Lecture 4: Multi-period Binomial Models, Options and Option Pricing Economics of Finance
School of Economics, UNSW
Extending into multi-periods
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Time: Present (time 0); Future time periods (times 1 and 2)
State: Two possible realizations of uncertainty: good times and bad times
Multi-periods
• Notation:
• Number of letters (g, gg) indicates time period;
• Sequence of letters indicates the path taken to reach the
• Notice the states of gb and bg can be identical.
• Two-period zero-coupon bond (no coupon payments)
• Its initial value is $1.00.
• Its price increases 5% of its prior value in every period.
gg 1 1.1025
1 1.1025 H b
PqP 1.1025
• Its initial value is $1.00. It pays no dividends.
• Its price increases 26% of its prior value in good times. • Its price falls to 96% of its prior value in bad times.
gg 1 1.5876
1 1.2096 H b
PPq 0.9216
Security revisited
The number of future states of the world equals six.
• But seems we have only two ”securities”, bond and stock.
• What can we do?
• Recall our definition of ”security”, i.e., state contingent contract.
• Now that the time span has been extended into more than one periods, we need to extend the security space to accommodate them.
Planned Acquisitions
Consider the following set of planned acquisitions
B0: Buy a Bond at period 0, sell it at the end of the next
S0: Buy a Stock at period 0, sell it at the end of the next period;
Bg: At period 1, if the state is g, buy a Bond, sell it at the end of the next period;
Sg: At period 1, if the state is g, buy a Stock, sell it at the end of the next period;
Bb: At period 1, if the state is b, buy a Bond, sell it at the end of the next period;
Sb: At period 1, if the state is b, buy a Stock, sell it at the end of the next period.
0 1.05 1.26 0 0 gg 0 1.05 0.96 0 0 gb
Matrix Notation
We write down the payment of these acquisitions in a matrix: B0 S0 Bb Sb
1.05 1.26 −1 −1 0 0 g
1.05 0.96 0 0 −1 −1 b
0 0 0 0 1.05 0.96 bb
1.05 1.26 bg
• A set of 6 column vectors;
• Each presenting a payment stream for a planned
acquisition;
• Notice they are linearly independent, det(Q) ̸= 0
• Such linearly independent vector set is not unique, just like bond and stock is not the unique set of linearly independent securities in the one period world.
Price Vector
Price Vector entails:
B0 S0 BgSgBbSb
pS = 1.00 1.00 0.0 0.0 0.0 0.0
• Why are the strategies Bg, Sg, Bb, and Sb priced as 0? • Any insights on the meaning of ”price”?
Pricing a state
To price a unit of payment at each state, we can now use the formula we are familiar with: patom = pS · Q−1 :
g b gg gb bg bb
patom = 0.2857 0.6666 0.0816 0.1904 0.1904 0.4444
• Extending time span necessarily extends the space of states;
• This, however, does not necessarily mean we need more than two securities;
• Instead, by manipulating with existing securities in various periods, we expand the action space;
• These actions creates linearly independent planned acquisitions. We call them “elementary strategies”;
• Each state can be price in a similar way to the atomic securities;
• Notice the set of elementary strategies may not be unique.
Definition
• Call option vs. Put option:
• Call option: entitles the right to buy an underlying asset
(say shares, foreign currency or commodity) at a specified
strike price, or, exercise price(X).
• Put option: entitles the right to sell the underlying asset at
a specified strike price X.
• European option vs. American Option
• European put or call option: can be exercised only on expiry date.
• American put or call option: can be exercised on any date up to and including its expiration date.
Terminology
In-the-money, at the money and out-of the money: Denote p as the market price:
Call Option
Put Option
In-the-money
At-the-money
Out-of-the-money
Example: General Electric options (March 2013)
Call option payment
Example: Consider a call option that entitles the right to buy the stock at $55. Strike price: X = $55
• Case 1: If the actual stock price is less than the strike price, p < X, then the option holder will not exercise the Call option. The payoff of exercising this Call option would be zero.
• Case 2: If the actual stock price in a year is more than the strike price, p > X, then it pays to exercise the Call option.
For example, if p = $75, then the Call option’s payoff if exercised is 75 − 55 = $20.
Note: No need to actually buy the stock to receive this payoff.
Long Call Payoff = M ax{p − X, 0}
We plot the payoff of a call option with a given strike price as a function of price of the underlying security (“underlier”).
