# Reading 10-MONTE CARLO SIMULATION

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2018-02-18 17:47

## Section 1

Question | Answer |
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The Lognormal Distribution | widely used for modeling the probability distribution of share and other asset prices. For example, the lognormal appears in the Black–Scholes–Merton option pricing model |

Black–Scholes–Merton | assumes that the price of the asset underlying the option is lognormally distributed. |

most noteworthy observations about the lognormal distribution | that it is bounded below by 0 and it is skewed to the right (it has a long right tail) |

price relative, S1/S0, | is an ending price, S1, over a beginning price, S0; it is equal to 1 plus the holding period return on the stock from t = 0 to t = 1: |

The continuously compounded return | associated with a holding period is the natural logarithm of 1 plus that holding period return, or equivalently, the natural logarithm of the ending price over the beginning price (the price relative) |

continuously compounded return from t to t + 1 | rt,t+1 = ln(St+1/St) = ln(1 + Rt,t+1) |

We can also express ST/S0 as the product of price relatives: | ST/S0 = (ST/ST–1)(ST–1/ST–2)…(S1/S0) |

E(r0,T) = E(rT–1,T) + E(rT–2,T–1) + … + E(r0,1) = μT (we add up μ for a total of T times) and | σ2(r0,T) = σ2T |

## Section 2

Question | Answer |
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MONTE CARLO SIMULATION | An approach to estimating a probability distribution of outcomes to examine what might happen if particular risks are faced. This method is widely used in the sciences as well as in business to study a variety of problems. |

Asian call option | European-style option with a value at maturity equal to the difference between the stock price at maturity and the average stock price during the life of the option, or $0, whichever is greater. |

steps of MC simulation | Specify the quantities of interest (option value, for example, or the funded status of a pension plan) in terms of underlying variables. The underlying variable or variables could be stock price for an equity option, the market value of pension assets, or other variables relating to the pension benefit obligation for a pension plan. Specify the starting values of the underlying variables. To illustrate the steps, we are using the case of valuing an Asian call option on stock. We use CiT to represent the value of the option at maturity T. The subscript i in CiT indicates that CiT is a value resulting from the ith simulation trial, each simulation trial involving a drawing of random values (an iteration of Step 4). Specify a time grid. Take the horizon in terms of calendar time and split it into a number of subperiods, say K in total. Calendar time divided by the number of subperiods, K, is the time increment, Δt. Specify distributional assumptions for the risk factors that drive the underlying variables. For example, stock price is the underlying variable for the Asian call, so we need a model for stock price movement. Say we choose the following model for changes in stock price, where Zk stands for the standard normal random variable: Δ(Stock price) = (μ × Prior stock price × Δt) + (σ× Prior stock price × Zk) In the way that we are using the term, Zk is a risk factor in the simulation. Through our choice of μ and σ, we control the distribution of stock price. Although this example has one risk factor, a given simulation may have multiple risk factors. Using a computer program or spreadsheet function, draw K random values of each risk factor. In our example, the spreadsheet function would produce a draw of K values of the standard normal variable Zk: Z1, Z2, Z3, …, ZK. Calculate the underlying variables using the random observations generated in Step 4. Using the above model of stock price dynamics, the result is K observations on changes in stock price. An additional calculation is needed to convert those changes into K stock prices (using initial stock price, which is given). Another calculation produces the average stock price during the life of the option (the sum of K stock prices divided by K). Compute the quantities of interest. In our example, the first calculation is the value of an Asian call at maturity, CiT. A second calculation discounts this terminal value back to the present to get the call value as of today, Ci0. We have completed one simulation trial. (The subscript i in Ci0 stands for the ith simulation trial, as it does in CiT.) In a Monte Carlo simulation, a running tabulation is kept of statistics relating to the distribution of the quantities of interest, including their mean value and standard deviation, over the simulation trials to that point. Iteratively go back to Step 4 until a specified number of trials, I, is completed. Finally, produce statistics for the simulation. The key value for our example is the mean value of Ci0 for the total number of simulation trials. This mean value is the Monte Carlo estimate of the value of the Asian call. |

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