I'm very glad if you use or at least read sometimes my blog. However, I'd like to give you a list of wonderful blogs which are useful if you are interested in applied mathematics and programming for simulation.
My favorite one is written by Arthur Charpentier, a French mathematician. His blog is called Freakonometrics and is a wonderful store of R codes and explanations of statistics:
http://freakonometrics.blog.free.fr/
R websites:
-I can't do a reasonable list of websites without talking about R-cran the website of R. http://cran.r-project.org/. You can find nearly everything about R on this website, especially an important documentation about the software.
-R-bloggers : http://www.r-bloggers.com/. This blog is a useful summary of many blogs related to R. If you are interested in R in general, or if you want to learn how to code in R or if you are interested in a particular mathematical issue, this website is a great hub of knowledge.
-R-statistics (http://www.r-statistics.com/) is another valuable resource for examples of codes in R.
-SAS and R (http://sas-and-r.blogspot.co.uk/). Even though this website is not only in R but also in SAS, I love this website. It made my life better when I was coding in SAS or in R.
-Stackoverflow (http://stackoverflow.com/). This website is about programming in general. It is useful when you have any bug you can't solve. It's a forum where people help or ask for help about any programming problems. When you can't already find someone who had experienced the same problem than you, you can ask for help. The answers are very reactive and you don't have to wait too long before being helped.
Mathematics websites:
-A good blog about probability and its applications is http://probabilityandstats.wordpress.com/. I like it because funny topics are analyzed with a serious approach.
-Gower's weblog is certainly the most well known blog about mathematics. Therefore it has to be in my list. http://gowers.wordpress.com/
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")
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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