Put/Call Parity and Binomial Model
(McDonald, Chapters 3, 5, 10)
Synthetic Forwards
• A synthetic long forward contract
– Buying a call and selling a put on the same underlying asset, with
each option having the same strike price and time to expiration
– Example: buy the $1,000strike S&R call and sell
the $1,000-strike S&R
put, each with 6 months
to expiration
Figure 3.6 Purchase of a 1000 strike
S&R call, sale of a 1000-strike S&R
put, and the combined position. The
combined position resembles the
profit on a long forward contract.
4-2
Synthetic Forwards (cont’d)
• Differences between a synthetic long forward
contract and the actual forward
– The forward contract has a zero premium, while the
synthetic forward requires that we pay the net option
premium
– With the forward contract, we pay the forward price,
while with the synthetic forward we pay the strike price
4-3
Equation 3.1: Put/Call Parity
4-4
Insuring a Long Position: Floors
• A put option is combined with a position in the
underlying asset
• Goal: to insure against a fall in the price of the
underlying asset
4-5
Table 3.1 Payoff and profit at expiration from purchasing the S&R
index and a 1000-strike put option. Payoff is the sum of the first two
columns. Cost plus interest for the position is ($1000 + $74.201) ×
1.02 = $1095.68. Profit is payoff less $1095.68.
4-6
Insuring a Long Position: Floors (cont’d)
• Example: S&R index and a S&R put option with a
strike price of $1,000 together
Figure 3.1 Panel (a) shows the payoff
diagram for a long position in the index
(column 1 in Table 3.1). Panel (b) shows
the payoff diagram for a purchased index
put with a strike price of $1000 (column 2
in Table 3.1). Panel (c) shows the
combined payoff diagram for the index
and put (column 3 in Table 3.1). Panel (d)
shows the combined profit diagram for
the index and put, obtained by subtracting
$1095.68 from the payoff diagram in
panel (c)(column 5 in Table 3.1).
Buying an asset and a put
generates a position that looks
like a call!
4-7
Alternative Ways to Buy a Stock
• Four different payment and receipt timing combinations
–
–
–
–
Outright purchase: ordinary transaction
Fully leveraged purchase: investor borrows the full amount
Prepaid forward contract: pay today, receive the share later
Forward contract: agree on price now, pay/receive later
• Payments, receipts, and their timing
Table 5.1 Four different
ways to buy a share of
stock that has price S0 at
time 0. At time 0 you
agree to a price, which is
paid either today or at
time T. The shares are
received either at 0 or T.
The interest rate is r.
4-8
Pricing Prepaid Forwards
• Pricing by analogy
– In the absence of dividends, the timing of delivery is irrelevant
– Price of the prepaid forward contract same as current stock price
– F P 0,T
 S0
(where the asset is bought at t = 0, delivered at t = T)
4-9
Pricing Prepaid Forwards (cont’d)
• Pricing by arbitrage
– If at time t=0, the prepaid forward price somehow exceeded the stock
price, i.e.,
, an arbitrageur could do the following
F P 0,T  S0
Table 5.2 Cash flows and
transactions to undertake
arbitrage when the prepaid
forward price, FP 0,T , exceeds
the stock price, S0.
F P 0,T  S0
4-10
Pricing Prepaid Forwards (cont’d)
• What if there are dividends? Is F P 0,T  S0
still valid?
– No, because the holder of the forward will not receive
dividends that will be paid to the holder of the stock 
F P 0,T  S0  PV (all dividends paid from t=0 to t=T)
F P 0,T  S0
– For discrete dividends Dti at times ti, i = 1,…., n
• The prepaid forward price: F P 0,T  S0  ni1PV0,ti (Dti )
• For continuous dividends with an annualized yield d
• The prepaid forward price:
F P 0,T  S0 ed T
4-11
Pricing Prepaid Forwards (cont’d)
• Example 5.1
– XYZ stock costs $100 today and is expected to pay a
quarterly dividend of $1.25. If the risk-free rate is 10%
compounded continuously, how much does a 1-year
prepaid forward cost?
