Futures and the Binomial Tree Model

1. Introduction to Financial Derivatives
Future Contracts & the Binomial Tree Model
Natalia Lopez - Reg No: 120013401
BSc Economics
April 2015

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Abstract
Introduction
The data of Apple’s stock price between the 4th
of January 2010 and the 12th
of February
2015 is the starting point of this report. The underlying asset (Apple’s stock) provides a
known yield which is expressed as a percentage of its price. The six-month future price
observed in the market during that same period is provided as well as the US risk free rate.
The task is to calculate the theoretical six-month forward price and compare it with the six-
month future price with the support of a graphical analysis. This section of the report also
includes an analysis of Facebook’s stock future and spot prices between the 24th
of March
2014 and the 8th
of April 2015. The intention is to demonstrate how future and spot prices
converge at maturity. The second part of the paper focuses on calculating the prices as well
as the delta values of the European call and put options with the binomial tree. The
European put option will be calculated using the binomial formula for different maturities with
the purpose of analysing how option prices change as the number of periods increase.
Finally, the American put option is also calculated using the binomial tree; the result is then
compared to the values obtained for the European put option. The report concludes with a
discussion about the limitations of the binomial formula when applied to early exercise
options.
The marking to market mechanism creates discrepancies between the prices of forward
and future contracts when the risk free rate is uncertain. If there is a positive correlation
between the stock price and the interest rate then the futures contract will tend to be
higher than the forwards contract; the converse also holds true. Furthermore, the futures
price converges to the spot price as the maturity of the contract approaches. If this
condition does not hold there will be arbitrage opportunities. The binomial tree model can
be used to calculate the value of American/European put and call options as well as its
delta values which are then used for hedging positioning. As the number of periods to
maturity increase the price of the option is higher because there is a higher probability
that the option will be exercised at maturity. Conversely, the price decreases as the
option comes closer to expiration day. Although the binomial formula proofs to be a good
method to calculate European options it cannot be applied to early exercised options.
This is because it is not possible to know whether exercising earlier is optimal or not
using this method.

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Section 1 - Future Contracts
1.1. Comparing the theorical six-month forward price to the six-month future price of
Apple’s stock.
Prior comparing the six-month forward and future prices, it is of great importance to
understand the main differences underpinning both types of contracts.
Future Contracts Vs Forward Contracts
Although the objective of future and forward contracts is the same: to purchase or sell an
asset at a specified future time at an agreed price (the delivery price will be denoted with (K)),
there exist significant differences in the way they are carried out. Unlike forward contracts,
which are privately arranged between two parties, future contracts are traded in central
exchanges like futures markets meaning that the former is not as strict regarding its terms
and conditions as the latter. The nature of forward contracts imply that a party might default
to deliver its part of the agreement; conversely, future contracts are safer in that sense as
they are safeguarded by clearing houses that guarantee the delivery of transactions through
the operation of margin accounts. With future contracts, gains and losses are realised every
day: a process known as marking to market. This means that variation of prices is settled
daily until the end of the contract whilst settlement with forward contracts occurs only once
being the last day of the contract (or settlement date). Finally, forward contracts are normally
used to hedge against the volatility of asset prices thus the delivery of the asset will often
occur. On the other hand, investors usually enter future contracts for speculation purposes
for which there will be a range of delivery dates and often delivery does not actually take
place. That is, future contracts are often closed prior to maturity.
The differences between forward and future prices
It has been stablished that the marking to market mechanism allows agents agreeing on
future contracts to gain or lose on a daily basis. Then, does this make the future price
different from the forward price? In theory, future and forward prices for contracts with the
same delivery dates are equal when the risk free rate (r) is certain and constant throughout
different maturities. Furthermore, the no arbitrage condition has to hold for this statement to
be true; consider the following scenario:
Strategy 1:
- Go long a forward contract with delivery price F (0, T) and invest a quantity of cash
F (0, T) at the assumed to be certain and constant (r). See algebraic expression below:
𝐹(0, 𝑇)𝑒 𝑟𝑇
+ 𝑒 𝑟𝑇[𝑆(𝑇) − 𝐹(0, 𝑇)] = 𝐹(0, 𝑇)𝑒 𝑟𝑇
+ 𝑒 𝑟𝑇
𝑆(𝑇) − 𝑒 𝑟𝑇
𝐹(0, 𝑇) = 𝒆 𝒓𝑻
𝑺(𝑻)

