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The process of pinning is frequently mentioned in stock traders
lore, but its
effect upon the price of an option is not well understood. Some
traders be-
lieve that on days when equity options expire (typically the
third Friday of a
given month), many stocks seem to close near a multiple of $5.
In this paper, we
give statistical evidence for the existence of pinning, and
develop an option-pricing
model that incorporates this phenomenon. We conclude that, near
expiration, there
is a discrepancy between the Black-Scholes price of an option
and the price of anoption whose underlying stock has a higher than
normal probability of being pinned.
We analyse the various cases (eg, when the pinned price is
cheaper than the Black-
Scholes price) and provide intuition for the price
discrepancies.
What is pinning?
Many stock traders have observed that, on expiration
Fridays(days when exchange-
traded equity options expire, typically the third Friday of a
calendar month), an un-
usually large number of stocks seem to close near a multiple of
$5 (eg, near $95,
$100, $105, ).
These $5 multiples correspond to strike values for equity
options. A possible
reason for this phenomenon is that a few market participants
have large short
gamma positions. For example, they may have sold a large number
of straddles
(combinations of puts and calls at the same strike). These
traders delta hedging ac-
tivities cause the stock price to move towards the strike. For
example, consider a
trader who is short straddles at struck at $100. As the stock
moves from $102 to$101, the trader is inclined to sell more stock,
thus pushing the stock price even
lower. If the stock continues to drop and reaches $99, the
trader will be inclined to
purchase shares, driving the stock price up.
There has been much academic work on expiration-day effects. For
example,
Stoll and Whaley (1987) report large trading volumes in the last
hour of trading
on expiration Fridays. This research suggests that stock price
dynamics may be
different on expiration Fridays than on other trading days.
Other papers, such
as Roder,K and Bamberg (1996), Schlag (1996) and Merrick also
investigate price
dynamics on expiration days.
Statistical evidence of pinning
In this section, we provide a concrete example where pinning
demonstrably occurs.
Our example is fairly typical for optionable stocks. We notice
that pinning seems
to have increased in recent years along with option trading
volume. To simplify our
analysis, we assume that strike prices occur at $5 increments.
In reality, there areoften fractional strike prices due to stock
splits and other corporate actions.
To check our assumptions, we define various degrees of pinning
for a given
stock:
A. stock closes within 1/16 of a strike
B. stock closes within 1/8 of a strike
C. stock closes within 0.25 of a strike
D. stock closes within 0.5 of a strike
In the table below, we consider Microsoft (MSFT) closing prices
over the in-
terval January 1990June 2001, distinguishing expiration Fridays
from non-expira-
tion days (all trading days except for expiration Fridays). We
have chosen MSFT
since it is a very liquid stock whose options have been traded
very actively in recent
years. For the purposes of illustration, we describe scenario C
in detail. If a stock
were to evolve randomly and prices were continuous, then the
stock would be e
pected to close within 0.25 of a $5 strike 2.25/5 of the
time.
First, let us consider non-expiration days. As can be seen
below, MSFT clos
within 0.25 of a strike price 13.52% of the time. This is higher
than the expect
10%. A number of plausible explanations may be given for
this:
stock prices move discretely, in increments of 1/32;
stop-loss trades are typically executed at integer values;
a stock with a reasonably high spot price (greater than $10,
say) tends to mo
in increments of 1/16 or even 0.25.
Suppose, then, that the change in a given stock were never
smaller than 0.
and that the initial stock price were an integer multiple of
0.25. Then the sto
would trade of 3/20 = 15% of the time within 0.25 of a strike
price, which mo
closely corresponds to the MSFT data.
It is instructive to compare the expiration Fridays (the columns
on the right
table 1) with other trading days. On expiration Fridays, MSFT
landed within 0.
of a strike 23.29% of the time, nearly twice as frequently as
usual.
These results are not unique to MSFT, and indeed pinning seems
to occur f
a variety of stocks which have actively traded options.
We can show that pinning is a statistically significant
phenomenon using a si
ple price transformation and the Wilcoxon rank-sum test. If the
terminal pr
lands within 25 cents of a strike, we assign a 1 to the price,
otherwise a 0. In t
way, we can compare the closing prices on expiration Fridays and
other days.pinning is truly observable, the mean of the transformed
prices should be la
er on expiration Fridays than other days.
Using the Wilcoxon rank-sum test, we generate apvalue which is
smaller th0.01, so that the alternative hypothesis gives more than
a 99% chance of bei
correct. (We have not applied the more common Student t-test
since our pr
data is discrete and the data sets are not normally
distributed.)
