsheet by using the function =NORM.S.INV(RAND()).
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Stock Price in First Trial
• Suppose that the standard normal random number that we generate in the first trial of the simulation is 0.4557.
• Using the parameter values for the example, the simulated stock price at the option expiry date one year from now in this first trial is:
𝑆 𝑇 = 𝑆 0 e x p 𝑟 − 12 𝜎 ‘ 𝑇 + 𝜎 𝜖 T
=100exp 0.06−120.2′ 1 +0.2 0.4557 1 =114.01
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Option Payoff and Further Trials
• This results in a payoff to the European call option of: max 𝑆 𝑇 −𝐾,0 = max 114.01−100,0 = 14.01
• We keep track of this value and go back to generate a standard normal random number for the second trial.
• Suppose that this time the number is -0.2227. This gives a simulated stock price at the option expiry date of:
𝑆 𝑇 =100exp 0.06−120.2’ 1 +0.2 −0.2227 1 =99.55
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Payoffs For Many Trials
• This time, the terminal stock price is below the strike price of the option, so the payoff is zero:
max 𝑆 𝑇 −𝐾,0 = max 99.55−100,0 = 0.00
• Again, we keep track of this and go back to simulate a stock price for the third trial.
• We continue like this until we have completed 𝑀 trials, where 𝑀 is a large number (e.g. 10,000 or 100,000).
• For each trial, we have the payoff to the option that would result from the simulated stock price for that trial.
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Expected Payoff and Option Value
• We add up all the payoffs and divide by 𝑀 to calculate their average; this is our estimate of the expected payoff.
• Suppose that the average payoff across all the paths turns out to be 11.95. Discounting this expected payoff at the risk-free rate gives an estimate of the option value:
𝑀
• Our Monte Carlo estimate of the option price is $11.25.
𝑀𝑜𝑛𝑡𝑒 𝐶𝑎𝑟𝑙𝑜 𝑂𝑝𝑡𝑖𝑜𝑛 𝑉𝑎𝑙𝑢𝑒 = 𝑒%() #&”
# = 𝑒%!.!, ” 11.95 = 11.25
∑* 𝑃𝑎𝑦𝑜𝑓𝑓
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Expected Payoff and Discounting
• In the case of the European call, since all the payoffs occur on the same date, we could instead have discounted the
payoff in each trial individually and then averaged over the discounted payoffs; this gives the same result.
• For derivatives where payoffs may occur on different dates, depending on the path followed by the asset price, we should in fact discount before averaging the payoffs.
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Accuracy of Monte Carlo
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Accuracy of Monte Carlo
• The accuracy of the result given by Monte Carlo simulation depends on the number of trials.
• In general, a larger number of trials should give more accurate results.
• It is usual to report the standard deviation of the sample estimates (i.e. the standard deviation of the prices for each simulated path) when using Monte Carlo methods.
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Confidence Interval for Price
• The standard error of the estimated price is:
𝑠 𝑀
and 𝑀 is the number of trials.
• We can use standard methods from statistical inference to
calculate a 95% confidence interval for the price: 𝑃𝑟𝑖𝑐𝑒 = 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 ± 1.96 𝑆𝐸
𝑆𝐸 =
where 𝑠 is the sample standard deviation of the estimate
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Confidence Interval Example
• Suppose that a Monte Carlo simulation with 1,000 trials results in an estimated price of 11.25 with a standard deviation of 14.87. The standard error of the estimate is:
𝑆𝐸= 𝑠 = 14.87 =0.47 𝑀 1000
• This gives a 95% confidence interval of: 11.25±1.96 0.47 → 10.33,12.17
• Based on our estimate, we are 95% confident that the true price lies between 10.33 and 12.17.
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Monte Carlo and BSM Price
• For the European call option in this example, the Black- Scholes-Merton price is 10.99, which lies within the 95% confidence interval.
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Increasing Accuracy
• Notice that our uncertainty about the price is inversely related to the square root of the number of trials.
• To double the accuracy, we would need to quadruple the number of trials. To increase the accuracy by a factor of 10, we would need 100 times as many trials.