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Strategy 2:
- Under the same conditions, go long a future contract with delivery price Ƒ (0, T) and invest a
quantity of cash Ƒ (0, T) at (r). Here the marking to market mechanism allows for regular
adjustments of the margin account meaning that the investor makes a daily earning of:
𝑒 𝑟𝑡
[ Ƒ(t, T) − Ƒ(t − 1 , T)] [Which is then invested at r]
The total payoff at maturity is:
Ƒ(0, T)𝑒 𝑟𝑇
+ 𝑒 𝑟𝑇
[ Ƒ (T, T) − Ƒ(0, T)] [Where Ƒ(T, T) = 𝑆(𝑇)]
Ƒ(0, T)𝑒 𝑟𝑇
+ 𝑒 𝑟𝑇[𝑆(𝑇) − Ƒ(0, T)] = Ƒ(0, T)𝑒 𝑟𝑇
+ 𝑒 𝑟𝑇
𝑆(𝑇) − 𝑒 𝑟𝑇
Ƒ(0, T) = 𝒆 𝒓𝑻
𝑺(𝑻)
In other words:
𝐹(0, 𝑇) = Ƒ(0, T) If this condition does not hold there would be an arbitrage opportunity.
However, in the real world, the risk free rate is not certain or constant and there will be
differences between the two. If the interest rate changes and it does in a way that is
correlated with changes in the underlying asset then the marking to market mechanism
creates discrepancies between the two types of contracts. In order to show this, the theorical
forward rate of Apple’s stock has been calculated using the formula:
𝐹(𝑡, 𝑇) = 𝑆(𝑡)𝑒(𝑟−𝑞)(𝑇−𝑡)
[Where q is the dividend yield]
And it is compared with the future price during the same period. The first graph shows how
the prices of the six-month future and forward contracts of Apple move in the same direction.

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The spread is the difference between the theorical forward price and the future price and,
taking a closer look, it becomes apparent that they are not the same. For example, the
spread fluctuates more from August 2012 when Apple introduces dividend yield payments.
It is also interesting to see with a histogram how the difference between the forward and
future prices converge to the normal distribution as t becomes larger.
Interest rates play a fundamental role in the relationship between forward and future prices.
For example, consider the scenario where interest rates are positively correlated with stock
prices. If stock prices increase, agents in a long position on a future contract will benefit
through an increase of their margin account. Furthermore, interest rates will also have

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increased due to the positive correlation between the two. This means that the agent will be
able to invest the amount by which his/er margin account has increased at a higher than
average rate. On the other hand, when stock prices decrease, the agent with the long
position will be worse of; however, this will be compensated by a lower than average interest
rate payments. Conversely, if there is a negative correlation between interest rates and the
stock price, the same agent with a long position on a future contract will be less favourably
affected. This is because the amount by which the margin account increases when stock
prices rise will be invested at a lower than average rate (the interest rates will have
decreased). Equally, when stock prices decrease, the agent with the long position will be
negatively affected by having to pay higher than average interest rates (the interest rates will
have increased). Having said this, it is expected that the price of future contracts will be
higher than the price of forward contracts when interest rates and the stock price are
positively correlated. Then, it is also expected that the price of future contracts will be lower
than the price of forward contracts when the same variables are negatively correlated.
The correlation between Apple’s stock price and the risk free rate is negative (- 0.4966148).
Given the theory, it would be expected that the price of Apple’s forward contracts was overall
higher than future contracts. However, this is not true in the example as the sum of the
differences between the theorical forward and the futures price of Apple is a negative
number. Nonetheless, according to Hull (2011) p.114, the differences between forward and
future prices of short term contracts are “sufficiently small” not to be considered. There are
also a number of factors that are not taken into consideration when calculating the theorical
forward price such as taxes or transaction costs.

