Validation of new Pricing Model for Exotic Options

“Final Year Report” 

Sept 2010 

Validation of New Pricing Model for Exotic Options 

Trainee: Dima HAMZE 

Tutor: Nadia ISRAEL 

“Pioneering Again” 

Responsible: Alain CHATEAUNEUF

Acknowledgements 

I would like to take this opportunity to pay tribute and express my deepest appreciation 

to everyone who was by my side till the completion of my graduate studies. 

To Professor Alain CHATEAUNEUF, I attribute the level of my Masters degree to your 

encouragement, effort and advices. One simply could not wish for a better and friendlier 

responsible. I am heartily thankful for being by my side. 

In my daily work I have been blessed with helpful and cheerful group of colleagues 

whose contribution in assorted ways to the research and the making of this paper 

deserves special mention. It is a pleasure to convey my gratitude especially to Nadia 

ISRAEL without you this paper, would not have been completed on due time. Moreover, 

in this humble acknowledgment, collective and individual thanks go to you all FXO team 

members. 

Words fail me to express my appreciation to my family and their non-ending attention 

and prayers. I owe you my deepest gratitude for providing me with the moral support I 

required throughout all my studies. My Father, Ghassan HAMZE, in the first place is 

the person whose encouragement, guidance and support from the initial to the final step 

of my university studies helped in shaping the person I am now. My Mother, Hala 

TABSH, whose persistent confidence in me ever since I was a child always elevated my 

self-esteem during the hardest times. Nadine, Razan and Nour, you made your blessings 

available in a number of ways thanks for being supportive and caring sisters. 

Finally, I gratefully acknowledge all my professors who supported me in any aspect and 

contributed to the successful realization of my paper and the completion of my 

internship. Last but not least I am indebted to many of my friends who were a great 

source of encouragement and assistance along my academic journey. 

Dima Hamze , Final year Internship Report (IRFA 2009/2010) 

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Contents 

Acknowledgement 

Outline 

I. About Murex, FXD and Basic definition 

A. General Presentation 

B. Brief Description of FXD Market 

a. Quotations and conventions of the FOREX Market 

b. Main traded exotic products of the FOREX Market 

C. Volatility and Smile Construction and Particularity of the FX Market 

a. Volatility 

b. Some Sensitivity measure 

c. Smile 

i.Delta 

ii.Vega 

iii.Volga 

iv.Vanna 

i. Strike scale 

ii. Delta scale 

iii. Smile construction in FXO 

d. Pricing models and volatility 

i. Local 

ii. Stochastic 


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iii. Hybrid 

II. Pricing models for Exotic options 

A. Existing Pricing models and tools for FX Options 

a. Replication Models (Skew model) 

b. Diffusion Models (Heston Model) 

c. Hybrid Model (Tremor I) 

B. Need for a more accurate model to price the partial barriers (Tremor II) 

III. Validation of Tremor II Model (To be continued) 

IV. Conclusion (To be continued) 

Bibliographies 


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A. General Presentation 

Murex, established in 1986, is a software company specialized in developing integrated front 

to back office financial software for the majority of derivative and physical assets, including 

but not limited to Interest rates, Foreign Exchange, Commodities, Equities, and Credit 

derivatives. With London constituting the largest concentration of users, Murex systems 

equip trading rooms of approaching 100 major financial institutions worldwide. To mention, 

the leading clients include ABN-AMRO, HSBC, J.P. Morgan Chase, etc. . . . 

Every day, thousands of users in banks, asset managers, corporations and utilities rely on 

Murex people and Murex solutions to support their capital markets activities across asset 

classes. For many years Murex has consistently been recognised as a solid leader in 

software development for trading, risk management and processing. 

Murex mainly focuses on five segments of clients: 

1) Global banks that are the leading market makers. 

2) Processing centres that are financial institutions that implement multi-business and multientity 

horizontal processing platforms. 

3) Integrated trading and processing solutions .Those are financial institutions looking for 

capital markets solution. 

4) Asset managers and hedge funds. 

5) Enterprise risk management that are the risk departments. 

Besides, Murex serves other segments of clients that include treasury centres of 

corporations and utilities. 

More than 800 staffs work with dedication and passion in Paris, New York, Beirut, Tokyo, 

Singapore, Dublin, Luxembourg, London, Beijing and Sydney. 

