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- Herausgeber: John Wiley & Sons
- Kategorie: Fachliteratur
- Serie: Wiley Trading Series
- Sprache: Englisch
An essential guide to real-world derivatives trading FX Derivatives Trader School is the definitive guide to the technical and practical knowledge required for successful foreign exchange derivatives trading. Accessible in style and comprehensive in coverage, the book guides the reader through both basic and advanced derivative pricing and risk management topics. The basics of financial markets and trading are covered, plus practical derivatives mathematics is introduced with reference to real-world trading and risk management. Derivative contracts are covered in detail from a trader's perspective using risk profiles and pricing under different derivative models. Analysis is approached generically to enable new products to be understood by breaking the risk into fundamental building blocks. To assist with learning, the book also contains Excel practicals which will deepen understanding and help build useful skills. The book covers of a wide variety of topics, including: * Derivative exposures within risk management * Volatility surface construction * Implied volatility and correlation risk * Practical tips for students on trading internships and junior traders * Market analysis techniques FX derivatives trading requires mathematical aptitude, risk management skill, and the ability to work quickly and accurately under pressure. There is a tremendous gap between option pricing formulas and the knowledge required to be a successful derivatives trader. FX Derivatives Trader School is unique in bridging that gap.
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Veröffentlichungsjahr: 2015
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Table of Contents
Title Page
Copyright
Dedication
Preface
Acknowledgments
Part I: The Basics
Chapter 1: Introduction to Foreign Exchange
Practical Aspects of the FX Market
What Do FX Traders Call Different Currency Pairs?
Chapter 2: Introduction to FX Derivatives
Vanilla Call and Put Options
Practical Aspects of the FX Derivatives Market
Chapter 3: Introduction to Trading
Bids and Offers
Market Making
Price Making and Risk Management Overview
Practical A: Building a Trading Simulator in Excel
Task A: Set Up a Ticking Market Price
Task B: Set Up a Two-Way Price and Price-Taking Functionality
Task C: Introduce Price-Making Functionality
Extensions
Chapter 4: FX Derivatives Market Structure
Client Types
Bank FX Derivatives Trading Desk Structure
Tips for a Trading Internship
Tips for a Junior Trader
FX Derivatives Interbank Direct Market
FX Derivatives Interbank Broker Market
Chapter 5: The Black-Scholes Framework
Black-Scholes Stochastic Differential Equation (SDE)
Solving the Black-Scholes SDE
Calculating Option Values Using Terminal Spot Distributions
The Black-Scholes Formula
Practical B: Building a Numerical Integration Option Pricer in Excel
Task A: Set Up the Terminal Spot Distribution
Task B: Set Up the Option Payoff and Calculate the Option Price
Testing
Chapter 6: Vanilla FX Derivatives Greeks
Option Value
Delta
Gamma
Vega
Summary
Practical C: Building a Black-Scholes Option Pricer in Excel
Task A: Set Up a Simple Black-Scholes Options Pricer
Task B: Set Up a VBA Pricing Function
Task C: Generate First-Order Greeks
Task D: Plot Exposures
Chapter 7: Vanilla FX Derivatives Pricing
Maintaining Volatility Surfaces
Vanilla Price Making
Chapter 8: Vanilla FX Derivatives Structures
Straddle
Strangle
Butterfly (Fly)
Risk Reversal (RR)
Leveraged Forward
ATM Calendar Spread
Call/Put Spreads
Seagull
Chapter 9: Vanilla FX Derivatives Risk Management
Trading Gamma
Trading the Short-Date Position
Trading the ATM Position
FX Derivatives Trading P&L
FX Derivatives Market Language
Chapter 10: Vanilla FX Derivatives Miscellaneous Topics
Present Valuing and Future Valuing
Market Tenor Calculations
Option Premium Conversions
Practical D: Generating Tenor Dates in Excel
Part II: The Volatility Surface
Chapter 11: ATM Curve Construction
Variance
Core ATM Curve Construction
ATM Curve Construction: Short-Dates
Practical E: Constructing an ATM Curve in Excel
Task A: Constructing an ATM Curve Using Interpolation
Task B: Constructing an ATM Curve Using a Model
Task C: Adding Weights to an ATM Curve
Chapter 12: Volatility Smile Market Instruments and Exposures
Market Instrument Vega Exposures
Risk Reversal Contract
Butterfly Contract
Volatility Smile Risk Management
Volatility Smile Construction Methods
Practical F: Constructing a Volatility Smile in Excel
Task A: Set Up the Malz Smile Model
Task B: Plot Implied Volatility versus Delta and Investigate Parameters
Task C: Use Black-Scholes to Get Strike from Delta
Task D: Switch to VBA Functions and Plot Implied Volatility versus Strike
Task E: Investigate Volatility Smile Strike Placement
Chapter 13: Probability Density Functions
Fat-Tailed Distributions
Confidence Intervals
Limitations of Volatility Smile Parameterization
Practical G: Generating a Probability Density Function from Option Prices in Excel
Part III: Vanilla FX Derivatives Trading
Chapter 14: Vanilla FX Derivatives Trading Exposures
Delta
Gamma and Theta
Vega and Weighted Vega
Adapted Greeks
Zeta
Interest Rate Risk (Rho)
Chapter 15: Vanilla FX Derivatives Trading Topics
Understanding the FX Derivatives Market
Trading the Overnight
Gartman's Rules of Trading
Vega Positioning
Short-Date Trading: Long ATM versus Short Wings
Client Option Orders
Quoting Vanilla Spreads
Trading Long-Dated FX Derivatives
Trading Pegged Currency Pairs
Positive Vanilla Spreads
Writing-Off Vanilla Risk
Vanilla Pin Risk
Low Delta Vanilla Options
Agreeing Broker Market Data
Chapter 16: ATM Volatility and Correlation
ATM Volatility Triangles
Dephased Vega
Managing Cross-Currency Positions
Chapter 17: FX Derivatives Market Analysis
Calculating Breakevens
Implied versus Realized Analysis
Market Instrument Analysis
Carry Trades
Part IV: Exotic FX Derivatives
Chapter 18: Exotic FX Derivatives Pricing
Exotic Pricing Example
Pricing the Volatility Smile
VVV Pricing Example: Part 1
Stopping Time
VVV Pricing Example: Part 2
Path Dependence
Chapter 19: FX Derivatives Pricing Models