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1.2. The future and spot prices of Facebook’s stock
As the expiration date of a futures contract approaches it will naturally converge to the spot
price of the underlying asset. This statement can be shown by taking an already expired
futures contract of any stock or commodity (Facebook) and compare it to the spot price at
the maturity of the contract. See below:
There is a noticeable positive correlation between the spot and futures price:
The difference between the futures and spot prices should be equal to zero at maturity and
so it is: 8.6 (is the futures price at T) - 8.6 (being the spot price) = 0

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This relationship can be explained using the no arbitrage argument. Consider the case
where the futures contract for Facebook’s stock is higher than the spot price as the
expiration date approaches. That is:
Ƒ0 > 𝑆0 𝑒 𝑟𝑇
This represents an arbitrage opportunity because agents would short futures contracts and
buy the asset today at S0. This is a risk free investment with a positive payoff as the profit
earned by shorting the contract exceeds the amount paid for the asset:
Ƒ0 − 𝑆0 𝑒 𝑟𝑇
> 0 [This is the profit at maturity]
Now consider the opposite situation where the spot price is higher than the futures contract
for Facebook’s stock as maturity approaches. That is:
Ƒ0 < 𝑆0 𝑒 𝑟𝑇
In this situation, arbitrageurs would short sell the asset whilst simultaneously go long the
futures contract today (they would repurchase the asset in the future). Then:
𝑆0 𝑒 𝑟𝑇
− Ƒ0 > 0 [This is the profit at maturity]
Summarising, in the first case, arbitrageurs selling future contracts will create excess supply
for which the price of future contracts will decrease. Equally, the price of the underlying asset
will increase due to the increased demand. In the second case, the excess supply of the
underlying asset will result in a decrease of the spot price whilst the demand for futures
contracts will cause an increase of the contract’s price. Arbitrageurs will close arbitrage
opportunities bringing the market to equilibrium through demand and supply forces and, for
this reason, futures prices converge to the spot price at the maturity of the contract.
Section 2 - Binomial Tree Model
The option pricing binomial tree model is widely used by financial institutions to price
derivatives. It involves the construction of a binomial tree on Excel or other advanced
software packages where the stock price follows different paths during the life of the option.
The intuition is that the value of the asset can only take two values in the next period: either
up or down, but as the number of periods increase some outcomes become more likely than
others. The following diagram depicts a two-step model where it is easy to appreciate that
there are two ways to reach S ud / du and only one way to reach either Suu or Sdd.

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Adding more combinations increases the number of steps and, as the intervals become
shorter, it converges to the Black-Scholes model which assumes that the returns of the
underlying asset are normally distributed. To demonstrate how it works the following
parameters have been provided:
S0 = 50 The stock price at t0.
K = 58 The strike price.
u = 1.2 Factor by which the stock price goes up derived from: 𝑢 = 𝑒 𝜎√𝛿𝑡
where σ
represents volatility.
d = 0.83 Factor by which the stock price goes down derived from: 𝑑 = 1/ 𝑒 𝜎√𝛿𝑡
.
r = 5% Risk free rate.
Furthermore, the martingale probability by which the stock price goes up or down is
calculated using the following formulas:
𝒑∗
=
1 + 𝑟 − 𝑑
𝑢 − 𝑑
=
1 + 0.05 − 0.83
1.2 − 0.83
= 0.59 𝟏 − 𝒑∗
=
𝑢 − (1 + 𝑟)
𝑢 − 𝑑
=
1.2 − (1 + 0.05)
1.2 − 0.83
= 0.41
That is, the probability by which the stock goes up is 0.59 and the probability by which the
stock goes down is 0.41. Given these parameters it is now possible to construct the binomial
tree model to price options.
2.1. The binomial tree model for a European call option and its possible delta values.
The Binomial Tree Model of the European call option
A call option gives the holder the right but not the obligation to buy an asset at a specific time
in the future at an agreed price known as the strike price (K). European call options can only