The front line interaction with clients, design testing and quality assurance, support and 

implementation. Moreover, the technical/architectural teams converge to facilitate clients’ 

requirements at the cutting edge of financial markets. They are represented by the financial 

consultancy activity that includes highly skilled financial professionals. It covers working with 

clients through the life cycle of Murex solutions that goes as follows: Pre-sale analysis and 

presentation, design and testing, implementation and configuration, and system extension. 

Murex’s motto “Pioneering Again” sums it all up. Murex has always proved itself as a 

pioneering company that escort capital markets revolutions by offering innovating software 

solutions to the financial industry. 


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B. Brief Description of FXD Market 

Some symbols: 

a. Conventions and Quotations of the FOREX Market 

The Foreign Exchange options’ market is one of the largest and most liquid financial 

markets. This market has some particularities that will be discussed in the context of this 

report. 


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Exchange rates quotation: 

The exchange rates are quoted as (XXX-YYY) = (FOREIGN-DOMESTIC) = 

(DEN- Numéraire) = (UNDERLYING-BASE) 

The exchange rate is the quantity needed of domestic currency to buy one single unit of 

foreign currency. For example, EUR/USD currency pair is quoted as EUR-USD. Based on 

the explanation above, USD is the domestic currency and EUR is the foreign currency. It is 

worth to mention here that the term domestic is not related to the country of trading and that 

the slash (/) does not mean a division. It just symbolizes the currency pairs. 

Currency pair Default quotation Sample quote 

GBP/USD GBP-USD 1.8000 

GBP/CHF GBP-CHF 2.2500 

EUR/USD EUR-USD 1.2392 

EUR/GBP EUR-GBP 0.9600 

EUR/JPY EUR-JPY 135.00 

EUR/CHF EUR-CHF 1.5500 

USD/JPY USD-JPY 108.25 

USD/CHF USD-CHF 1.2800 

Exchange rates are usually quoted with five figures; taking the EUR-USD example again; an 

observed quote can be 1.2392. The last digit ‘2’ is called the pip, and the middle digit ‘3’ is 

called the big figure. Consider an exchange rate of 108.25 for USD-JPY pairs so an increase 

by 20 pips will give us a rate of 108.45 and a rise by 2 big figures will give us a rate of 

110.25. 

Quotation of option prices: 

There are six ways to quote the Values and prices of vanilla options 

The Black-Scholes formula quotes d pips. The others can be computed using the following 

formulas. 


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Table taken from “FX Options and Structured Products” book by “Uwe Wystup” 

b. Main Exotic Options 

First generation of exotics Second 

exotics 

generation of Third generation of exotics 

Barriers Asians Accumulators 

Touch Baskets Tarns 

Best of/Worst of Compounds 

KIKO Choosers 

Double average rate option 

Volatility Products 

Multi barrier 

In this paper, we will mainly focus on barrier options. 

Barrier options are vanilla options that either come to existence or are terminated when a 

barrier level is reached. 

There are two types of barriers: 

1) Simple barriers that have one barrier 

2) Double barriers that have two barriers. 

Each kind has its own sub-types listed in the table below. 


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Simple barrier Double barrier 

Up and In Knock In 

Up and Out Knock Out 

Down and In 

Down and Out 

In Options: If the barrier level is reached (when we hit the barrier) a simple option is 

obtained with the same characteristics as the barrier. 

Out Options: If the barrier level is reached (when we hit the barrier) the option is terminated 

and a rebate is obtained if specified. 

In Options Out Options 

If the barrier is reached during the 

option’s life, the option is exercised. 

The exercise creates a new simple option 

at barrier’s knock-time. 

If the barrier is not reached up to 

maturity, the option is early terminated 

and a Rebate is paid if specified. 


If the barrier is reached during the 

option’s life, the option is early 

terminated 

After early termination; the option is dead 

and a rebate is paid if specified. 

If up to maturity the barrier is not 

reached; the option is either exercised if 

it was ITM or expired if it was OTM. 

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Simple barrier Option behaviour throughout its life: Example of “Up and Out call”: 

1) When the spot goes beyond the Strike at a level far from the barrier, the option’s Market 

Value (MV) increases until it gets close to the barrier. 

2) Near the barrier, the MV decreases rapidly. 

3) The MV becomes zero when the Spot hits the barrier. 

This barrier feature can be applied over specified time of the option’s life rather than over the 

whole life. This application changes the option’s style in to a window barrier according to 

the following criteria: 

1) European Option: The window start date =window end date=maturity date. 

2) American Option: The window start date =trading date and 

The window end date =maturity date. 