Stochastic Volatility Models
Local Volatility Models
Mixed Volatility Models
Jump Diffusion Models
Stochastic Interest Rate Models
Chapter 20: Exotic FX Derivatives Product Classification
First-Generation Exotics
Second-Generation Exotics
Third-Generation Exotics
Chapter 21: European Digital Options
European Digital Replication
European Digital Pricing
European Digital Bid–Offer Spread
European Digital Greeks
European Digital Range
Chapter 22: European Barrier Options
European Knock-out Replication
Intrinsic Value
European Knock-in Replication
European Barrier Greeks
European Barrier Pricing
European Barrier Bid–Offer Spread
Chapter 23: Touch Options
Delta Risk
Vega Risk
Gamma and Pin Risk
Touch Barrier Delta Gap
One-Touch Pricing
One-Touch Variations
No-Touch Options
Chapter 24: American Barrier Options
Regular American Barrier Options
Reverse American Barrier Options
Double American Barrier Options
Knock-in/Knock-out Option Replication
Strike-out Options
Chapter 25: Exotic FX Derivatives Trading Topics
Exotic Risk Management Overview
Exotic Bid–Offer Spreading
Exotic Interbank Broker Market
Structured FX Hedging Strategies
FX Derivatives Investment Products
Shadow Barriers
Recycling Exotic Risk
Why Market Participants Prefer to Sell American Barriers
Chapter 26: Window Barrier and Discrete Barrier Options
Front-Window Barrier Options
Rear-Window Barrier Options
Generic Window Barrier Options
Window Barrier Risk Management
Discrete Barrier Options
Practical H: Building a Monte Carlo Option Pricer in Excel
Task A: Set Up the Simulation
Task B: Set Up a Vanilla Option Payoff and the Monte Carlo Loop
Task C: Set Up Multiple Payoffs
Task D: Pricing Barrier Options
Task E: Multi-Asset Simulation
Extensions
Chapter 27: Vanilla Variations
Late-Delivery Vanilla Options
American Vanilla Options
Self-Quanto Vanilla Options
Chapter 28: Accrual and Target Redemption Options
Accrual Options
Target Redemption Options
Chapter 29: Asian Options
Average Rate Options
Average Strike Options
Double Average Rate Options
Chapter 30: Multi-Asset Options
Multi-Asset Trading Risks
Multi Asset Bid–Offer Spreading
Basket Options
Dual Digital Options
Best-of and Worst-of Options
Quanto Options
Chapter 31: Miscellaneous Options
Volatility and Variance Swaps
Forward Volatility Agreements
Forward Start Options
Compound Options
Further Reading
About the Companion Website
Index
End User License Agreement
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Guide
Cover
Table of Contents
Preface
Part I: THE BASICS
Begin Reading
List of Exhibits
Chapter 1: Introduction to Foreign Exchange
Exhibit 1.1 G10 Currencies
Exhibit 1.2 P&L from long USD10m USD/CAD spot at 0.9780
Exhibit 1.3 P&L from short USD10m USD/CAD spot at 0.9780
Exhibit 1.4 P&L from long USD100m USD/JPY spot at 101.00
Exhibit 1.5 Sample G10 spot rates
Exhibit 1.6 Selected G10 Currency Pair Names
Exhibit 1.7 Exhibit Selected EM Currency Pair Names
Chapter 2: Introduction to FX Derivatives
Exhibit 2.1 P&L per apple at maturity from call option with 10p strike
Exhibit 2.2 P&L per apple at maturity from put option with 10p strike
Exhibit 2.3 Converting between CCY1 and CCY2 notionals
Exhibit 2.4 P&L at maturity from long USD10m USD/CAD call option with 0.9780 strike
Exhibit 2.5 P&L at maturity from long USD10m USD/CAD put option with 0.9780 strike
Exhibit 2.6 FX derivatives pricing tool showing a USD/JPY vanilla contract
Chapter 3: Introduction to Trading
Exhibit 3.1 Apple market order book
Exhibit 3.2 Trader A two-way price
Exhibit 3.3 Trader A two-way price in scenario 1
Exhibit 3.4 Trader A two-way price in scenario 2
Exhibit 3.5 Trader A two-way price in scenario 3
Exhibit 3.6 Market two-way price in scenario 3
Chapter 4: FX Derivatives Market Structure
Exhibit 4.1 Client requesting prices from banks
Exhibit 4.2 Interbank broker market interactions
Exhibit 4.3 Roles on an FX derivatives trading desk
Exhibit 4.4 Interbank direct call from July 23, 2014
Exhibit 4.5 Interbank broker structure
Exhibit 4.6 Trader A requesting a price from their broker
Exhibit 4.7 Brokers going to the market requesting a price on the contract
Exhibit 4.8 Brokers collecting prices from the market and reporting the best rate back to trader A
Exhibit 4.9 The interest negotiating with the trader who made the best price
Exhibit 4.10 Brokers showing the rate out to the market
Exhibit 4.11 Brokers printing the trade to the market
Chapter 5: The Black-Scholes Framework
Exhibit 5.1 Short-term forward path
Exhibit 5.2 Long-term forward path
Exhibit 5.3 USD/TRY model versus market forward path
Exhibit 5.4 Excel setup for generating a realization of a Wiener process
Exhibit 5.5 A sample realization of a Wiener process
Exhibit 5.6 Representation of the Black-Scholes framework
Exhibit 5.7 Terminal spot distribution at short tenor and low volatility
Exhibit 5.8 Terminal spot distribution at long tenor and high volatility
Exhibit 5.9 Sample Implied Volatility Term Structure
Exhibit 5.10 Realizations of a Wiener process using different implied volatility term structures
Exhibit 5.11 Valuing vanilla options using the terminal spot distribution
Exhibit 5.12 Cumulative normal distribution function
Chapter 6: Vanilla FX Derivatives Greeks
Exhibit 6.1 Value at maturity of long vanilla call option with 100.00 strike
Exhibit 6.2 Value at maturity of short vanilla call option with 100.00 strike
Exhibit 6.3 Total P&L at maturity (including initial premium) of long vanilla call option with 100.00 strike
Exhibit 6.4 Value at maturity of long vanilla put option with 100.00 strike
Exhibit 6.5 Value at maturity of short vanilla put option with 100.00 strike
Exhibit 6.6 Value of long vanilla call option with 100.00 strike and 10% volatility
Exhibit 6.7 Time value of long vanilla call option with 100.00 strike and 10% volatility
Exhibit 6.8 Value of long vanilla call option with 100.00 strike and 20% volatility
Exhibit 6.9 Delta of long vanilla call option with 100.00 strike
Exhibit 6.10 Value of long vanilla put option with 100.00 strike and 10% volatility
Exhibit 6.11 Delta of long vanilla put option with 100.00 strike
Exhibit 6.12 Value at maturity of long call option
Exhibit 6.13 Value at maturity of short forward
Exhibit 6.14 Value at maturity of long put option