Call option
Short Call Payoff = −M ax{p − X, 0} Payoffs of selling call option.
Payoff −$20
Long Put Payoff = Max{X − p, 0} Consider a put option, where X = $55 and P = $45. For the
option buyer:
Put option
Short Put Payoff = −M ax(X − S, 0)
Similarly, payoffs of a seller of put option $45 $55
Payoff −$10
Overall profit
The overall profit/loss will also include the price of the option. For example, if Call option price is $5.75.
Profit/loss
Option price ($5.75)
$55 $60.75 P Break-even price
The Setup: Three Period Binomial Model
• Two-period zero-coupon bond with initial value of $1.00. Its price increases 5% of its prior value in every period.
• The Stock pays no dividends. Its initial value is $1.00.
• Its price increases 26% of its prior value in good times.
• Its price falls to 96% of its prior value in bad times.
1 1.2096 H b
gg 1 1.5876
PqP 0.9216
Computing atomic (state) prices
• The Payment Matrix:
B0 S0 Bb Sb
1.05 1.26 −1 −1 0 0 g
1.05 0.96 0 0 −1 −1 b
0 1.05 1.26 0 0 gg 0 1.05 0.96 0 0 gb
0 0 0 0 1.05 0.96 bb
• The Price Vector:
B0 S0 BbSb pS = 1.00 1.00 0.0 0.0 0.0 0.0
• The atomic prices patom= pS · Q−1:
g b gg gb bg bb
patom = 0.2857 0.6666 0.0816 0.1904 0.1904 0.4444 23/46
1.05 1.26 bg
Alternative way to compute atomic (state) prices
• The Payment Matrix:
B0 S0 Bb Sb
−1.05 −1.26 0
0 g −0.96 b
0gb 0 0 0 0 1.1025 1.2096 bg 0 0 0 0 1.1025 0.9216 bb
• The Price Vector:
B0 S0 BbSb pS = 1.00 1.00 0.0 0.0 0.0 0.0
• The atomic prices patom= pS · Q−1:
g b gg gb bg bb
patom = 0.2857 0.6666 0.0816 0.1904 0.1904 0.4444 24/46
1.05 1.26 1.05 0.96 0 0
1.1025 1.5876 0 Q=0 01.10251.2096 0
European Call Option
The matrix c can be derived from the payoff of call options at the end of each state by using Max (S-X,0) where X is given by $1.1 in the example. Since the European call option will be likely exercised at T=2 (i.e:expiry date), payoff at T=1 (ie: g and b states) will be zero.
Call option payoff gg
Max(1.5876-1.1,0)=0.4876
Max(1.2096-1.1,0)=0.1096
Max(1.2096-1.1,0)=0.1096
Max(0.9216-1.1,0)=0
Example: European Call Option
Consider a European Call option that gives the holder a right to buy the Stock at Period 2 at the Exercise Price, X = 1.10.
Pricing a European Call Option
The cash flow associated with the Call option:
0g 0b 0.4876 gg
c=0.1096 gb 0.1096 bg 0 bb
The atomic prices are still the same:
g b gg gb bg bb
patom = 0.2857 0.6666 0.0816 0.1904 0.1904 0.4444 The value of the Call option is:
pCall = patom · c =0.0816
Example: European Put Option
The matrix c can be derived from the payoff of put options at the end of each state by using Max (X-S,0) where X is given by $1.2 in the example. Since the European put option will be likely exercised at T=2 (i.e:expiry date), payoff at T=1 (ie: g and b states) will be zero.
Call option payoff gg
Max(1.2-1.5876,0)=0
Max(1.2-1.2096,0)=0
Max(1.2-1.2096,0)=0
Max(1.2-0.9216,0)=0.2784
Example: European Put Option
Consider a European Put option that gives the holder a right to sell the Stock in Period 2 at the Exercise Price, X = 1.20.
Pricing a European Put Option
The cash flow associated with the Put option: 0g
0b 0 gg
c= 0 gb 0 bg
0.2784 bb The atomic prices are still the same:
g b gg gb bg bb patom = 0.2857 0.6666 0.0816 0.1904 0.1904 0.4444
The value of the Put option is:
pPut = patom · c =0.4444 · 0.2784 = 0.1237
American Put Option
Consider an American Put option that allows the holder to sell the Stock at a price of X = 1.20 at either Period 1 or Period 2.
