Put/Call Parity and Binomial Model

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• Put/Call Parity and Binomial Model
• (McDonald, Chapters 3, 5, 10)

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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 1,000-strike SR call and sell
  the 1,000-strike SR put, each with 6 months
  to expiration

Figure 3.6 Purchase of a 1000 strike SR call,
sale of a 1000-strike SR put, and the combined
position. The combined position resembles the
profit on a long forward contract.
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Synthetic Forwards (contd)

• 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

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Equation 3.1 Put/Call Parity
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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

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Table 3.1 Payoff and profit at expiration from
purchasing the SR 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.
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Insuring a Long Position Floors (contd)

• Example SR index and a SR 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!
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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.
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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
• (where the asset is bought at t 0,
  delivered at t T)

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Pricing Prepaid Forwards (contd)

• Pricing by arbitrage
• If at time t0, the prepaid forward price somehow
  exceeded the stock price, i.e., , an
  arbitrageur could do the following
• Since, this sort of arbitrage profits are traded
  away quickly, and cannot persist, at equilibrium
  we can expect

Table 5.2 Cash flows and transactions to
undertake arbitrage when the prepaid forward
price, FP 0,T , exceeds the stock price, S0.
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Pricing Prepaid Forwards (contd)

• What if there are dividends? Is still valid?
• No, because the holder of the forward will not
  receive dividends that will be paid to the holder
  of the stock ?
• For discrete dividends Dti at times ti, i 1,.,
  n
• The prepaid forward price
• For continuous dividends with an annualized yield
  d
• The prepaid forward price

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Pricing Prepaid Forwards (contd)

• 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 (contd)

• 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
• Continuous dividends

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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
• Note that the total payoff at expiration is same
  as forward payoff

Table 5.3 Demonstration that borrowing S0e-dT to
buy e-dT shares of the index replicates the
payoff to a forward contract, ST - F0,T .
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Table 5.4 Demonstration that going long a
forward contract at the price F0,T S0e(r-d)T
and lending the present value of the forward
price creates a synthetic share of the index at
time T .
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Table 5.5 Demonstration that buying e-dT shares
of the index and shorting a forward creates a
synthetic bond.
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Creating a Synthetic Forward (contd)

• 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.
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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 distributionthat 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-Ross-Rubinstein pricing model

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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

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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

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Computing the Option Price (contd)

• Payoffs
• Portfolio A Stock Price in 1 Year
• 30.0 60.0
• Payoff 0 20.0
• Portfolio B Stock Price in 1 Year
• 30.0 60.0
• 2/3 purchased shares 20.000 40.000
• Repay loan of 18.462 20.000 20.000
• Total payoff 0 20.000

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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

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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

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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 ?, which is reinvested in the stock.
  Thus, if you buy one share at time t, at time th
  you will have e?h 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

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The Binomial Solution (contd)

• Stock price tree ? Corresponding tree for
  the value of the option
• uS0 Cu
• S0 C0
• dS0 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

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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

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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

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A Graphical Interpretation of the Binomial
Formula (contd)
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 ?.
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Pricing with Dividends
Equation 10.5
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Constructing a Binomial Tree (contd)

• With uncertainty, the stock price evolution is
• (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
• (10.11)
• We refer to a tree constructed using equation
  (10.11) as a forward tree.

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Figure 10.3 Binomial tree for pricing a European
call option assumes S 41.00, K 40.00, s
0.30, r 0.08, T 1.00 years, d 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.
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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.

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A Two-Period European Call

• We can extend the previous example to price a
  2-year 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, s
0.30, r 0.08, T 2.00 years, d 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.
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Pricing the Call Option (contd)

• 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 options ? 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

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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)

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Many Binomial Periods (contd)

• 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, s
0.30, r 0.08, T 1.00 year, d 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.
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Many Binomial Periods (contd)

• 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
• 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

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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)

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Put Options (contd)

• 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, s
0.30, r 0.08, T 1.00 year, d 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.
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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, s 0.30, r 0.08, T 1.00 year, d
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.
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Options on Other Assets

• The model developed thus far can be modified
  easily to price options on underlying assets
  other than nondividend-paying stocks.
• The difference for different underlying assets is
  the construction of the binomial tree and the
  risk-neutral probability.

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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, s 0.30, r 0.05, T
1.00 year, d 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.
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Options on Currency

• With a currency with spot price x0, the forward
  price is
• Where rf is the foreign interest rate, and it
  replaces in the dividend case.

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Options on Currency (contd)

• 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