Flaws of Black-Scholes & Exotic Greeks

Flaws with Black Scholes & Exotic Greeks
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Treasury Perspectives
Flaws with Black Scholes
& Exotic Greeks

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Dear Readers:-
It’s been a difficult and volatile year for companies across the Globe. We have seen numerous
risk management policies failures. To name a few... UBS, JPM Morgan, Libor manipulations by
European, US and Japanese banks and prominent accounting scandals like Lehman…
As rightly said by Albert Einstein “We can't solve problems by using the same kind
of thinking we used when we created them.” and when you can't solve the
problem, then manage it and don’t be dependent upon science as Science is
always wrong, it never solves a problem without creating ten more.
The same is the case with Foreign Exchange Risk Management Policies (FXRM) which if can’t be
managed properly then would lead to either systematic shocks or negative implications at the
bottom line of the corporate, banks, FI and trading houses P&L A/cs.
That is something risk management struggles with, say the experts. In Richard Meyers’
estimation, risk managers or traders do not socialize enough. “It’s all about visibility,” he said.
Meyers, chairman and CEO of Richard Meyers & Associates, a talent acquisition and
management firm in New Jersey, relates the story of a firm that decided to adopt an Enterprise
Risk Management (ERM) strategy. Instead of appointing its risk manager to head ERM, the
company brought in someone else. Why?
Time has come when organizations across the world have to do deep amendments in their
Enterprise Risk Management (ERM) policies covering foreign exchange hedging programs,
diversification in derivatives portfolio, Enterprise risk management policies and deeper and
deeper understanding towards financial models.
With this background paper would like to appraise you on the “Flaws with Black Scholes &
Exotic Greeks” and take you through various Options strategies, Flaws, Greeks and appropriate
thoughts towards the diversification in the derivatives portfolio.
Thanks You,
Rahul Magan
Author, Flaws with Black Scholes & Exotic Derivatives
LinkedIn- Rahulmagan8@gmail.com
Twitter: - Rahulmagan8
Face book: - Rahulmagan8@gmail.com

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Flaws with Black Scholes Model (BSM) & Exotic Greeks
Rahul Magan
Sydney, Australia
ABSTRACT
In 1973, Fisher Black, Myron Scholes and separately Robert Merton derived the Black-Scholes-
Merton (BSM) model, which was rewarded the Nobel Prize in 1997. Despite its limitations, the
model has survived until today as the dominant pricing model for standard and exotic
European style options.
The model owes its success to its simplicity, high intuition and versatility. In 1997, the
importance of their model was recognized worldwide when Myron Scholes and Robert Merton
received the Nobel Prize for Economics. Unfortunately, Fisher Black died in 1995, or he would
have also received the award [Hull, 2000]. The Black-Scholes model displayed the importance
that mathematics plays in the field of finance. It also led to the growth and success of the new
field of mathematical finance or financial engineering.
This paper is all about flaws with Black Scholes and subsequent linkages with Exotic Greeks.
Directly Black Scholes is linked with six plain vanilla options Greeks and numerous exotics
linked with each of these plain vanilla Greeks.
The paper is trying to establish relationship between plain vanilla and their linked exotics
besides highlighting various thoughts on flaws with Black Scholes. As per author biggest flaw
with Black Scholes is assumption of constant implied volatility & non applicability of
principle of Skewness which is not true today due to huge monetization programs running by
almost all central banks across the world. Such monetization programs would give rise to
implied volatility and swan shocks and continue to stay for longer periods of time unless
balance sheet deleveraging starts which do have its own positive and negative repercussions.
Paper also takes various references of plain vanilla Greeks, exotic Greeks, respective
formulations and last but not the least effective hedging strategies. At respective point’s paper
using various references pertaining to statistical data distributions like Normal Distribution,
Poisson distribution, Weibull Distribution and none the less Extreme value Theory (EVT)
which in turn linked with swan events data shocks. Additional references are also taken to
establish link between FX volatility w.r.t various markets parameters.
Key words: Black Scholes, Options derivatives, Exotic derivatives, Extreme Value Theory and
Statistical distributions.

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Table of Contents
Part 1:Central banks monetization programs & Volatility in FX markets
(Topology of economic shocks & Mark to Market)
Page No 7
Part 1[A]: Option Structure and M2M hierarchy – US GaaP (FAS 157)
(Levels in M2M hierarchy)
Page No 10
Part 1[B]: Option Structures & Effectiveness
(Intrinsic & Extrinsic Valuation)
Page No 11
Part 2:Current assumptions with Black Scholes Model
(Nine most famous Black Scholes assumptions)
Page No 13
Part 3: Flaws with Black Scholes Model
(Four most famous Black Scholes flaws)
Page No 14
Part 4: Current Black Scholes Methodology
(Black Scholes & pricing mechanism)
Page No 16
Part 5: Delta vs. Dynamic Hedging
(Types of Delta Hedging & formula)
Page No 19
Part 6: Options Plain vanilla
(Description & understanding)
Page No 19
Part 7(A): Options Plain vanilla & exotic Greeks topology
(Options topology)
Page No 24
Part 7(B): Options Plain vanilla & exotic Greeks topology
(Formula & Derivations)
Page No 25
Part 8: Volatility Skewness & Frown
(Principle of Skewness)
Page No 29
Part 9: Options flaws with practical applicability
(Principle of Skewness)
Page No 31
Part 10: Conclusion
(Conclusion)
Page No 41
Part 11: About the author
(Professional & Social Networking)
Page No 42
Part 12: References & Citations
(References & linkups)
Page No 43
Part 13: Readers Feedback
(Technical Feedback)
Page No 44
Part 14: Notes
(Technical Notes)
Page No 45

