My previous post is about a method to simulate a Brownian motion. A friend of mine emailed me yesterday to tell me that this is useless if we do not know how to simulate a normally distributed variable.

My first remark is: use the rnorm() function if the quality of your simulation is not too important (Later, I'll try to explain you why the R "default random generation" functions are not perfect). However, it may be fun to generate a normal distribution from a simple uniform distribution. So, yes, I lied, I won't create the variable from scratch but from a uniform distribution. 

The method proposed is really easy to implement and this is why I think it is a really good one. Besides, the result is far from being trivial and is really unexpected. This method is called the Box-Muller method. You can find the proof of this method here. The proof is not very complicated, however, you will need a few mathematical knowledges to understand it.

Let u and v be two independent variables uniformly distributed.  Then we can define:

x = sqrt(-2log(u))sin(2 PI v)
y = sqrt(-2log(u))cos(2 PI v)

x and y are two independent and normally distributed variables. The interest of this method is its extreme simplicity in term of programming (We only need 9 lines if we don't want to test the normality of the new variables neither plot the estimation of the density).

We can obtain a vector of variables normally distributed. The Lillie test doesn't reject the null hypothesis of normal distribution. Besides, we can plot the estimation of the density of the variables. We obtain the following plot that looks indeed similar to the Gaussian density.



The program (R):

 # import the library to test the normality of the distribution
library(nortest)

size = 100000

u = runif(size)
v = runif(size)

x=rep(0,size)
y=rep(0,size)

for (i in 1:size){
  x[i] = sqrt(-2*log(u[i]))*cos(2*pi*v[i])
  y[i] = sqrt(-2*log(u[i]))*sin(2*pi*v[i])
}

#a test for normality
lillie.test(c(x,y))

#plot the estimation of the density
plot(density(c(x,y)))


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The financial market is not only made of stock options. Other financial products enable market actors to target specific aims. For example, an oil buyer like a flight company may want to cover the risk of increase in the price of oil. In this case it is possible to buy on the financial market what is known as a "Call" or a "Call Option".

A Call Option is a contract between two counterparties (the flight company and a financial actor). The buyer of the Call has the opportunity but not the obligation to buy a certain  quantity of a certain product (called the underlying) at a certain date (the maturity) for a certain price (the strike).

If the flight company doesn't want to suffer from the risk of an increase of oil on the market within the next year, a Call Option of maturity 1 year, with a certain strike K may be bought. In this case, if the oil price on the market is over the strike, the flight company will use the Call option to buy oil for the price of the strike. If the price of the oil is lower than the strike, then the flight will not use the option and will buy oil on the market.

As you can see, this contract is not symmetric  The flight company has an option (to buy or not to buy) while the second counterparty just follows the decision of the flight company. Therefore, the flight company will have to pay something to the second counterparty. One of the great question is how much should it be charged.

We will therefore use a pricing algorithm to estimate the price of this Call Option. We first build an algorithm to simulate the value of an asset in the model of Black-Scholes. Then we simulate the historical value of this asset in order to simulate the final value of the Call Option.

Indeed, if we know the value of the asset at the maturity (A(T)), we get directly the value of the Call Option of strike K at maturity : max(A(T) - K , 0).

Our asset is randomly distributed as:
A(t+dt) = A(t) (1 + r dt + v(B(t+dt) -B(t))

Where r is the interest rate, v the volatility of the asset and B a Brownian process (for more detail you can look at the post on Brownian motion).

After repeating the process many times, we estimate the mean of the Call Option for different strike in order to estimate the price of the Call Option.


As we can see the price of the Call decreases with the strike. Actually it converges towards 0. Indeed, the higher the strike is, the lower the chance are such that the asset goes over the strike.

What is done here can be done for many, many, many other financial products in order to price them by Monte Carlo.



The code (R):


sample.size <- 365
mu <- 0.1
sigma <- 0.2

a0 <- 1
Asset <- function(sample.size = 365, mu = 0.1, sigma = 0.2, a0 = 1){
  dt <- 1/sample.size
  sdt =sigma*sqrt(dt)
  gauss <- rnorm(sample.size)
  asset <- NULL
  asset[1] <- a0 + mu + sigma * gauss[1]
  test.default <- FALSE
  for(i in 2:365){
    if (!test.default){
      asset[i] <- asset[i-1] * (1 + dt*mu + sdt * gauss[i])
    }
    else{
      asset[i] <- 0
    }
    if (asset[i] <= 0){
      asset[i] <- 0
      test.default <- TRUE
    }
  }
  return(asset)
}


PriceEstimation <- function(t, tf = 365, r = 0.1, strike = 1, n = 1000, mu = 0.1, sigma = 0.2, a0 = 1){
  mean <- 0
  for(i1 in 1:n){
    mean <- mean + exp(-r*(tf-t)/tf) * max(0, (Asset(tf, mu, sigma, a0)-strike))
  }
  mean <- mean/n
  return(mean)
}

res <- NULL
for(i in 1:length(seq(0, 3, 0.05))){
  res[i] <- PriceEstimation(t = 0, tf = 365, r = 0.1, strike = seq(0, 3, 0.05)[i], n = 1000, mu = 0.1, sigma = 0.2, a0 = 1)
}

plot(seq(0,3, 0.05), res,type = 'l', xlab = "Strike Value",  ylab = "Price of the Call")



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