Option Pricing
Submitted by
Eugene Bala
1. Identifying arbitrage opportunities given a certain interest rates.
From the historical data for the past 12 months - downloaded from moneycentral.msn.com - for the
Nortel Networks' stock price movements the volatility has been estimated based on (1):
(1)
2S2 /  
(2)
S2 = 1/n-1 x ui - u) 2
where
Historical volatility = 58% ( Exhibit 1) which correspond to the figure quoted on
Bloomberg (60% for a 100 days sample).
Furthermore, Nortel's indicated annual dividend yield is 0.12% and $0.0375 dividend are
being paid in cash quarterly - February, May, July and October negatively affecting the call
options ( and positively the put options).
Nortel Networks Corporation (NT)
Daily prices (4/5/99 to 4/4/00)
Last 12 months
Last 30 days
Sigma Ui
1.267886
Sigma Ui
0.034334
Sigma U**i
0.341619
Sigma U**i
0.034342
(Sigma Ui)**21.607535
(Sigma Ui)**20.001179
nb tr days
254
nb tr days
254
stdev
0.036331
stdev
0.033814
volatility = 0.579021
volatility = 0.185206
Last 90 days
Last 5 days
Sigma Ui
0.463173
Sigma Ui -0.178418
Sigma U**i
0.196954
Sigma U**i
0.011466
(Sigma Ui)**20.214529
(Sigma Ui)**20.031833
nb tr days
254
nb tr days
254
stdev
0.046493
stdev
0.006682
volatility = 0.44107
volatility = 0.106491
Exhibit 1
Using the Black-Scholes-Merton model (Winopa software)- considering the above implied
volatility - the risk free rate have been calculated for the NTDR April 21, 2000 Call option and the
NTPR April 21, 2000 Put option. While the call option is in the money the put option is out-of-themoney -Annexes /Exhibit 2.
Eugene Bala
Option Pricing
The NTPR April 21, 2000 Put option
The implied annual Rf = 4.9% continuously compounded when considering the option
price $0.1 which is lower than the Bid or the Last Price (0.125, respectively 0.75) - Exhibit 3. The
yield curve is inverted as the Fed increased the short-term interest rate several times in the recent past
and decided to rise it another 25 BPS several times until August 2000.
(3)
Rf 1-Month (nominal) =
5.67%
Inflation =
1.5%
Real Rf = 5.67%-1.5% =
4.17% (discrete)
Real Rf continuously compounded = 3.58%.
As the put option is way out-of-the-money it is less likely that in a short time span there would be a
major shift in the underlying stock price movement (not impossible just highly improbable under
normal circumstances - and hoping that John Roth is fine and dandy…). When adjusting the
volatility to a shorter recent period of time (answer #3) at 18.5% the option price resulting from the
BSM is 0.
The NTDR April 21, 2000 Call option
The implied annual Rf = 199% continuously compounded considering the option price $30.375,
which is the average of the Bid/Ask Price (Annexes/Exhibit 3). In fact, the call option is largely inthe-money and probably not trading. Working backwards using the Rf from (3) the option price
should be $27.91. The current option price is much higher reflecting the Bid/Ask premium. Although
it might be interesting for some hedging strategies this call option is less attractive for pure
speculation (no 'free' lunch!).
Lower bound for calls and puts
(4)
(5)
c > S - D - X e-r * t
p > D + X e-r * t - S
where :
D is the PV of the dividends = $0.105
S = $117.594
X= $90
r = 3.58%
For the call option
30.375 > 117.594 - 0.105 - 89.772
30.375 > 27.71
For the put option
0.1875 > 0.105 + 89.772 - 117.994
0.1875 > -27.71
There is no arbitrage opportunity as the put price and the call price is greater than the lower respective
bound.
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Eugene Bala
Option Pricing
The Put-Call parity
(6)
S - D - X < C -P < S - X e-r * t
117.594 - 0.105 - 90 < 30.375 -0.1875 < 117.594 - 89.772
27.489 < 30.1875 < 27.82
The put-call parity does not hold and creates an arbitrage opportunity.
In this case the call is overpriced compared to the put option. The arbitrage strategy would be to short
the call, buy the put and buy the stock.
This strategy involves an investment of
$117.594 + $ 0.185 - $ 30.374 = $87.405
When the investment is financed with the risk-free interest rate, a repayment of
$81.405 e.048 * .0515 = $87.606
The possible situations are as follow:
1. stock price is greater than $90. The counterparty exercises the call.
Net profit = 90 - 87.405 = $2.595
2. stock price is less than $90. The investor exercises the put and the share is sold.
Net profit = 90- 87.405 = $2.595
It is important to notice that there is an implementation issue. The above strategy holds if there is
enough liquidity and trades are currently available on those options. In this particular case, last trading
data and volumes are N/A. Again - it's the role of the market makers to provide
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Eugene Bala
Option Pricing
2. Finding the implied volatility and significant differences across options?
3.
The volatility smile plotted based on the BSM implied volatility as a function of the options' strike
price is shown in the Exhibit 4 & 5. Out-of-the-money and in-the-money options have tend to have
higher implied volatilities than at-the-money options.
Calls
Puts
Volatility[%] Strike Price
Volatility[%]
85
146
85
91
90
123
90
64
95
114
95
53
100
90
100
54
105
24
105
48
110
64
110
46
115
65
115
44
120
61
120
41
125
56
125
37
130
54
130
36
135
53
135
59
140
55
140
50
145
56
145
59
150
57
150
97
Strike Price
Exhibit 4
Volatility
160
140
120
%
100
Call Options
80
Put Options
60
40
20
0
Strike Price
Exhibit 5
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Eugene Bala
Option Pricing
The model rely on strong assumptions about the returns random walk without considering significant
phenomena observed on the stock market - that is the rebound effect after a drop in the asset price and
the momentum built on high trading volumes, pushing the market even further upwards.