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Black Scholes abbreviations:-
BSM – Black Scholes Model
EVT – Extreme Value Theory
RR – Risk Reversals, Zero Cost Collar, Range Forwards, Fences, Cylindrical
LTFX – Long Term Foreign Exchange Hedging
Bfly – Butterfly Spreads
OM – Options Moneyness
ATM - At the money
ITM – In the money
OTM – Out of the money
Call/Put – Call option, Put option
ZCSP –Zero Coupon Swap Pricing
Statistical abbreviations:-
σx /σ – Standard Deviation
σ2 – Variance
ND – Normal Distribution
RFIR – Risk free Interest rates
IV – Implied Volatility
US GaaP abbreviations:-
M2M – Mark to Market
M2M (L1/L2/L3) – M2M level hierarchy, US GaaP
Central banking abbreviations:-
FED – Federal Reserve
ECB – European Central Bank
BOJ – Bank of Japan
SNB – Swiss national Bank
ZIRP – Zero Interest Rate Policy

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Part 1:- Central banks Monetization program & Volatility in FX markets
Today all cross currency exchange pairs are facing huge implied volatility due to
excessive monetization program run by almost all central banks across the world. Due
to this FX markets across the world are flush with huge USD liquidity which in turn
creates either systematic economic or swan shocks.
Today Bank of Japan (BOJ) and Federal Reserve (FED) are linking their monetization
programs with real time economic variables like Inflation and employment
respectively. At present Fed holds the balance sheet size of over $ 4 Trillion and
growing by ~ $ 1 Trillion/ Year which creates huge dollar liquidity in worldwide FX
markets. European Central Bank (ECB) and Bank of Japan (BOJ) are holding almost
same position when it comes to size of their balance sheets which are not only
ballooning in nature but also growing in leaps and bounds.
With that level of implied volatility which is due to aforesaid reasons it is pertinent for
Treasurers to protect their bottom line from forecasted or non-forecasted FX risks,
volatility and economic shocks. The present FX world is no more lead by normally
distributed economic environment rather working under extreme value theory
(EVT) where in any sort of economic shocks or swan events are pretty common with
periodic velocity.
These swan shocks can be further divided into four parts – White, Grey, Black and
Neon Swan events based upon ascending order of severity. Treasurers have to take
conscious call and try and make sure that their derivatives portfolio won’t go in sudden
gains/ (losses) because of sudden shift in economic variables due to swan shocks.
The aforesaid swan events can’t be covered or hedged by just creation of derivatives
portfolio having plain vanilla forward contracts or Options (exotic or non-exotic).
Organizations have to have diversified their derivatives portfolio with deep level of
understanding towards derivatives pricing models especially Black Scholes for Options
Pricing and Zero Coupon Swap Pricing (ZCSP) for LTFX hedging.
Today there is a high time when derivatives portfolio should appropriately diversified
using options (plain vanilla or exotic Greeks) covering various assets classes. The days
of zero currency volatility is gone henceforth plain vanilla derivatives are not effective.
Time has come when organizations have to have amended their risk management or
foreign exchange hedging policies and make them in line with the markets else they
are prone to huge M2M (Mark to Market) gains/ (losses) with even simplest white
swan event/shock in the world.

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Few glimpses on BOJ Monetization program:-
Source: - JP Morgan Research, IMF and WB research
Relative size of
the monetary
base and
USD/JPY
Japan’s
monetary base
and CPI
Difference in
balance sheet
expansion and
USD/JPY

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Principle of Skewness, Kurtosis & data distribution:-
 Skewness tells you the amount and direction of skew (departure from
horizontal symmetry), and kurtosis tells you how tall and sharp the central
peak is, relative to a standard bell curve.
 If Skewness is positive, the data are positively skewed or skewed right,
meaning that the right tail of the distribution is longer than the left. If
Skewness is negative, the data are negatively skewed or skewed left,
meaning that the left tail is longer.
 If Skewness is less than −1 or greater than +1, the distribution is highly
skewed.
Topology of Economic/ Systematic Shocks
Normally Distributed
Economic/ Systematic Shocks
Extreme Value Theory (EVT)
Economic / Systematic Shocks
Fat tail distributions (Fat tail)
or Heavy tail distribution
Black Swan Event Theory
(Swan Events)
Leptokurtic
distribution
s
Mesokurtic
distribution
Platykurtic
distribution
Black Swan
Events
Grey Swan
Events
White Swan
EventsKurtosis Skewness
Neon Swan
Events

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 If Skewness is between −1 and −1/2 or between +1/2 and +1, the
distribution is moderately skewed.
 If Skewness is between −1/2 and +1/2 the distribution is approximately
symmetric.
Principle of Kurtosis & fat tail data distribution:-
 Kurtosis is a measure of extreme observations. How likely will the returns be
extreme, either positive or negative. Though the sign of Skewness is enough
to tell us something about the data, kurtosis is often expressed relative to
that of a normal distribution.
 Data that has more kurtosis than the normal is sometimes called fat-tailed,
because its extremes extend beyond that of the normal. By definition, and
according to the formulas used, the kurtosis of a normal distribution is 3.0.
Fat-tailed distributions have values of Kurtosis that are greater than this.
Part 1[A]: Option Structure and M2M hierarchy – US GaaP (FAS 157)
Accounting world also facing big shifts in valuation methodologies especially M2M and
derivatives standards. We have seen radical shifts in fair valuation and derivatives
accounting standards like FAS 157 (Fair value principles) & FAS 133 (valuation of
derivatives) in last couple of years. The M2M valuations cover all three types under
US GaaP FAS 157 “Fair Value Measurements”.
US GaaP FAS 157
(Fair Value Measurements)
Mark to Market (M2M)
Valuations
Mark to Market
(L1)
Mark to Model
(L3)
Mark to Matrix
(L2)
Level 1 input are quoted prices
(unadjusted) in active markets for
identical assets or liabilities that
The reporting entity has the ability to
access at the measurement date
Level 2 inputs are inputs other than
quoted Prices included within Level 1
that are observable for the asset or
liability, either directly or indirectly
through corroboration with
observable (market-corroborated
inputs) market data
Level 3 inputs are
unobservable inputs for the
Assets or liability, that is,
inputs that reflect the
reporting entity’s own
assumptions about the
assumptions
market participants would
use in pricing the asset or
liability

