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 ContentsAbstract.......................................................................................................................................................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.........................................................................................................................................12

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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.

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

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uS, Φ(u)

dS, Φ(d)S0

the change on the price of the stock option to the change in the price of the underlying stock”

(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 (q¿¿u)¿ or down (q¿¿d)¿ by a certain amount

represented by the u factor when going up and the d factor when going down.

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Figure 1 Representation of one step binomial tree

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 wit h probability qu

d ∙ s wit h 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 w h ere Z is a stoc hastic 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=(1+r )−d

u−dqd=

u−(1+r )u−d

In the continuous compounding interest case: qu=er−du−d

qd=u−er

u−d

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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:

For American put options with simple interest: P¿Φ=max ¿

For American call options with simple interest: C¿Φ=max ¿

For European put options with simple interest: P¿Φ=max ¿

For European call options with simple interest: C¿Φ=max ¿

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+u S0 ∙ y=Φ (u )¿ (1+r ) ∙ x+d S0 ∙ y=Φ (d )

And solving for x and y:

{x= 11+r

u ∙ Φ (d )−d ∙Φ (u )u−d

¿ y= 1S0

Φ (u )−Φ ( d )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=ϑ VϑS

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

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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:

ΔC=Ν (d1) ΔP=Ν (d ¿¿1)−1¿

where Ν is the cumulative probability distribution function and d1=

ln ( S0

K )+[ r+σ2

2 ]Tσ √Τ

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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∂ S

=∂2 V∂2 S

=0

Furthermore, a delta-gamma neutral portfolio can be respectively formulated by the delta

and gamma neutrality equation:

{ ∆P=N s∗Δs−Δc∗N c+ ΔP∗N P=0¿ ΓP=N s∗Γs−Γc∗N c+Γ P∗NP=0

In addition, we know that Δs=1 and Γ s=0.

{NS=Δc∗N c−ΔP∗NP

¿ N P=Γc∗N c

Γ P

Thus, we can solve the aforementioned equations in order to find the call and put option for our

delta-gamma portfolio:

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{ NC=NS∗Γ P

Γ P∗ΔC−ΔP Γ C

¿ NP=N S∗ΓC

Γ P∗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 Delta-Gamma 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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