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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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).

I found a golden website. The blog of Esteban Moro. He uses R to work on networks. In particular he has done a really nice code to make some great videos of networks. This post is purely a copy of his code. I just changed a few arguments to change colors and to do my own network.

To create the network, I used the  Barabási-Albert algorithm that you can find at the end of the post on the different algorithms for networks. Igraph is the library which has been used.
3

As you have certainly seen now, I like working on artificial neural networks. I have written a few posts about models with neural networks (Models to generate networks, Want to win to Guess Who and Study of spatial segregation).

Unfortunately, I missed so far a nice and pleasant aspect of networks : its graphical approach. Indeed, plots of neural networks are often really nice and really useful to understand the network.

Sometimes such a graph can point out some characteristics of the network.
1

I already talked about networks a few times in this blog. In particular, I had this approach to explain spatial segregation in a city or to solve the Guess Who? problem. However, one of the question is how to generate a good network. Indeed, I aim to study strategy to split a network, but I need first to work with a realistic neural network. I could have downloaded data of a network, but I'd rather study the different models proposed to generate neural networks.

The function apply() is certainly one of the most useful function. I was scared of it during a while and refused to use it. But it makes the code so much faster to write and so efficient that we can't afford not using it. If you are like me, that you refuse to use apply because it is scary, read the following lines, it will help you. You want to know how to use apply() in general, with a home-made function or with several parameters ? Then, go to see the following examples.
1

Have you ever played the board game "Guess who?". For those who have not experienced childhood (because it might be the only reason to ignore this board game), this is a game consisting in trying to guess who the opponent player is thinking of among a list of characters - we will call the one he chooses the "chosen character". These characters have several characteristics such as gender, having brown hair or wearing glasses.

If you want to choose randomly your next holidays destination, you are likely to process in a way which is certainly biased. Especially if you choose randomly the latitude and the longitude. A bit like they do in this lovely advertising (For those of you who do not speak French, this is about a couple who have won the national gamble prize and have to decide their next travel. The husband randomly picks Australia and the wife is complaining : "Not again!").
4

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.

The Brownian motion is certainly the most famous stochastic process (a random variable evolving in the time). It has been the first way to model a stock option price (Louis Bachelier's thesis in 1900).

The reason why is easy to understand, a Brownian motion is graphically very similar to the historical price of a stock option.
1

The merge of two insurance companies enables to curb the probability of ruin by sharing the risk and the capital of the two companies.

For example, we can consider two insurance companies, A and B. A is a well known insurance company with a big capital and is dealing with a risk with a low variance. We will assume that the global risk of all its customers follow a chi-square distribution with one degree of freedom.
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