• Two-period zero-coupon bond with initial value of $1.00. Its price increases 5% of its prior value in every period.
• The Stock pays no dividends. Its initial value is $1.00.
• Its price increases 26% of its prior value in good times.
• Its price falls to 96% of its prior value in bad times.
1 1.2096 H b
Stock payment
gg 1 1.5876
PqP 0.9216
American Put Option
Put payoff is max{X − S, 0}, where X = 1.2
American Put Option
Do we exercise the option in Period 1 when the stock price has risen?
• The answer: keep it till Period 2! Exercising in state g will result in a negative cash flow.
Do we exercise the option in Period 1 when the stock price has fallen?
• Consider the cash flows that correspond to the two decisions:
0gg cexercise = 0 gb
0 bg 0 bb
0 gg ckeep = 0 gb
0 bg 0.2784 bb
American Put Option
The atomic prices are as before:
g b gg gb bg bb
patom = 0.2857 0.6666 0.0816 0.1904 0.1904 0.4444 The present values of the two cash flows are:
patom · cexercise = 0.6666 · 0.24 = 0.16 patom · ckeep = 0.4444 · 0.2784 = 0.1237
The decision: Exercise the option in Period 1 if the stock price has fallen.
American Put Option
The simplified decision tree include only optimal paths is:
American put option pricing using atomic security
The cash flow associated with the Put option: 0g
0.24 b 0 gg
c= 0 gb 0 bg
0 bb The atomic prices are still the same:
g b gg gb bg bb
patom = 0.2857 0.6666 0.0816 0.1904 0.1904 0.4444
Option price
The value of the Put option is:
pput =patom ·c=0.6666·0.24=0.16
When does American option early exercise?
We know that American options provide more ”options” of early exercising;
• In general, this implies that an American option is more worthy than European option with identical underlier and strike price;
• How general is this intuition?
• The fact is quite disappointing.
• In most cases, the American option value is exactly the same as the corresponding European option.
• Early exercise is rare
Put-Call Parity
Put-Call Parity is a relationship, first identified by Stoll (1969), that must exist between the prices of European Put and Call options that both have:
• the same underlying stock; • the same strike price;
• the same expiration date.
The relationship is derived using arbitrage arguments. Consider two portfolios consisting of:
• The Call option and an amount of cash equal to the present value of the strike price.
• The Put option and the underlying stock.
Put-Call Parity: Cash and Call
Put-Call Parity: Underlier and Put
Put-Call Parity
• The two portfolios (call + cash and put + underlier) have identical expiration values.
• Irrespective of the value of the underlier at expiration, each portfolio will have the same value as the other.
• If the two portfolios are going to have the same value at expiration, then they must have the same value today. Otherwise, an investor could make an arbitrage profit
Put-Call Parity
Accordingly, we have the price equality:
pcall + PV (X) = pput + punderlier (1)
• pcall is the current market value of the call;
• P V (X) is the present value of the strike price, X;
• pput is the current market value of the put;
• punderlier is the current market value of the underlying stock.
Note: “Current” refers to Period 0 since you are evaluating today prices
Put-Call Parity: An example
• We have priced a European Call option that gives the holder a right to Buy the Stock at Period 2 at the Exercise Price, X = 1.10. We found its price to be pCall = 0.0816.
• Consider a European Put option that gives the holder a right to sell the Stock at Period 2 at the Exercise Price, X = 1.10.
The cash flow associated with the Put option: c=0 0 0 0 0 0.1784′
The atomic prices are still the same:
g b gg gb bg bb
patom = 0.2857 0.6666 0.0816 0.1904 0.1904 0.4444 The value of the Put option is:
pPut = patom · c =0.4444 · 0.1784 = 0.0793
Put-Call Parity: An example (cont’d)
According to the Put-Call parity we have
pcall + PV (X) = pput + punderlier
Notice that PV (X) = df(2) · X, where df(2) is the discount factor for Period 2. df(2) is the present value of one certain dollar received at Period 2. It must equal to the sum of atomic security prices for states: gg, gb, bg and bb.
df (2) = 0.0816 + 0.1904 + 0.1904 + 0.4444 = 0.9070 PV(X)=df(2)·X =0.9070·1.10=0.9977
pcall =pput +punderlier −PV(X)
= 0.0793 + 1 − 0.9977 = 0.0816
This is the same value as the one we found before.
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