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Part 1[B]: Option Structures & Effectiveness (Extrinsic / Intrinsic Valuation)
Options structures are amongst highly effective tools to hedge organizational forecasted
cash flows, revaluations risks due to fair valuation of foreign currency assets and liability
in balance sheets of respective legal entities and net investments hedge exposures
(intercompany loan from one legal entity to another.)
Options are also pretty cost effective in nature subject to risk management policies of
corporate. Organizations have to take appropriate call whether to hedge their foreign
currency cash flows in flows using zero cost collar, risk reversals or paid collars and
subsequent amortization in there profit & loss segment.
Treasurers need to take conscious call whether to hedge their forecasted receivables or
payables using plain vanilla forwards contracts, Options or exotic derivatives. There are
millions of options exotic structures available to hedge your foreign exchange risk.
Options Intrinsic Valuation (Option Moneyness)
This represents the amount of money, if any, that could currently be realized by
exercising an option with a given strike price. For example, a call option has intrinsic
value if its strike price is below the spot exchange rate. A put option has intrinsic value
if its strike price is above the spot exchange rate.
In-The-Money: This term is applied to an option that has intrinsic value. That is when a
profit can be realized upon exercising it. For a call option, it is the case when the spot
Exchange rate is higher than the strike price of the option, and for a put option, when
the spot exchange rate is below the strike price.
Options Strategies
(Pricing using Black Scholes)
Range Forwards/
Risk reversals /
Zero Cost Collars
Seagull (Buy
Call + Risk
reversals)
Call spreads
(Bullish/
Bearish)
Put Spreads
(Bullish /
Bearish)
Box/Condor/
Calendar
Spreads

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Out-the-Money: A call option is said to be “out-of-the-money” if the underlying spot
exchange rate is currently less than the strike price of the option. A put option is said to
be “out-of-the-money” if the underlying spot exchange rate is currently more than the
strike price of the option. An option that is “out-of-the-money” at expiry will have no
value, and the holder of the option will allow it to expire worthless.
At-The-Money: This means that the strike price and the spot exchange rate are the
same. Like the “out-of-the-money” option, the holder would allow the option to expire.
Options Extrinsic Valuation (Time Value)
Time value is a little more complex. When the price of a put or call option is greater than
its intrinsic value, it is because the option has time value.
Time value is determined by: the spot price; the volatility of the underlying currency; the
exercise price; the time to expiration; and the difference in the ‘risk-free’ rate of interest
that can be earned by the two currencies. The time value of the option contract will
diminish over the life of the option and at expiration will be zero.
The time value portion of an option is at its greatest when the option is “at-the-money:,
that is the strike (exercise) rate is equal to the market rate. This is because the entire
premium is equal to time value, as the option has no intrinsic value.
Options Fair value = Intrinsic Valuation + Extrinsic (Time value)
Options fair value is the sum of intrinsic valuation and extrinsic valuation or time value.

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Part 2:- Current assumptions with Black Scholes Model
 Exercise Timings. Options will be exercised in the European model, meaning
no early exercise is possible. In fact, U.S. listed stocks are exercised in the
American model, meaning exercise may occur at any time prior to expiration.
This makes the original calculation inaccurate, since exercise is one of the key
attributes of valuation.
 Dividends. The underlying security does not pay a dividend. Today, many
stocks pay dividends and, in fact, dividend yield is one of the major components
of stock popularity and selection, and a feature affecting option pricing as well.
 Calls but not puts. Modeling was based on analysis of call options values only.
At the time of publication, no public trading in puts was available. Once puts
began to trade, the formula was again modified. However, if traders continue
relying on the original BSM, even for put valuation, they may be missing a
fundamental inaccuracy in the price attributes.
 Taxes. Tax consequences of trading options are ignored or non-existent. In fact,
option profits are taxed at both federal and state levels and this affects net
outcome directly.
In some instances, holding the underlying over a one-year period may
lead to short-term capital gains taxation due to the nature of options
activity, for example. The exclusion of tax rules makes the model
applicable as a pre-tax pricing model, but that is not realistic. In fairness
to the model, everyone pays different tax rates combining federal and
state, that any model has to assume pre-tax outcomes.
 Transaction costs. No transaction costs apply to options trades. This is another
feature affecting net value, since it’s impossible to escape the brokerage fees for
both entry and exit into any trade.
This is a variable, of course; fee levels are all over the place and, making it
even more complex, the actual options fee is reduces as the number of
contracts traded rises. The model just ignored the entire question, but
every trade knows that commissions can turn a marginally profitable
trade into a net loss.
 Unified Risk free Interest rates. A single interest rate may be applied to all
transactions and borrowing; interest rates are unchanging and constant over the
life span of the option. The interest component of B-S is troubling for both of
these assumptions. Single interest rates do not apply to everyone, and the
effective corresponding rates, risk-free or not, are changing continually.

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 Constant Implied Volatility (IV) Volatility remains constant over the life span
of an option. Volatility is also a factor independent of the price of the
underlying security. This is among the most troubling of the BSM assumptions.
Volatility changes daily, and often significantly, during the option life span.
It is not independent of the underlying and, in fact, implied volatility is
related directly to historical volatility as a major component of its change.
Furthermore, as expiration approaches, volatility collapse makes the
broad assumption even more inaccurate.
 Trading is continuous. Trading in the underlying security is continuous and
contains no price gaps. Every trader recognizes that price gaps are a fact of life
and occur frequently between sessions.
It would be difficult to find a price chart that did not contain many
common gaps. It is understandable that in order to make the pricing
model work, this assumption was necessary as a starting point. But the
unrealistic assumption further points out the flaws in the model.
 Price movement is normally distributed. Price changes in the short term in
the underlying security are normally distributed. This statistical assumption is
based on averages and the behavior of price; but studies demonstrate that the
assumption is wrong. It is one version of the random walk theory, stating that all
price movement is random.
Influences like earnings surprises, merger rumors, and sector, economic
and political news, all affect price in a very non-random manner. The
stock price process in the Black-Scholes model is lognormal, that is, given
the price at any time, the logarithm of the price at a later time is normally
distributed.
It is also known how to do option pricing for a continuous-time model
with normally distributed prices, but the lognormal model is more
reasonable because stocks have limited liability and cannot go negative.
Part 3: Flaws with Black Scholes Model
 Exercise Timings. Black Scholes model should consider all three possible
exercise timings scenarios using options– European, American and Bermudian.
This would help traders to price options in a better way considering
reversal of trades at favorable fair valuation in live markets.

