Financial Economics
Spring 2009
Chapter. 15 Options and Contingent Claims
ü 15-3 Put-Call Parity Relation
In the following, we construct portfolio A and B. These two portfolios will have the same value on the maturity. If the
values will be the same in the future, costs to construct A and B must be the same today. Meanwhile, A and B contain
position of put or call. To have the same cost to construct A and B, the premiums of put and call must satisfy the parity
relation. We consider call and put with the same strike price. Let K be strike price.
Portfolio A (called protective put): Long position of a share of stock and long position of a put option with strike price
K.
Portfolio B: Long position of a call option with strike price K and long position of a discount bond with face value of
K.
Let VA and VB be values of portfolio A and B on the expiration date;
VA = Max ( K-S1, 0 ) + S1
VB = Max ( S1- K,0 ) + K
We show that VA and VB are equal, first by graph and second by rewriting equations.
ü Derivation by Payoff Diagrams
ü Payoff Diagram of Portfolio A
Let's see the graphs of VA's components; values of put and stock. Payoff Diagram of Protective Put
ClearVA, VB, put, gput, stock, gstock, gva, S1; K  100;
putS1_ : MaxK  S1, 0;
 payoff of put 
stockS1_ : S1 ;
 value of stock 
VAS1_ : putS1  stockS1;  value of portfolio A 
gput  PlotputS1, S1, 0, 200, AspectRatio  1, AxesLabel  "S1", "$",
PlotRange  0, 200, PlotLabel  "Value of Put", ImageSize  140;
gstock  PlotstockS1, S1, 0, 200,
AxesLabel  "S1", "$", AspectRatio  1,
PlotStyle  Dashing0.01, 0.02,
PlotLabel  "Value of Stock", ImageSize  140;
ü About Mathematica
" PlotRangeØ{0,200} " is to draw y axis from 0 to 200.
"ImageSize Ø 160" is to make graph smaller than default size.
"AspectRatio Ø 1" is to set ratio of graph's vertical and horizontal lines being one.
" ; " By using semicolon at the end of drawing graph command, you can suppress a graph.
gva  PlotVAS1, S1, 0, 200, ImageSize  140, PlotRange  0, 200,
AspectRatio  1, AxesLabel  "S1", "$",
PlotStyle  Thickness0.02, PlotLabel  "Portfolio A";
Clearshow3; show3  Showgva, gput, gstock, PlotLabel  "Portfolio's Payoff" ;
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Financial Economics
Spring 2009
Rowgput, gstock, show3
Value of Put
Portfolio's Payoff
Value of Stock
$
$
$
200
200
200
150
150
150
100
100
100
50
50
50
0
50
100
150
200
S1
50
100
150
200
S1
S1
0
50
100
150
200
VA's graph looks the same as a graph of call plus constant value.
ü Payoff Diagram of Portfolio B
Portfolio B's value is given by the following equation;
VB=Max (S1-K,0)+K
Clearcall, vb, vK, gcall, gbond, gvb; callS1_ : MaxS1  K, 0
vKS1_ : K  face value of pure discount bond 
vbS1_ : callS1  vKS1  value of portfolio B 
Cleargcall, gbond, show3
gcall  PlotcallS1, S1, 0, 250, AxesLabel  "S1", "value of call",
PlotRange  0, 250, AspectRatio  1, ImageSize  140;
gbond  PlotvKS1, S1, 0, 250, AxesLabel  "S1", "value of bond",
AspectRatio  1, PlotStyle  Dotted, ImageSize  140;
gvb  PlotvbS1, S1, 0, 250, PlotRange  0, 250, AspectRatio  1,
PlotStyle  Thickness0.02, ImageSize  140;
show3  Showgcall, gbond, gvb,
PlotLabel  "Payoff of Portfolio B", AxesLabel  "S1", "$";
Rowgcall, gbond, show3
Payoff of Portfolio B
value of call
250
value of bond
200
200
250
200
150
150
150
100
100
100
50
50
$
50
S1
0
50 100 150 200 250
50 100 150 200 250
S1
0
50
100 150 200 250
S1
The graphs of portfolios show that A and B have the same value on the maturity for any value of S1.
