選擇權定價實作：American put options on zero-coupon bonds under the CIR model

cir model pricing 來結束這學期的財務演算法課程^^  這個故事告訴我們小時候數學要學好，不然長大會被霸凌XD

感謝耕竹大大的指導!

一、問題描述：American put options on zero-coupon bonds under the CIR model

price an x-year American-style put option on a zero-coupon bond that matures at year y with a par value of 1 dollar. Use trees for the CIR model.

Inputs:

• x (year), 
• y (year), 
• r (%) (initial short rate), 
• b (%) and 
• m (%), 
• s (%), 
• n (the number of steps during the option's life), and 
• strike price X (% of par). 

For example,  when

• x = 1, y = 2, r = 4 (%), b = 20 (%), m = 4 (%), s = 10 (%), n = 30, and X = 95 (%). 

the option price is about 2.65173 (% of par)

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R語言執行方式：直接使用 R 直譯器執行即可

R語言執行結果：

二、設計與實作

1. 產生 short rate tree

approximate the short rate process

• direct discretization of the process is problematic because the resulting binomial tree will not combine
• using transformed process to generate short rate tree

R code

# generate short rate tree
shortrate_tree <- matrix(data = 0, nrow = n+1, ncol = n+1)
shortrate_tree <- initShortRateTree(shortrate_tree,
                                    x = 2*sqrt(r_0)/sigma,
                                    delta_t)
shortrate_tree <- transformR(shortrate_tree, sigma)

2. 產生 prob. tree

由  short rate tree 代入以下公式即可得 prob. tree。

R code

# generate prob tree
prob_tree <- matrix(data = 0, nrow = n, ncol = n)
prob_tree <- initProbTree(prob_tree, shortrate_tree, mu, beta, delta_t)

3. 產生 price tree

由 "Numerical pricing of American put options on zero-coupon bonds" 這篇paper提供的cir model公式可獲得cir model price tree。

R code

# generate price tree
price_tree <- matrix(data = 0, nrow = n+1, ncol = n+1)
price_tree <-initCIRPriceTree( price_tree, shortrate_tree,
                               sigma = sigma, r_inf = mu, kapa = beta,
                               E = 1, T_star = y,
                               delta_t = delta_t)

4. 產生 value tree

用 classical American put 的架構，把binomial tree的price值改成cir model預估的值，並把機率也改成cir model預估的機率即可。

R code

# generate value tree (pricing)
valueTree <- matrix(data = 0, nrow = n+1, ncol = n+1)
X = X*0.01

for(i in 1:(n+1)){
  valueTree[i,n+1] = max(X-price_tree[i,n+1], 0);
}

for(j in n:1){
  for(i in 1:j){
    q = prob_tree[i,j]
    valueTree[i,j] =  max(X-price_tree[i,j] ,
                          q*valueTree[i,j+1] + (1-q)*valueTree[i+1,j+1])
  }
}

cat("price under cir: ", valueTree[1,1])

References

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

"Numerical pricing of American put options on zero-coupon bonds" August 2003
Walter Allegretto, Yanping Lin, Hongtao Yang