Hello (New World!), 

My name is Edwin, I’m a 22 year-old French student in applied mathematics. In particular, I study probability, statistics and risk theory. We learn interesting things but there is still one question I feel stupid about: Is statistics, actually, useful? I mean not theoretically but in the real life. Many friends of mine asked me this, and I vainly struggle in answering with understandable and cogent examples. I can’t get rid of the stereotypes such as the over optimistic forecasts of GDP gross given by government neither of the very easy statistics which are about collecting data rather than proper data analyses.


Because, mathematics should never be like that... (source:http://lovestats.wordpress.com/dman/)

This question is the reason of this blog, I would like to explain by examples how statistics and probability could offer a useful perception of our environment. Indeed, according to the Oxford Dictionary, perception is the ability to see, hear, or become aware of something through the senses, or in a second meaning, the way in which something is regarded, understood, or interpreted. This is how probability and statistics are useful. They give a perception of our environment, which might be wrong, as well as our sight can suffer from an optical illusion, but the perception may help us to understand and become aware of phenomena of our environment.

To answer “the” question I will try to use examples as easy as possible so that not only the conclusions but also the process are understandable. As you will certainly see, I like sport, finance and risk issues, and I will use many examples from these areas to answer the question. I will use the software R to illustrate the different topics. Although, I’m not a great programmer, feel free to use my programs if you think they could be useful.  They are certainly not as efficient as possible and, again, feel free to comment any of my programs if you have any ideas to improve the program or the method.

Finally, before blogging for the first time in my life, I’d like to thank all the people who settle in my mind the unfathomable question of the utility of statistics. In particular Claire, Justine, Arthur, Clement and Rudy (even though the three last ones certainly know better than I do how useful is statistics) for the countless, long and unfinished delicious discussions we had about statistics and probability. Claude for his unconditional but fair question: “You like mathematics, but what kind of work can you do with mathematics?” and many other people, who, I am sure will recognize they have left their print on this blog.

Edwin.
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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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