examplesBinomTree.Rmd

---
title: "Pricing options via Binomial Trees in R"
author: "Nicolai Vicol"
date: "August 24, 2017"
output: html_document
---

\
```{r, echo=FALSE}
source("E:/interviews/pandawerk/task/getBinomTree.R")
```

#Examples of pricing   
**Parameters of the tree:**   
A binomial tree model with N time steps, e.g. N={10, 20, 50}.      
Suppose that $u=e^{(\sigma*\sqrt{t})}$, where $\sigma=0.15$, and $d=1/u$, $S_0=100$.  
The continuously compounded risk-free interest rate is $r=5%$ p.a. for all maturities.      
\
a) Price a European call option with maturity T-t = 1 year, and strike K=105.
```{r}
dta <- getBinomTree(S0=100, K=105, vol=0.15, dT=1, r=0.05, qdiv=0, N_steps=50, isPut=F)
dta$P[1]
```
...show head and tail of the data frame containing the tree mapped into it:      
```{r}
head(dta, 10)
tail(dta, 55)
```
\
b) Price a European put option with maturity T-t = 1 year, and strike K=105
```{r}
dta <- getBinomTree(S0=100, K=105, vol=0.15, dT=1, r=0.05, qdiv=0, N_steps=50, isPut=T)
dta$P[1]
```
\
c) Price an American put option with maturity T-t = 1 year, and strike K=105 
(assume that the option can be exercised at any node after t). 
```{r}
dta <- getBinomTree(S0=100, K=105, vol=0.15, dT=1, r=0.05, qdiv=0, N_steps=50, isPut=T, isAmerican=T)
dta$P[1]
```
Show in which nodes of the tree is it optimal to exercise this option.
```{r}
dta <- getBinomTree(S0=100, K=105, vol=0.15, dT=1, r=0.05, qdiv=0, N_steps=10, isPut=T, isAmerican=T)
head(dta[dta$P > 0 & dta$dP_Exerc >= 0, ], sum(dta$P > 0 & dta$dP_Exerc >= 0))
```
\
d) Price an average-strike put option of European-type (can only be exercised at maturity) with maturity T-t = 1 year.    
The payoff of the option at maturity is $max(S_{avg}-S_T,0)$, where $S_{avg}=1/N*\sum\limits_{i=1}^{N}S_{t+i*dt}$       
For faster computation: let's take N=10, the number of possible paths grows as function factorial of N.     
```{r}
dta <- getBinomTree.avgK(S0=100, vol=0.15, dT=1, r=0.05, qdiv=0, N_steps=10, isPut=T)
dta$P[1]
```
\
e) Price an "up-and-out" put option (a knock-out put option) of European type (can only be exercised at maturity)    
with maturity T-t = 1 year, strike K=105 and the barrier H=110    
(the option ceases to exist when the barrier is reached at least once).   
```{r d}
dta <- getBinomTree.ko(S0=100, K=105, vol=0.15, dT=1, r=0.05, qdiv=0, N_steps=50, isPut=T, H=110)
dta$P[1]
```
\
f) - i) Price a chooser option with the following characteristics:    
The holder can choose whether the option is a call with maturity T-t = 1 year and strike Kc=105,     
or a put with maturity T-t = 1 year and strike Kp = 95.     
Both these options are European-type (can only be exercised at maturity).     
The holder must announce his choice at T1-t = 6 months.   
```{r f-i}
dta <- getBinomTree.chooser(S0=100, K=105, vol=0.15, dT=1, r=0.05, qdiv=0, N_steps=50, Kc=105, Kp=95, choose_t1=0.5)
P_choose_050 <- dta$P[1]
dta$P[1]
```
\
f) - ii) Price a chooser option with the following characteristics:    
The holder can choose whether the option is a call with maturity T-t = 1 year, or a put with maturity T-t = 1 year.     
Both these options are European-type (can only be exercised at maturity). The holder must announce his choice at    
either T1=t+3 months (take the node closest to 3 months), or T2=t+6 months. Once the choice is announced, the type     
of the option cannot be changed. If the holder announces his choice at T1, then the strike prices are given by Kc=105,    
and Kp=95. If the holder announces his choice at T2, then the strike prices are given by Kc=108, and Kp=92.      
```{r f-ii}
dta <- getBinomTree.chooser(S0=100, K=105, vol=0.15, dT=1, r=0.05, qdiv=0, N_steps=50, Kc=105, Kp=95, choose_t1=0.25)
P_choose_025 <- dta$P[1]
dta$P[1]
```
Find the best time for the holder to announce his choice.    
```{r, echo=F}
print(paste("P(choose at t=0.50):", P_choose_050))
print(paste("P(choose at t=0.25):", P_choose_025))
print("better to exercise at t=0.5 than at t=0.25")
print("value of option with two choose times (0.25, 0.50) but only one choice is equal to value of choice at t=0.50")
```