Some Practical Issues in FX and Equity Derivatives

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Some Practical Issues in FX and Equity Derivatives"

Transcription

1 Some Practical Issues in FX and Equity Derivatives

2 Phenomenology of the Volatility Surface The volatility matrix is the map of the implied volatilities quoted by the market for options of different strikes and different maturities. Implied volatility is the parameter σ needed to calculate the B&S formula. In practice the matrix can be built according to two main rules: Sticky Delta: the matrix of implied volatilities is mapped, for each expiry, with respect to the of the option; this rule is usually adopted in the over the counter markets (e.g.: fx options). Options are priced depending on their. That has subtle implications for the running of a book of options. Sticky Strike: the matrix of implied volatilities is mapped, for each expiry, with respect to the strike prices; this the rule is usually adopted in official markets (e.g.: equity options and futures options). Implied volatilities remain constant for each strike, even if the underlying asset price changes.

3 Phenomenology of the Volatility Surface Sticky Delta Rule: For each strike K and expiry T the implied volatility of any option is dependent on the level of its and consequently on the movements of the underlying asset (and on the elapsing of time).

4 Phenomenology of the Volatility Surface Sticky Strike Rule:For each strike K and expiry T the implied volatility of any option is independent of the movements of the underlying asset.

5 Sticky Strike Arbitrage Consider the P&L of a perfectly hedged portfolio in a short period dt: P&L = dπ + f = dc ds + f where f is the financing cost of the position. Let s assume the underlying asset s evolution is commanded by a Brownian motion: ds = µsdt + σ t SdZ and apply Ito s lemma (note that σ t is the actual and not the implied volatility) dc = (Θ σ2 ts 2 Γ + µ )dt + σ t SdZ

6 Sticky Strike Arbitrage Since f is: f = ( r d (t)c + r d (t) S r f (t) S)dt = ( r d (t) S + Df d r d (t)kφ(d 2 ) + r d (t) S r f (t) S)dt we get Θdt + f = 1 2 σ2 KS 2 Γdt and hence (recalling Θ definition): P&L = 1 2 S2 Γ[σ 2 t σ 2 K]dt

7 Sticky Strike Arbitrage Now, let s assume we have two call option C 1 (K 1, σ 1 ) and C 2 (K 2, σ 2 ) with σ 1 > σ 2. We build a portfolio such as: long Γ 1 options C 2 and short Γ 2 options C 1 The P&L in the interval dt is. which is a positive amount. P&L = Γ 1Γ 2 2 S2 [σ 1 σ 2 ]dt Thus Sticky Strike rule imply an arbitrage if it is realized in the market

8 Sticky Delta Arbitrage The Sticky Delta rule imply arbitrage as well, if it is realized in the market, though in less evident way. One should consider the shape of the surface in terms of slope and convexity and then build complex portfolios Generally speaking, with relatively symmetric matrices (such as in the Eur/Usd FX market), a position short butterfly grants a positive Γ and positive Θ portfolio. A short butterfly = long an ATM straddle and short a symmetric strangle (e.g.: short a 25 Call and a 25 Put), in an amount such that the total Vega of the portfolio is nil

9 Conclusions on the Sticky Strike and Sticky Delta Rules Matrices with different implied volatility for different levels of strike show that the constant volatility assumption of the B&S model is not realistic. The two Sticky Strike and Sticky Delta rules imply arbitrage should they actually be realized: so they both cannot be considered as a feasible model of the evolution of the volatility surface. They can be considered just as two convention for quoting volatility surfaces and they are respectively chosen according to their suitability to different markets.

10 Phenomenology of the Volatility Surface The evolution of the volatility surface can be decomposed in three main movements, for each expiry: Parallel Shift Convexity Increase/Decrease Slope Increase/Decrease. To represent these movements in terms of market instruments, one can consider: The ATM straddle volatility as an indicator of the level The Vega Weighted Butterfly as an indicator of the convexity The Risk Reversal as an indicator of the slope.

11 Phenomenology of the Volatility Surface Parallel

12 Phenomenology of the Volatility Surface Convexity

13 Phenomenology of the Volatility Surface Slope

14 Phenomenology of the Volatility Surface Composite

15 Model Risk in Hedging Derivatives Exposures Every time we choose a model to price and to hedge a derivative, we make more or less realistic assumptions about the risk factors we want to take into consideration. In practice, markets never behave in the way predicted by the model, so that the risk one incurs in by using a misspecified model is very high. In order to minimize the model risk, one can analyze which is the hedging error arising from the non-realistic modeling of the factors included in the model; subsequently, one tries to minimize this error. In what follows we will analyze which is the hedging error produced by the false assumption of a constant volatility (as in the B&S model). We also analyze which is the error produced by the more realistic assumption of a changing implied volatility, in case we are not able to correctly describe its evolution.

16 Model Risk: Hedging by B&S (constant volatility) We have shown above that the P&L in short interval dt of a perfectly hedged (under B&S s assumptions) portfolio is: P&L = 1 2 S2 Γ[σ 2 t σ 2 K]dt We make a profit if the realized volatility σ t higher than the implied volatility σ K, and the magnitude of this profit is directly linked to the level of the Γ (which is always positive in the case of a plain vanilla call option). If we integrate over the entire option s life, we obtain the total P&L resulting from running a -hedged book at constant implied volatility: P&L = T S2 t Γ(S t, σ K, t)[σ 2 t σ 2 K]dt

17 Model Risk: Hedging by B&S (constant volatility) From the formula above we can infer that: Continuous -hedging of a single option revalued at a constant volatility generates a P&L directly proportional to the Γ of the option; In general the P&L of a long position in the option, continuously rehedged, is positive if the realized volatility is, on average during the option s life, higher than the constant implied volatility; it is negative in the opposite case; The previous statement is not always true since the total P&L is dependent on the path followed by the underlying: if periods of low realized volatility are experienced when the Γ is high, whereas periods of high realized volatility are experienced when the Γ is negligible, then the total P&L is negative, though the realized volatility can be higher than implied volatility for periods longer then those when it is lower.

18 Model Risk: Hedging by a Floating Implied Volatility In practice, every day the trader s book is marked to the market, so to have a revaluation as near as possible to the true current value of the assets and other derivatives. That means that the book is revaluated at current market conditions regarding the price of the underlying asset and the implied volatility (we drop for the moment the fact that also the interest rates are updated to the current level). We would like to explore now the impact on the hedging performance when the implied volatility is floating and continuously updated to the market levels.

19 Model Risk: Hedging by a Floating Implied Volatility Under the real probability measure P, the underlying asset s price evolves according to the following SDE: ds = µsdt + σ t dz 1 The implied volatility σ K is now considered as a new stochastic factor and model its evolution, under the real probability measure P, as: dσ K = αdt + ν t dz 2 where dz 1 and dz 2 are two correlated Brownian motion. Under the equivalent martingale measure Q: ds = (r d r f )Sdt + σ t dw 1 dˆσ K = ˆαdt + ν t dw 2 where dw 1 and dw 2 are again two correlated Brownian motion with correlation parameter ρ.