3) Partial barrier option that includes 3 subtypes: 

The forward/late start: window start date> trading date and window end 

date=maturity date. 

The early end: window start date =trading date and window end date < maturity date. 

The mixed: window start date> trading date and window end date< maturity dates. 

C. Volatility and Smile Construction and Particularity of the FX Market 

a. Volatility 

Volatility usually refers to the standard deviation of the continuously compounded returns of a 

financial instrument over a specific time horizon. It is often used to quantify the risk of the 


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instrument over that time period. It is a main input parameter to compute the value of the 

option. 

For Non liquid options(no market quotes available), historical volatility is used: 

The historical volatility for an asset relates to a past period of time where volatility is 

observed at. For example we look at the figures for the past week, for the past month, and so 

forth. 

How to calculate the Historical Volatility? 

Step 1: Create a sequence of log-returns 

Step 2: Compute the average: 

Step 3: Compute the Variance: 

Step 4: Compute the standard deviation: 

This historical volatility does not have any predictive capabilities. 

On the other hand, for Liquid options (Market quotes available and quoted by traders, 

brokers and real time data providers), implied volatility can be used: 

Implied volatility is the calculated value of volatility by inverting the relevant option-pricing 

model .The prices are the inputs, observed on the market and the volatilities are the implied 

outputs. Those inputs reflect the market participants’ view and their predictions of the future. 

Therefore, instead of using the model to solve for the option's price, it is used to solve for the 


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option's volatility curve. By having this curve implied from several market quotes, other nonquoted 

options can be priced. 

b. Main Sensitivity measures discussed in the scope of this report: 

i. Delta: Is the first order derivative of the option price for a 1-percentage point 

increase in the spot. 

In FX market the Moneyness of vanilla options is expressed in terms of delta. Generally 

speaking, a 50 delta corresponds to an ATM option (K=S).More details on this subject will be 

tackled in the smile section. 

ii. Vega 

Vega is the change in the value of an option for a 1-percentage point increase in the implied 

volatility of the underlying asset price. 


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Cases Vega 

Long option position ( Buying) Always positive 

ATM Greatest Vega 

Further ITM or OTM Decreases 

Time Vega is greater for long dated options 


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iii. Volga: 

Volga measures second order sensitivity to implied volatility (the rate of change to Vega as 

volatility changes) 

iv. Vanna: 

Vanna is a second order derivative of the option value, once to the underlying spot price and 

once to the volatility. It is mathematically equivalent to Dvega/Dspot or Ddelta/Dvol. Vanna 

can be a useful sensitivity measure to monitor when maintaining a delta- or a Vega-hedged 

portfolio, as it will help the trader to anticipate changes to the effectiveness of a delta-hedge 

as volatility changes or the effectiveness of a Vega-hedge against change in the underlying 

spot price. 

c. Smile 

The Black-Scholes model assumes a constant volatility throughout. However, market prices 

of traded options imply different volatilities for different maturities and different deltas or 

strikes. 

The cumulative normal density function underestimates the probability of extreme 

occurrences. The implied distribution has fatter tail thus more probability for extreme 

occurrences, thus higher volatilities on the tails that leads to the existence of the smile. The 

volatility of exchange rate is far from constant. 


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Smile is defined as the plot of the options volatility as a function of its strike price or delta. It 

is the markets’ estimation of where the volatility will be should the market move. 

There are two common views of smile behaviour: 

i. Fixed (Sticky Strike) in which the volatility depends on the option 

strike for a certain maturity. Whatever the spot level is, for the same strike, one gets 

the same volatility .Based on this assumption, the smile quoted in strike terms is 

indifferent to spot changes. 

If 2 FX options have the same strike they are priced with the same vol. 

ii. Floating (Sticky Delta) in which the volatility depends on the option 

delta for a certain maturity. This effectively centres the smile around the current 

spot/fwd value or the ATM. The implied smile floats with the spot level. Based on this 

assumption, the smile quoted in terms if delta is indifferent to the spot changes. (ATM 

volatility or 50 delta volatility is constant whatever the spot level). 


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With ‘sticky delta’ we know that as the market moves a different volatility is applied to each 

strike and if 2 FX options have the same delta they are priced with the same vol. 

In FX Market, the smile is quoted on a delta scale, taking into account the most liquid points 

(5 points 10D put, 25D put, ATM, 25D call and 10D call). This kind of quotation allows: 

- Stickiness to the moneyness 

- Introduction of a common convention ( common language) 

- Replication of market behaviour/logic (An option with an ATM strike should not have 

the same volatility as an option with the same strike being OTM (out of the money). 

