Flaws with 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- [email protected] Twitter: - Rahulmagan8 Face book: - [email protected]

mailto:[email protected]


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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

Events Kurtosis 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.

http://en.wikipedia.org/wiki/Statistics

http://en.wikipedia.org/wiki/Deviation_(statistics)

http://en.wikipedia.org/wiki/Median

http://en.wikipedia.org/wiki/Probability_distribution

http://en.wikipedia.org/wiki/Sample_(statistics)

http://en.wikipedia.org/wiki/Probability_theory

http://en.wikipedia.org/wiki/Statistics

http://en.wikipedia.org/wiki/Probability_distribution

http://en.wikipedia.org/wiki/Extreme_value_theory

http://en.wikipedia.org/wiki/Gumbel_distribution

http://en.wikipedia.org/wiki/Fr%C3%A9chet_distribution

http://en.wikipedia.org/wiki/Weibull_distribution

http://en.wikipedia.org/wiki/Fisher%E2%80%93Tippett%E2%80%93Gnedenko_theorem


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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.

http://en.wikipedia.org/wiki/Call_option

http://en.wikipedia.org/wiki/Put_option

http://en.wikipedia.org/wiki/Put%E2%80%93call_parity


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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-

http://en.wikipedia.org/wiki/Delta_neutral

http://en.wikipedia.org/wiki/Short_(finance)

http://en.wikipedia.org/wiki/Gamma_(letter)

http://en.wikipedia.org/wiki/Derivative


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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

http://en.wikipedia.org/wiki/Convexity_(finance)

http://en.wikipedia.org/wiki/Volatility_(finance)

http://en.wikipedia.org/wiki/Volatility_(finance)

http://en.wikipedia.org/wiki/Kappa


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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.

http://en.wikipedia.org/wiki/Straddle

http://en.wikipedia.org/wiki/Theta_(letter)

http://en.wikipedia.org/wiki/Option_time_value

http://en.wikipedia.org/wiki/Intrinsic_value_(finance)

http://en.wikipedia.org/wiki/Intrinsic_value_(finance)


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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:-

http://en.wikipedia.org/wiki/Rho_(letter)


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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


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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


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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.


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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


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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)


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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.


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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.


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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.


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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


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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.


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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


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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


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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


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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.


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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


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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.


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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.


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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.


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Part 13: Readers Feedback Dear Reader – You are most welcome to share your feedback in technical context at below given details. Email: - [email protected]/ [email protected] Handheld: 91 -9899242978/9868281769

LinkedIn- [email protected] Twitter: - Rahulmagan8 Face book: - [email protected]

mailto:[email protected]/




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Notes:-


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Notes:-


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Flaws with Black Scholes & Exotic Greeks

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