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i/t 0 1 2 3 4 5 6 7 8 9 10
0 50.00 60.00 72.00 86.40 103.68 124.42 149.30 179.16 214.99 257.99 309.59
1 41.67 50.00 60.00 72.00 86.40 103.68 124.42 149.30 179.16 214.99
2 34.72 41.67 50.00 60.00 72.00 86.40 103.68 124.42 149.30
3 28.94 34.72 41.67 50.00 60.00 72.00 86.40 103.68
4 24.11 28.94 34.72 41.67 50.00 60.00 72.00
5 20.09 24.11 28.94 34.72 41.67 50.00
6 16.74 20.09 24.11 28.94 34.72
7 13.95 16.74 20.09 24.11
8 11.63 13.95 16.74
9 9.69 11.63
10 8.08
Stock Price
i/t 0 1 2 3 4 5 6 7 8 9 10
0 0.00 2.00 14.00 28.40 45.68 66.42 91.30 121.16 156.99 199.99 251.59
1 0.00 0.00 2.00 14.00 28.40 45.68 66.42 91.30 121.16 156.99
2 0.00 0.00 0.00 2.00 14.00 28.40 45.68 66.42 91.30
3 0.00 0.00 0.00 0.00 2.00 14.00 28.40 45.68
4 0.00 0.00 0.00 0.00 0.00 2.00 14.00
5 0.00 0.00 0.00 0.00 0.00 0.00
6 0.00 0.00 0.00 0.00 0.00
7 0.00 0.00 0.00 0.00
8 0.00 0.00 0.00
9 0.00 0.00
10 0.00
Payoff Table of Call Option
be exercised at its maturity. To construct the tree the first step is to calculate the stock price
at each node of the tree.
To calculate the stock price:
𝑆𝑡
𝑖
= 𝑆0 𝑢 𝑇−𝑖
𝑑 𝑖
Where (i) is the number of downward movements.
To simplify the demonstration the table below shows in the first row the different stock prices
for upward movements only whilst the rest of values represent downward movements. The
stock price then ranges from a minimum of £8.08 to a maximum of £309.59.
Given the stock values, it is possible to calculate the payoff at each node as well as the value
of the European call option at t0.
The payoff of the European call option is: 𝐶 𝑇 = [𝑆 𝑇 − 𝐾]+
When ST < K at maturity CT is negative for which the option is not exercised and the payoff is
zero. As the table shows, the payoff for the holder of the European call option ranges from a
maximum of £251.59 to a minimum of £0.00. The payoff cannot be negative simply because
the holder of the option will not exercise if ST < K is satisfied.

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i/t 0 1 2 3 4 5 6 7 8 9 10
0 18.49 25.39 34.41 46.03 60.72 79.04 101.58 129.06 162.38 202.75 251.59
1 10.78 15.45 21.84 30.42 41.68 56.15 74.31 96.69 123.92 156.99
2 5.35 8.11 12.13 17.87 25.88 36.77 51.07 69.18 91.30
3 2.03 3.30 5.31 8.47 13.32 20.61 31.16 45.68
4 0.44 0.79 1.40 2.50 4.43 7.88 14.00
5 0.00 0.00 0.00 0.00 0.00 0.00
6 0.00 0.00 0.00 0.00 0.00
7 0.00 0.00 0.00 0.00
8 0.00 0.00 0.00
9 0.00 0.00
10 0.00
European Call Option
The range of payoffs at maturity is then used to calculate in a backward fashion the present
value of the call; a technique known as backwards induction. Since we are using the risk
neutral measures of probability, the value at each node is the expected value from the
previous node discounted at the rate of 5%. This approach is algebraically expressed as:
𝑉𝑡
𝑖
=
1
1+𝑟
( 𝑝𝑉𝑡+1
𝑖+1
+ (1 − 𝑝)𝑉𝑡+1
𝑖
)
For example:
1
1+0.05
( 0.59(251.59) + 0.41(156.99)) ≈ 𝟐𝟎𝟐. 𝟕𝟓
1
1+0.05
( 0.59(156.99) + 0.41(91.30)) ≈ 𝟏𝟐𝟑. 𝟗𝟐 And so on.
Then working backwards through the “branches” of the tree all the values are reduced to the
present value of the option; that is, at time zero, the value of the call is £18.49.
Using delta for hedging
The delta value of an option is the rate at which the price of the option changes with respect
to the rate at which the price of the underlying stock changes. That is:
𝛿𝑡 =
∆𝑉𝑡+1
∆𝑆𝑡+1
=
𝑉𝑢 − 𝑉𝑑
𝑆 𝑢 − 𝑆 𝑑
=
𝑉𝑢 − 𝑉𝑑
𝑆0(𝑢 − 𝑑)
Delta has a very important meaning because it is the amount of stock the party shorting the
call needs to have in the portfolio to hedge the derivative contract. In the case of a call option
this is the same party that needs to long the underlying asset so at each point in time the
amount of the stock held in the portfolio has to be readjusted. The table of the delta for the
call option shows how at t0 the writer has to purchase 80% of the position which is underlying