It is a fact that options prices tend to be higher in environments with high interest rates and high
volatility. In other words, the Black-Scholes-Merton model underprices both deep in-the-money and
deep out-of-the-money options relative to near at-the-money options (when the strike price = forward
price of the asset). Therefore, arbitrage opportunity might arise - as discussed in the previous section.
However, the biases in the BSM model have in average little economical significance as the actual
deviation from the model is in the order of 2%.
4. Computing historic volatility for your stock based on historic stock prices.
Preparing the study "Influence of Beta on the Price of Equity and the Cost Capital" that I have
prepared last year, I have been confronted with a classical sampling issue:
'..When using weekly observations vs monthly observations, the value of betas are smaller as they may
introduce some irrelevant information, like unchanged return on stock and/or the market portfolio.
Nevertheless it increases the size of the sample and thus , the reliability of the estimate..'.
Company
Market Index
Walt
Disney
Company
Unisys Corp.
Dow Jones
S&P500
S&P500
Beta
Monthly obs.
1.07
.85
1.38
Weekly obs.
1.02
.83
1.31
The same apply when using historical data for estimating the volatility. Other biases that come into
play are:


stock splits - Nortel proceeded to a 2:1 stock split in July 29, 1999.
company spin off - Nortel spun off from BCE in January.
The historical volatility have been calculated for different sample sizes of historical data, as shown in
the Exhibit 1. The volatility is lower as the size of the sample is smaller (no surprises).
However, working with larger samples increases the accuracy and the power of the prediction, under
the condition that outlyers and stalled or irrelevant data are adjusted or eliminated. The volatility
closest to the at-the-money options is 40% for the put options and 50% for the call options. These
figures correspond to the estimated value for the past 90 days which and is the lowest point on the
chart in Exhibit 5.
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Eugene Bala
Option Pricing
Annexes
Nortel Networks Corporation
Last: 117.594 Change: -2.156 Volume: 15,708,500
Shaded options are in-the-money
April 21, 2000 Calls
Strike Price
85.000
90.000
95.000
100.000
105.000
110.000
115.000
120.000
125.000
130.000
135.000
140.000
145.000
150.000
Symbol
Last
.NTDQ 36.250
.NTDR 30.125
.NTDS 47.750
.NTVDT 17.625
.NTVDA 29.875
.NTVDB 7.250
.NTVDC 10.250
.NTVDD 3.750
.NTVDE 2.250
.NTVDF 3.750
.NTVDG 2.188
.NTVDH 1.750
.NTVDI 1.000
.NTVDJ 0.750
Chg
%Chg
NA
NA
NA
-7.750
NA
-5.125
NA
-3.625
-3.000
NA
NA
unch
NA
NA
NA
NA
NA
-30.54%
NA
-41.41%
NA
-49.15%
-57.14%
NA
NA
unch
NA
NA
Time
Value
-6.094
-5.719
-5.344
-3.594
-1.719
0.656
3.156
3.875
2.500
2.125
1.375
0.938
0.750
0.438
Bid
Ask
25.500
20.875
16.250
13.125
10.375
7.750
5.375
3.625
2.625
1.875
1.125
0.688
0.375
0.250
26.500
21.875
17.250
14.000
10.875
8.250
5.750
3.875
2.500
2.125
1.375
0.938
0.750
0.438
Vol
NA
NA
NA
45
NA
31
NA
43
6
NA
NA
1
NA
NA
Shaded options are in-the-money
April 21, 2000 Puts
Strike Price
85.000
90.000
95.000
100.000
105.000
110.000
115.000
120.000
125.000
130.000
135.000
140.000
145.000
150.000
Symbol
Last
Chg
%Chg
.NTPQ 0.250
.NTPR 0.125
.NTPS 1.875
.NTVPT 3.125
.NTVPA 5.750
.NTVPB 3.250
.NTVPC 5.250
.NTVPD 14.875
.NTVPE 11.250
.NTVPF 13.000
.NTVPG 28.250
.NTVPH 19.875
.NTVPI 26.000
.NTVPJ 30.375
NA
NA
+1.250
+1.688
+3.250
NA
unch
+7.750
NA
NA
+11.000
NA
NA
NA
NA
NA
+200.00%
+117.39%
+130.00%
NA
unch
+108.77%
NA
NA
+63.77%
NA
NA
NA
Time
Value
0.813
1.375
1.938
4.000
5.625
8.125
11.000
11.594
10.594
9.469
8.844
8.344
7.844
7.844
Bid
Ask
0.563
1.125
1.688
3.750
5.250
7.625
10.500
13.500
17.500
21.125
25.500
30.000
34.500
39.500
0.813
1.375
1.938
4.000
5.625
8.125
11.000
14.000
18.000
21.875
26.250
30.750
35.250
40.250
Vol
NA
NA
12
15
20
NA
122
15
NA
NA
1
NA
NA
NA
Quotes supplied by Standard & Poor's ComStock, Inc.
Exhibit 2
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Eugene Bala
Option Pricing
Last
Change
% Change
Volume
Day's High
Day's Low
Nortel Networks Corporation (Call Apr 00 90)
Open
34.000
+6.000
Previous Close
+21.43%
Bid
75
Ask
34.000
Strike Price
34.000
Expiration Date
34.000
28.000
35.875
36.875
90.000
April 21,2000
Last
Change
% Change
Volume
Day's High
Day's Low
Nortel Networks Corporation (Put Apr 00 90)
Open
0.750
NA
Previous Close
NA
Bid
NA
Ask
NA
Strike Price
NA
Expiration Date
NA
0.750
0.125
0.250
90.000
April 21,2000
Exhibit 3
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