Exhibit 6.15 Gamma of long vanilla option with 100.00 strike
Exhibit 6.16 Vega of long vanilla option with 100.00 strike
Chapter 7: Vanilla FX Derivatives Pricing
Exhibit 7.1 Example volatility surface
Exhibit 7.2 Example ATM run
Exhibit 7.3 Example ATM curve
Exhibit 7.4 Parallel ATM curve shift
Exhibit 7.5 Weighted ATM curve shift
Exhibit 7.6 Example volatility smile
Exhibit 7.7 Example downward-sloping volatility smile
Exhibit 7.8 Example upward-sloping volatility smile
Exhibit 7.9 Example symmetric volatility smile
Exhibit 7.10 Example extreme volatility smile
Exhibit 7.11 Changing the ATM curve to match new volatility market information
Exhibit 7.12 Existing volatility smile plus new volatility market information
Exhibit 7.14 Changing the ATM to match new volatility market information
Exhibit 7.13 Changing the wings of the volatility smile to match new volatility market information
Exhibit 7.15 Changing the skew of the volatility smile to match new volatility market information
Exhibit 7.16 Standard ATM Bid–Offer Volatility Spreads
Exhibit 7.17 Standard ATM Bid–Offer Volatility Spreads and Premium Spreads
Exhibit 7.18 Vega versus put strike delta
Exhibit 7.19 Volatility spread versus put strike delta
Exhibit 7.20 Premium spread versus put strike delta
Exhibit 7.21 Vanilla option pricing with and without delta hedge
Exhibit 7.22 Traded vanilla option contracts
Chapter 8: Vanilla FX Derivatives Structures
Exhibit 8.1 Value at maturity of long 100.00 straddle
Exhibit 8.2 Vega profile of long 100.00 straddle
Exhibit 8.3 Value at maturity of long 90.00/110.00 strangle
Exhibit 8.4 Vega profile of long 90.00/110.00 strangle
Exhibit 8.5 Value at maturity of long 90.00/100.00/110.00 equal notional butterfly
Exhibit 8.6 Vega profile of long 90.00/100.00/110.00 vega-neutral butterfly
Exhibit 8.7 Value at maturity of 90.00/110.00 risk reversal (buying topside)
Exhibit 8.8 Hedging an FX exposure with a collar (risk reversal)
Exhibit 8.9 Vega profile of a 90.00/110.00 risk reversal
Exhibit 8.10 Bucketed vega profile from long 3mth/6mth vega-neutral ATM calendar spread
Exhibit 8.11 Value at maturity of long 100.00/110.00 call spread
Exhibit 8.12 Value at maturity of long 90.00/100.00/110.00 seagull
Chapter 9: Vanilla FX Derivatives Risk Management
Exhibit 9.1 Gamma profile of long vanilla option with 100.00 strike
Exhibit 9.2 Initial EUR/USD spot ladder with long gamma exposure
Exhibit 9.3 EUR/USD spot ladder with spot higher
Exhibit 9.4 EUR/USD spot ladder after rebalancing delta
Exhibit 9.5 EUR/USD spot ladder with spot lower
Exhibit 9.6 EUR/USD spot ladder after partially rebalancing delta
Exhibit 9.7 EUR/USD spot ladder at maturity
Exhibit 9.8 Initial EUR/USD spot ladder with short gamma
Exhibit 9.9 AUD/USD spot ladder
Exhibit 9.10 AUD/USD spot ladder with delta jump highlighted
Exhibit 9.11 AUD/USD trade query
Exhibit 9.12 AUD/USD strike topography
Exhibit 9.13 AUD/USD spot ladder with P&L balance highlighted
Exhibit 9.14 AUD/USD spot ladder with better P&L balance highlighted
Exhibit 9.15 Vega profile of long vanilla option with 100.00 strike
Exhibit 9.16 EUR/USD long vega trading position
Exhibit 9.17 EUR/USD trading position with higher ATM implied volatility
Exhibit 9.18 EUR/USD trading position with 3mth vega hedge
Exhibit 9.19 AUD/USD bucketed vega profile
Chapter 10: Vanilla FX Derivatives Miscellaneous Topics
Exhibit 10.1 Timeline of the four key dates within market tenor calculations
Exhibit 10.2 Formulas for converting options premiums
Chapter 11: ATM Curve Construction
Exhibit 11.1 ATM curve A defined at market tenors
Exhibit 11.2 ATM curve A generated using linear volatility interpolation
Exhibit 11.3 Variance profile for ATM curve A generated using linear volatility interpolation
Exhibit 11.4 ATM curve B defined at market tenors
Exhibit 11.5 Variance profile for ATM curve B generated using linear volatility interpolation
Exhibit 11.6 Daily variance profile for ATM curve B generated using linear volatility interpolation
Exhibit 11.7 Variance profile for ATM curve B using linear variance interpolation
Exhibit 11.8 ATM curve B generated using linear variance interpolation
Exhibit 11.9 ATM curve A generated using linear variance interpolation
Exhibit 11.10 Daily variance profile for ATM curve A generated using linear variance interpolation
Exhibit 11.11 Function used within a simple ATM curve model
Exhibit 11.12 ATM curve output at market tenors
Exhibit 11.13 Short-date variance examples framework
Exhibit 11.14 Impact of forward drift on option prices
Exhibit 11.15 Monday to Friday ATM saw-toothing
Exhibit 11.16 Stylized intraday hourly realized variance
Exhibit 11.17 USD/JPY spot over Non-Farm Payroll data release from May 2013
Exhibit 11.18 Stylized intraday hourly realized variance on Non-Farm Payroll day
Exhibit 11.19 Average daily spot variance for G10 pairs in 2012
Exhibit 11.20 Pricing a same-day vanilla option
Chapter 12: Volatility Smile Market Instruments and Exposures
Exhibit 12.1 Example EUR/USD market instruments at market tenors
Exhibit 12.2 Deltas quoted within the volatility smile
Exhibit 12.3 25 delta market instruments within the volatility smile
Exhibit 12.4 Volatility smile with zero risk reversal and zero butterfly
Exhibit 12.5 Volatility smile with zero risk reversal and positive butterfly
Exhibit 12.6 Volatility smile with positive risk reversal
Exhibit 12.7 Volatility smile with negative risk reversal
Exhibit 12.8 Vega profile from long ATM at different implied volatility levels
Exhibit 12.9 Vanna profile from long ATM
Exhibit 12.10 Volga profile from long ATM
Exhibit 12.11 Vega profile from risk reversal (buying topside) at different implied volatility levels
Exhibit 12.12 Vega versus spot profile from risk reversal (buying downside)
Exhibit 12.13 Vega versus log spot profile from risk reversal (buying downside)
Exhibit 12.14 Vanna profile from risk reversal (buying topside)
Exhibit 12.15 Vega profile from long butterfly at different implied volatility levels
Exhibit 12.16 Volga profile from long butterfly
Exhibit 12.17 Vega exposures from market instruments
Exhibit 12.18 AUD/JPY volatility smile
Exhibit 12.19 1yr interest rate differential versus 1yr 25d risk reversal scatter plot