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i/t 0 1 2 3 4 5 6 7 8 9 10
0 0.80 0.86 0.92 0.96 0.98 1.00 1.00 1.00 1.00 1.00 1.00
1 0.66 0.75 0.83 0.90 0.96 0.99 1.00 1.00 1.00 1.00
2 0.48 0.58 0.68 0.79 0.89 0.96 1.00 1.00 1.00
3 0.27 0.36 0.46 0.59 0.74 0.88 1.00 1.00
4 0.09 0.13 0.20 0.29 0.43 0.64 1.00
5 0.00 0.00 0.00 0.00 0.00 0.00
6 0.00 0.00 0.00 0.00 0.00
7 0.00 0.00 0.00 0.00
8 0.00 0.00 0.00
9 0.00 0.00
10 0.00
Delta of European Call Option
the option contract; at t1 will hedge 86% in the up-state and 66% in the down-state and so on.
The reasoning behind this approach is that the agent shorting the call will not purchase 100%
of the stock at t0 for £50.00 and risk to end up at t10 with the unexercised stock worth £8.08
on the portfolio. Equally, it would not be a good decision waiting until t10 and risk having to
purchase the asset for £309.59 when it could have been acquired at a cheaper price in
previous periods. Instead, the amount of stock held in the portfolio is readjusted in
accordance to market movements.
At maturity (t10), if the payoff is larger than 0, the writer will be forced
to purchase 100% of the asset hence delta equals to 1. Conversely, if
the option is not exercised, the payoff to the holder is 0 then the
amount of the asset the writer will need to deliver is 0%. The value of
the call option increases when the stock price increases; there is a
positive correlation hence it is always positive. It is not greater than 1
simply because the call option cannot lose or gain value more rapidly
than the underlying asset. If that was the case, there would be an
arbitrage opportunity.
2.2. The binomial tree model for a European put option and its possible delta values.
The Binomial Tree Model of the European put option
A put option gives the holder the right but not the obligation to sell an asset at a specific time
in the future at the agreed price K. European put options can only be exercised at its maturity.
Payoff at
Maturity
Delta
Values
10 10
251.59 1.00
156.99 1.00
91.30 1.00
45.68 1.00
14.00 1.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00

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i/t 0 1 2 3 4 5 6 7 8 9 10
0 4.09 2.77 1.67 0.84 0.32 0.07 0.00 0.00 0.00 0.00 0.00
1 6.50 4.71 3.06 1.70 0.73 0.18 0.00 0.00 0.00 0.00
2 9.89 7.66 5.41 3.31 1.60 0.47 0.00 0.00 0.00
3 14.31 11.86 9.09 6.19 3.43 1.21 0.00 0.00
4 19.61 17.30 14.40 10.93 7.04 3.12 0.00
5 25.35 23.60 21.17 17.89 13.57 8.00
6 30.97 30.01 28.50 26.30 23.28
7 36.15 35.86 35.14 33.89
8 40.98 41.28 41.26
9 45.55 46.37
10 49.92
European Put Option
i/t 0 1 2 3 4 5 6 7 8 9 10
0 8.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
1 16.33 8.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
2 23.28 16.33 8.00 0.00 0.00 0.00 0.00 0.00 0.00
3 29.06 23.28 16.33 8.00 0.00 0.00 0.00 0.00
4 33.89 29.06 23.28 16.33 8.00 0.00 0.00
5 37.91 33.89 29.06 23.28 16.33 8.00
6 41.26 37.91 33.89 29.06 23.28
7 44.05 41.26 37.91 33.89
8 46.37 44.05 41.26
9 48.31 46.37
10 49.92
Payoff Table of Put Option
Constructing the tree of stock prices for the put option yields the same results as those
obtained for the call option; that is a range of stock prices between £8.08 and £309.59.
The payoff of the European put option is: 𝐶 𝑇 = [𝐾 − 𝑆 𝑇]+
When ST > K at maturity CT is negative for which the option is not exercised and the payoff is
zero. As the table shows, the payoff for the holder of the European put option ranges from a
maximum of £49.92 to a minimum of £0.00. The payoff cannot be negative because the
holder of the option will not exercise if ST > K is satisfied.
The same methodology previously used to calculate the value of the European call is applied
to find the value of the put. However, if the value of the call at t0 is known, it would not be
necessary to go through all the calculations to obtain the value of the put at t0. Instead, the
European put-call parity formula can be used:
𝑝(𝑡) = 𝑐(𝑡) + 𝐾𝐵(𝑡, 𝑇) − 𝑆(𝑡) Where B (t, T) is the discreet discount factor
1
(1+𝑟) 𝑛
18.49 + 58
1
(1+0.05)10 − 50 = 𝟒. 𝟎𝟗