20 Model Risk: Hedging by a Floating Implied Volatility Under the real probability measure P, the option s price evolves as: ( C dc = t C 2 S 2σ2 ts 2 + C C µs + α S σ K C 2 σ 2 K ν 2 t + 2 C σ K S ρσ tsν t ) dt + C S σ tsdz 1 + C σ K ν t dz 2 or, writing the partial derivatives as Greeks: dc = ( Θ Γσ2 ts 2 + µs + Vα Wν2 t + Xρσ t Sν t ) dt + σt SdZ 1 + Vν t dz 2 where V is the Vega, X is the vanna and the W is the volga.

21 Model Risk: Hedging by a Floating Implied Volatility Under equivalent martingale measure Q the dynamics of the call option is: dĉ = Θ Γσ2 ts 2 + (r d r f )S + V ˆα Wν2 t + Xρσ t Sν t = r d Ĉ Let s build a portfolio of: long one call option and short quantity of the underlying; it s P&L over a small period dt is: dπ = dc ds + f where f is the cost born to finance the position: f = ( r d (t)c + r d (t) S r f (t) S)dt

22 Model Risk: Hedging by a Floating Implied Volatility After a few substitutions we get dπ = ( Θ+ 1 2 Γσ2 ts 2 + (r d r f )S+Vα+ 1 2 Wν2 t +Xρσ t Sν t ) dt r d Cdt+Vν t dz 2 Adding and subtracting V ˆα and by means of previous equations we have dπ = Vν t dz 2 + (Vα V ˆα)dt = V(dσ k ˆαdt) By integrating over the option s life: P&L = T 0 dπ = T 0 V(dσ k ˆαdt)

23 Model Risk: Hedging by a Floating Implied Volatility From the formula above we can infer that: Continuous -hedging of a single option revalued at a running implied volatility generates a P&L proportional to the Vega of the option. In general the P&L of a long position in the option, continuously rehedged, is positive if the realized variation in implied volatility is, on average during the option s life, higher than its expected (risk-neutral) variation, it is negative in the opposite case. The previous statement is not always true since the total P&L is dependent on the path followed by the underlying: if periods of low realized variations of the actual implied volatility are experienced when the Vega is high, whereas periods of high realized variations of the actual implied volatility are experienced when the Vega is negligible, then the total P&L is negative, though the realized variations could be on average grater then expected implied volatility s variation.

24 Hedging Volatility Risk in a B&S World In practice, the a trader s book is frequently updated in terms of the underlying asset price and implied volatility. If the book is re-valued and hedged as in a B&S world, then we know from the previous analysis that we have to minimize the model risk by minimizing the Vega exposure. Then the book will be -hedged against the movements of the underlying asset; it will be Vega-hedged against the change in the implied volatility. Vega-hedging must be considered in a very extended meaning: the portfolio must remain Vega-hedged even after movements in the implied volatility and/or the underlying asset.

25 Hedging Volatility Risk in a B&S World So, hedging a book in a B&S world implies setting to zero the following Greeks: V X (Vanna or DvegaDspot) W (Volga or DvegaDvol) The exposure is (usually) easily set to zero by trading in the underlying asset s cash market. The volatility-related Greeks are set to zero by trading (combinations of) other options.

26 Hedging Volatility Risk in a B&S World Tools to cancel Vega exposures are: ATM straddle: this structure has a strong Vega exposure, low Volga exposure and nil Vanna. Risk Reversal 25 : no Vega and Volga exposures, strong Vanna exposure. Vega Weighted Butterfly 25 : no Vega and Vanna exposures, strong Volga exposures. By combining the three structures above, traders make their book Vega-hedged, and the keep this hedging stable to implied volatility ad underlying asset movements.

27 Hedging Volatility Risk in a B&S World Exotic option (e.g.: barriers and One Touch) often show more sensitivity to the Volga and to the Vanna than to the Vega. As an example we consider an Up&Out Eur Call Usd Put option: Spot Ref.: Expiry: 6M Strike: Barrier Up&Out:

28 Hedging Volatility Risk in a B&S World Value spot time to ( 1.05

29 Hedging Volatility Risk in a B&S World Vega spot time to 1.05

30 Hedging Volatility Risk in a B&S World Volga spot time to 1.05

31 Hedging Volatility Risk in a B&S World Vanna spot time t 1.05

32 Hedging Volatility Risk in a B&S World Another example We consider a Down&Out Eur Put Usd Call option: Spot Ref.: Expiry: 3M Strike: Barrier Down&Out:

33 Hedging Volatility Risk in a B&S World Value spot time to (d 1.10

34 Hedging Volatility Risk in a B&S World Vega spot time to ( 1.10

35 Hedging Volatility Risk in a B&S World Volga spot time to ( 1.10

36 Hedging Volatility Risk in a B&S World Vanna spot time to 1.10

37 Hedging Volatility Risk in a B&S World Given the Vega, Vanna and Volga of an option, we calculate the equivalent position in terms of three basic options (ATM, 25 Call and 25 Put): these quantities can be easily converted in amounts of the hedging instruments we have shown above. The U&O Eur Call Usd Put and the D&O Eur Put Usd Call have a volatility exposure as presented in the following table: 25 put 25 call ATM put Up&Out call 79,008,643 54,195, ,556,533 Down&Out put -400,852, ,348, ,163,095 Table 1: Quantities of plain vanilla options to hedge the barrier options according to B&S model.

38 Hedging Volatility Risk in a Stoch Vol World Managing the volatility risk on the B&S s assumption is inconsistent and incomplete. All the volatility related Greeks are zeroed, but the model assumes that the impled volatility is constant, so they should not be hedged. The book is revalued with one implied volatility (typically the ATM), whereas on the market a whole volatility surface is quoted and it changes over time (the three movements for any expiry have been analyzed before). The pricing of exotic options is not consistent with a volatility surface.