Exhibit 12.20 USD/JPY 1yr 25d risk reversals from May 2002 to November 2012
Exhibit 12.21 25d risk reversal vega profile versus 10d risk reversal vega profile
Exhibit 12.22 25d and 10d risk reversals on the volatility smile
Exhibit 12.23 AUD/USD 1yr outright strike vega and 1yr long risk reversal vanna
Exhibit 12.24 AUD/USD 1yr risk reversal multipliers
Exhibit 12.25 Broker fly strike placement
Exhibit 12.26 Pricing tool showing broker fly premiums
Exhibit 12.27 AUD/JPY Volatility Surface Instruments
Exhibit 12.28 25d butterfly vega versus 10d butterfly vega profiles
Exhibit 12.29 25d butterfly volga versus 10d butterfly volga profiles
Chapter 13: Probability Density Functions
Exhibit 13.1 Volatility smile with flat 10% implied volatility
Exhibit 13.2 Probability density function from flat 10% volatility smile
Exhibit 13.3 Volatility smiles with flat 10% and flat 15% implied volatility
Exhibit 13.4 Probability density functions from flat 10% and flat 15% volatility smiles
Exhibit 13.5 Volatility smiles with flat 10% and positive wing implied volatility
Exhibit 13.6 Probability density functions from flat 10% and positive wing volatility smiles
Exhibit 13.7 Volatility smiles with flat 10% and negative skew implied volatility
Exhibit 13.8 Probability density functions from flat 10% and negative skew volatility smiles
Exhibit 13.9 USD/JPY realized versus theoretical log daily change distribution (2003 to 2013)
Exhibit 13.10 USD/JPY realized versus theoretical log daily change distribution (2003 to 2013/log scale)
Exhibit 13.11 Three-state volatility model pdfs
Exhibit 13.12 Three-state volatility model average pdf
Exhibit 13.13 GBP/USD confidence intervals
Exhibit 13.14 EUR/CHF spot in 2011 and 2012
Exhibit 13.15 EUR/CHF 1mth and 1yr Market Instruments at June 1, 2011
Exhibit 13.16 EUR/CHF 1yr implied volatility smile and pdf at June 1, 2011
Exhibit 13.17 EUR/CHF 1yr intuitive pdf
Exhibit 13.18 EUR/CHF 1mth and 1yr Market Instruments at June 1, 2012
Exhibit 13.19 EUR/CHF 1yr actual and intuitive pdfs at June 1, 2012
Chapter 14: Vanilla FX Derivatives Trading Exposures
Exhibit 14.1 Long vanilla call delta (premium paid in CCY2) with 1.0000 strike
Exhibit 14.2 Long vanilla call option premium with 1.0000 strike
Exhibit 14.3 Long vanilla call option delta (premium paid in CCY1 or CCY2) with 1.0000 strike
Exhibit 14.4 Long vanilla put option delta (premium paid in CCY1 or CCY2) with 1.0000 strike
Exhibit 14.5 Vanilla call option delta with 100.00 strike over time
Exhibit 14.6 Long vanilla option with 100.00 strike theta over time
Exhibit 14.7 ATM curve roll
Exhibit 14.8 Stylized P&L/theta distribution from long gamma position
Exhibit 14.9 Stylized P&L/theta distribution from short gamma position
Exhibit 14.10 AUD/JPY ATM vega
Exhibit 14.11 Weighted Vega Multipliers (1mth reference)
Exhibit 14.12 Adapted delta finite difference calculation shown on volatility smile
Exhibit 14.13 Symmetric volatility smile
Exhibit 14.14 Adapted gamma finite difference calculation for a symmetric volatility smile
Exhibit 14.15 Adapted gamma versus Black-Scholes gamma in a symmetric volatility smile
Exhibit 14.16 Maximum curvature points for symmetric and non-symmetric volatility smiles
Exhibit 14.17 Adapted gamma versus Black-Scholes gamma profiles for a non-symmetric volatility smile
Exhibit 14.18 Gamma adaption effect profile for a non-symmetric volatility smile
Exhibit 14.19 Volatility smile adjustment at higher ATM volatility
Exhibit 14.20 Spot ladder from selling USD/JPY 1yr risk reversal (Black-Scholes exposures)
Exhibit 14.21 Spot ladder from selling USD/JPY 1yr risk reversal (adapted exposures)
Exhibit 14.22 AUD/JPY 1yr volatility smile
Exhibit 14.23 AUD/JPY 1yr zeta profile
Exhibit 14.24 Volatility smile with positive wings and no skew
Exhibit 14.25 Zeta profile for volatility smile with positive wings and no skew
Exhibit 14.26 Long 1yr call option with 1.3100 strike rho
Exhibit 14.27 Long 1yr call option with 1.3100 strike forward delta
Exhibit 14.28 Long 1yr call option with 1.3100 strike rho over time
Exhibit 14.29 Long 1yr put option with 1.3100 strike rho
Exhibit 14.30 Long 1yr ATM straddle with 1.3100 strike rho
Exhibit 14.31 Long 1yr ATM straddle with 1.3100 strike rho at different implied volatility levels
Chapter 15: Vanilla FX Derivatives Trading Topics
Exhibit 15.1 USD/KRW spot and implied volatility from 2006 to 2009
Exhibit 15.2 P&L from buying topside call option
Exhibit 15.3 Inception trading position from buying O/N topside call option
Exhibit 15.4 Trading position from long topside call option on expiry date
Exhibit 15.5 Inception trading position from selling topside call option and buying spot
Exhibit 15.6 Trading position from short topside call option and long delta on expiry date
Exhibit 15.7 P&L profiles from different trading strategies
Exhibit 15.8 Trading position from buying 1yr ATM and selling a gamma-neutral amount of 3mth ATM
Exhibit 15.9 USD/TRY volatility smile pre– and post–spot jump
Exhibit 15.10 Trading position from long ATM and short wings
Exhibit 15.11 Trading position containing short downside wing strike
Exhibit 15.12 Trading position containing short downside wing strike and a long strike in front
Exhibit 15.13 Pricing tool showing low delta 1wk EUR/USD vanilla option
Chapter 16: ATM Volatility and Correlation
Exhibit 16.1 Cosine rule triangle
Exhibit 16.2 ATM implied volatility triangle
Exhibit 16.3 Realized AUD/USD versus USD/CAD historical spot correlation
Exhibit 16.4 ATM implied volatility with zero correlation
Exhibit 16.5 Correlation and inverse cosine function
Exhibit 16.6 ATM implied volatility triangle with correlation above zero
Exhibit 16.7 ATM implied volatility triangle with 100% correlation
Exhibit 16.8 ATM implied volatility triangle with correlation below zero
Exhibit 16.9 ATM implied volatility triangle with –100% correlation
Exhibit 16.10 Cross-currency ATM volatility versus correlation profile
Exhibit 16.11 ATM implied volatility triangle
Exhibit 16.12 1yr ATM implied volatility triangle containing EUR, USD, and CNH
Exhibit 16.13 Dephased vegas from a 1yr EUR/CNH vega exposure
Chapter 17: FX Derivatives Market Analysis
Exhibit 17.1 Call option breakeven
Exhibit 17.2 Call option breakevens
