Download Hedging and rebalancing options in a binomial tree.

MMA707 – Analytical Finance I
Jan Röman
Hedging and rebalancing options in a binomial tree
Boulougari Andromachi
Matute Pablo
Straβ Belinda
19th October 2016
Division of Applied Mathematics
School of Education, culture and Communication
Mälardalen University
Box 883, SE-721 23 Västerås, Sweden
Abstract
The purpose of this report is to critically analyze different stock options, the Greeks measures
in order to evaluate the portfolio performance as well as the hedging and its methods used to
neutralize the risk involved. We determine the importance and the usage of the aforementioned
financial components and implement in Python a program for hedging and rebalancing European
options using the binomial tree model to make valid conclusions.
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Table of Contents
Abstract ......................................................................................................................................................... 2
1.
Introduction ........................................................................................................................................... 4
2.
Options .................................................................................................................................................. 4
3.
Binomial Trees ...................................................................................................................................... 6
4.
Hedging ................................................................................................................................................. 9
4.1.
Delta hedging ................................................................................................................................ 9
4.2.
Delta-Gamma hedging ................................................................................................................ 10
5.
Conclusion .......................................................................................................................................... 11
6.
Appendix ............................................................................................................................................. 12
7.
References ........................................................................................................................................... 15
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1. Introduction
In the financial sector, the most significant concept is hedging. Hedging is an investment
position to eliminate any solid losses or gains that an individual or an organization may
experience (Taleb, 1997, p.v). However, because option prices are constantly changing and the
cost of hedging is high, individuals or organizations should use the appropriate type of hedging
so as to neutralize the risk involved in exercising the option. This means that they should keep
the Greeks (Delta, Gamma Theta, Vega, Rho) equal to zero. In this project, we thoroughly
analyze the American and European options, the Greeks measures as well as the usage of a
binomial tree to price these options. Moreover, as the option price is constantly changing, traders
should rebalance their portfolios periodically in order to neutralize the risk involved so as the
managers can choose the appropriate scenario. Therefore, this can be successfully done by using
the most crucial and useful types of hedging; the Delta and Delta-Gamma hedging. Finally, we
implement in Python a program for hedging and rebalancing European options using the
binomial tree model (see Appendix).
2. Options
Options are financial derivatives meaning they are financial instruments in the form of
contracts that depend on an underlying asset’s price. There are two types of options: Call options
and Put options. A call option gives the holder the right to buy the underlying asset at a certain
time for a certain price while a put option gives the holder the right to sell the underlying asset at
a certain time for a certain price (Hull, 2008, p.6). In fact, there are some basic terms that both
the seller and the buyer must agree on in order to establish the contract: Strike price (the price at
which the underlying asset can be bought or sold) and expiration date (the date in the contract,
also called maturity).
Moreover, based on the option’s exercising date, they can be divided into American options
and European options. American options can be exercised at any time before or at the expiration
date while European options can only be exercised at the expiration date (Hull, 2008, p.6). These
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names have nothing to do with the location or origin of the options, they only differentiate the
option’s exercising date property.
The price of an option (denoted as C for call and P for put) is affected primarily by six factors:
1. The current stock price, which can be denoted as 𝑆0
2. The strike price denoted as K
3. The time to expiration (maturity) denoted as T
4. The volatility of the stock price denoted as σ
5. The risk-free interest rate denoted as r
6. The dividends expected during the life of the option
The following table (Table 1) shows the changes in the option’s price when one of the above
changes while the rest remain fixed.
Table 1 Changes in the option's prices due to the increasing value of one variable.
Variable
European call European put American call American put
Current stock price
+
-
+
-
Strike price
-
+
-
+
Time to expiration
?
?
+
+
Volatility
+
+
+
+
Risk-free rate
+
-
+
-
Amount of future dividends -
+
-
+
(+) indicates that an increase in the variable causes the option to increase
(-) indicates that an increase in the variable causes the option price to decrease
(?) indicates that the relationship is uncertain
Source: John C. Hull, 2008
One way of evaluating the change in the options value respect to the underlying price is to
use the measure called delta (Δ). As Hull (2008) states “the delta of a stock option is the ratio of
the change on the price of the stock option to the change in the price of the underlying stock”
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(p.247) This property can be used to hedge a portfolio by using the change in the ratio through
the life of the option in order to balance the amount of money market account and the number of
stocks. Related to delta and the other arrangements used in hedging, are the other Greek letters.
These measures, as shown with delta, are useful for evaluating the portfolio in different ways
(Hull, 2008, pp.360-375):

Theta θ: rate of change of the value of the portfolio with respect to time.