39 Hedging Volatility Risk in a Stoch Vol World Need for a model to capture smile effects Non-lognormal models (e.g.: CEV) Local-volatility models (e.g.: Dupire) Stochastc Volatility models (e.g.: Heston, SABR) Lognormal Mixture models (e.g.: Brigo & Mercurio; Brigo Mercurio & Rapisarda)

40 Hedging Volatility Risk in a Stoch Vol World Example: we try to hedge the volatility risk by the Brigo, Mercurio & Rapisarda (2004) model. The hedging procedure is based on the concept of sensitivity bucketing and reflects what a trader is willing to do in practice. This is possible thanks to the model capability of exactly reproducing the fundamental volatility quotes (at least for the three basic instruments). One shifts such a volatility by a fixed amount σ, say ten basis points. One then fit the model to the tilted surface and calculate the price of the exotic, π NEW, corresponding to the newly calibrated parameters. Denoting by π INI the initial price of the exotic, its sensitivity to the given implied volatility is thus calculated as: π NEW π INI σ

41 Hedging Volatility Risk in a Stoch Vol World In practice, it can be more meaningful to hedge the typical movements of the market implied volatility curves. To this end, we start from the three basic data for each maturity (the ATM and the two 25 call and put volatilities), and calculate the exotic s sensitivities to: i) a parallel shift of the three volatilities; ii) a change in the difference between the two 25 wings; iii) an increase of the two wings with fixed ATM volatility. This is actually equivalent to calculating the sensitivities with respect to the basic market quotes. In this way we capture the effect of a parallel, a twist and a convexity movement of the volatility surface. Once these sensitivities are calculated, it is straightforward to hedge the related exposure via plain vanilla options, namely the ATM calls, 25 calls and 25 puts for each expiry.

42 Hedging Volatility Risk in a Stoch Vol World We use the following volatility surface and interest rate data σ ATM σ RR σ V WB P d (0, T) P f (0, T) 1W 13.50% 0.00% 0.19% W 11.80% 0.00% 0.19% M 11.95% 0.05% 0.19% M 11.55% 0.15% 0.21% M 11.50% 0.15% 0.21% M 11.30% 0.20% 0.23% M 11.23% 0.23% 0.23% Y 11.20% 0.25% 0.24% Y 11.10% 0.20% 0.25% Table 2: Market data for EUR/USD as of 31 st March 2004.

43 Hedging Volatility Risk in a Stoch Vol World The hedging quantities calculated according to UVUR model with the scenario approach are shown below. The expiry of the hedging plain vanilla options is once again the same as the corresponding barrier options. It is noteworthy that both the sign and order of magnitude of the hedging options is the similar to those of the BS model we calculated before. 25 put 25 call ATM put Up&Out call 76,409,972 42,089, ,796,515 Down&Out put -338,476, ,078, ,195,436 Table 3: Quantities of plain vanilla options to hedge the barrier options according to UVUR model with the scenario approach.

44 Analogies between B&S and Stoch Vol Hedging The sign and magnitude of the hedging quantities show that some analogies exist between the exposures of an option priced by a B&S model and a model which consider the smile effects. The B&S Vega can be though of as the equivalent of the sensitivity of the option price to a parallel shift of the volatility surface. The B&S Volga is the equivalent of the sensitivity to a change in the convexity of the volatility surface, i.e.: an upward or downward movement of the wings with respect to the ATM level. The B&S Vanna is the equivalent of the sensitivity to a change in the slope of the volatility surface, i.e.: a twist of the wings with respect to the ATM level, considered as a pivot point.

45 More Risks Other risks, related to plain vanilla and exotic options, have to be managed P (Rho) and Φ exposure, i.e.: the sensitivity of the option price to the domestic interest rate and the foreign interest rate, or dividend yield in case of equity option. Risks related to some exotics, e.g.: gap at the breach of the barrier. Correlation risk: many exotic options (especially in the equity market) have as underlying basket of stocks, or the pay-off is contingent on the future evolution of a given number of stocks. In these cases, correlation between the single assets become a main risk.

46 Correlation Risk As an example of correlation risk, we discuss three different option types with the following payout structures An at-the-money (ATM) call option on an equally weighted basket of n stocks. An option on the maximum performance of n assets An option on the minimum performance of n assets. Payouts are defined relative to S i, i = 1,.., n i.e.: the asset price at the expiry T = 0.

47 Correlation Risk The risk management of multi-asset options implies the canceling of the first and second order spot and volatility sensitivities, though in this case we have to deal with matrices of sensitivities : The vector C S i The Γ matrix C S i S j The Vega vector C σ i The Volga matrix The Vanna matrix C σ i σ j C σ i S j First-order correlation risk (correlation Vega) can be calculated as a triangular matrix C ρ ij with i < j.

48 Correlation Risk Single stocks and plain vanilla options on single stocks hedge only the the Vega and the diagonal elements of the Γ, Volga and Vanna matrix. The remaining risks, i.e. the nondiagonal elements (cross Γ, cross Volga and cross Vanna) and the correlation Vega, can be hedged only by other multi-asset options. It can be shown that in a B&S world the following relationship holds: C ρ ij = S i S j σ i σ j T C S i S j So that by hedging all the cross Γ exposure one hedges also the correlation Vega exposures.

49 Correlation Risk The correlation risk affects the price of an options in two ways, depending also on the kind of pay-off of the structure: It impacts on the volatility of the entire basket of underlying stocks. It impacts on the dispersion of the single stocks within the basket. We make some intuitive considerations on these two effects with respect to the three kind of exotic options we listed above.

50 Correlation Risk Basket options: The value of the option is affected only by the basket volatility. The dispersion of individual assets does not influence the option price, because the payout only depends on the sum of the asset prices at maturity. Higher correlations increase basket volatility and thus the option price. Hence basket options are long in correlation.

51 Options on the Maximum: Correlation Risk Increasing correlations imply higher volatility of the basket Increasing dispersion of the single stocks increases the probability of any stock to reach a very high value at maturity. This effect grows with declining correlations. So for max options, the two effects operate in opposite directions. From moderate to high correlation, option prices decrease with increasing correlation: hence, the dispersion effect is stronger than the basket volatility effect and the max option is short in correlation. It should be stressed that this is the initial exposure when the S i (0) are fixed. Situations can occur where the max option is both short and long in correlation depending on the specific levels of correlation and spot prices.

52 Options on the Minimum: Correlation Risk The min option is affected by both effects, but both take the same direction in this case. The dispersion effect increases option prices as correlations become higher, since this minimizes the probability that any asset reaches a very low level at maturity and maximizes the value of the min option Higher correlation implies also a higher volatility of the basket and this increase the option value. Since both effects operate in the same direction, the correlation sensitivity is positive and especially high for this option type.