Exhibit 17.3 ATM straddle option breakeven
Exhibit 17.4 Realized volatility versus implied volatility
Exhibit 17.5 Realized volatility versus implied volatility in early January
Exhibit 17.6 High-frequency AUD/USD spot trades
Exhibit 17.7 AUD/USD daily spot samples
Exhibit 17.8 AUD/USD daily spot log returns
Exhibit 17.9 AUD/USD 6mth realized spot volatility
Exhibit 17.10 1mth realized volatility with an extreme spot jump
Exhibit 17.11 EWMA weights
Exhibit 17.12 0.97 EWMA realized volatility with an extreme spot jump
Exhibit 17.13 AUD/USD daily spot and 5yr forward outright samples
Exhibit 17.14 AUD/USD spot and 5yr forward realized volatility with 6mth rolling calculation window
Exhibit 17.15 Spot and 5yr forward framework
Exhibit 17.16 Spot and 5yr forward framework with a higher forward
Exhibit 17.17 Spot and 5yr forward framework with a lower forward
Exhibit 17.18 AUD/USD versus USD/JPY spot correlation
Exhibit 17.19 AUD versus USD 5yr daily deposit rate samples
Exhibit 17.20 AUD/USD spot versus interest rate correlation
Exhibit 17.21 AUD and USD realized interest rate volatility
Exhibit 17.22 ATM volatility triangle for trading correlation
Exhibit 17.23 Simulation P&L from gamma hedging
Exhibit 17.24 AUD/USD 1yr implied volatility cone on Dec. 2013
Exhibit 17.25 1yr ATM implied volatility in various currency pairs during the 2008 financial crisis
Exhibit 17.26 Historic USD/JPY 1yr 25d risk reversals
Exhibit 17.27 Time series of EUR/USD 2M risk reversal versus 30-day log spot/log implied volatility changes covariance
Exhibit 17.28 Historic AUD/JPY market instruments
Exhibit 17.29 Historic AUD/JPY key strikes on the volatility smile
Exhibit 17.30 P&L from a long volga position
Exhibit 17.31 USD/JPY 1yr ATM implied volatility versus spot scatter plot showing daily samples from 2007 and 2008
Exhibit 17.32 AUD/USD spot from 2006 to 2009
Exhibit 17.33 Call spread carry trade
Exhibit 17.34 Put spread carry trade
Chapter 18: Exotic FX Derivatives Pricing
Exhibit 18.1 Vanilla and exotic pricing methodologies
Exhibit 18.2 One-touch option contract details
Exhibit 18.3 One-touch option market data
Exhibit 18.4 One-touch option pricing outputs
Exhibit 18.5 Long risk reversal vega profile
Exhibit 18.6 Long topside one-touch vega profile
Exhibit 18.7 Long butterfly vega profile
Exhibit 18.8 Vega profile of long DNT
Exhibit 18.9 Vega profiles of long DNT and ATM vega hedge
Exhibit 18.10 Aggregate vega profile of long DNT plus ATM vega hedge
Exhibit 18.11 AUD/USD 3mth volatility smile
Exhibit 18.12 Vega profile of long AUD/USD downside one-touch contract
Exhibit 18.13 Stopping time of a EUR/USD 1yr 1.2500 American barrier at different implied volatility levels
Exhibit 18.14 Stopping time versus no-touch TV of a 1yr 1.2500 EUR/USD American barrier
Chapter 19: FX Derivatives Pricing Models
Exhibit 19.1 FX derivatives valuation framework
Exhibit 19.2 Volatility smile from Heston model with zero vol-of-vol and correlation parameters
Exhibit 19.3 Volatility smile from Heston model with zero correlation parameter and positive vol-of-vol parameters
Exhibit 19.4 Volatility smile from Heston model with positive correlation and vol-of-vol parameters
Exhibit 19.5 Local volatility surface construction
Exhibit 19.6 USD/JPY implied volatility and local volatility smiles
Exhibit 19.7 USD/BRL implied volatility and local volatility smiles
Exhibit 19.8 Sample Merton model spot path
Chapter 20: Exotic FX Derivatives Product Classification
Exhibit 20.1 European knock-out call option payoff at maturity
Exhibit 20.2 European knock-in call option payoff at maturity
Exhibit 20.3 Knock-out option structure
Exhibit 20.4 Reverse knock-out option structure
Exhibit 20.5 Double knock-out option structure
Exhibit 20.6 ITM knock-out option structure
Exhibit 20.7 Transatlantic option structure
Chapter 21: European Digital Options
Exhibit 21.1 100.00 European digital call payoff at maturity
Exhibit 21.2 100.00 European digital call value over time
Exhibit 21.3 100.00 European digital call vanilla call spread replication
Exhibit 21.4 100.00 European digital call vanilla call spread replication with tighter strikes
Exhibit 21.5 Typical EUR/USD European Digital Bid–Offer Spreads
Exhibit 21.6 100.00 European digital call vega over time
Exhibit 21.7 European digital call TV adjustment and volatility smile in a currency pair with a downside risk reversal
Exhibit 21.8 100.00 European digital call gamma over time
Exhibit 21.9 USD1m 100.00 European digital call theta into expiry date
Exhibit 21.10 Long USD1m 95.00/105.00 European digital range payoff at maturity
Exhibit 21.11 Long USD1m 95.00/105.00 European digital range vega profile
Exhibit 21.12 Long USD1m 95.00/105.00 European digital range volga profile
Chapter 22: European Barrier Options
Exhibit 22.1 European knock-out payoff at maturity
Exhibit 22.2 European knock-in payoff at maturity
Exhibit 22.3 European knock-out barrier versus European vanilla value profiles
Exhibit 22.4 European knock-out barrier value over time
Exhibit 22.5 Long vanilla call spread payoff at maturity
Exhibit 22.6 Short European digital payoff at maturity
Exhibit 22.7 Intrinsic value in a European knock-out barrier option
Exhibit 22.8 Long vanilla call payoff at maturity
Exhibit 22.9 Long European digital payoff at maturity
Exhibit 22.10 Intrinsic value in a European knock-in barrier option
Exhibit 22.11 European knock-out barrier vega profile with strike and barrier far apart
Exhibit 22.12 European knock-out barrier vega profile with strike and barrier close together
Exhibit 22.13 European knock-in barrier versus vanilla vega profiles
Chapter 23: Touch Options
Exhibit 23.1 100.00 one-touch TV over time
Exhibit 23.2 100.00 one-touch delta over time
Exhibit 23.3 130.00 one-touch vega over time
Exhibit 23.4 120.00 vanilla vega over time
Exhibit 23.5 Vega of 130.00 one-touch hedged with 120.00 vanilla in proportions such that vega risk at 6mth tenor is minimized
Exhibit 23.6 AUD/USD 5yr 0.8000 one-touch vega
Exhibit 23.7 100.00 one-touch gamma over time
Exhibit 23.8 One-touch vega versus TV
Exhibit 23.9 One-touch vanna versus TV (topside barrier)
Exhibit 23.10 One-touch volga versus TV
Exhibit 23.11 1yr EUR/USD topside one-touch TV adjustment under various smile pricing models