To highlight, the delta scale can be translated into a strike scale. It is an iterative process 

since: 

Delta= dc/dS and c = f (S,K,t,r, rf, vol). 

So for every strike we have a delta and for each delta we have volatility. 

iii. Smile construction in FXO 

To price accurately and be able to manage the risk, looking at the volatility smile is a must. 

FX derivative ‘s market participants are confronted with the indirect observation of the smile 

in the market since it is built with specific input parameters that decompose it into symmetric 

part that reflects convexity (flies) and skew part (riskies) using 


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• Risk Reversal (RR)=call – put 

• Strangle or Butterfly strategies(BF) =(call + put)/2 

• ATM options with strike = forward i.e. value of call= value of put. 

This is done on a delta scale following the market conventions for FXD. 

In summary, three volatility quotes are extracted from the market data for a given delta 

(Ex. Delta =+ or – 0.25). 

• ATM volatility. 

• Risk reversal volatility for 25 delta. 

• Strangle volatility for 25 delta. 

In terms of ATM volatility of call and put (0), OTM volatility of call (+) and put (-) we get 

This gives: 


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Smile Curve: 

d.Pricing models and volatility 

The Black-Scholes starting point is that all options are valued at the same volatility, 

regardless of strike level. 

In smile world, Black Scholes considers the probable distribution of prices centred around the 

forward price (F) on the expiry date. 


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i. Local volatility models 

The most well known local volatility model is the binomial tree. The binomial tree with returns 

following the log distribution are used to calculate the local volatility that is the standard 

deviation of the returns at the node (n, i) 

Consider ln(X) evolves to ln(Y) with probability p and to ln(Z) with probability (1-p) 

Then 

E[lnX] = p ln(Y) + (1-p) ln(Z). 

And, 

V[lnX] = (standard deviation) ² 

Local volatility models expect the underlying price to be variable and the level of volatility to 

be wholly determined by the level of the underlying price 


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A local volatility model sees the probable distribution of volatility is 100% correlated with the 

level of the Forward price (see graphic). 

Another Local volatility model based on the diffusion process is Dupire Model which widely 

used in equity market. 

ii. Stochastic volatility 

Stochastic volatility models assume both the volatility and the stock price to follow a 

Brownian motion. 

Stochastic volatility models expect both the underlying price and volatility to be random. 

A Stochastic volatility model sees the probable distribution of (V) proportionate to the 

volatility of the Brownian variance (VoVol) and T, with the central point determined given a 

starting volatility (V0) and a terminal volatility (Θ) 


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iii. Hybrid, Universal or “Mixed” models are a mixture of the latter two 

A Stochastic/Local volatility hybrid model sees the probable distribution of (V) proportionate 

to the volatility of the Brownian variance (VoVol) and T, with the central point determined 

given a starting volatility (V0) and a terminal volatility (Θ) combined with a correlation 

between F & V. 


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Some notes on local and stochastic volatilities: 

-We observe that steep skew indicates locality. 

-The Correlation describes the skewness of the smile so: 

o If the correlation =0 the smile is completely convex and can be explained by 

stochastic volatility. 

o If the correlation is different than 0 the smile is has higher risk reversals. 

o If the correlation =1 the vovol is probably equal to 0 and the smile can be 

explained by local volatility. 

-The more local volatility is the less stochastic it is. 

II. Pricing models for Exotic options 

A. Existing Pricing models and tools for FX Exotics. 

To price exotics consistently with smile several models have been introduced: 

a. Replication Models (Skew model) 

Barrier and Touch options can be priced in the Black-Scholes framework (assuming constant 

volatility and rates throughout the life of the option). The Black-Scholes price uses only the 

ATM volatility at the expiry date of the option and doesn't take into account the smile curve. 

Because Barrier and Touch options are path-dependent products (the final payoff depends 

on the path followed by the spot price throughout the life of the option, not only on the value 

of the spot on maturity date), we cannot simply plug the smile volatility into the Black-Scholes 

price. One solution to get a better price for these exotic options would be to use a model 

where the volatility is not assumed constant throughout the life of the option (stochastic 

volatility model for example). 


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The Skew methodology allows traders to compute quickly an adjustment to the Black 

Scholes price of an exotic option that takes into account the presence of a smile curve. 