Consistent pricing and hedging of an FX options book

Consistent pricing and hedging of an FX options book Consistent pricing and hedging of an FX options book L. Bisesti, A. Castagna and F. Mercurio 1 Introduction In the foreign exchange (FX) options market away-from-the-money options are quite actively traded,

More information

Vanna-Volga Method for Foreign Exchange Implied Volatility Smile. Copyright Changwei Xiong 2011. January 2011. last update: Nov 27, 2013

Vanna-Volga Method for Foreign Exchange Implied Volatility Smile. Copyright Changwei Xiong 2011. January 2011. last update: Nov 27, 2013 Vanna-Volga Method for Foreign Exchange Implied Volatility Smile Copyright Changwei Xiong 011 January 011 last update: Nov 7, 01 TABLE OF CONTENTS TABLE OF CONTENTS...1 1. Trading Strategies of Vanilla

More information

FX Options and Smile Risk_. Antonio Castagna. )WILEY A John Wiley and Sons, Ltd., Publication

FX Options and Smile Risk_. Antonio Castagna. )WILEY A John Wiley and Sons, Ltd., Publication FX Options and Smile Risk_ Antonio Castagna )WILEY A John Wiley and Sons, Ltd., Publication Preface Notation and Acronyms IX xiii 1 The FX Market 1.1 FX rates and spot contracts 1.2 Outright and FX swap

More information

THE MAIN RISKS OF AN FX OPTIONS PORTFOLIO

THE MAIN RISKS OF AN FX OPTIONS PORTFOLIO THE MAIN RISKS OF AN FX OPTIONS PORTFOLIO Abstract. In this document we study the risks of an FX options portfolio; we will focus on which, we think, are the main sources of risk. Besides, we analyze also

More information

Consistent Pricing of FX Options

Consistent Pricing of FX Options Consistent Pricing of FX Options Antonio Castagna Fabio Mercurio Banca IMI, Milan In the current markets, options with different strikes or maturities are usually priced with different implied volatilities.

More information

Hedging Illiquid FX Options: An Empirical Analysis of Alternative Hedging Strategies

Hedging Illiquid FX Options: An Empirical Analysis of Alternative Hedging Strategies Hedging Illiquid FX Options: An Empirical Analysis of Alternative Hedging Strategies Drazen Pesjak Supervised by A.A. Tsvetkov 1, D. Posthuma 2 and S.A. Borovkova 3 MSc. Thesis Finance HONOURS TRACK Quantitative

More information

Pricing Barrier Options under Local Volatility

Pricing Barrier Options under Local Volatility Abstract Pricing Barrier Options under Local Volatility Artur Sepp Mail: artursepp@hotmail.com, Web: www.hot.ee/seppar 16 November 2002 We study pricing under the local volatility. Our research is mainly

More information

Exotic Options Trading

Exotic Options Trading Exotic Options Trading Frans de Weert John Wiley & Sons, Ltd Preface Acknowledgements 1 Introduction 2 Conventional Options, Forwards and Greeks 2.1 Call and Put Options and Forwards 2.2 Pricing Calls

More information

IL GOES OCAL A TWO-FACTOR LOCAL VOLATILITY MODEL FOR OIL AND OTHER COMMODITIES 15 // MAY // 2014

IL GOES OCAL A TWO-FACTOR LOCAL VOLATILITY MODEL FOR OIL AND OTHER COMMODITIES 15 // MAY // 2014 IL GOES OCAL A TWO-FACTOR LOCAL VOLATILITY MODEL FOR OIL AND OTHER COMMODITIES 15 MAY 2014 2 Marie-Lan Nguyen / Wikimedia Commons Introduction 3 Most commodities trade as futures/forwards Cash+carry arbitrage

More information

Hedging Barriers. Liuren Wu. Zicklin School of Business, Baruch College (http://faculty.baruch.cuny.edu/lwu/)

Hedging Barriers. Liuren Wu. Zicklin School of Business, Baruch College (http://faculty.baruch.cuny.edu/lwu/) Hedging Barriers Liuren Wu Zicklin School of Business, Baruch College (http://faculty.baruch.cuny.edu/lwu/) Based on joint work with Peter Carr (Bloomberg) Modeling and Hedging Using FX Options, March

More information

How the Greeks would have hedged correlation risk of foreign exchange options

How the Greeks would have hedged correlation risk of foreign exchange options How the Greeks would have hedged correlation risk of foreign exchange options Uwe Wystup Commerzbank Treasury and Financial Products Neue Mainzer Strasse 32 36 60261 Frankfurt am Main GERMANY wystup@mathfinance.de

More information

On the Valuation of Power-Reverse Duals and Equity-Rates Hybrids

On the Valuation of Power-Reverse Duals and Equity-Rates Hybrids On the Valuation of Power-Reverse Duals and Equity-Rates Hybrids Oliver Caps oliver.caps@dkib.com RMT Model Validation Rates Dresdner Bank Examples of Hybrid Products Pricing of Hybrid Products using a

More information

Mathematical Finance

Mathematical Finance Mathematical Finance Option Pricing under the Risk-Neutral Measure Cory Barnes Department of Mathematics University of Washington June 11, 2013 Outline 1 Probability Background 2 Black Scholes for European

More information

Market Risk for Single Trading Positions

Market Risk for Single Trading Positions Chapter 6 Market Risk for Single Trading Positions Market risk is the risk that the market value of trading positions will be adversely influenced by changes in prices and/or interest rates. For banks,

More information

Using the SABR Model

Using the SABR Model Definitions Ameriprise Workshop 2012 Overview Definitions The Black-76 model has been the standard model for European options on currency, interest rates, and stock indices with it s main drawback being

More information

European Options Pricing Using Monte Carlo Simulation

European Options Pricing Using Monte Carlo Simulation European Options Pricing Using Monte Carlo Simulation Alexandros Kyrtsos Division of Materials Science and Engineering, Boston University akyrtsos@bu.edu European options can be priced using the analytical

More information

VALUATION IN DERIVATIVES MARKETS

VALUATION IN DERIVATIVES MARKETS VALUATION IN DERIVATIVES MARKETS September 2005 Rawle Parris ABN AMRO Property Derivatives What is a Derivative? A contract that specifies the rights and obligations between two parties to receive or deliver

More information

Hedging Exotic Options

Hedging Exotic Options Kai Detlefsen Wolfgang Härdle Center for Applied Statistics and Economics Humboldt-Universität zu Berlin Germany introduction 1-1 Models The Black Scholes model has some shortcomings: - volatility is not

More information

Black-Scholes Equation for Option Pricing

Black-Scholes Equation for Option Pricing Black-Scholes Equation for Option Pricing By Ivan Karmazin, Jiacong Li 1. Introduction In early 1970s, Black, Scholes and Merton achieved a major breakthrough in pricing of European stock options and there

More information

On Black-Scholes Equation, Black- Scholes Formula and Binary Option Price

On Black-Scholes Equation, Black- Scholes Formula and Binary Option Price On Black-Scholes Equation, Black- Scholes Formula and Binary Option Price Abstract: Chi Gao 12/15/2013 I. Black-Scholes Equation is derived using two methods: (1) risk-neutral measure; (2) - hedge. II.