Exhibit 23.12 1yr EUR/USD downside one-touch TV adjustment under various smile pricing models
Exhibit 23.13 CCY1 versus CCY2 one-touch options in pricing tool
Exhibit 23.14 Double-no-touch vega profile
Exhibit 23.15 Double-no-touch vega profile over time
Exhibit 23.16 Symmetric-barrier double-no-touch volga against theoretical value profile
Chapter 24: American Barrier Options
Exhibit 24.1 CCY1 call knock-out barrier option structure
Exhibit 24.2 Knock-out barrier and vanilla TV profiles
Exhibit 24.3 Knock-out barrier and vanilla delta profiles
Exhibit 24.4 TV of knock-out barrier option with different barrier levels
Exhibit 24.5 Vega profiles of knock-out barrier option with different barrier levels
Exhibit 24.6 Volga profiles of knock-out barrier option with different barrier levels
Exhibit 24.7 Vega profile of a vanilla option versus a knock-in barrier option with the same strike
Exhibit 24.8 Volga profile of a vanilla option verses a knock-in barrier option with the same strike
Exhibit 24.9 CCY1 put reverse knock-out structure
Exhibit 24.10 Vanilla call and reverse knock-out TV profiles
Exhibit 24.11 Long 1yr 1.3000 strike/1.5000 barrier CCY1 call reverse knock-out vega profile
Exhibit 24.12 Reverse knock-out “replication” in pricing tool
Exhibit 24.13 Long 1.3000 strike/1.5000 barrier CCY1 call reverse knock-out vega profile over time
Exhibit 24.14 Long reverse knock-in and long one-touch vega profiles
Exhibit 24.15 Reverse knock-out and equivalent one-touch option within a pricing tool
Exhibit 24.16 CCY1 call double knock-out structure
Exhibit 24.17 Knock-in/knock-out replication in pricing tool
Exhibit 24.18 AUD/USD strike-out option structure with 0% AUD rates and 0% USD rates
Exhibit 24.19 AUD/USD strike-out option with 0% AUD rates and 0% USD rates
Exhibit 24.20 AUD/USD strike-out option payoff with 5% AUD rates and 0% USD rates
Exhibit 24.21 AUD/USD strike-out option with 5% AUD rates and 0% USD rates
Exhibit 24.22 AUD/USD 0.8000 strike-out vega profile (AUD rates = 5%/USD rates = 0%)
Exhibit 24.23 AUD/USD strike-out option payoff with 0% AUD rates and 5% USD rates
Exhibit 24.24 AUD/USD strike-out option with 0% AUD rates and 5% USD rates
Exhibit 24.25 AUD/USD 0.8000 strike-out vega (AUD rates = 0%/USD rates = 5%)
Exhibit 24.26 AUD/USD 0.8000 strike-out vega over time (AUD rates = 0%/USD rates = 5%)
Chapter 25: Exotic FX Derivatives Trading Topics
Exhibit 25.1 Underlying client FX exposure
Exhibit 25.2 Forward extra payoff
Exhibit 25.3 Forward extra and client FX exposure net position
Exhibit 25.4 USD/JPY ATM volatility run
Chapter 26: Window Barrier and Discrete Barrier Options
Exhibit 26.1 Front-window double knock-out barrier structure
Exhibit 26.2 Estimating barrier risk on a front-window barrier option
Exhibit 26.3 Front-window barrier TV adjustment approximation in pricing tool
Exhibit 26.4 Front-window barrier option in pricing tool
Exhibit 26.5 Front-window barrier vega exposure
Exhibit 26.6 Rear-window up-and-out barrier structure
Exhibit 26.7 Generic window barrier structure
Chapter 27: Vanilla Variations
Exhibit 27.1 Pricing tool showing two vanilla option contracts, one with late delivery
Exhibit 27.2 Pricing tool showing late cash vanilla option
Exhibit 27.3 Pricing tool showing option on forward
Exhibit 27.4 Pricing tool showing late delivery vanilla
Exhibit 27.5 European option value versus early exercise value: zero interest rates
Exhibit 27.6 European option value versus early exercise value: rCCY1 = 0%/rCCY2 = 5%
Exhibit 27.7 European option value versus early exercise value: rCCY1 = 1%/rCCY2 = 0%
Exhibit 27.8 European option value versus early exercise value: rCCY1 = 5%/rCCY2 = 0%
Exhibit 27.9 1yr 1.3650 EUR call/USD put American vanilla versus European vanilla price
Exhibit 27.10 1yr 1.3650 EUR call/USD put American vanilla versus European vanilla delta
Exhibit 27.11 1yr 1.3650 EUR call/USD put American vanilla versus European vanilla gamma
Exhibit 27.12 1yr 1.3650 EUR call/USD put American vanilla versus European vanilla vega
Exhibit 27.13 1yr 1.3650 EUR call/USD put American vanilla versus European vanilla volga
Exhibit 27.14 1yr 1.3650 EUR call/USD put American vanilla versus European vanilla vanna
Exhibit 27.15 European vanilla versus self-quanto call option payoff at maturity
Exhibit 27.16 European vanilla to self-quanto call option adjustment discrete replication
Exhibit 27.17 European vanilla versus self-quanto call option vega profile
Exhibit 27.18 European vanilla versus self-quanto put option payoff at maturity
Exhibit 27.19 European vanilla versus self-quanto put option vega profile
Chapter 28: Accrual and Target Redemption Options
Exhibit 28.1 European range accrual structure
Exhibit 28.2 European range accrual vega profile
Exhibit 28.3 American keep range accrual vega profile
Exhibit 28.4 American keep double accrual forward structure
Exhibit 28.5 Double accrual forward with American keep barriers vega (trading desk perspective)
Exhibit 28.6 Target redemption vega exposures
Exhibit 28.7 Target redemption versus equivalent vanilla structure vega profiles
Exhibit 28.8 Target redemption vega profiles for different targets
Exhibit 28.9 Target redemption spot ladder
Chapter 29: Asian Options
Exhibit 29.1 Example single average fixing schedule A
Exhibit 29.2 Example single average fixing schedule B
Exhibit 29.3 EUR/GBP realized spot, fixings, and the cumulative average
Exhibit 29.4 EUR/GBP average rate option versus vanilla option price
Exhibit 29.5 EUR/GBP average rate option in pricing tool
Exhibit 29.6 EUR/GBP average rate option versus equivalent vanilla vega profile
Exhibit 29.7 EUR/GBP average rate option versus equivalent vanilla bucketed vega profile
Exhibit 29.8 EUR/GBP average rate option versus equivalent vanilla gamma profile
Exhibit 29.9 EUR/GBP average rate option versus equivalent vanilla vega profile after six monthly fixings
Exhibit 29.10 EUR/GBP average rate option versus equivalent vanilla gamma profile after six monthly fixings
Exhibit 29.11 AUD/USD average rate option forward drift
Exhibit 29.12 EUR/GBP average strike option vega profile as fixings occur
Exhibit 29.13 Example double average fixing schedule
Exhibit 29.14 Double average rate option bucketed vega profile