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Today majority of the traders are keeping options in their derivatives
portfolio under Available for Sale (AFS) or Held for Trading (HFT)
categories hence forth they prefer American over European options.
The former can be realized any moment depends upon intrinsic &
extrinsic valuation of the options however later might not be unless
subject to reversal or cancellation.
There are hardly any traders left who keep Options as an derivative under
Held till Maturity (HTM) hence forth restriction towards exercise timings
is all flawed w.r.t current market structure.
 Unified Risk free Interest rates (RFIR). There is no single index or any G sec
bond which can act as a risk free interest rate for all FX pricing models.
All G7 currencies are having their respective risk free interest rates hence
forth no single interest rate can act as a universal risk free interest rate for
respective currency pairs. As of now UST (United States Treasuries)
yields are acting as unified interest rates to price any USD denominated
options w.r.t G7 currencies where in USD is acting as base or termed
currency.
Central banks are doing huge monetization along with maintenance of
zero interest rates policy (ZIRP) for both shorter and longer period of
debt portfolio. Considering that there should be multiple rates for
multiple periods to do options valuations for respective currency pairs.
 Constant Implied Volatility Implied, Historical and realized volatility can never
be constant as it keeps changing. That change depends upon level of shocks in FX
markets across the world as volatility is a Meta measure.
Any volatility measure can’t be constant for longer tenors hence forth
options pricing models should consider moving or ranged volatility to
price contracts. Black Scholes should also have ranged volatility as an
input variable to price contracts in a better way.
Traders have to decide whether they would like to go with implied,
realized, historical volatility (with or without outliers) or statistical
volatility.
There is a great probable chance that Traders would use statistical
volatility which is further derived using statistical data distributions.
It may or might not have any outliers and all depends upon input
valuation parameters taken by traders along with current valuation of
stocks or currency pair in respective markets.

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 Price movement is normally distributed We are living in the world of
“Extreme Value Theory(EVT)” where in FX markets are always suspect to any
kinds of swan events like white, grey, black and neon swan events.
Implied volatility can go either ways depends upon the shocks and their
resistance. There are various technical or fundamental indictors
available to assess the valuation of these black swan events but these
indicators nowhere support any form of normal distribution.
Extreme value theory deals with the stochastic behavior of the extreme
values in a process. For a single process, the behavior of the maxima can
be described by the three extreme value distributions–Gumbel, Fréchet
and negative Weibull–as suggested by Fisher and Tippett (1928).
The key to EVT is extreme value theory which a cousin of better known
central limit theorem which tells us what distribution of extreme value
should look like in the limit as our sample size increases. Extreme value
theory or extreme value analysis (EVA) is a branch of statistics dealing
with the extreme deviations from the median of probability distributions.
It seeks to assess, from a given ordered sample of a given random
variable, the probability of events that are more extreme than any
previously observed.
In probability theory and statistics, the generalized extreme value (GEV)
distribution is a family of continuous probability distributions developed
within extreme value theory to combine the Gumbel, Fréchet and Weibull
families also known as type I, II and III extreme value distributions. By the
extreme value theorem the GEV distribution is the limit distribution of
properly normalized maxima of a sequence of independent and identically
distributed random variables. Because of this, the GEV distribution is used
as an approximation to model the maxima of long (finite) sequences of
random variables.
Part 4: Current methodology of Black Scholes
 The Black-Scholes formula can be derived as the limit of the binomial pricing
formula as the time between trades shrinks, or directly in the continuous time
model using an arbitrage argument. The option value is a function of the stock
price and time, and the local movement in the stock price can be computed using
a result called Itô's lemma, which is an extension of the chain rule from
calculus.
 Once Itô's lemma is used to calculate the local change in the option value in
term of derivatives of the function of stock price and time, absence of arbitrage
implies a restriction on the derivatives of the function (in economic terms, risk
premium is proportional to risk exposure), essentially similar to the per-period
hedge in the binomial model.

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 The two terms in Black-Scholes call formula are prices of digital options. The
term SN(x1) is the price of a digital option that pays one share of stock at
maturity when the stock price exceeds X: this is a digital option if we measure in
terms of the stock price (this is called using the stock as numeraire and is like a
currency conversion).
 The second term XN(x2) is the price of a short position in a digital option that
pays X at maturity when the stock price exceeds X. There is a slightly mystical
result that the two terms also represent the portfolio we hold to replicate the
option if we want to create the call option at the end by holding SN(x1) long in
stocks and BN(x2) short in bonds (with trading to vary this continuously as time
passes and the stock price evolves).
 Option traders call the formula they use the “Black-Scholes-Merton” formula
without being aware that by some irony, of all the possible options formulas that
have been produced in the past century is the one the furthest away from what
they are using. In fact of the formulas written down in a long history it is the only
formula that is fragile to jumps and tail events.
 The Black-Scholes-Merton argument, simply, is that an option can be hedged
using a certain methodology called “dynamic hedging” and then turned into a
risk-free instrument, as the portfolio would no longer be stochastic.
 The Black-Scholes-Merton argument and equation flow a top-down general
equilibrium theory, built upon the assumptions of operators working in full
knowledge of the probability distribution of future addition to a collection of
assumptions that, we will see, are highly invalid mathematically, the main one
being the ability to cut the risks using continuous trading which only works in
the very narrowly special case of thin-tailed distributions.
 But it is not just these flaws that make it inapplicable: option traders do not “buy
theories”, particularly speculative general equilibrium ones, which they find too
risky for them and extremely lacking in standards of reliability.
 A normative theory is, simply, not good for decision-making under
uncertainty (particularly if it is in chronic disagreement with empirical
evidence). People may take decisions based on speculative theories, but avoid
the fragility of theories in running their risks. This discussion will present our
real-world; ecological understanding of option pricing and hedging based on
what option traders actually do and did for more than a hundred years. This is a
very general problem.

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There should not be
unified risk free
interest rates to price
options in various cross
currency pairs.
There should not be
constant implied
volatility to price
options in various cross
currency pairs.
FX markets are
following EVT than
any form of normal
distribution.