More information

Properties of the SABR model

Properties of the SABR model U.U.D.M. Project Report 2011:11 Properties of the SABR model Nan Zhang Examensarbete i matematik, 30 hp Handledare och examinator: Johan Tysk Juni 2011 Department of Mathematics Uppsala University ABSTRACT

More information

FX Derivatives Terminology. Education Module: 5. Dated July 2002. FX Derivatives Terminology

FX Derivatives Terminology. Education Module: 5. Dated July 2002. FX Derivatives Terminology Education Module: 5 Dated July 2002 Foreign Exchange Options Option Markets and Terminology A American Options American Options are options that are exercisable for early value at any time during the term

More information

The real P&L in Black-Scholes and Dupire Delta hedging

The real P&L in Black-Scholes and Dupire Delta hedging International Journal of Theoretical and Applied Finance c World Scientific Publishing Company The real P&L in Black-Scholes and Dupire Delta hedging MARTIN FORDE University of Bristol, Department of Mathematics,

More information

Valuing Stock Options: The Black-Scholes-Merton Model. Chapter 13

Valuing Stock Options: The Black-Scholes-Merton Model. Chapter 13 Valuing Stock Options: The Black-Scholes-Merton Model Chapter 13 Fundamentals of Futures and Options Markets, 8th Ed, Ch 13, Copyright John C. Hull 2013 1 The Black-Scholes-Merton Random Walk Assumption

More information

Return to Risk Limited website: www.risklimited.com. Overview of Options An Introduction

Return to Risk Limited website: www.risklimited.com. Overview of Options An Introduction Return to Risk Limited website: www.risklimited.com Overview of Options An Introduction Options Definition The right, but not the obligation, to enter into a transaction [buy or sell] at a pre-agreed price,

More information

Validation of New Pricing Model for Exotic Options Pioneering Again

Validation of New Pricing Model for Exotic Options Pioneering Again Final Year Report Sept 2010 Validation of New Pricing Model for Exotic Options Pioneering Again Trainee: Dima HAMZE Tutor: Nadia ISRAEL Responsible: Alain CHATEAUNEUF Acknowledgements I would like to take

More information

The Black-Scholes Formula

The Black-Scholes Formula FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 The Black-Scholes Formula These notes examine the Black-Scholes formula for European options. The Black-Scholes formula are complex as they are based on the

More information

DIGITAL FOREX OPTIONS

DIGITAL FOREX OPTIONS DIGITAL FOREX OPTIONS OPENGAMMA QUANTITATIVE RESEARCH Abstract. Some pricing methods for forex digital options are described. The price in the Garhman-Kohlhagen model is first described, more for completeness

More information

A Day in the Life of a Trader

A Day in the Life of a Trader Siena, April 2014 Introduction 1 Examples of Market Payoffs 2 3 4 Sticky Smile e Floating Smile 5 Examples of Market Payoffs Understanding risk profiles of a payoff is conditio sine qua non for a mathematical

More information

Foreign Exchange Symmetries

Foreign Exchange Symmetries Foreign Exchange Symmetries Uwe Wystup MathFinance AG Waldems, Germany www.mathfinance.com 8 September 2008 Contents 1 Foreign Exchange Symmetries 2 1.1 Motivation.................................... 2

More information

4. ANNEXURE 3 : PART 3 - FOREIGN EXCHANGE POSITION RISK

4. ANNEXURE 3 : PART 3 - FOREIGN EXCHANGE POSITION RISK Annexure 3 (PRR) - Part 3, Clause 18 - Foreign Exchange Position Risk Amount 4 ANNEXURE 3 : PART 3 - FOREIGN EXCHANGE POSITION RISK (a) CLAUSE 18 - FOREIGN EXCHANGE POSITION RISK AMOUNT (i) Rule PART 3

More information

Write clearly; the grade will also take into account the quality of the presentation and the clarity of the explanations

Write clearly; the grade will also take into account the quality of the presentation and the clarity of the explanations Name: Student-ID number: Write clearly; the grade will also take into account the quality of the presentation and the clarity of the explanations Question Points Score 1 29 2 17 3 19 4 2 5 2 6 1 Total:

More information

Review of Basic Options Concepts and Terminology

Review of Basic Options Concepts and Terminology Review of Basic Options Concepts and Terminology March 24, 2005 1 Introduction The purchase of an options contract gives the buyer the right to buy call options contract or sell put options contract some

More information

Beyond Black- Scholes: Smile & Exo6c op6ons. Professor André Farber Solvay Brussels School of Economics and Management Université Libre de Bruxelles

Beyond Black- Scholes: Smile & Exo6c op6ons. Professor André Farber Solvay Brussels School of Economics and Management Université Libre de Bruxelles Beyond Black- Scholes: Smile & Exo6c op6ons Professor André Farber Solvay Brussels School of Economics and Management Université Libre de Bruxelles 1 What is a Vola6lity Smile? Rela6onship between implied

More information

Invesco Great Wall Fund Management Co. Shenzhen: June 14, 2008

Invesco Great Wall Fund Management Co. Shenzhen: June 14, 2008 : A Stern School of Business New York University Invesco Great Wall Fund Management Co. Shenzhen: June 14, 2008 Outline 1 2 3 4 5 6 se notes review the principles underlying option pricing and some of

More information

w w w.c a t l e y l a k e m a n.c o m 0 2 0 7 0 4 3 0 1 0 0

w w w.c a t l e y l a k e m a n.c o m 0 2 0 7 0 4 3 0 1 0 0 A ADR-Style: for a derivative on an underlying denominated in one currency, where the derivative is denominated in a different currency, payments are exchanged using a floating foreign-exchange rate. The

More information

The Behavior of Bonds and Interest Rates. An Impossible Bond Pricing Model. 780 w Interest Rate Models

The Behavior of Bonds and Interest Rates. An Impossible Bond Pricing Model. 780 w Interest Rate Models 780 w Interest Rate Models The Behavior of Bonds and Interest Rates Before discussing how a bond market-maker would delta-hedge, we first need to specify how bonds behave. Suppose we try to model a zero-coupon

More information

DIFFERENT PATHS: SCENARIOS FOR n-asset AND PATH-DEPENDENT OPTIONS II

DIFFERENT PATHS: SCENARIOS FOR n-asset AND PATH-DEPENDENT OPTIONS II DIFFERENT PATHS: SCENARIOS FOR n-asset AND PATH-DEPENDENT OPTIONS II David Murphy Market and Liquidity Risk, Banque Paribas This article expresses the personal views of the author and does not necessarily

More information

Or part of or all the risk is dynamically hedged trading regularly, with a. frequency that needs to be appropriate for the trade.

Or part of or all the risk is dynamically hedged trading regularly, with a. frequency that needs to be appropriate for the trade. Option position (risk) management Correct risk management of option position is the core of the derivatives business industry. Option books bear huge amount of risk with substantial leverage in the position.

More information

Lecture 11: The Greeks and Risk Management

Lecture 11: The Greeks and Risk Management Lecture 11: The Greeks and Risk Management This lecture studies market risk management from the perspective of an options trader. First, we show how to describe the risk characteristics of derivatives.