Option pricing and pinning: an idealised model
In this section, we propose a diffusion model for stocks which
incorporates p
ning. We will use this model to price options. Our model makes
two simplifying
sumptions:
The stock is certain to get pinned on expiration.
When the stock gets pinned, it goes to the strike price exactly
(eg, it does n
close within 0.25 of the strike).
In the Black-Scholes formulation, the price Stof a stock evolves
according
the equation
S
WWW.RISK.NET DECEMBER 2001 RISK EQUITY RISK S
Options
The effect of stock pinning
upon option pricesThe existence of standardised expiration dates
for listed equity options affects the prices ofunderlying stocks
close to these dates due to hedging activity. Here, Hari Krishnan
and Izzy
Nelken demonstrate that the effect is significant in US markets
and show how to account for it
in pricing models
MSFT Non-expiration day Expiration Friday(2959 days) (146
days)
90-Now Pinned % Pinned Pinned % Pinned
1/16 123 4.16% 13 8.90%1/8 235 7.94% 20 13.70%
0.25 400 13.52% 34 23.29%
0.5 715 24.16% 47 32.19%
1. Expiration Fridays comparison
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where Stis the current stock price, is the drift rate, is an
increment of time,
is the stock volatility and 0 ~ N(0,1) is a normally distributed
random variable. Ina risk-neutral world, the drift is given by = r
q 2/2, where ris the risk-free(continuously compounded) interest
rate and q is the dividend rate. Note that the
traditional Black-Scholes equations do not take pinning into
account, so that is
both time- and price-independent.In order to force the price
path S
tto land on a strike (here assumed to be a mul-
tiple of $5), we need to constrain the drift of the above
process. In our case, the
drift = (t,T,P,Pt,, 0) depends on price, time and volatility,
and it is stochastic.
Our model is constructed in several steps:
1. Starting at St, we draw a random number 0 ~ N(0,1) and
calculate the termi-
nal price
Each time we run a simulation,we generate a different 0 and
hence a different ST.
2. For a given 0, we choose the closest strike to ST. We denote
this strike byF(ST).Since 0 is a random variable,a stock can
theoretically get pinned at any strike. How-ever, Step 1 ensures
that the probability that S
Twill be pinned at a nearby strike is
large (relative to a strike that is far away), for a reasonable
. (It is easiest for a mar-
ket participant to drive the price of a stock to the nearest
possible strike.)
3.We next create a (simulation-dependent) drift which connects
ST
to F(ST).
Thus, we evolve ST
according to the process
where
and 0 ~ N(0,1) is a random variable independent of 0.It is
important to realise that we need two random variables, 0,1, to
evolve S
from t to t+. 0 decides which strike the price should shoot at
and 1 generates
a diffusion around this line. Our process can be loosely thought
of as a generalisa-tion of a Brownian bridge with a lattice of
boundary conditions.
Open interest
Our model does not incorporate the open interest of options with
differing strikes.
Several market participants as well as our anonymous referee
correctly mentioned
that stock prices are more likely to be pinned to strikes with
large open interest.
However, in our opinion:
It is difficult to obtain historical open interest data going
back ten years, and
consequently, we cannot test our model over a large data
set.
Pinning activity occurs only during the last few trading hours
on an expiration
Friday. It would be difficult and expensive to move the stock by
more than $5 in
such a short amount of time.
It is hard to say for any one particular strike whether a given
market maker is
long or short gamma. As we do not know on which side the market
makers are on,
we cannot say with any certainty whether pinning would, in fact,
occur to that strike.For these reasons, we feel that adding open
interest information is difficult, and
may not contribute substantively to the model.
Computational results
Charts 1 and 2 show option prices for a call option with two
days to maturity. We
assume that the volatility is 30%, the risk free rate is 6% and
that the underlying
stock pays no dividends.
The option prices in charts 1 and 2 have been obtained using
three processes:
A Monte Carlo simulation of a traditional stock price diffusion
that does not
incorporate pinning.
The Black-Scholes price, calculated analytically.
The process described in the section above (generating what we
will from now
on call a five-round price, since we are forcing the stock to a
strike).
As expected, we get an excellent agreement between the
traditional Monte Carlo
simulation process (1) and the Black-Scholes formula
(2).However, the pinned sim-
ulation process (3) gives results that are significantly
different.