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Part 5: Delta vs. Dynamic Hedging
 Delta Hedging In this form of hedging traders tries to neutralize the portfolio
delta w.r.t movement in the underlying price almost on daily basis. The size of
the derivatives portfolio is so large that even 10 Bps shift (II shift or non II shift)
would lead to windfall gains or losses hence forth almost daily intervention is
required.
 Absolute movement in underlying vs. delta
 Relative movement in underlying vs. delta - Delta or Gamma Cash
 Elasticity of the Options ( applicable for both puts and calls )
 Dynamic hedging In this form of hedging traders tries to neutralize the change
in portfolio valuation on or at specific period of times vs. on daily basis in case of
delta hedging. The difference b/w Delta neutral and dynamic hedging is former
is done almost on almost daily basis while later is done at periodic intervals
(unless huge movements in FX markets).
Part 6: Options Plain Vanilla Greeks
Options
Payoffs
Delta Gamma Theta Vega Rho
Long Call +Ve +Ve -Ve +Ve +Ve
Short Call -Ve -Ve +Ve -Ve -Ve
Long Put -Ve +Ve -Ve +Ve -Ve
Short Put +Ve -Ve +Ve -Ve +Ve
 Delta Greek - Delta is the option's sensitivity to small changes in the underlying
asset price. Delta is positive for calls and negative for puts. For a vanilla option,
delta will be a number between 0.0 and 1.0 for a long call (and/or short put) and
0.0 and −1.0 for a long put (and/or short call) – depending on price, a call option
behaves as if one owns 1 share of the underlying stock (if deep in the money), or
owns nothing (if far out of the money), or something in between, and conversely
for a put option.
The difference of the delta of a call and the delta of a put at the same strike
is close to but not in general equal to one, but instead is equal to the
inverse of the discount factor. By put–call parity, long a call and short a
put equals a forward F, which is linear in the spot S, with factor the
inverse of the discount factor, so the derivative dF/dS is this factor.

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The sign and percentage are often dropped – the sign is implicit in the
option type (negative for put, positive for call) and the percentage is
understood. The most commonly quoted are 25 Delta put, 50 Delta put/50
Delta call, and 25 Delta call. 50 Delta put and 50 Delta call are not quite
identical, due to spot and forward differing by the discount factor, but
they are often conflated. Delta is always positive for long calls and
negative for long puts (unless they are zero).
The total delta of a complex portfolio of positions on the same
underlying asset can be calculated by simply taking the sum of the
deltas for each individual position – delta of a portfolio is linear in the
constituents. Since the delta of underlying asset is always 1.0, the trader
could delta-hedge his entire position in the underlying by buying or
shorting the number of shares indicated by the total delta.
For example, if the delta of a portfolio of options in XYZ (expressed as
shares of the underlying) is +2.75, the trader would be able to delta-hedge
the portfolio by selling short 2.75 shares of the underlying. This portfolio
will then retain its total value regardless of which direction the price of
XYZ moves.
 Gamma Greek - Gamma is the delta's sensitivity to small changes in the
underlying asset price. Gamma is identical for put and call options. Gamma,
measures the rate of change in the delta with respect to changes in the
underlying price. Gamma is the second derivative of the value function with
respect to the underlying price. All long options have positive gamma and all
short options have negative gamma. Gamma is greatest approximately at-the-

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money (ATM) and diminishes the further out you go either in-the-money (ITM)
or out-of-the-money (OTM).
Gamma is important because it corrects for the convexity of value.
When a trader seeks to establish an effective delta-hedge for a
portfolio, the trader may also seek to neutralize the portfolio's
gamma, as this will ensure that the hedge will be effective over a wider
range of underlying price movements. However, in neutralizing the
gamma of a portfolio, alpha (the return in excess of the risk-free rate) is
reduced.
 Vega Greek - Vega is the option's sensitivity to a small change in the volatility of
the underlying asset. Vega is identical for put and call options. Vega measures
sensitivity to volatility. Vega is the derivative of the option value with respect
to the volatility of the underlying asset. Vega is not the name of any Greek
letter. However, the glyph used is the Greek letter .
Presumably the name Vega was adopted because the Greek letter nu
looked like a Latin vee, and Vega was derived from vee by analogy with
how beta, eta, and theta are pronounced in English. The symbol kappa, is
sometimes used (by academics) instead of Vega (as is tau ( ), though this
is rare).
Vega is typically expressed as the amount of money per underlying
share that the option's value will gain or lose as volatility rises or falls
by 1%. Vega can be an important Greek to monitor for an option trader,
especially in volatile markets, since the value of some option strategies
can be particularly sensitive to changes in volatility. The value of an

22
option straddle, for example, is extremely dependent on changes to
volatility.
 Theta Greek - Theta is the option's sensitivity to a small change in time to
maturity. As time to maturity decreases, it is common to express theta as minus
the partial derivative with respect to time. Theta , measures the sensitivity of
the value of the derivative to the passage of time (see Option time value): the
"time decay." Theta is almost always negative for long calls and puts and
positive for short (or written) calls and puts. An exception is a deep in-the-
money European put. The total theta for a portfolio of options can be determined
by summing the thetas for each individual position.
The value of an option can be analyzed into two parts: the intrinsic
value and the time value or extrinsic value. The intrinsic value is the
amount of money you would gain if you exercised the option immediately,
so a call with strike $50 on a stock with price $60 would have intrinsic
value of $10, whereas the corresponding put would have zero intrinsic
value. The time value or extrinsic value is the value of having the option of
waiting longer before deciding to exercise.

