Lesson 4: Multiple Periods, Option Pricing and Put-call Parity
Economics of Finance
School of Economics, UNSW
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Extending into multi-periods
Time: Present (time 0); Future time periods (times 1 and 2)
State: Two possible realizations of uncertainty: good times and bad times
• 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 unity 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 exercised on any date up to and including its expiration date.
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
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 17/24
1.05 1.26 bg
Alternative and better 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
1.05 1.26 1.05 0.96 0 0
1.1025 1.5876 0 Q=0 01.10251.2096 0
patom = 0.2857 0.6666 0.0816 0.1904 0.1904 0.4444 18 / 24
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
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.
• 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
• Therefore, 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
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