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i/t 0 1 2 3 4 5 6 7 8 9 10
0 -0.20 -0.14 -0.08 -0.04 -0.02 0.00 0.00 0.00 0.00 0.00 0.00
1 -0.34 -0.25 -0.17 -0.10 -0.04 -0.01 0.00 0.00 0.00 0.00
2 -0.52 -0.42 -0.32 -0.21 -0.11 -0.04 0.00 0.00 0.00
3 -0.73 -0.64 -0.54 -0.41 -0.26 -0.12 0.00 0.00
4 -0.91 -0.87 -0.80 -0.71 -0.57 -0.36 0.00
5 -1.00 -1.00 -1.00 -1.00 -1.00 1.00
6 -1.00 -1.00 -1.00 -1.00 1.00
7 -1.00 -1.00 -1.00 1.00
8 -1.00 -1.00 1.00
9 -1.00 1.00
10 1.00
Delta of European Put Option
As expected it yields the same result as using the European put-call parity formula; £4.09.
Delta of the put option
The interpretation of the delta value for a put option is not different from the call option: the
rate at which the price of the option changes with respect to the rate the price of the stock
changes. Using the same formula:
Conversely to the example previously stated, the party shorting the put needs to short the
underlying stock in order to hedge the position. Equally, the delta values will indicate the
amount of stock by which the agent needs to go short. On this occasion, however, the delta
values are negative. This is because the value of the put
option increases when the price of the underlying stock
decreases; there is a negative correlation hence it is always
negative. The value of delta lies between 0 and -1. If the price
of the put increases by the same ratio the price of the
underlying stock decreases delta will equal -1. Again, it
cannot go out of these boundaries because the put cannot
lose or earn value quicker than the underlying asset.
Otherwise, there would be an arbitrage opportunity. The table
on the left shows that the put will only be exercised if there is
a positive payoff at maturity hence delta equals 1 (the party
with the short position will have to purchase the asset).
2.3 The binomial formula.
The binomial formula compared to the value achieved using the tree
The Bernoulli distribution of a random variable (the stock price) is a probability distribution
that takes value 1 if the state of the world (p) occurs and value 0 if state of the world (1-p)
Payoff at
Maturity Delta Values
10 10
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
8.00 1.00
23.28 1.00
33.89 1.00
41.26 1.00
46.37 1.00
49.92 1.00

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occurs. The binomial distribution is a Bernoulli distribution with a pdf of an upward
movement:
𝑓(𝑥) = (
𝑇
𝑥
) 𝑝 𝑥(1 − 𝑝) 𝑇−𝑥
𝐼{0,1,2,…,𝑇}(𝑥)
The value of the European put option using the binomial formula:
𝑉0 =
1
(1+𝑟) 𝑇
∑ (
𝑇
𝑖
) 𝑝 𝑇−𝑖𝑇
𝑖=0 (1 − 𝑝)𝑖
𝑉𝑇
𝑖
[where i is the number of downward movements]
Table breakdown:
Frequency: Combin (T, i)
Probability: freq pT – i
(1-p) i
Stock Price: S0u T- i
di
Payoff: max (K- ST, 0)
Payoff*Prob: SUMPRODUCT
(Discounted)
Pbinomial_formula = Ptree = 4.09
The value obtained using the binomial formula is the same as the value obtained using the
tree.
The binomial formula for different maturities
The probability that an option ends up in the money increases the further away the option is
from expiration. Then the option time value will decrease as expiration approaches.
Considering a put option with 5, 10 and 15 periods until maturity, it is expected that the price
of the option will decrease as we move along the line closer to expiration. The mathematical
reasoning behind this trend is that T is used for discounting as shown in the formula:
𝑉0 =
1
(1+𝑟) 𝑇
∑ (
𝑇
𝑖
) 𝑝 𝑇−𝑖𝑇
𝑖=0 (1 − 𝑝)𝑖
𝑉𝑇
𝑖
That is, the binomial formula will yield a lower value as T increases. The following graph
depicts the path of the price of the put option throughout time. The underlying stock has
more time to reach the strike price at period 5 (and a greater chance) than at period 10 then
the price of the option is higher.
i Frequency Probability Stock Price Payoff Payoff*Prob
0 1 0.01 309.59 0.00 0.00
1 10 0.04 214.99 0.00 0.00
2 45 0.11 149.30 0.00 0.00
3 120 0.21 103.68 0.00 0.00
4 210 0.25 72.00 0.00 0.00
5 252 0.21 50.00 8.00 1.66
6 210 0.12 34.72 23.28 2.79
7 120 0.05 24.11 33.89 1.61
8 45 0.01 16.74 41.26 0.51
9 10 0.00 11.63 46.37 0.09
10 1 0.00 8.08 49.92 0.01
P0 = 4.09