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Pricing Prepaid Forwards (cont’d)
• Example 5.2
– The index is $125 and the dividend yield
is 3% continuously compounded. How
much does a 1-year prepaid forward cost?
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Pricing Forwards on Stock
• Forward price is the future value of
the prepaid forward
– No dividends
F0,T  FV (F P 0,T )  FV (S0 )  S0 erT
– Continuous dividends
F0,T  S0e
( r d )T
4-14
Creating a Synthetic Forward
• One can offset the risk of a forward by creating a synthetic
forward to offset a position in the actual forward contract
• How can one do this? (assume continuous dividends at rate d)
– Recall the long forward payoff at expiration: = ST – F0, T
– Borrow and purchase shares as follows
Table 5.3 Demonstration
that borrowing S0e−δT to
buy e−δT shares of the
index replicates the payoff
to a forward contract,
ST − F0,T .
– Note that the total payoff at expiration is same as forward payoff
4-15
Table 5.4 Demonstration that going long a forward contract at the
price F0,T = S0e(r−δ)T and lending the present value of the forward
price creates a synthetic share of the index at time T .
4-16
Table 5.5 Demonstration that buying e−δT
shares of the index and shorting a forward
creates a synthetic bond.
4-17
Creating a Synthetic Forward (cont’d)
• The idea of creating synthetic forward leads to following
– Forward = Stock – zero-coupon bond
– Stock = Forward + zero-coupon bond
– Zero-coupon bond = Stock – forward
• Cash-and-carry arbitrage: Buy the index, short the forward
Figure 5.6 Transactions
and cash flows for a
cash-and-carry: A
marketmaker is short a
forward contract and
long a synthetic forward
contract.
4-18
Introduction to Binomial Option Pricing
• Binomial option pricing enables us to determine the price
of an option, given the characteristics of the stock or other
underlying asset
• The binomial option pricing model assumes that the price
of the underlying asset follows a binomial distribution—
that is, the asset price in each period can move only up or
down by a specified amount
• The binomial model is often referred to as the “Cox-RossRubinstein pricing model”
4-19
A One-Period Binomial Tree
• Example
– Consider a European call option on the stock of XYZ, with a $40
strike and 1 year to expiration
– XYZ does not pay dividends, and its current price is $41
– The continuously compounded risk-free interest rate is 8%
– The following figure depicts possible stock prices over 1 year, i.e.,
a binomial tree
$60
$41
$30
4-20
Computing the Option Price
• Next, consider two portfolios
– Portfolio A: buy one call option
– Portfolio B: buy 2/3 shares of XYZ and borrow $18.462
at the risk-free rate
• Costs
– Portfolio A: the call premium, which is unknown
– Portfolio B: 2/3  $41 – $18.462 = $8.871
4-21
Computing the Option Price (cont’d)
• Payoffs:
– Portfolio A:
Stock Price in 1 Year
$30.0
Payoff
0
– Portfolio B:
2/3 purchased shares
Repay loan of $18.462
Total payoff
$60.0
$20.0
Stock Price in 1 Year
$30.0
$60.0
$20.000 $40.000
– $20.000 –$20.000
0
$20.000
4-22
A One-Period Binomial Tree
• Another Example
– Consider a European call option on the stock of XYZ, with a $40
strike and 1 year to expiration
– XYZ does not pay dividends, and its current price is $41
– The continuously compounded risk-free interest rate is 8%
– The following figure depicts possible stock prices over 1 year, i.e.,
a binomial tree
4-23
Computing the Option Price
• Next, consider two portfolios
– Portfolio A: buy one call option
– Portfolio B: buy 0.7376 shares of XYZ and borrow
$22.405 at the risk-free rate
• Costs
– Portfolio A: the call premium, which is unknown
– Portfolio B: 0.7376  $41 – $22.405 = $7.839
4-24
The Binomial Solution
• How do we find a replicating portfolio consisting of 
shares of stock and a dollar amount B in lending, such that
the portfolio imitates the option whether the stock rises or
falls?