More information

Black-Scholes and the Volatility Surface

Black-Scholes and the Volatility Surface IEOR E4707: Financial Engineering: Continuous-Time Models Fall 2009 c 2009 by Martin Haugh Black-Scholes and the Volatility Surface When we studied discrete-time models we used martingale pricing to derive

More information

第 9 讲 : 股 票 期 权 定 价 : B-S 模 型 Valuing Stock Options: The Black-Scholes Model

第 9 讲 : 股 票 期 权 定 价 : B-S 模 型 Valuing Stock Options: The Black-Scholes Model 1 第 9 讲 : 股 票 期 权 定 价 : B-S 模 型 Valuing Stock Options: The Black-Scholes Model Outline 有 关 股 价 的 假 设 The B-S Model 隐 性 波 动 性 Implied Volatility 红 利 与 期 权 定 价 Dividends and Option Pricing 美 式 期 权 定 价 American

More information

Study on the Volatility Smile of EUR/USD Currency Options and Trading Strategies

Study on the Volatility Smile of EUR/USD Currency Options and Trading Strategies Prof. Joseph Fung, FDS Study on the Volatility Smile of EUR/USD Currency Options and Trading Strategies BY CHEN Duyi 11050098 Finance Concentration LI Ronggang 11050527 Finance Concentration An Honors

More information

WHS FX options guide. Getting started with FX options. Predict the trend in currency markets or hedge your positions with FX options.

WHS FX options guide. Getting started with FX options. Predict the trend in currency markets or hedge your positions with FX options. Getting started with FX options WHS FX options guide Predict the trend in currency markets or hedge your positions with FX options. Refine your trading style and your market outlook. Learn how FX options

More information

Caput Derivatives: October 30, 2003

Caput Derivatives: October 30, 2003 Caput Derivatives: October 30, 2003 Exam + Answers Total time: 2 hours and 30 minutes. Note 1: You are allowed to use books, course notes, and a calculator. Question 1. [20 points] Consider an investor

More information

1 The Black-Scholes model: extensions and hedging

1 The Black-Scholes model: extensions and hedging 1 The Black-Scholes model: extensions and hedging 1.1 Dividends Since we are now in a continuous time framework the dividend paid out at time t (or t ) is given by dd t = D t D t, where as before D denotes

More information

Barrier Options. Peter Carr

Barrier Options. Peter Carr Barrier Options Peter Carr Head of Quantitative Financial Research, Bloomberg LP, New York Director of the Masters Program in Math Finance, Courant Institute, NYU March 14th, 2008 What are Barrier Options?

More information

Underlying (S) The asset, which the option buyer has the right to buy or sell. Notation: S or S t = S(t)

Underlying (S) The asset, which the option buyer has the right to buy or sell. Notation: S or S t = S(t) INTRODUCTION TO OPTIONS Readings: Hull, Chapters 8, 9, and 10 Part I. Options Basics Options Lexicon Options Payoffs (Payoff diagrams) Calls and Puts as two halves of a forward contract: the Put-Call-Forward

More information

Equity derivative strategy 2012 Q1 update and Trading Volatility

Equity derivative strategy 2012 Q1 update and Trading Volatility Equity derivative strategy 212 Q1 update and Trading Volatility Colin Bennett (+34) 91 28 9356 cdbennett@gruposantander.com 1 Contents VOLATILITY TRADING FOR DIRECTIONAL INVESTORS Call overwriting Protection

More information

CS 522 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options

CS 522 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options CS 5 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options 1. Definitions Equity. The common stock of a corporation. Traded on organized exchanges (NYSE, AMEX, NASDAQ). A common

More information

ENGINEERING AND HEDGING OF CORRIDOR PRODUCTS - with focus on FX linked instruments -

ENGINEERING AND HEDGING OF CORRIDOR PRODUCTS - with focus on FX linked instruments - AARHUS SCHOOL OF BUSINESS AARHUS UNIVERSITY MASTER THESIS ENGINEERING AND HEDGING OF CORRIDOR PRODUCTS - with focus on FX linked instruments - AUTHORS: DANIELA ZABRE GEORGE RARES RADU SIMIAN SUPERVISOR:

More information

Open issues in equity derivatives modelling

Open issues in equity derivatives modelling Open issues in equity derivatives modelling Lorenzo Bergomi Equity Derivatives Quantitative Research ociété Générale lorenzo.bergomi@sgcib.com al Outline Equity derivatives at G A brief history of equity

More information

τ θ What is the proper price at time t =0of this option?

τ θ What is the proper price at time t =0of this option? Now by Itô s formula But Mu f and u g in Ū. Hence τ θ u(x) =E( Mu(X) ds + u(x(τ θ))) 0 τ θ u(x) E( f(x) ds + g(x(τ θ))) = J x (θ). 0 But since u(x) =J x (θ ), we consequently have u(x) =J x (θ ) = min

More information

Cash-settled swaptions How wrong are we?

Cash-settled swaptions How wrong are we? Cash-settled swaptions How wrong are we? Marc Henrard Quantitative Research - OpenGamma 7th Fixed Income Conference, Berlin, 7 October 2011 ... Based on: Cash-settled swaptions: How wrong are we? Available

More information

Delta-hedging Vega Risk?

Delta-hedging Vega Risk? Delta-hedging Vega Risk? Stéphane Crépey June 10, 2004 Abstract In this article we compare the Profit and Loss arising from the delta-neutral dynamic hedging of options, using two possible values for the

More information

Introduction to Equity Derivatives

Introduction to Equity Derivatives Introduction to Equity Derivatives Aaron Brask + 44 (0)20 7773 5487 Internal use only Equity derivatives overview Products Clients Client strategies Barclays Capital 2 Equity derivatives products Equity

More information

Risk Management and Governance Hedging with Derivatives. Prof. Hugues Pirotte

Risk Management and Governance Hedging with Derivatives. Prof. Hugues Pirotte Risk Management and Governance Hedging with Derivatives Prof. Hugues Pirotte Several slides based on Risk Management and Financial Institutions, e, Chapter 6, Copyright John C. Hull 009 Why Manage Risks?