23
 Rho Greek - Rho is the option's sensitivity to small changes in the risk-free
interest rate. Rho, measures sensitivity to the interest rate: it is the derivative
of the option value with respect to the risk free interest rate (for the relevant
outstanding term). Except under extreme circumstances, the value of an option
is less sensitive to changes in the risk free interest rate than to changes in other
parameters. For this reason, rho is the least used of the first-order Greeks. Rho is
typically expressed as the amount of money, per share of the underlying, that the
value of the option will gain or lose as the risk free interest rate rises or falls by
1.0% per annum (100 basis points).
 Greeks Linkup:-

24
Part 7: Topology of plain vanilla & exotic options Greeks
Topology of Plain Vanilla and
Exotic Option Greeks
Delta Greek Gamma Greek Theta Greek Vega Greek Phi/Rho/ carry
Rho
DdeltaDvol,
Dvega Dspot,
DvannaDvol,
DdeltaDtime
(Charm)
DgammaDvol,
Zomma,
DgammaDspot,
Speed,
DgammaDtime,
Color
DvegaDvol
(Vomma),
DvommaDvol
(Ultima),
DvegaDtime,
DdeltaDvar
Drift less Theta,
Bleed-Offset
Volatility, Theta
Gamma Greek
Volatility in local
or foreign
currency interest
rates w.r.t
underlying
Dzeta Dvol = Zeta sensitivity/Implied Volatility
DzetaDtime = In the money risk neutral volatility/
Theta
Implied volatility plays a critical role in valuation of all
exotic Greeks.
ITM ATM OTM

25
Part 7(B): Plain vanilla & exotic options Greeks – Formula & Derivations
 Exotic Greeks on Delta (DdeltaDvol):- DdeltaDvol is mathematically the same
as Dvega- Dspot, defined as (aka vanna). They both measure approximately how
much delta will change due to a small change in the volatility, and how much
Vega will change due to a small change in the asset price where n(x) is the
standard normal density
Assumption of constant implied volatility is amongst the biggest flaws
with Black Scholes henceforth any Greek having delta as a numerator and
implied volatility as a denominator would never be able to act as a right
measure in pricing options contracts.
Delta in itself is a measure of change in option price to change in
underlying and change in underlying is 100% linked with principle of
Skewness hence forth DdeltaDvol would not act as a right measure.
Principle of Skewness suggests that options with lower strike would
have high implied volatility and options with higher strike price are
having low implied volatility. The same would act vice versa in case of
any swan events. Alternatively traders have to define whether they would
like to go with ATM, ITM or OTM implied volatility.
DdeltaDvol = Change in Delta / Change in Implied Volatility
Delta = Change in option price / change in underlying

26
 Exotic Greeks on Delta (DvannaDvol):- The second-order partial derivative of
delta with respect to volatility, also known as DvannaDvol.
This exotic Greek is about change in Gamma w.r.t to change in implied
volatility. Gamma is nothing but second order partial derivative for delta.
Gamma considers all nonlinear movements in delta and with the
assumption of constant implied volatility this exotic Greek won’t work.
DvannaDvol = Change in Gamma/ Change in Implied Volatility
Gamma = Change in Delta / Change in Underlying
 Exotic Greeks on Delta (DdeltatDtime, Charm):- DdeltatDtime, also known as
charm or Delta Bleed, a term used in the excellent book by Taleb (1997),
measures the sensitivity of delta to changes in time. This Greek indicates what
happens with delta when we move closer to maturity.
The exotic Greek is about change in delta w.r.t change in theta. Theta
supports the writers in options as with the decrease in the time value
there would be support to option writers.

27
 Exotic Greeks on Gamma (DgammaDvol):- DgammaDvol (aka zomma) is the
sensitivity of gamma with respect to changes in implied volatility. DgammaDvol
is in my view one of the more important Greeks for options trading.
The exotic Greek is about change in Gamma / Change in Implied Volatility.
Gamma is second order derivatives of Delta and delta is change in option
price w.r.t to change in underlying.
Zomma= Change in Gamma/Change in Implied Volatility
Gamma = Change in Delta/ Change in underlying
Delta = Change in option price / change in underlying

28
 Exotic Greeks on Gamma (DgammaDspot):- The third derivative of the option
price with respect to spot is known as speed. Speed was probably first
mentioned by Garman (1992).
The exotic Greek is about change in Gamma w.r.t to change in spot price of
the exchange pair. Gamma is the second order partial derivative of the
delta which is change in delta to change in underlying
This Greek would tell you the non linear impact of the change in spot
w.r.t Gamma. Change in spot is always linked with principle of skewness
which is “change in spot is always linked with sudden change in volatility
for near, medium or far terms for both put and call options.
 Exotic Greeks on Gamma (DgammaDtime):- The change in gamma with
respect to small changes in time to maturity, DGammaDtime—also called Gamma
Theta or color (Garman, 1992)—is given by (assuming we get closer to maturity)

29
Part 8: Volatility Skew & Frown
 Volatility Skew/Frown / Principle of Skewness:-
The Black and Scholes assume that volatility is constant. This is at
odds with what happens in the market where traders know that the
formula misprices deep in-the-money and deep out-the-money options.
 The mispricing is rectified when options (on the same underlying with
the same expiry date) with different strike prices trade at different
volatilities - traders say volatilities are skewed when options of a given
asset trade at increasing or decreasing levels of implied volatility as
you move through the strikes.
 The empirical relation between implied volatilities and exercise prices
is known as the “volatility skew”. The volatility skew can be
represented graphically in 2 dimensions (strike versus volatility).
 The volatility skew illustrates that implied volatility is higher as put
options go deeper in the money. This leads to the formation of a curve
sloping downward to the right
 The implied volatility is the one which when input into the Black-
Scholes option pricing formulae gives the market price of the option.
 It is often described as the market’s view of the future actual volatility
over the lifetime of the particular option.
 The actual volatility is very difficult to measure and can be thought of
as the amount of randomness in an asset return at any particular time.

30
 If we take options with the same maturity on a certain foreign
currency that varies only in strike price we can calculate the implied
volatility for each one.
 Keep in mind that since they share the same underlying asset we
expect the volatility to remain constant regardless of the strike price.
 The volatility is relatively low for at-the-money options and gets
progressively higher as an option moves either into or out of the
money.
 We gain some analytical insight into why this occurs if we compare the
implied volatility distribution with the lognormal one with the same
mean and standard deviation.
 Consider a deep out-of-the-money call with strike price above K2. This
derivative will only pay off if the exchange rate closes above K2, and
according to the above figure the probability of this happening is
higher for the implied distribution than the lognormal one.
 A higher probability will generate a higher price, which in turn means
a higher implied volatility. The same is true of a deep out-of-the-
money put with strike price below K1.