ü costs to construct portfolios
Two portfolios have the same value on the expiration date.By law of one price, costs to construct these portfolios
should be the same.
Cost of A: price of stock + put premium; S + p
Cost of B: call premium + price of discount bond; c +
2011-0706Ch15a.nb
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K
;
1+r
Financial Economics
Spring 2009
Hence,
S+p=c+
K
1+r
This is called put-call parity relation.
ü Derivation of Put-Call Parity by Equations
We can also have the same result by rewriting equations. If we rewrite equation of VA, then we can derive that
for VB.
VA= Max ( K-S1, 0 ) + S1
= Max ( K, S1 )
= Max( 0, S1- K) + K
=VB
ü Arbitration ?
If S + p < c +
K
,
1+r
then how can you arbitrage?
ü Derivation by Covered Call
Portfolio called “Covered call” consists of call short position and stock long position. Payoff diagram of covered
call is equivalent to that of a portfolio which consists of put short position and pure discount bill long position.
ü 15.5 Two-State Option Pricing
We assume that the stock price can take only one of the two possible values at the expiration date of the option.
We construct a synthetic call using only stocks and riskless borrowing. Then, by the Law of One Price, we know
that the price of the call must equal the cost of the synthetic call.
We take an example of one-year call with K=100. Suppose that S0=100. One year later stock price is either
S1=120 or 80. So, payoff, i.e., expiration date value of option is given Max( S1-K, 0 ) = Max( S1-100, 0 ) .
∫
S1 call payoff
up 120
20
down 80
0
ü Construction of Replicating Portfolio
We want to construct a synthetic call option which replicates the original call option.Replicating call implies it
will have the same values as the original call option. Suppose we have portfolio A which consists of x unit of
stock and amount y of borrowing. We want to determine values of x and y so that portfolio A has the same value
of call option whether S1=120 or 80. It implies that x and y satisfy the following equation;
120 x – (1+r) y = 20
80 x– (1+r) y = 0
Clearx, y, r, ans, costA; r  0.05;
ans  NSolve120 x  1  r y  20, 80 x  1  r y  0, x, y;
x  x . ans1, 1; y  y . ans1, 2;
Print"x", x, " and y", y
x0.5 and
y38.0952
If portfolio A consists of 0.5 unit of stock and $38.0952 of borrowing, then portfolio A will have the same value
of call option in the next period. Portfolio A replicates call option. Since portfolio A has the same value as the call
option at t = 1, Construction of portfolio A at t = 0 should cost as much as call option. Cost to construct A is
equal to call's premium.
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Financial Economics
Spring 2009
cost of portfolio A premium of call
0.5  100  38.0952
C
 11.9048
costA  100 x  y; Print"Cost to construct Portfolio is $", costA
Cost to construct Portfolio is $11.9048
By law of one price, call's premium is equal to $11.9048.
ü About Mathematica; solving linear equation
Clearans, a, b, c, d, v1, v2, x, y
ans  Solvea x  b y  v1, c x  d y  v2 , x, y
x  
d v1  b v2
bcad
,y
 c v1  a v2
bcad

Solution stays inside of braces. We want to take out values of x and y. Variable x is shown as { { x , y } }. So it
is the first element of the first brace. Solutions of Mathematica takes the following form in general; { {x1,y1 }, {
x2, y2}, ..., { xn, yn } }. In case of this question, we have {x1, y1} only. So we identify x as ans[ [1,1 ] ] and y as
ans[ [ 1, 2] ].
x  x . ans1, 1
y  y . ans1, 2


d v1  b v2
bcad
 c v1  a v2
bcad
ü Homework No. 11, Due July 13, 2011
Q1. Ch15, Problem 5
Q2. Ch15, Problem 7
Q3. Ch15, Problem 13. Hedge ratio is the same as value of x. Interest rate is over 91 days, i.e. APR = 4%.
Q4. Ch15, Problem 14. Let S1 be stock price in the next period. Assume S1 = (1+s )S0 or (1-s )S0 .
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