Chapter 30: Multi-Asset Options
Exhibit 30.1 Realized component spot and basket spot returns
Exhibit 30.2 Dual digital TV versus correlation
Exhibit 30.3 Dual digital TV versus cross volatility
Exhibit 30.4 Dual digital EUR/USD vega for different GBP/USD digital levels
Exhibit 30.5 Dual digital GBP/USD vega profile
Exhibit 30.6 TV versus correlation profiles for best-of and worst-of options: Same payoff direction on common currency
Exhibit 30.7 TV versus cross-volatility profiles for best-of and worst-of options: Same payoff direction on common currency
Exhibit 30.8 TV versus correlation profiles for best-of and worst-of options: Different payoff direction on common currency
Exhibit 30.9 TV versus cross-volatility profiles for best-of and worst-of options: Different payoff direction on common currency
Chapter 31: Miscellaneous Options
Exhibit 31.1 Volatility swap and variance swap payoffs
Exhibit 31.2 Volatility swap value prior to any fixings
Exhibit 31.3 Volatility swap value after fixing
Exhibit 31.4 Volatility swap delta after fixing
Exhibit 31.5 Volatility swap gamma after fixing
Exhibit 31.6 Volatility swap vega over time
Exhibit 31.7 Volatility swap vega at different levels of implied volatility
Exhibit 31.8 Variance swap value after fixing
Exhibit 31.9 Variance swap delta after fixing
Exhibit 31.10 Variance swap vega over time
Exhibit 31.11 Variance swap vega at different levels of implied volatility
Exhibit 31.12 Log Contract Vanilla Replication
Exhibit 31.13 FVA dates structure
Exhibit 31.14 FVA vega profile change at first fixing
Exhibit 31.15 Forward start option dates structure
Exhibit 31.16 Forward start vega profile
Exhibit 31.17 Compound option dates structure
Exhibit 31.18 Compound option vega profile
Copyright © 2015 by Giles Jewitt. All rights reserved.
Published by John Wiley & Sons, Inc., Hoboken, New Jersey.
Published simultaneously in Canada.
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For my wife and daughters: Laura, Rosie, and Emily.
Preface
Acknowledgments
Thanks are due to
…those who taught me:
Fred Boillereau
Jeff Wang
Mike Killen
Rob Ross
Hossein Zaimi
…those who supported me in writing the book and helped make it happen:
Howard Savery
Caroline Prior
Vincent Craignou
Selene Chong
…all colleagues who helped me with content, especially:
Chris Potter
Charlie Chamberlain
Daniela Asikian
Allen Li
…Marouane Benchekroun and his Quants for the tools I used to produce most of the charts in the book.
…those who looked after my girls and I while the book was completed:
Frances, Tony, Phil, Gerry, Tim, Jod and Mark.
…my family for their support, encouragement and assistance: Mum, Dad, and Anna.
This publication reflects the views of the author only.
This publication is intended to be educational in nature and should be used for information purposes only.
Any opinions expressed herein are given in good faith, but are subject to change without notice.
Any strategies discussed are strictly for illustrative and educational purposes only and are not to be construed as an endorsement, recommendation, or solicitation to buy or sell any financial securities.
All rates and figures used are for illustrative purposes only and do not reflect current market rates.
Part IThe Basics
Chapter 1
Chapter 2Introduction to FX Derivatives
The FX market can be split into three main product areas with increasing complexity:
Spot: guaranteed currency exchange occurring on the spot date.
Swaps / Forwards: guaranteed currency exchange(s) occurring on a specified date(s) in the future.
Derivatives
: contracts whose value is
derived
in some way from a reference FX rate (most often spot). This can be done in many different ways, but the most common FX derivative contracts are
vanilla call options
and
vanilla put options
, which are a
conditional
currency exchange occurring on a specified date in the future.
Vanilla Call and Put Options
Vanilla FX call option contracts give the right-to-buy spot on a specific date in the future while vanilla FX put option contracts give the right-to-sell spot on a specific date in the future. The term vanilla is used because calls and puts are the standard contract in FX derivatives. The vast majority of derivative transactions executed by an FX derivatives trading desk are vanilla contracts as opposed to exotic contracts. Exotic FX derivatives (covered in Part IV) have additional features (e.g., more complex payoffs, barriers, averages).
To understand how call and put options work, forget FX for the moment and think about buying and selling apples (not Apple Inc. stock, but literally the green round things you eat). Apples currently cost 10p each. I know that I will need to buy 100 apples in one month's time. If I simply wait one month and then buy the apples, perhaps the prevailing price will be 5p and hence I can buy the apples cheaper than they currently are or perhaps the price will be 15p and hence more expensive or perhaps they will cost 10p, 1p, or 999p. The point is that there is uncertainty about how much the apples will cost and this uncertainty makes planning for the future of my fledgling apple juice company more difficult. Call and put options allow this uncertainty to be controlled.
One possible contract that could be purchased to control the risk is a one-month (1mth) call option with a strike of 10p and a notional of 100 apples. Note the different elements within the contract: the date in the future at which I want to complete the transaction (maturity: one month), the direction (I want to buy apples; therefore, I purchase a call option), the level at which I want to transact (strike: 10p) and the amount I want to transact (notional: 100 apples). After buying this call option, one month hence, at the maturity of the contract, if the price of apples is above the strike (e.g., at 15p) I will exercise the call option I bought and buy 100 apples at 10p from the seller (also known as the writer) of the option contract. Alternatively, if the price of apples is below the strike (e.g., at 5p), I don't want or need to use my right to buy them at 10p; hence the call option contract expires. Instead I will buy 100 apples directly in the market at the lower rate.