Gamma Γ: rate of change of the portfolio’s delta with respect to the price of the
underlying asset

Vega ν: rate of change of the value of the portfolio with respect to the volatility of the
underlying asset.

Rho ρ: rate of change of the value of the portfolio with respect to the interest rate.
Delta, gamma and hedging in general will be discussed with more detail in section 4 of this
report.
3. Binomial Trees
There are several techniques for pricing an option; however, the most useful one is the
binomial tree. It represents the different paths that the stock value S(t) can take ergo the option
value Φ(u) or Φ(d). It is assumed that these paths follow a random walk, in other words, in each
step there is a certain probability of moving up (𝑞𝑢 ) or down (𝑞𝑑 ) by a certain amount
represented by the u factor when going up and the d factor when going down.
uS, Φ(u)
S0
dS, Φ(d)
Figure 1 Representation of
one step binomial tree
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Nevertheless, before pricing the option there are some assumptions that must be made: i)
the process is done under the risk-neutral probability measure 𝑄 = (𝑞𝑢 , 𝑞𝑑 ), ii) there are two
possible investments or securities in the market: the money-market account B(t) and the stock
S(t), iii) it is possible to buy or sell the stock and invest or lend in the money-market account, iv)
the interest rate r for saving and lending money is the same and v) the market is 100% liquid
(Röman, 2016, p.42).
The representation of the money-market account at time t=0 and time t=1 would be:
𝐵(0) = 1
{
𝐵(1) = 1 + 𝑟
And the representation of the stock price at time t=0 and time t=1 would be:
𝑆(0) = 𝑠
{
𝑆(1) = {
𝑢 ∙ 𝑠 𝑤𝑖𝑡ℎ 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑞𝑢
𝑑 ∙ 𝑠 𝑤𝑖𝑡ℎ 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑞𝑑
Moreover, a portfolio h (B, S) can be considered as a vector h = (x, y), where x and y are
the number of bonds or money-market account and the number of stocks respectively. With these
assumptions, the value process of the portfolio h can be defined as:
𝑉 (𝑡, ℎ) = 𝑥 ∙ 𝐵(𝑡) + 𝑦 ∙ 𝑆(𝑡)
For t=0,1
𝑉(0, ℎ) = 𝑥 + 𝑦 ∙ 𝑠
{
𝑉(1, ℎ) = 𝑥 ∙ (1 + 𝑟) + 𝑦 ∙ 𝑠 ∙ 𝑍 𝑤ℎ𝑒𝑟𝑒 𝑍 𝑖𝑠 𝑎 𝑠𝑡𝑜𝑐ℎ𝑎𝑠𝑡𝑖𝑐 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒
As stated above, all this process is under the risk-neutral probability 𝑄 = (𝑞𝑢 , 𝑞𝑑 ), which,
Röman (2016) describes as “the probability of a future event or state that both trading parties in
the market agree upon” (p.46). 𝑄 = (𝑞𝑢 , 𝑞𝑑 ) is given by:
In the simple compounding interest case:
𝑞𝑢 =
In the continuous compounding interest case:
(1+𝑟)−𝑑
𝑢−𝑑
𝑞𝑢 =
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𝑒 𝑟 −𝑑
𝑢−𝑑
𝑞𝑑 =
𝑢−(1+𝑟)
𝑞𝑑 =
𝑢−𝑑
𝑢−𝑒 𝑟
𝑢−𝑑
Once the value of the stock in each node of the binomial tree is known, it is possible to
obtain the value of the option in each node using the following equations:
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For American put options with simple interest: P= 𝛷 = 𝑚𝑎𝑥 {𝐾 − 𝑆, 1+𝑟 (𝑞𝑢 ∙ 𝛷(𝑢) + 𝑞𝑑 ∙ 𝛷(𝑑)}
1
For American call options with simple interest: C= 𝛷 = 𝑚𝑎𝑥 {𝑆 − 𝐾, 1+𝑟 (𝑞𝑢 ∙ 𝛷(𝑢) + 𝑞𝑑 ∙ 𝛷(𝑑)}
1
For European put options with simple interest: P= 𝛷 = 𝑚𝑎𝑥 {1+𝑟 (𝑞𝑢 ∙ 𝛷(𝑢) + 𝑞𝑑 ∙ 𝛷(𝑑)}
1
For European call options with simple interest: C= 𝛷 = 𝑚𝑎𝑥 {1+𝑟 (𝑞𝑢 ∙ 𝛷(𝑢) + 𝑞𝑑 ∙ 𝛷(𝑑)}
It is good to notice that at the ending nodes (at maturity), the value of the American and
European options is the same and can be calculated by:
𝑃 = 𝛷 = 𝑚𝑎𝑥{𝐾 − 𝑆}
𝐶 = 𝛷 = 𝑚𝑎𝑥{𝑆 − 𝐾}
With the known values of the stock and option in each node and the value process described
above, it is possible to obtain the replicated portfolio (the number of stocks and amount of
money-market account equivalent to the option value) by constructing the following pair of
equations:
(1 + 𝑟) ∙ 𝑥 + 𝑢𝑆0 ∙ 𝑦 = 𝛷(𝑢)
{
(1 + 𝑟) ∙ 𝑥 + 𝑑𝑆0 ∙ 𝑦 = 𝛷(𝑑)
And solving for x and y:
𝑥=
{
1 𝑢 ∙ 𝛷(𝑑) − 𝑑 ∙ 𝛷(𝑢)
1+𝑟
𝑢−𝑑
𝑦=
1 𝛷(𝑢) − 𝛷(𝑑)
𝑆0
𝑢−𝑑
Where x and y, as described above, are the amount of money in the money-market account and
the number of stocks respectively.
