美式選擇權定價實作：American puts pricing using least-squares Monte Carlo method

這次是比較簡單的蒙地卡羅演算法選擇權定價，學期快進入尾聲了有點不知所措啊~

一、問題描述：American puts pricing using least-squares Monte Carlo method

Write a least-squares Monte Carlo program to price American puts.

Output:

Price and
Standard derivation(sample)

Inputs:

S (stock price at time 0),
X (strike price),
T (maturity in years),
s (%) (annual volatility),
r (%) (continuously compounded annual interest rate),
n (number of periods), and
k (number of simulation paths).

For example, when

S = 101, X = 105, T = 1 (years), s = 15%, r = 2%, n = 50, and k = 100,000,

the price is about 7.3642 and the standard derivation is about 0.9975.

二、執行方式與執行結果

直接使用 R 直譯器執行 LSMC_implement.R 即可

LSMC_implement.R : 本次實作的主程式，可於參數區塊調整測試參數，答案(price, sd)將以標準輸出印於terminal
LSMC_package.R : 第三方套件 LSMonteCarlo，用於驗證答案（需要安裝 LSMonteCarlo 套件）
textbook_example.txt : 投影片講義之測試資料 http://www.csie.ntu.edu.tw/~lyuu/finance1.html

使用題目測試資料跑出來的結果如下：（目前還不知為何sd不太準）

用 IDE Rstudio 直接執行之畫面：

三、設計與實作

1. Create two table : Stock Price Path Table and Stock Path Table

Generate the multivariate normal distribution for the Monte Carlo pricing of multivariate derivatives. using the formula below :

then we can generate a stock price paths table : StockPath[k,n]

and initial a cash flow table like this :

R code

# Create Stock Price Path Table
StockPath <- matrix(NA, nrow=k, ncol=n)

for(i in 1:k) {
  front_para <- (r - s^2/2) * deltaT
  back_para <- s * (sqrt(deltaT)) * rnorm(n, mean=0, sd=1)
  StockPath[i,] <- S * exp(cumsum(front_para+back_para))
}
StockPath[ StockPath>X ] <- 0

# Create Cash Flow Table
CashFlow <- matrix(0, nrow=k, ncol=n)
CashFlow[,n] <- ifelse(StockPath[,n]!=0, X-StockPath[,n], 0)

2. Backward pricing with regression

將 Cash Flow Table 從最後一期一直推回第一期

Select in-the-money path : The cash flows at each year are the exercise value if the put is in the money.
Regression : Only in-the-money paths will be used in the regression
Compare : Compare Exercise value and Continuation value

R code

# Backward pricing with regression
for(m in (n-1):1){ 
  sel_path <- which(StockPath[,m] > 0)

  x_reg <- StockPath[sel_path,m]
  y_reg <- vector(mode = "numeric", length = length(sel_path))
  
  for(i in 1:length(sel_path)){
      d <- min(which(CashFlow[ sel_path[i], ] > 0)) - m
      y_reg[i] <- CashFlow[ sel_path[i], m+d] * discount_fac^d
  }
  
  contiuation <- predict( lm(y_reg ~ x_reg + I(x_reg^2)), x=x_reg)
  exercise <- X - StockPath[which(StockPath[,m]>0), m]

  CashFlow[sel_path,m] <- ifelse(exercise > contiuation, exercise, 0 )
}

3. Discount to calculate the price

最後，把 CahFlow表格的各個路徑的最後保留的值折現即可得到選擇權價格。

四、結語：善用現成的工具

其實蒙地卡羅方法的選擇權定價在R中有第三方的套件可以直接使用，在實際應用的場合就不用「重造輪子」囉！

R code : using LSMonteCarlo package

library("LSMonteCarlo")
AmerPutLSM()
put<-AmerPutLSM(Spot = 101, Strike= 105, 
                mT = 1, sigma = 0.15, r = 0.02,
                m = 50, n = 100000)
summary(put)

References

Y.-D. Lyuu - Principles of Financial Computing
http://www.csie.ntu.edu.tw/~lyuu/finance1.html

Pages

2016年5月15日星期日