31
Part 9: Options flaws / limitations vs. Practical applicability
 Exercise Timings. Black Scholes model should consider all three possible
exercise timings scenarios using options– European, American and Bermudian.
This would help traders to price options in a better way considering
reversal of trades at favorable fair valuation in live markets.
Today majority of the traders are keeping options in their derivatives
portfolio under Available for Sale (AFS) or Held for Trading (HFT)
categories hence forth they prefer American over European options.
The former can be realized any moment depends upon intrinsic &
extrinsic valuation of the options however later might not be unless
subject to reversal or cancellation which is further subject to risk
management policies and effective implementation.
There are hardly any traders left who keep Options as an derivative
under Held till Maturity (HTM) hence forth restriction towards
exercise timings is all flawed w.r.t current market structure. The below
chartings taken from Reuters which clearly indicate the real time Greeks,
Implied Volatility and Zero Coupon Swap pricing (ZCSP).
In American Options traders are having right to reverse the trade any
time depends upon options fair value which is nothing sum of Intrinsic
and Extrinsic value.
American Options can be reversed using 10 Delta to ATM or till 100 Delta
and all depends upon the levels of Delta and Gamma trading in the
market. These options can also be trade w.r.t to volatility trading in the
markets.
10 D ATM 100 D
Chart 1: - USD American Put in OTC Markets with plain vanilla Greeks
The below chart depicts the valuation methodology of American Put along
with its plain vanilla Greeks. The same is not possible in European Put
options because trades are unable to reverse their trades.
Zero Deltas to ATM is
considered as OTM trade.
ATM Delta to 100 Delta is
considered as ITM trade.

32
In European trades you have to wait till the maturity dates to get it
realized and park realized gains/ (losses) in profit & loss a/c which hit
bottom line for the organization.
The below chart also shares 5 plain vanilla Greeks like Delta, Gamma,
Theta, Vega, Rho and extended Greeks like 7 Days Theta, break-even price
and break-even delta which hold no value in case of European put options.
Chart 2: - Realized Volatility of USD American Put in OTC Markets with IV
The below charts depicts two years full range volatility for Euro along
with historical volatilities for one, two, three and six months.
Options Greeks – From
Delta to Break even delta
Input variables for
American OTC Put

33
The charts also shares realized volatility pertaining to ATM Bid/Ask,
25 D RR, 25 Bfly, and 10 D RR for Euro/USD currency pairs. The same
charts also shares ATM volatility pertaining to Eur/USD, USD/JPY,
USD/CHF and other G7 currency pairs.
It is apparent that volatility keeps changing on daily basis hence and if we
link this with “Principle of Option Skewness” then the change in implied
volatility would have impact on the strike price of the options. The
change in strike price due to change in implied volatility change implied
options fair valuation as well.
Periodic Volatility and
full range volatility
Principle of Options
Skewness – Options
strike price changes
with change in Implied
Volatility.

34
 Constant Implied Volatility Implied, Historical and realized volatility can never
be constant as it keeps changing. That change depends upon level of shocks in FX
markets across the world as volatility is a Meta measure.
Any volatility measure can’t be constant for longer tenors hence forth
options pricing models should consider moving or ranged volatility to
price contracts. Black Scholes should also have ranged volatility as an
input variable to price contracts in a better way.
Traders have to decide whether they would like to go with implied,
realized, historical volatility (with or without outliers) or statistical
volatility.
Options ATM to 10 Bfly
volatility
ATM volatility
for G7 currency
pairs – SW
(Spot Week) till
1 Yr

35
There is a great probable chance that Traders would use statistical
volatility which is further derived using statistical data distributions.
It may or might not have any outliers and all depends upon input
valuation parameters taken by traders along with current valuation of
stocks or currency pair in respective markets.
Chart 3: Eur/ USD Options Implied Volatility & Risk Reversals
The chart depicts the various volatility surfaces for Eur/USD from ATM till
25 D RR. It also shares the volatility surfaces for Eur/USD Butterfly
spreads.
The chart also shares volatility surfaces from SW (Spot Week) to 10 YRR
(10 Years Risk Reversals) for both Bid/Ask spreads. This volatility can be
used to price options using Black Scholes for various maturities periods.
This volatility can also be used to price various option strategies like risk
reversals, zero cost collars, fences, call and put spreads (bullish or bearish
strategies) and Seagulls.
Eur/USD 10% Delta
RR from SW to 10 Yr
Eur/USD Bfly spreads
from SW to 10 Yr

36
Chart 4: Option FX, Volatility Matrix and Volatility surfaces
The chart depicts about FX vols, ATM FX Vols, FX Smiles and volatility
surfaces for Euro USD. These charts clearly depicts that volatility can’t be
constant as it keeps moving in either ways. This is amongst most
fundamental flaws in BSM regarding options pricing.
The charts depicts various charts indicating implied/ATM volatility,
realized volatility, underlying spot rate and spreads b/w (realized and
implied volatility) from Q1’12 – Q2’13.
Implied ATM Vol,
Realized Vol,
Underlying spot
rate and spreads
Composite
Volatility from
SW till 10 Yr

37
Chart 5: Option FX, Volatility Matrix and Volatility surfaces
The chart depicts the volatility curves b/w real time ATM and Historical
ATM. It clearly states that the assumption of either constant or no ranged
volatility is wrong and gives you no realistic call and put prices. These
prices are subject to M2M even with the simplest swan shock and lead to
huge M2M gains/ (losses) in profit & loss a/c.
Real time vs.
Historical ATM
vols
Historical vs. real
time Implied
volatility

38
Chart 6: G7 volatility matrix
The chart depicts volatility matrix for following G7 currencies covering
below currencies
Commodities pairs:-
 AUS/USD, USD/CAD, NZD/USD
G7 Cross Currencies pairs:-
 USD/JPY, GBP/USD, USD/CHF, USD/CZK, EUR/USD, GBP/EUR,
GBP/AUD and GBP/EUR
Most volatility currency pairs:-
 EUR/USD, USD/JPY, USD/INR, GBP/AUD, EUR/AUD
Cross currency
Volatility
Matrix

39
 Unified Risk free Interest rates (RFIR). There is no single index or any G sec
bond which can act as a risk free interest rate for all FX pricing models.
Chart 7: USDOIS & INRIRS as interest rates
The below charts depicts the interbank interest rates for USD OIS and
INRIRS. The period selected for USDOIS is SW (Spot week) till2 Yrs and for
INRIRS is from 1 Yrs till 10 Yrs.
USDOIS Interest rates
from SW till 2 Yrs.
INRIRS Interest rates
from 1Yrs till 10 Yrs to
price swaps in various
currency pairs.