More information

Learn. Commodity Options Trading & Risk Management ORM2. Commodity Derivatives Strategy

Learn. Commodity Options Trading & Risk Management ORM2. Commodity Derivatives Strategy Learn Commodity Options Trading & Risk Management ORM2 2 days of intensive and comprehensive options/risk management training with practical analysis of option trading strategies and risk management with

More information

Caps and Floors. John Crosby

Caps and Floors. John Crosby Caps and Floors John Crosby Glasgow University My website is: http://www.john-crosby.co.uk If you spot any typos or errors, please email me. My email address is on my website Lecture given 19th February

More information

Implied Volatility Surface

Implied Volatility Surface Implied Volatility Surface Liuren Wu Zicklin School of Business, Baruch College Options Markets (Hull chapter: 16) Liuren Wu Implied Volatility Surface Options Markets 1 / 18 Implied volatility Recall

More information

OPTIONS EDUCATION GLOBAL

OPTIONS EDUCATION GLOBAL OPTIONS EDUCATION GLOBAL TABLE OF CONTENTS Introduction What are FX Options? Trading 101 ITM, ATM and OTM Options Trading Strategies Glossary Contact Information 3 5 6 8 9 10 16 HIGH RISK WARNING: Before

More information

OPTIONS. FINANCE TRAINER International Options / Page 1 of 38

OPTIONS. FINANCE TRAINER International Options / Page 1 of 38 OPTIONS 1. FX Options... 3 1.1 Terminology... 4 1.2 The Four Basic Positions... 5 1.3 Standard Options... 7 1.4 Exotic Options... 7 1.4.1 Asian Option (Average Rate Option, ARO)... 7 1.4.2 Compound Option...

More information

No-arbitrage conditions for cash-settled swaptions

No-arbitrage conditions for cash-settled swaptions No-arbitrage conditions for cash-settled swaptions Fabio Mercurio Financial Engineering Banca IMI, Milan Abstract In this note, we derive no-arbitrage conditions that must be satisfied by the pricing function

More information

Forward Price. The payoff of a forward contract at maturity is S T X. Forward contracts do not involve any initial cash flow.

Forward Price. The payoff of a forward contract at maturity is S T X. Forward contracts do not involve any initial cash flow. Forward Price The payoff of a forward contract at maturity is S T X. Forward contracts do not involve any initial cash flow. The forward price is the delivery price which makes the forward contract zero

More information

Model Independent Greeks

Model Independent Greeks Model Independent Greeks Mathematical Finance Winter School Lunteren January 2014 Jesper Andreasen Danske Markets, Copenhagen kwant.daddy@danskebank.com Outline Motivation: Wots me Δelδa? The short maturity

More information

Handbook FXFlat FX Options

Handbook FXFlat FX Options Handbook FXFlat FX Options FXFlat Trading FX Options When you open an FX options account at FXFlat, you can trade options on currency pairs 24- hours a day, 5.5 days per week. The FX options features in

More information

FX Barriers with Smile Dynamics

FX Barriers with Smile Dynamics FX Barriers with Smile Dynamics Glyn Baker, Reimer Beneder and Alex Zilber December 16, 2004 Abstract Our mandate in this work has been to isolate the features of smile consistent models that are most

More information

Sensex Realized Volatility Index

Sensex Realized Volatility Index Sensex Realized Volatility Index Introduction: Volatility modelling has traditionally relied on complex econometric procedures in order to accommodate the inherent latent character of volatility. Realized

More information

Computational Finance Options

Computational Finance Options 1 Options 1 1 Options Computational Finance Options An option gives the holder of the option the right, but not the obligation to do something. Conversely, if you sell an option, you may be obliged to

More information

Guaranteed Annuity Options

Guaranteed Annuity Options Guaranteed Annuity Options Hansjörg Furrer Market-consistent Actuarial Valuation ETH Zürich, Frühjahrssemester 2008 Guaranteed Annuity Options Contents A. Guaranteed Annuity Options B. Valuation and Risk

More information

Pricing complex options using a simple Monte Carlo Simulation

Pricing complex options using a simple Monte Carlo Simulation A subsidiary of Sumitomo Mitsui Banking Corporation Pricing complex options using a simple Monte Carlo Simulation Peter Fink Among the different numerical procedures for valuing options, the Monte Carlo

More information

1.2 Structured notes

1.2 Structured notes 1.2 Structured notes Structured notes are financial products that appear to be fixed income instruments, but contain embedded options and do not necessarily reflect the risk of the issuing credit. Used

More information

CURRENCY OPTION PRICING II

CURRENCY OPTION PRICING II Jones Grauate School Rice University Masa Watanabe INTERNATIONAL FINANCE MGMT 657 Calibrating the Binomial Tree to Volatility Black-Scholes Moel for Currency Options Properties of the BS Moel Option Sensitivity

More information

Lecture 4: The Black-Scholes model

Lecture 4: The Black-Scholes model OPTIONS and FUTURES Lecture 4: The Black-Scholes model Philip H. Dybvig Washington University in Saint Louis Black-Scholes option pricing model Lognormal price process Call price Put price Using Black-Scholes

More information

INTEREST RATES AND FX MODELS

INTEREST RATES AND FX MODELS INTEREST RATES AND FX MODELS 8. Portfolio greeks Andrew Lesniewski Courant Institute of Mathematical Sciences New York University New York March 27, 2013 2 Interest Rates & FX Models Contents 1 Introduction

More information

CRUDE OIL HEDGING STRATEGIES An Application of Currency Translated Options

CRUDE OIL HEDGING STRATEGIES An Application of Currency Translated Options CRUDE OIL HEDGING STRATEGIES An Application of Currency Translated Options Paul Obour Supervisor: Dr. Antony Ware University of Calgary PRMIA Luncheon - Bankers Hall, Calgary May 8, 2012 Outline 1 Introductory

More information

Option pricing. Vinod Kothari

Option pricing. Vinod Kothari Option pricing Vinod Kothari Notation we use this Chapter will be as follows: S o : Price of the share at time 0 S T : Price of the share at time T T : time to maturity of the option r : risk free rate

More information

Implementing CCR and CVA in a Primary International Bank

Implementing CCR and CVA in a Primary International Bank Implementing CCR and CVA in a Primary International Bank www.iasonltd.com 2011 Index Introduction 1 Introduction 2 3 Index Introduction 1 Introduction 2 3 Counterparty Credit Risk and Credit Value Adjustment

More information

A SNOWBALL CURRENCY OPTION

A SNOWBALL CURRENCY OPTION J. KSIAM Vol.15, No.1, 31 41, 011 A SNOWBALL CURRENCY OPTION GYOOCHEOL SHIM 1 1 GRADUATE DEPARTMENT OF FINANCIAL ENGINEERING, AJOU UNIVERSITY, SOUTH KOREA E-mail address: gshim@ajou.ac.kr ABSTRACT. I introduce

More information

A Vega-Gamma Relationship for European-Style or Barrier Options in the Black-Scholes Model

A Vega-Gamma Relationship for European-Style or Barrier Options in the Black-Scholes Model A Vega-Gamma Relationship for European-Style or Barrier Options in the Black-Scholes Model Fabio Mercurio Financial Models, Banca IMI Abstract In this document we derive some fundamental relationships

More information

Introduction Pricing Effects Greeks Summary. Vol Target Options. Rob Coles. February 7, 2014

