The Olympic Games have finished a couple of days ago. Two entire weeks of complete devotion for sport. Unfortunately I hadn’t got any ticket but I didn’t fail to watch many games on TV and internet. I was looking at decathlon men competition and I was very impressed by the general quality of these athletes. They have to be able to do everything: sprinting (100 m), jumping high, jumping fast (110 m hurdles), long, throw heavy (put shot) and light (javelin) things, running longer (400 m) and even longer (1500 m)… It became obvious in my mind that it was the quintessence of the sport, every athlete has to find the perfect balance between those different performances to compete efficiently. This sport induces all the quality of a strong man: power, endurance, flexibility, sprint…

Is it really true? Is it really the most balanced athlete who win the decathlon competition?

I decided to test this assumption with the results of the previous Olympic Games (Beijing, 2008). I only kept the athletes who have completed all the disciplines so that I can do the study on a data set without any missing values. I used the observations of the scores for each discipline which are calculated according to the time of the distance done by the athlete. If you are interested in those details, you can have a look at the way it is calculated on: http://www.iaaf.org/mm/Document/Competitions/ ... _Tables_of_Athletics_2011_23299.pdf.

I have been very surprised to see that the winner, Bryan Clay who has an average of 879 points per discipline, did very poorly in 400 meter (865 points), high jump (794 points) and in the 1500 meters race (522 points). On the contrary, he performed very well in 100 meters, 110 meters hurdle and long jump disciplines. Thus, I started wondering if the decathlon was not about power rather than about my so-called balance capacity in all the different areas.

Sir Prasanta Chandra Mahalanobis answered to this question some decades ago. In 1936 he decided to create a new function to measure the distance separating two observations. The most common distance is the Euclidian distance. However, this distance does not take into account two important elements. The first element is the variance of the different variables. Indeed, let's consider the high jump discipline and the pole vault, a gap of 30 centimeters between two athletes is huge in high jumping whereas it is a reasonable difference in pole vault. The reason is easy to understand, the variance in pole vaulting discipline is higher than in high jumping. Fortunately, most of the robustness to the variance is taken into account by the international athletic association (the federation who sets the scores) – although we will see that this is not perfectly true. But there is another problem which is even more important. The correlation of the different disciplines. For example the following graphic shows a positive correlation between shot put and disc throw, which, if we think about it, makes sense! Thus, if we look for the most complete athlete, there should be no cumulative rewards - we don’t want to give athletes too many points when they have performed well in two very similar disciplines. On the contrary, if two disciplines are negatively correlated such as 1500 meters and 100 meters we want to give extra points to athletes who perform well in both of the disciplines. The Mahalanobis distance has been created in this purpose.

If S is the matrix of variance-covariance of the data set, we can formally write the Mahalanobis distance between the vectors x and y as:
           

Once the matrix S is computed, we can calculate the Mahalanobis score for every athlete - say the distance between zero and the scores of the athlete in the different disciplines. It was unexpected to see that the gold medal would be claimed by Oleksiy Kasyanov who has finished 7th during the Olympic Games. On the contrary, Bryan Clay the Olympic champion would now rank 5th. You can find below two tables, the first one is the ranking of the athletes according to the Mahalanobis distance, and the second one is the official decathlon ranking. As you can see they are many differences. Therefore, decathlon is not the ultimate sport of complete athlete.

Mahalanobis Ranking
Athlete
Mahalanobis score
1
Oleksiy Kasyanov
790.60
2
Andrei Krauchanka
789.16
3
Maurice Smith
767.85
4
Leonel Suárez
754.27
5
Bryan Clay
742.40
6
Yordanis Garciá
737.40
7
Michael Shrade
723.31
8
Romain Barras
709.31
9
Aleksandr Pogorelov
701.18
10
Andres Raja
696.00
11
Roman Sebrle
693.79
12
Aleksey Drozdov
690.95
13
André Niklaus
687.12
14
Massimo Bertocchi
681.92
15
Jangy Addy
681.16
16
Mikk Pahapill
677.04
17
Mikalai Shubianok
667.82
18
Hadi Sepehrzad
653.71
19
Damjan Sitar
651.63
20
Eugene Martineau
637.66
21
Haifeng Qi
631.22
22
Aliaksandr Parkhomenka
630.64
23
Slaven Dizdarevic
607.92
24
Daniel Awde
607.78


Decathlon Ranking
Athlete
Decathlon Score
1
Bryan Clay
8791
2
Andrei Krauchanka
8551
3
Leonel Suárez
8527
4
Aleksandr Pogorelov
8328
5
Romain Barras
8253
6
Roman Sebrle
8241
7
Oleksiy Kasyanov
8238
8
André Niklaus
8220
9
Maurice Smith
8205
10
Michael Shrade
8194
11
Mikk Pahapill
8178
12
Aleksey Drozdov
8154
13
Andres Raja
8118
14
Eugene Martineau
8055
15
Yordanis Garciá
7992
16
Mikalai Shubianok
7906
17
Aliaksandr Parkhomenka
7838
18
Haifeng Qi
7835
19
Massimo Bertocchi
7714
20
Jangy Addy
7665
21
Daniel Awde
7516
22
Hadi Sepehrzad
7483
23
Damjan Sitar
7336
24
Slaven Dizdarevic
7021


The code (R):

#data and data3 are randomly generated for the example

a = rnorm(24)
data=data.frame(shotPut=a, discusThrow=0.5*a + 0.5 * rnorm(24))
data3=data.frame(X1=a, X2=0.5*a + 0.5 * rnorm(24), X3 = rnorm(24), X4 = rnorm(24), , X5 = rnorm(24), X6 = rnorm(24))

lm.shotPut = lm(data$shotPut~data$discusThrow)

plot(data$discusThrow, data$shotPut, axes=TRUE, ann=FALSE)
abline(lm.shotPut)
title(ylab="Score at shot put", xlab = 'Score at discus throw', col.lab=rgb(0,0,0))

Sigma = cov(data3)
distance = mahalanobis(data3,0 , Sigma, inverted = FALSE)

0

Add a comment

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")



0

Add a comment

Blog Archive
Translate
Translate
Loading