40
Chart 8: USD Interest rates (LIBOR – OIS)
The below charts depicts the Interest rates pertaining to USD from LIBOR
to OIS. The chart also shares FRA (Forwards Rate Agreements) rates from
0x3 till 9X12 periods.
It also shares Basis swaps interest rates from 1 Yrs till 10 Yrs on basis.
Basis swaps are the swaps where in both the parties pay floating rate
interest rates.
Interest rates
from Libor - OIS

41
Part 10: Conclusion of the paper
Throughout the paper we discussed multiple strategic flaws with current
methodology of Black Scholes. The present methodology is having deep impact on the
FX markets although unnoticed by traders and regulators every time. These
methodologies require deeper amendments because of huge and tactical shift in
implied volatility structures in today FX markets which is far more structural than pre
or post cyclical in nature.
Financial terminals like Reuters and Bloomberg should also need to amend their
current methodologies regarding fixing of various Interest rates which further acts as
benchmarks for pricing of cross currency swaps for respective currencies pairs. These
formula-based pricing assumptions rest on the belief under BSM that a risk-free
interest rate is available and, furthermore, that it applies to everyone and remains
unchanged. These assumptions are untrue. Consequently, a user of such terminals may
not rely on outcomes without the ability to adjust the assumptions to suit their
individual needs.
Black Scholes should have added tenured interest rates (may or might not be risk
free) and ranged volatility (excluding or including outliers) as input variables to
leverage its valuation methodology with markets.
Corporates are also diversifying their derivatives portfolio by adding options for
various maturities periods and make their risk management policies in line with
recent developments in markets. Corporates need strategic shift from plain vanilla
derivatives contracts like forwards to options for both longer and shorter versions of
the hedging tenor.
World FX markets are facing periodic swan events hence forth we need strong
resolutions to fix these structural issues which are yet to be resolved. The time has
come to either amend these obsolete assumptions or make a shift to live with
updated pricing model having better assumptions.

42
Part No 11: About the author
Professional Front:-
At present author is working as Manager Treasury - Front & Middle Office - FX,
Derivatives, ISDA & Global Investments in EXL Service. He holds ~ 6 Yrs of work
experience in corporate treasuries of top Indian IT & ITES companies like HCL
Technologies Limited and presently with EXL Service.com (I) Pvt. Ltd.
Author holds well diversified experience in respective functions of Front Office of
corporate Treasuries:-
 Foreign Exchange Hedging - Cash Flow & Fair Value Hedging Program
 Offshore & Onshore Treasury Risk Management
 National & International Treasury compliances - RBI, SEC and FSA
 End to End Designing of Treasury Management Systems – SAP & Oracle
 Global investments managements & trapped cash funding
 ISDA compliances , Dodd Frank
Social Networking Front:-
Author is pretty active on LinkedIn and holds networking base of over 35 Million
professionals across the world. He also actively participates in various issues pertaining
to FX, derivatives, macro & micro prudential and structural issues with eminent
professionals across the world. His participation is with over 100 top and eminent
LinkedIn Groups.
LinkedIn nominated his profile amongst top 1% for three years in a row (2009-
2012).He holds his own FX Group –“Foreign Exchange Maverick Thinkers” and also
acting as manager for “Italian Options Traders” having membership base of +530 &
+800 members respectively.
Foreign Exchange Maverick Thinkers:-
The Group dedicates to all those who not only think but also acts different in Foreign
Exchange markets. The current membership stands over 530 which includes International
FX Brokers, Italian & Australian Options Traders, International Business consultants,
Worldwide Investments Bankers, FX Research heads of various eminent Financial
Institutions, Worldwide Foreign Exchange consultants & Trainers, Chicago Mercantile
Exchange Traders (CME ), International Govt. Budgeting bodies, Central banks members
and last but not the least Corporate Treasurers of various International big corporate
working across the Globe.

43
Part No 12: References & Citations
Reference Source:- Reference Type
FX Manuals Option Greeks
LinkedIn Thoughts on Macro prudential & monetization programs
JP Morgan Research Charts on BOJ monetization programs
IMF/WB/IFC Central banks monetization data and balance sheets
Option pricing
formulas
The complete guide to Option Pricing formulas , Espen Gaarder
Haug
FX Charts Reuters EIKON
Greek Charts Numerous FX & derivatives books
Citation on ThomsettOptions.com:-
ThomsettOptions.com is an options educational site. Author Michael C. Thomsett has
published many books about options, including the best-selling Getting Started in
Options (John Wiley & Sons, currently in its 9th edition and with over 250,000 copies
sold). On this website, the author presents daily free articles about options topics,
notably on the problems of relying on Black Scholes. He also operates a virtual portfolio
for members, in which he transactions options-based trades using real-time stock and
option values, for the purpose of demonstrated how a range of different options trades
works and the rationale for entry and exit. The site also publishes a free weekly
newsletter. Thomsett also posts daily articles on LinkedIn groups, and belongs to 560
groups including Foreign Exchange Maverick Thinkers where he became associated
with Rahul Magan.

44
Part 13: Readers Feedback
Dear Reader – You are most welcome to share your feedback in technical context at
below given details.
Email: - Rahulmagan8@gmail.com/ Rahulmagan80@gmail.com
Handheld: 91 -9899242978/9868281769
LinkedIn- Rahulmagan8@gmail.com
Twitter: - Rahulmagan8
Face book: - Rahulmagan8@gmail.com

45
Notes:-

46
Notes:-

47

48

Flaws of Black-Scholes & Exotic Greeks

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