Introduction Pricing Effects Greeks Summary. Vol Target Options. Rob Coles. February 7, 2014 February 7, 2014 Outline 1 Introduction 2 3 Vega Theta Delta & Gamma Hedge P& L Jump sensitivity The Basic Idea Basket split between risky asset and cash Chose weight of risky asset w to keep volatility

More information

Hedging. An Undergraduate Introduction to Financial Mathematics. J. Robert Buchanan. J. Robert Buchanan Hedging

Hedging. An Undergraduate Introduction to Financial Mathematics. J. Robert Buchanan. J. Robert Buchanan Hedging Hedging An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2010 Introduction Definition Hedging is the practice of making a portfolio of investments less sensitive to changes in

More information

Convenient Conventions

Convenient Conventions C: call value. P : put value. X: strike price. S: stock price. D: dividend. Convenient Conventions c 2015 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 168 Payoff, Mathematically Speaking The payoff

More information

The Black-Scholes-Merton Approach to Pricing Options

The Black-Scholes-Merton Approach to Pricing Options he Black-Scholes-Merton Approach to Pricing Options Paul J Atzberger Comments should be sent to: atzberg@mathucsbedu Introduction In this article we shall discuss the Black-Scholes-Merton approach to determining

More information

Understanding Options and Their Role in Hedging via the Greeks

Understanding Options and Their Role in Hedging via the Greeks Understanding Options and Their Role in Hedging via the Greeks Bradley J. Wogsland Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996-1200 Options are priced assuming that

More information

Bootstrapping the interest-rate term structure

Bootstrapping the interest-rate term structure Bootstrapping the interest-rate term structure Marco Marchioro www.marchioro.org October 20 th, 2012 Bootstrapping the interest-rate term structure 1 Summary (1/2) Market quotes of deposit rates, IR futures,

More information

Option Valuation. Chapter 21

Option Valuation. Chapter 21 Option Valuation Chapter 21 Intrinsic and Time Value intrinsic value of in-the-money options = the payoff that could be obtained from the immediate exercise of the option for a call option: stock price

More information

On Market-Making and Delta-Hedging

On Market-Making and Delta-Hedging On Market-Making and Delta-Hedging 1 Market Makers 2 Market-Making and Bond-Pricing On Market-Making and Delta-Hedging 1 Market Makers 2 Market-Making and Bond-Pricing What to market makers do? Provide

More information

Derivation of Local Volatility by Fabrice Douglas Rouah www.frouah.com www.volopta.com

Derivation of Local Volatility by Fabrice Douglas Rouah www.frouah.com www.volopta.com Derivation of Local Volatility by Fabrice Douglas Rouah www.frouah.com www.volopta.com The derivation of local volatility is outlined in many papers and textbooks (such as the one by Jim Gatheral []),

More information

Institutional Finance 08: Dynamic Arbitrage to Replicate Non-linear Payoffs. Binomial Option Pricing: Basics (Chapter 10 of McDonald)

Institutional Finance 08: Dynamic Arbitrage to Replicate Non-linear Payoffs. Binomial Option Pricing: Basics (Chapter 10 of McDonald) Copyright 2003 Pearson Education, Inc. Slide 08-1 Institutional Finance 08: Dynamic Arbitrage to Replicate Non-linear Payoffs Binomial Option Pricing: Basics (Chapter 10 of McDonald) Originally prepared

More information

EQUITY LINKED NOTES: An Introduction to Principal Guaranteed Structures Abukar M Ali October 2002

EQUITY LINKED NOTES: An Introduction to Principal Guaranteed Structures Abukar M Ali October 2002 EQUITY LINKED NOTES: An Introduction to Principal Guaranteed Structures Abukar M Ali October 2002 Introduction In this article we provide a succinct description of a commonly used investment instrument

More information

The Black-Scholes Model

The Black-Scholes Model The Black-Scholes Model Liuren Wu Zicklin School of Business, Baruch College Options Markets (Hull chapter: 12, 13, 14) Liuren Wu The Black-Scholes Model Options Markets 1 / 19 The Black-Scholes-Merton

More information

Options: Definitions, Payoffs, & Replications

Options: Definitions, Payoffs, & Replications Options: Definitions, s, & Replications Liuren Wu Zicklin School of Business, Baruch College Options Markets Liuren Wu (Baruch) s Options Markets 1 / 34 Definitions and terminologies An option gives the

More information

The Black-Scholes pricing formulas

The Black-Scholes pricing formulas The Black-Scholes pricing formulas Moty Katzman September 19, 2014 The Black-Scholes differential equation Aim: Find a formula for the price of European options on stock. Lemma 6.1: Assume that a stock

More information

Vega Risk in RiskManager

Vega Risk in RiskManager Model Overview Lisa R. Goldberg Michael Y. Hayes Ola Mahmoud (Marie Cure Fellow) 1 Tamas Matrai 1 Ola Mahmoud was a Marie Curie Fellows at MSCI. The research leading to these results has received funding

More information

Forwards, Swaps and Futures

Forwards, Swaps and Futures IEOR E4706: Financial Engineering: Discrete-Time Models c 2010 by Martin Haugh Forwards, Swaps and Futures These notes 1 introduce forwards, swaps and futures, and the basic mechanics of their associated

More information

EXP 481 -- Capital Markets Option Pricing. Options: Definitions. Arbitrage Restrictions on Call Prices. Arbitrage Restrictions on Call Prices 1) C > 0

EXP 481 -- Capital Markets Option Pricing. Options: Definitions. Arbitrage Restrictions on Call Prices. Arbitrage Restrictions on Call Prices 1) C > 0 EXP 481 -- Capital Markets Option Pricing imple arbitrage relations Payoffs to call options Black-choles model Put-Call Parity Implied Volatility Options: Definitions A call option gives the buyer the

More information

Option Pricing. Chapter 4 Including dividends in the BS model. Stefan Ankirchner. University of Bonn. last update: 6th November 2013

Option Pricing. Chapter 4 Including dividends in the BS model. Stefan Ankirchner. University of Bonn. last update: 6th November 2013 Option Pricing Chapter 4 Including dividends in the BS model Stefan Ankirchner University of Bonn last update: 6th November 2013 Stefan Ankirchner Option Pricing 1 Dividend payments So far: we assumed

More information

The interest volatility surface

The interest volatility surface The interest volatility surface David Kohlberg Kandidatuppsats i matematisk statistik Bachelor Thesis in Mathematical Statistics Kandidatuppsats 2011:7 Matematisk statistik Juni 2011 www.math.su.se Matematisk

More information

FINANCIAL ECONOMICS OPTION PRICING

FINANCIAL ECONOMICS OPTION PRICING OPTION PRICING Options are contingency contracts that specify payoffs if stock prices reach specified levels. A call option is the right to buy a stock at a specified price, X, called the strike price.

More information