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4. Hedging
In the financial sector, the most significant concept is hedging. Hedging is an investment
position to eliminate any solid losses or gains that an individual or an organization may
experience (Taleb, 1997, p.v). In fact, hedging can be formulated by a wide range of financial
instruments such as stocks, derivatives, swaps as well as future contracts. Nevertheless, in
finance, traders should manage to keep the Greeks equal to zero even if in practice it is difficult
(Hull, 2008, p.376). Moreover, if and when the counterparty practices the option, the trader has
the commitment to deliver the underlying stock. However, “the trading costs per option being
hedge are high” (Hull, 2008, p.376). Therefore, individuals or organizations should use the
appropriate type of hedging so as to neutralize the risk involved in exercising the option. The
most crucial and useful types of hedging is the Delta and Delta-Gamma hedging.
4.1. Delta hedging
Delta hedging is a derivative hedging strategy that attempts to eliminate or reduce the risk
occurred by the price changes in the underlying asset. The process involves taking long (buy the
stock) or short (sell the stock) position in the underlying asset. “Delta means the sensitivity of a
derivative price to the movement in the underlying asset” (Taleb, 1997, p.115). To put it
succinctly, delta shows the reciprocal change of the option price V (C or P for call and put option
respectively) caused by small changes in the underlying stock price S. For instance, in a put
option, as the price of the option goes down the underlying stock price will respectively go up.
The correlation between the two variables is given by:
𝐷𝑒𝑙𝑡𝑎, 𝛥 =
𝜗𝑉
𝜗𝑆
Nevertheless, the prices usually change in a short period of time (Hull, 2008, p.361).
Therefore, traders should rebalance their portfolios periodically in order to neutralize the risk
involved so as the managers can choose the appropriate scenario. Moreover, Delta is similar to
the Black-Scholes-Merton analysis. This means that a portfolio with zero risk can be created in
terms of option -1 and number of shares of the stock +Δ (Hull, 2008, p.362). Delta can be
formulated under a call and put European option respectively:
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𝛥𝐶 = 𝛮(𝑑1 )
𝛥𝑃 = 𝛮(𝑑1 ) − 1
where Ν is the cumulative probability distribution function and 𝑑1 =
𝑆
𝑟+𝜎2
ln( 0 )+[
]𝑇
𝐾
2
𝜎√𝛵
.
4.2. Delta-Gamma hedging
Another important and useful trading strategy is the delta-gamma hedging. This strategy
is a combination of delta and gamma hedging used to eliminate the risk involved in the option
that is caused by the changes in the underlying asset as well as the variances (Investopedia,
2016). The essence of this strategy is to keep neutral the Γ and Δ hedge meaning that the first and
the second derivative of the portfolio value must be equal to zero:
𝜕𝑉 𝜕 2 𝑉
=
=0
𝜕𝑆 𝜕 2 𝑆
Furthermore, a delta-gamma neutral portfolio can be respectively formulated by the delta
and gamma neutrality equation:
∆𝑃 = 𝑁𝑠 ∗ 𝛥𝑠 − 𝛥𝑐 ∗ 𝑁𝑐 + 𝛥𝑃 ∗ 𝑁𝑃 = 0
{
𝛤𝑃 = 𝑁𝑠 ∗ 𝛤𝑠 − 𝛤𝑐 ∗ 𝑁𝑐 + 𝛤𝑃 ∗ 𝑁𝑃 = 0
In addition, we know that 𝛥𝑠 = 1 and 𝛤𝑠 = 0.