– Suppose that the stock has a continuous dividend yield of d, which
is reinvested in the stock. Thus, if you buy one share at time t, at
time t+h you will have edh shares
– If the length of a period is h, the interest factor per period is erh
– uS0 denotes the stock price when the price goes up, and dS0
denotes the stock price when the price goes down
4-25
The Binomial Solution (cont’d)
• Stock price tree:
uS0
S0
dS0
 Corresponding tree for
the value of the option:
Cu
C0
Cd
– Note that u (d) in the stock price tree is interpreted as one plus the rate of
capital gain (loss) on the stock if it foes up (down)
• The value of the replicating portfolio at time h, with stock
price Sh, is
Sh  e B
rh
4-26
Arbitraging a Mispriced Option
• If the observed option price differs from its
theoretical price, arbitrage is possible
– If an option is overpriced, we can sell the option.
However, the risk is that the option will be in the money
at expiration, and we will be required to deliver the
stock. To hedge this risk, we can buy
a synthetic option at the same time we sell the actual
option
– If an option is underpriced, we buy the option. To hedge
the risk associated with the possibility of the stock price
falling at expiration, we sell a synthetic option at the
same time
4-27
A One-Period Binomial Tree
• Example
– Consider a European call option on the stock of XYZ, with a $40
strike and 1 year to expiration
– XYZ does not pay dividends, and its current price is $41
– The continuously compounded risk-free interest rate is 8%
– The following figure depicts possible stock prices over 1 year, i.e.,
a binomial tree
$60
$41
$30
4-28
A Graphical Interpretation of
the Binomial Formula (cont’d)
Figure 10.2 The payoff to
an expiring call option is
the dark heavy line. The
payoff to the option at the
points dS and uS are Cd
and Cu (at point D). The
portfolio consisting of ∆
shares and B bonds has
intercept erh B and slope
∆, and by construction
goes through both points
E and D. The slope of the
line is calculated as
Rise/Run between points
E and D, which gives the
formula for ∆.
4-29
Pricing with Dividends
Equation 10.5
4-30
Constructing a Binomial Tree (cont’d)
• With uncertainty, the stock price evolution is
uSt  Ft ,t  h e 
h
dSt  Ft ,t  h e 
h
(10.10)
– Where  is the annualized standard deviation of the
continuously compounded return, and h is standard
deviation over a period of length h
• If we divide both sides by initial stock price, we can
rewrite (10.10) as
( r d) h   h
ue
d  e(r d)h
h
(10.11)
– We refer to a tree constructed using equation (10.11) as a
“forward tree.”
4-31
Figure 10.3 Binomial tree for pricing a European call option; assumes
S = $41.00, K = $40.00, σ = 0.30, r = 0.08, T = 1.00 years, δ = 0.00, and h =
1.000. At each node the stock price, option price, ∆, and B are given. Option
prices in bold italic signify that exercise is optimal at that node.
4-32
Summary
• In order to price an option, we need to know
–
–
–
–
–
Stock price
Strike price
Standard deviation of returns on the stock
Dividend yield
Risk-free rate
• Using the risk-free rate and , we can approximate the
future distribution of the stock by creating a binomial tree
using equation (10.11)
• Once we have the binomial tree, it is possible to price the
option using the regular equations.
4-33
A Two-Period European Call
• We can extend the previous example to price a 2year option, assuming all inputs are the same as
before
Figure 10.4 Binomial tree
for pricing a European call
option; assumes S =
$41.00, K = $40.00, σ
=0.30, r = 0.08, T = 2.00
years, δ = 0.00, and h =
1.000. At each node the
stock price, option price,
∆, and B are given. Option
prices in bold italic signify
that exercise is optimal at
that node.