In principle, we start by measuring the B-S Vega, Vanna and Volga of the exotic option and 

then find the weights of the 3 liquid vanillas (25 delta and ATM call and put) with a smile off 

mode that will perfectly hedge the Vega, Vanna and Volga of the exotic option. Then we 

proceed by computing the smile cost of this portfolio (price with smile – price with smile 

off).Finally, we multiply this over-hedge cost by the probability of survival of our exotic option 

and add this adjustment factor to our B-S price of the exotic option. 

The shortcomings of this model are: 

• It doesn’t include the stochastic nature of volatility. 

• It doesn’t factor in the smile term structure thus it is incapable of adjusting for 

complex volatility moves. 

• Doesn’t replicate the market prices 

Thus, better techniques should be used to overcome those limitations. 

b. Diffusion Models (Heston Model) 

Heston Model is a stochastic volatility model that assumes both the underlying price and the 

volatility to follow a Brownian motion. It is an extension of the B-S model taking in to 

consideration the stochastic characteristic of volatility. 

Heston typology 


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Heston Models and FXO market. 

We concluded that Heston model did not replicate the market prices of FX option. Moreover, 

it doesn’t take into account the well known correlation between the spot and the volatility and 

the spot and the risk reversals (also called smile dynamic). Therefore modelling the spot as a 

mixture between two models local and stochastic and monitoring the dynamics so that it 

matches spot with the correlation seems reasonable to replicate the market. 

d. Hybrid Model (Tremor I) 

Hybrid model is a mixture of the local volatility and the stochastic volatility model. It takes in 

to account the smile dynamics. The volatility used in the market ended up being neither 

completely stochastic nor completely local thus a logical guess would be a mix. 

Murex confirmed the aforementioned hypothesis and resulted in Tremor model that is a 

hybrid model used to price FX exotic options. The proportion of stochastic volatility is 

determined by the Cursor and the corresponding proportion of local volatility is deduced 

from the smile. The cursor explains the skewness and the convexity of the smile. 


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Tremor I Topology 

Tremor is a two factors model: 

The first diffusion process is the forward: it is a function of a stochastic volatility and 

a local volatility represented by the quadratic function of the forward g(f). 

The second process is the variance which is the volatility diffusion. 

The Kappa, also called the rate of mean reversion (controls the variance level over time 

prohibiting it from getting far from the mean).Since tremor is calibrated per maturity pillar and 

doesn’t take the term structure in to account; the kappa is considered as an input and is set 

to 1. 

The theta also called the long term variance requires a term structure to be calibrated so 

same as kappa it is set to the initial variance (actual volatility level). 

The volatility of volatility (vovol) describes the convexity of the smile. The more the 

convexity of the smile, the higher the vovol is. (More probability to be at the wings). 

The Correlation is already describes above. 

To recapitulate, we have six parameters to determine in order to match the five points of the 

smile: a, b, c, correl, vovol, and initial volatility. 


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This leaves us with six outputs and 5 inputs (5 vanilla prices of our calibration basket).So we 

need an additional input in order to calibrate the model. An extra input is introduced which is 

a factor that represents the proportion of stochastic volatility. This factor is called the cursor 

If the cursor =0% the model is purely local volatility one. 

If the cursor =100% the model is purely stochastic volatility one. 

The calibration is done in 3 steps (It will not be detailed due to confidentiality). 

We calibrate a reduced number of parameters of Heston parameters that fits 3 vanilla prices 

(10D’s and ATM) then we apply the cursor on the vovol by that Heston doesn’t explain the 

whole smile. 

We add the rest of the vanillas to our calibration basket to calibrate the local part. The local 

volatility will explain the rest of the smile. 

B. Need for a more accurate model to price the partial barriers (Tremor II) 

Tremor I didn’t serve as a good tool to price accurately partial barriers. As a result, 

ameliorations should be done by introducing term structure. This is Tremor II the gap filler of 

tremor I and the tool to price all the barriers. This introduced term structure takes in to 

account the state of the market in the window time where the option is alive and where the 

impacts on the price should be taken in to account. 

The calibrated parameters will be functions of time (3 dimensions). 

Test plan is being elaborated to validate Tremor II for partial barriers. 

TO BE CONTINUED…. 


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Bibliography: 

- Wystup, Uwe. FX Options and Structured Products. Chichester West Sussex, U.K: 

John Wiley and Sons, 2006. 

- Hull, John. Options,Futures and Other Derivatives. 6 th edition. Upper Saddle River, 

New Jersey: Pearson prentice hall, 2006. 

- Murex Internal documentation. 


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Validation of new Pricing Model for Exotic Options

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