For at-the-money options, the five-round process generates a
lower price than
the traditional Black-Scholes model. If a stock is trading at
$100 near expiration,
there is a large probability that it will close at $100, and
that the option will expire
worthless. On the other hand, if the option is slightly in the
money (eg, if the stock
price is $104), the five-round process implies that the option
is more expensive thanthe traditional model. If the share price is
at $104, the traditional model will price
the option at about $4 (since there is very little time
premium). On the other hand,
the five-round process will tend to drive the share price to
$105. Thus the option
should cost somewhere between $4 and $5.
Although put prices are not given in Chart 1, these can be
calculated directly by
=
ln ( ) ( )F SS
T tT
T
S S et t++=
1
S S eT tT t T t= + ( ) ( )0
S S et t++= 0
0
1
2
3
4
5
6
90 95 100 105 110
Normal
Black Scholes
Five round
0
0.5
1
1.5
2
2.5
97 98 99 100 101 102 103
Normal
Black Scholes
Five round
1. A 100 call option, two days to expiration, 30%
volatility across a range of stock prices
2. A magnified version of the central
part of chart 1
1 We can show that put-call parity holds for the pinned process
using the following argument (which can also be
found in Hull [H]):
Consider the two portfolios at timet: Portfolio A one European
call option plus an amount of cash equal to
xer(Tt). Portfolio B one European put option plus one share.
On expiration, at timeT, both of these portfolios will be
worthmax(ST,X), so that their values at timet must be
the same. Put-call parity does not depend upon the underlying
stock price process. We have also verified that put-call
parity holds in our numerical simulations
It is clear that the Black Scholes is very close to the
simulation. The five
round process, on the other hand, is significantly different
We compare the five round process with the Black Scholes price.
(Also
included is a normal simulation process to show its excellent
agreement
with Black Scholes.)
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simulation, or using put-call parity.1
It is also interesting to compare Black-Scholes and five-round
option prices for
a fixed initial price and a range of volatilities. As volatility
increases, one would ex-
pect the difference in prices to be proportionately closer,
since pinning has less of
an impact on the underlying stock evolution. Indeed, we have
used simulation to
verify that the difference in option prices converges to zero as
volatility increases.
We note that pinning has less of an impact on the price of an
option i f the
stock has a high volatility, or the option has a long time to
expiration.This small difference in option prices does not imply
that the stock is less likely
to get pinned. Rather, even if the stock is pinned, we do not
know to which strike.
The uncertainty causes the options price differential to
diminish. Thus, the pinned
price process with high volatility and long time to expiration
is less predictable, and
behaves like a highly discretised random walk, which results in
an option price close
to Black-Scholes.
The price discrepancies in Chart 1 are larger than would be
expected in real life,
since a particular stock will not always be pinned. However, if
there is a larger than
normal probability that a stock will be pinned, the true price
of an option will lie
somewhere in between the Black-Scholes and five round prices.
Thus, the differ-
ence between the Black-Scholes and five round prices given in
the graph should be
viewed as upper bounds, and our results should be interpreted
qualitatively.
We have also made the simplifying assumption that stocks are
pinned to the
strike price exactly.This does not account for the fact that a
pinned stock may close
near (but not at) the strike price. It would be interesting to
extend the work to coverthis.
Conclusions
In this paper we provided concrete statistical evidence for
pinning with certain
stocks and developed an alternate model governing the price
dynamics of a pinned
stock. Using this model, we simulated the payoff structure of a
call option for a
range of volatilities and spot prices. Our analysis suggests
that a low volatility stock
is more dramatically affected by pinning. There are situations
where the pinned
price is significantly different to the traditional
Black-Scholes price. We have also
shown how to extend our model to the case where a stock has a
positive probabil-
ity of being pinned, but is not guaranteed to land near a
strike.
Hari Krishnan is vice-president of the GWMS division, Morgan
Stanley,
Chicago. E-mail: [email protected]
Izzy Nelken is president of Super Computer Consulting,
USA.E-mail: [email protected], www.supercc.com
Hull, J
Options, Futures and Other Derivatives
Prentice Hall.
Merrick, J
Early unwindings and rollovers of stock index futures arbitrage
programs: analysis
and implications for predicting expiration day effects
87-16 / Federal Reserve Bank of Philadelphia
(RePEc:fip:fedpwp:87-16)
Schlag, C
Expiration Day Effects of Stock Index Derivatives in Germany
European Financial Management Journal 1 (1996), pp. 69-95.
Roder, K and Bamberg, G
Intraday-Volatility and Expiration Day Effects on the German
Stock Market
Kredit and Kapital, 2 (1996) , pp. 244-276
Stoll, HR and Whaley, REProgram Trading and Expiration-Day
Effects
Financial Analysts Journal, March/April 1987
References