4-34
Pricing the Call Option (cont’d)
• Notice that
– The option was priced by working backward
through the binomial tree
– The option price is greater for the 2-year than for
the 1-year option
– The option’s  and B are different at different
nodes. At a given point in time,  increases to 1 as
we go further into the money
– Permitting early exercise would make no difference.
At every node prior to expiration, the option price is
greater than S – K; thus, we would not exercise
even if the option was American
4-35
Many Binomial Periods
• Dividing the time to expiration into more periods
allows us to generate a more realistic tree with a
larger number of different values at expiration
– Consider the previous example of the 1-year European
call option
– Let there be three binomial periods. Since it is a 1-year
call, this means that the length of a period is h = 1/3
– Assume that other inputs are the same as before (so, r
= 0.08 and  = 0.3)
4-36
Many Binomial Periods (cont’d)
• The stock price and option
price tree for this option
Figure 10.5 Binomial tree for
pricing a European call option;
assumes S = $41.00, K = $40.00,
σ =0.30, r = 0.08, T = 1.00 year, δ
= 0.00, and h = 0.333. At each
node the stock price, option
price, ∆, and B are given. Option
prices in bold italic signify that
exercise is optimal at that node.
4-37
Many Binomial Periods (cont’d)
• Note that since the length of the binomial period is
shorter, u and d are smaller than before: u = 1.2212
and d = 0.8637 (as opposed to 1.462 and 0.803 with h
= 1)
– The second-period nodes are computed as follows
Su  $41e0.081/ 3 0.3 1/ 3  $50.071
Sd  $41e0.081/ 3 0.3
1/ 3
 $35.411
– The remaining nodes are computed similarly
• Analogous to the procedure for pricing the 2-year
option, the price of the three-period option is computed
by working backward using equation (10.3)
– The option price is $7.074
4-38
Put Options
• We compute put option prices using the same
stock price tree and in the same way as call
option prices
• The only difference with a European put option
occurs at expiration
– Instead of computing the price as max (0, S – K), we
use max (0, K – S)
4-39
Put Options (cont’d)
• A binomial tree for a European put
option with 1-year to expiration
Figure 10.6 Binomial tree for
pricing a European put option;
assumes S = $41.00, K = $40.00,
σ = 0.30, r = 0.08, T = 1.00 year,
δ = 0.00, and h = 0.333. At each
node the stock price, option
price, ∆, and B are given. Option
prices in bold italic signify that
exercise is optimal at that node.
4-40
American Put Options
• Consider an American version of the put
option valued in the previous example
Figure 10.7 Binomial tree for
pricing an American put option;
assumes S = $41.00, K =
$40.00, σ =0.30, r = 0.08, T =
1.00 year, δ = 0.00, and h =
0.333. At each node the stock
price, option price, ∆, and B
are given. Option prices in
bold italic signify that exercise
is optimal at that node.
4-41
Options on Other Assets
• The model developed thus far can be modified easily to
price options on underlying assets other than nondividendpaying stocks.
• The difference for different underlying assets is the
construction of the binomial tree and the risk-neutral
probability.
4-42
Options on a Stock Index
• A binomial tree for an
American call option
on a stock index:
Figure 10.8 Binomial tree for pricing
an American call option on a
stock index; assumes S = $110.00, K
= $100.00, σ = 0.30, r = 0.05, T =
1.00 year, δ = 0.035, and h = 0.333.
At each node the stock price, option
price, ∆, and B are given. Option
prices in bold italic signify that
exercise is optimal at that node.
4-43
Options on Currency
• With a currency with spot price x0, the forward
price is
(r rf )t
F0,t  x0 e
– Where rf is the foreign interest rate, and it replaces d
in the dividend case.
4-44
Options on Currency (cont’d)
• Consider a dollar-denominated American put
option on the euro, where
–
–
–
–
The current exchange rate is $1.05/€
The strike is $1.10/€
The euro-denominated interest rate is 3.1%
The dollar-denominated rate is 5.5%
4-45