• Increasing accuracy in Monte Carlo methods can be very expensive!
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Antithetic Variates
• One way to increase the accuracy of Monte Carlo estimates without doing a great deal of additional computation is through antithetic variates.
• Normally, for each trial we generate one or more standard normal random variables, which we use to compute the stock price or path.
• We then calculate the derivative payoff for that path.
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Antithetic Variates
• Before moving on to the next trial, however, we can change the signs of the standard normal variables that we
generated, recompute the stock price path, and calculate the payoff for this new path.
• In effect, this gives us two estimates of the payoff.
• We average these two estimates and move on to the next
trial. This is the method of antithetic variates.
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Antithetic Variates and Accuracy
• The advantage of antithetic variates is twofold: it reduces the number of standard normal samples that must be computed to generate a given number of paths and it reduces the variance of the sample paths.
• This can lead to a significant increase in the precision of the estimate that we obtain for the derivative price.
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Path-Dependent Options
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Monte Carlo and Binomial Trees
• How does valuation using Monte Carlo simulation compare with valuation using binomial trees?
• In Monte Carlo simulation, we generate 𝑀 random values for the terminal stock price 𝑆 𝑇 , calculate the option payoff for each simulated price, and estimate the expected payoff by averaging the 𝑀 resulting values.
• In effect, we approximate the distribution of 𝑆 𝑇 by an 𝑀-point distribution, with each point assigned equal probability.

Monte Carlo and Binomial Trees
• In the multi-period binomial model, we specifically choose the points and their probabilities by constructing a binomial tree.
• This allows us to use a much smaller number of points, while implicitly allowing many ‘paths’ through the tree.
• This means that we can use a much smaller number of points to obtain the same accuracy.
• This makes the binomial method faster.
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American Exercise
• The binomial method can also easily handle American exercise, as we saw when we used it to value an American put option in Lecture 5.
• Valuing options with American exercise using Monte Carlo methods is much more difficult (though not impossible), because we must assess at each point in time whether it is better to continue to hold the option or to exercise it now.
• Using the Monte Carlo method, we only look at one possible path in each trial, so it is harder to estimate the continuation value of the option at each point in time.
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Path Dependence
• But what about derivatives that are path-dependent, in the sense that the payoff depends on the actual path
followed by the underlying asset price during the derivative’s life, rather than just its terminal value?
• To value a path-dependent option using an 𝑛-step binomial tree, we would need to analyze all 2c paths through the tree. For a tree with 30 time steps, this is over one billion paths!
• For these options, Monte Carlo methods may be much more efficient.
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Path-Dependent Exotics
• Many exotic options have path-dependent payoffs. – Barrier options
– Lookback options
– Asian options
• In simple cases, it may be possible to find analytical formulas for the prices of these options in the BSM framework, as we discussed in Lecture 7.
• More complex path-dependent exotics, however, often require the use of Monte Carlo methods.
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Average Price Call
• In order to illustrate the application of Monte Carlo simulation to path-dependent options, consider the case of an average price call.
• The payoff to this option at expiry date 𝑇 is: max𝐴𝑇 −𝐾,0
where 𝐴 𝑇 is the average price of the underlying stock during the option’s life.
• In order to calculate the average, we must know the prices at which the stock traded during the option’s life.
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Monte Carlo Simulation
• We must therefore simulate the whole of the path for the stock price under the risk-neutral probability measure.
• We can do this by using the discrete-time approximation: 𝑆 𝑡 + Δ 𝑡 = 𝑆 𝑡 e x p 𝑟 − 12 𝜎 ‘ Δ 𝑡 + 𝜎 𝜖 Δ 𝑡
• For each of the 𝑀 trials, we generate a path of 𝑛 stock prices, starting at the beginning of the option’s life and moving forward in time to its expiry date.
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Monte Carlo Value for Option
• We then calculate the average of the prices observed over that path and use the average to determine the payoff to the average price option for that path.