Therefore, by buying the call option, the worst-case purchasing rate is known; under no circumstances will I need to buy 100 apples in one month at a rate higher than 10p (the strike). This reduction in uncertainty comes at a cost: the premium paid upfront to purchase the call option. It is not hard to imagine that the premium of the call option will depend on the details of the contract: How long it lasts, how many apples it covers, the transaction level, plus crucially the volatility of the price of apples will be a key factor. The more volatile the price of apples, the more the call option will cost.
Exhibit 2.1 shows the P&L profile from this call option at maturity, presented in familiar hockey-stick diagram terms but without the initial premium included.
Exhibit 2.1 P&L per apple at maturity from call option with 10p strike
At the option maturity, if the price of apples is below the strike (10p), the call option has no value because the underlying can be bought cheaper in the market. If the price of apples is above the strike at maturity, the call option value rises linearly with the value of the underlying.
Mathematically, the P&L at maturity from this call option is:
where is expressed in terms of number of apples, is the price of apples at the option maturity, and is the strike. Often is written .
It is worth noting that the P&L at maturity from the contract depends only on the price of apples at the moment the option contract matures; the path taken to get there is irrelevant.
Put options are the right-to-sell the underlying. This can be conceptually tricky to grasp at first—buying the right to sell. Imagine you own a forest of apple trees. You know that by the end of August you will harvest at least 1,000 apples, which you will then want to sell. Again, uncertainty arises from the fact that the future price of apples is unknown. To control this uncertainty, a put option maturing on August 31 could be bought with a notional of 1,000 apples and a strike of 10p.
This time, at the option maturity, if the price of apples is below the strike (e.g., at 5p), the put option will be exercised and 1,000 apples will be sold at 10p to the option seller. Alternatively, if the price of apples is above the strike (e.g., at 15p), the put option will expire
Chapter 3Introduction to Trading
Fundamentally,
Practical A
Building a Trading Simulator in Excel
This practical demonstrates how simple financial markets work and illustrates the differences between price-taking and price-making roles. Task A sets up a ticking (moving) midmarket price. Task B then introduces a two-way price (bid and offer) around the midmarket and price-taker controls whereby the trader can pay or give the market. Finally, Task C adds the ability for the trader to act as both price taker and price maker. This practical links closely to the material discussed in Chapter 3.
Task A: Set Up a Ticking Market Price
The trading simulator has one main VBA subroutine that updates the market price. The Application.OnTime command is used to pause between market ticks.
Step 1: Set Up a Ticking Midmarket Spot
Setting up the framework mainly requires VBA development. User inputs on the sheet are initial spot, time between ticks, and how much spot increments up or down at each tick. Outputs are the current time step and current spot. Control buttons for Go/Pause and Stop are also required:
The
Chapter 4FX Derivatives Market Structure
Market structure is a topic that is often skipped over. In practice though, it is vitally important because it defines how clients interact with the trading desk and how the trading desk accesses liquidity to hedge their risk.
In some financial markets all participants access a centralized market or exchange anonymously on the same terms. The FX derivatives market, however, is an over-the-counter (OTC) market, meaning that there is no centralized exchange and a clear distinction exists between banks and their clients. Note that “banks” here refers to large international banks with FX derivatives trading desks.
Fundamentally, bank FX derivatives trading desks transact with clients, aggregate and offset the risk where possible, and close out unwanted residual risk. More specifically:
Clients come to bank trading desks for prices, often via a sales desk within the bank. Usually the client simultaneously submits the same price request to multiple banks and deals on the best price as per
Exhibit 4.1
. Traders usually make two-way prices for clients because they do not know for certain whether a client is a buyer or a seller of a particular contract.
Bank trading desks transact with each other either via the
interbank broker market
or the
direct market
(a price request directly between a trader at one bank and the corresponding trader at another bank). The majority of bank-to-bank transactions occur in the interbank broker market. This structure is shown in
Exhibit 4.2
.
Exhibit 4.1 Client requesting prices from banks
Exhibit 4.2 Interbank broker market interactions
Client Types
Chapter 5The Black-Scholes Framework
Derivatives products have been traded in one form or another for centuries, but the development of the Black-Scholes model in the 1970s enabled financial derivatives markets to flourish by enabling volatility to be consistently priced.
Financial mathematics books generally give the derivation of the Black-Scholes formula and list the reasons why the assumptions underpinning it aren't correct in practice. Traders don't need to know how to derive the Black-Scholes formula from scratch. However, it is vital that they understand the features of the Black-Scholes framework since it is the foundation for all derivatives valuation.
Black-Scholes Stochastic Differential Equation (SDE)
The Black-Scholes framework assumes that the price of the underlying (i.e., the FX spot rate) follows a geometric Brownian motion. The Black-Scholes stochastic differential equation (SDE) is:
where is the price of the underlying (spot) at time , is the change in underlying at time , and are continuously compounded (see Chapter 10) CCY1 and CCY2 interest rates respectively, is the volatility of the underlying's returns, generally just called “volatility,” and is a Brownian motion. Sometimes, is called the foreign interest rate and the domestic interest rate because, as seen in Chapters 1 and 2, P&L on standard FX contracts is naturally generated in CCY2 terms.
The left-hand side of the SDE represents relative changes in the underlying (often called “returns”). Relative changes are used within the model because as the underlying gets smaller (closer to zero), changes get smaller in absolute terms. Therefore, spot in the model can never hit zero, as in real life for FX (note that an equity underlying could go to zero).
The right-hand side of the SDE has two parts:
Drift
from the interest rate differential
Uncertainty
from the volatility of the underlying
Drift
Drift is a predictable, deterministic component that depends on the interest rate differential and the time passed:
The drift gives the no-arbitrage expected future value of spot (i.e., the forward). Forward rates for different maturities in the future define the forward path.
If (i.e., no volatility), then:
which is solved by:
Plus recall from Chapter 1 that:
where is the forward to time and is current spot plus note the outrageous variable change from S (spot) to F (forward).
This is important: Zero volatility does not mean that spot is static; it means that spot perfectly follows the forward path.
Under Black-Scholes assumptions, the forward path is based on current spot and constant interest rates:
If CCY1 and CCY2 interest rates are equal, the forward path will equal spot.
If CCY2 interest rates are higher than CCY1 interest rates, the forward path moves higher as
increases. This is called
positive drift
.
If CCY1 interest rates are higher than CCY2 interest rates, the forward path moves lower as
increases. This is called
negative drift
.
Within this simplified framework, at a given maturity, either the forward plus one interest rate can be used to calculate the other interest rate or two interest rates can be used to calculate the forward. All issues regarding credit risk and basis risk are ignored within this analysis.
For example: = 0% and = 10%. CCY2 interest rates are higher than CCY1 interest rates and therefore there is positive drift. At shorter time-scales the forward path looks linear as shown in Exhibit 5.1.
Exhibit 5.1 Short-term forward path
Pushing the maturity out to ten years, the exponential nature of the function reveals itself in Exhibit 5.2.
Exhibit 5.2 Long-term forward path