𝑁𝑆 = 𝛥𝑐 ∗ 𝑁𝑐 − 𝛥𝑃 ∗ 𝑁𝑃
{
𝑁𝑃 =
𝛤𝑐 ∗ 𝑁𝑐
𝛤𝑃
Thus, we can solve the aforementioned equations in order to find the call and put option for our
delta-gamma portfolio:
𝑁𝐶 =
𝑁𝑆 ∗𝛤𝑃
𝛤𝑃 ∗𝛥𝐶 −𝛥𝑃 𝛤𝐶
{
𝑁𝑃 =
𝑁𝑆 ∗𝛤𝐶
𝛤𝑃 ∗𝛥𝐶 −𝛥𝑃 𝛤𝐶
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5. Conclusion
Options are versatile instruments that allow investors to hedge their portfolios through the
use of the different properties that each type of option – European call, European put, American
call, American put, and other options outside the scoop of these report – possesses. Their prices
depend on the underlying asset’s price which can be calculated using the binomial model.
When it comes to the topic of hedging, most people from the financial sector will readily
agree that the prices are constantly changing. Even if it is difficult, individuals or organizations
should aim at the risk-neutral probability involved in exercising the option. This means that they
should keep the Greeks (Delta, Gamma Theta, Vega, Rho) measures equal to zero (Hull, 2008,
p.357). Consequently, traders should use the appropriate hedging tools, namely Delta and DeltaGamma hedging, in order to successfully eliminate any solid losses or gains in the future.
The usage of a programming language simplifies the calculation of different scenarios for the
stock price and the referring option prices. In this project, we use the programing language
Python for European options. The model was constructed in a way that only the initial
parameters has to be changed to obtain the output for the corresponding scenarios. The
calculation of x and y, which represents the amount of the money-market-account and the
amount of stocks respectively, is also given as an output. In this case any change in the amount
of stocks is readily seen which simplifies the hedging process.
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6. Appendix
Python code for pricing and hedging European options:
import sys
import numpy as np
from math import pow
# create the Binomial Tree
def build_stock_tree_2(S,u,d,N):
tree=np.zeros((N+1,N+1))
for i in range(N+1):
for j in range(i+1):
tree[i][j]=S*pow(u,j)*pow(d,i-j)
return(tree)
# Calculate the risk-neutral Probability
def q_prob_2(r,u,d):
return((1+r-d)/(u-d))
# Calculate the Optionprice Tree
def value_binomial_option_2(tree,u,d,N,r,X,type):
q=q_prob_2(r,u,d)
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option_tree_2=np.zeros((len(tree),len(tree[0])))
if(type=="put"):
for i in range(len(tree[0])):
option_tree_2[len(option_tree_2)-1][i]=max(X-tree[len(tree)-1][i],0)
else:
for i in range(len(tree[0])):
option_tree_2[len(option_tree_2)-1][i]=max(tree[len(tree)-1][i]-X,0)
for i in reversed(range(len(tree)-1)):
for j in range(i+1):
option_tree_2[i][j]=((1-q)*option_tree_2[i+1][j]+q*option_tree_2[i+1][j+1])/(1+r)
return(option_tree_2)
# Define your Variables
def main():
S=100
#Stock price
u=1.4
# up factor
d=0.8
# down factor
r=0.5
# interest rate
N=2
# number of periods
X=100
# Strike price
type="put"
amount=1000 #Amount of options
# Execute the Problem
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tree=build_stock_tree_2(S, u,d,N)
print("Tree %r"%(tree))
value=value_binomial_option_2(tree,u,d,N,r,X,type)
print("Value %r"%(value))
x=(1/(1+r)*((u*value[1][0])-(d*value[1][1]))/(u-d))*amount
y=((1/tree[0][0])*((value[1][1])-value[1][0])/(u-d))*amount
print("x=%r"%(x))
print("y=%r"%(y))
if __name__ == "__main__":
sys.exit(int(main() or 0))
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7. References
Hull, J. (2008). Options, futures and other derivatives (7.th ed.). Harlow: Pearson Prentice Hall.
Investopedia. (2016 October 13). Delta-Gamma Hedging. Retrieved from
http://www.investopedia.com/terms/d/deltagamma-hedging.asp
Röman, R. M. J. (2016). Analytical Finance – The Mathematics of Equity Derivatives, Markets,
Risk and Valuation. Retrieved from
http://janroman.dhis.org/AFI/Analytical%20Finance%20I.pdfhuff/taytay/taytay.html
Taleb, N. (1997). Dynamic hedging: Managing vanilla and exotic options (Wiley series in
financial engineering). New York: Wiley.
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