• The average of the payoffs over the 𝑀 trials is an estimate of the expected payoff to the option under the risk- neutral probability measure.
• Discounting the expected payoff at the risk-free rate gives a Monte Carlo estimate of the option’s price.
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Options on Multiple Assets
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Options on Multiple Assets
• Monte Carlo simulation is also much better suited than binomial trees to derivatives that depend on several underlying assets.
• A common example is a basket option (i.e. a call or put on a portfolio of assets), which we discussed in Lecture 7.
• We can build a binomial tree in multiple dimensions to value the option, but the computational burden becomes unmanageable when there are three or more assets.

Monte Carlo vs. Binomial Trees
• This is because, in binomial methods, the work required increases exponentially with the number of variables.
• With Monte Carlo simulation, the main additional burden is simulating a set of 𝑀 paths for each additional asset.
• This means that the time taken for the simulation increases linearly with the number of underlying assets.
• This makes the Monte Carlo method much better than binomial trees for options with multiple underlying assets.
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Correlation
• For derivatives that depend on more than one underlying asset, however, the paths that we generate must take into
account the correlation or dependence between the assets, since this may affect the expected payoff.
• How can we do this?
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Simulating Correlated Assets
• Consider the case of two correlated assets 𝑆d and 𝑆d. The discrete time approximations used to generate simulated paths for the prices of these two assets are:
𝑆” 𝑡+Δ𝑡 =𝑆” 𝑡 exp 𝑟−12𝜎”‘ Δ𝑡+𝜎”𝜖” Δ𝑡 𝑆’ 𝑡+Δ𝑡 =𝑆’ 𝑡 exp 𝑟−12𝜎’ Δ𝑡+𝜎’𝜖’ Δ𝑡
• Now, however, 𝜖d and 𝜖2 must be correlated standard normal random variables, with correlation coefficient 𝜌.
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Simulating Correlated Assets
• We can construct these correlated standard normal random variables by starting with two independent standard normal random variables 𝑧d and 𝑧2.
• The correlated variables 𝜖d and 𝜖2 are then calculated as: 𝜖” = 𝑧”
𝜖’=𝑧”𝜌+𝑧’ 1−𝜌’
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Simulating Correlated Assets
• For each trial in the simulation, we first generate 𝑛 standard normal random numbers for each of the two assets independently.
• We then combine these as shown above to generate the correlated standard normal random numbers 𝜖d and 𝜖2 that are used to generate the paths for the asset prices.
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Cholesky Decomposition
• Equivalently, we could instead write the discrete-time approximations for the asset prices directly in terms of the independent standard normal random variables 𝑧d and 𝑧2:
𝑆” 𝑡+Δ𝑡 =𝑆” 𝑡 exp 𝑟−12𝜎”‘ Δ𝑡+𝜎”𝑧” Δ𝑡
𝑆’ 𝑡+Δ𝑡 =𝑆’ 𝑡 exp 𝑟−12𝜎’ Δ𝑡+𝜎’𝑧”𝜌 Δ𝑡+𝜎’𝑧’ 1−𝜌’ Δ𝑡
• This method of generating simulated paths for correlated assets, which can be extended to cases in which there are many assets, is based on the Cholesky decomposition of the covariance matrix.
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Control Variates
• We saw earlier that the efficiency of Monte Carlo simulation can be improved through variance reduction techniques such as antithetic variates.
• Another way in which Monte Carlo estimates can sometimes be improved is through control variates.
• Suppose, for example, that we want to price a discretely sampled average-price call.
• We will use Monte Carlo methods, as there is no analytical solution for the option price.
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Control Variates
• But there is an analytical solution for a related contract, a discretely sampled geometric-average-price call.
• We simulate the geometric-average-price call along with the option in which we are interested and compare its sample mean (i.e. the Monte Carlo estimate) with its known mean (i.e. the price from the analytical formula).
• We use the relationship between the simulated and analytical values for this option to adjust the estimate for our option upward or downward as appropriate.
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References Main reading:
• Hull, ch. 21 Background reading: • Hull, ch. 14
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