16. Introduction to Financial Derivatives
Natalia Lopez February 2015 Page 15
The frequency by which each node is achieved converges to a normal distribution as the
number of periods increase:
2.4 The American put option.
The American put option gives the holder the option to exercise prior to expiration. For this
reason, when calculating the price of the option at each period, it is necessary to take into
account the payoff on each particular period. The point is to compare which one is greater:
the expected value or the payoff of the option. If the payoff is greater than the expected
value then the option will be exercised at that period. This flexibility increases the value of
American put options with respect to European options in certain situations.
The American put option is then calculated:
𝑉𝑡
𝑖
= 𝑚𝑎𝑥 (𝑚𝑎𝑥(𝐾 − 𝑆 𝑇
𝑖
, 0);
1
(1 + 𝑟)
( 𝑝𝑉𝑡+1
𝑖−1
+ (1 − 𝑝)𝑉𝑡+1
𝑖
)
The stock prices and the payoff are calculated in the same way as before.

17. Introduction to Financial Derivatives
Natalia Lopez February 2015 Page 16
i/t 0 1 2 3 4 5 6 7 8 9 10
0 9.08 4.83 2.38 1.02 0.35 0.07 0.00 0.00 0.00 0.00 0.00
1 16.33 8.97 4.62 2.13 0.79 0.18 0.00 0.00 0.00 0.00
2 23.28 16.33 8.79 4.32 1.76 0.47 0.00 0.00 0.00
3 29.06 23.28 16.33 8.53 3.85 1.21 0.00 0.00
4 33.89 29.06 23.28 16.33 8.12 3.12 0.00
5 37.91 33.89 29.06 23.28 16.33 8.00
6 41.26 37.91 33.89 29.06 23.28
7 44.05 41.26 37.91 33.89
8 46.37 44.05 41.26
9 48.31 46.37
10 49.92
American Put Option
Applying the formula the American put option yields a value of £9.08 at t0 which, as expected,
is a value higher than that obtained for the European put; £4.09.
It is never optimal to exercise American call options before maturity (hence American and
European calls yield the same result) but it might be optimal to exercise American puts prior
to maturity and this is the added value of the American option.
However, the binomial formula has its limitations and it cannot be applied for early exercise
options like the American put. This is because by applying this formula it will not be possible
to know whether an early exercise is optimal or not at each node. For that reason, the
American put is only calculated using the tree.

18. Introduction to Financial Derivatives
Natalia Lopez February 2015 Page 17
Bibliography
John C. Hull., (2011). “Fundamentals of Futures and Option Markets” 7th
Edition.
USA: Pearson Education. Ch: 2, Ch: 3, Ch: 5, Ch: 12, and Ch: 16.
Temizsoy, A., (2015). “Options - Binomial Tree”
Moodle City University. Last Accessed [12.04.15]
Image for the Two-step Binomial Tree Model extracted from:
goddardconsulting.ca (2015). “Option Pricing Using the Binomial Model”
http://www.goddardconsulting.ca/option-pricing-binomial-index.html
Last Accessed [06.04.15]