Why Does the Mathematics of
Probability Have
Applicability and
Explanatory/Predictive Power for
Financial Derivatives Markets?
IPPP School on
Physics and Techniques of Event Generators
Durham, United Kingdom
18-20 April 2007
Bryan Lynn
Managing Director
Merrill Lynch
Global Markets and Investment Banking
1
Summary:
 Comparison of Applicability, Explanatory
Power and Predictive Power of
Mathematics of Probability
 Statistical Physics (Natural Law)
 Gambling, Financial Derivatives Markets
(Binding Contracts)
 Criminology (Sociology/Psychology)
 Finance Industry Standard Practice:
 Hedging of Derivatives Risk
 Implied Volatility
 Conclusions
Post Script:
 Finance Industry Standard Practice:
 Diversification of Cash Portfolio Risk
 Value at risk (VaR)
 Finance Innovation: Diversification of NonLinear Derivatives Risk
 Finance Industry Standard (Quadratic
Approximation)
 An Unsolved Monte Carlo Financial
Mathematical problem
2
Statistical Physics
 Applicability and Explanatory/Predictive
Power of Probability
 Conservation of Probability Current  Heat
Equation
 Brownian motion in non-relativistic gases
 Normal distribution emerges from
conservation of Energy-Momentum
during molecular collisions
 Probability predicts relationships among
thermodynamic (statistically averaged)
quantities
3
Gambling (Dice or Roulette)
 Probably (historically) invented mathematics
of probability
 Can analyze transparent complete-information
games:
 Applicability and explanatory/predictive
power comes from ultra-strong constraints
in gaming rules and consequent financial
agreements.
 Best Strategy: use probability to drive all
betting
4
Gambling (Poker)
 Non-transparent incomplete-information
games:
 Still significant explanatory/predictive
power because of weaker constraints due
to agreements/rules
 Probability still compellingly useful for
early (and most later) betting
 But human behaviour (e.g. bluffing, mindreading, body language) is very important
and the best hand does not always win
 If you don’t know the odds you will
certainly lose, but you must have human
skill in order to win
 “The race is not always to the swift, nor is the
victory to the strong. But that’s the way to
bet!”…Sergeant Bilko (~1960)
5
Criminology (Sociology/Psychology)
 Predictions of recidivism for individual
criminals chosen from among a large set of
prisoners (e.g. murderers) up for parole.
 Could use fancy mathematics (e.g.
probability), but this approach is known to
fail: i.e. by physics, gambling and
financial derivatives valuation/risk
standards
 Instead, examine (and try to evaluate) the
character and behaviour of individuals
 Mathematics is not applicable and has no
explanatory or predictive power
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Financial Derivatives derivable - by selfconsistent “no-arbitrage” - from existing
Contracts/Agreements (i.e. analogous with
Dice/Roulette)
 Building blocks
 $ Bonds  $ interest rate (compounding)
discount factor
P0$T  exp(  r $T )
 Yen Bonds  Yen interest rate
(compounding) discount factor
Yen
P0Yen

exp(

r
T)
T

r $ , r Yen are annualized continuous
interest rates
$ / Yen
S
 Spot FX: t  0 the number of $ bought
for 1 Yen at t=0 (now)
 Pricing is driven by self-consistency (noarbitrage) with related contracts
 $-Yen FX Forwards
Forward T$ / Yen  S 0$ / Yen PTYen / P0$T
 Obscure remark: Neglected $-Yen basis
 1000’s on self-consistent derivatives contracts
across all asset classes and regions
 Typical pricing accuracy
~ 10 4 (comparable to best LEP physics
accuracies)
7
Market Risk (i.e. analogy with odds-based
Poker)
 Statistical fluctuations in Spot FX over
differentially small time period
 Ito process for Spot FX:
dt  0
dS t$ / Yen / S t$ / Yen  (r $  r Yen   )dt   $ / Yen dWt $ / Yen
dWt $ / Yen   (0,1) dt
 (0,1)  NormallyDi stributed
r $ , r Yen ,  $ / Yen  Cons tan ts( Simplicity )
dt  dt  dt 3 / 2
 Connection to IPPP School:
 (0,1) introduces random event generation
together with a conserved probability
current.
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Delta Hedging and Option Pricing (i.e. analogy
with odds-based Poker)
FXOption(t , St$ / Yen ;T , K $ / Yen ; r $ , r Yen , $ / Yen )
t T
DeltaHedge dFXOptionPortfolio
$ / Yen FXOption
 FXOption  St
St$ / Yen
d ( DeltaHedge dFXOptionPortfolio )
FXOption 1  2 FXOption $ / Yen 2
 dt

dSt
2
$
/
Yen
t
2 St
 O (dt 3 / 2 )
$ / Yen
~
dW
t
 Statistical Fluctuations
have
cancelled!
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Remember the Heat (Probability) Kernal (with
slightly transformed variables)
xt  ln S t$ / Yen ;
xT  ln ST$ / Yen ;
t T
Kolmogorov( Backward ) Equation :
  1 $ / Yen 2  2
1 $ / Yen 2  
$
Yen



(
r

r


)

 Kernal  0
2
2
xt 
xt
 t 2
Fok ker Planck ( KolmogorovForward ) Equation :
  1 $ / Yen 2  2
1 $ / Yen 2  
$
Yen
 
 (r  r  
)

 Kernal  0
2

T
2
2

x
xT
T 

Kernal (t  T , xt ; T , xT ; r $ , r Yen ,  $ / Yen )   ( xt  xT )
Kernal (t , xt  ; T , xT ; r $ , r Yen ,  $ / Yen )  0
Kernal (t , xt ; T , xT  ; r $ , r Yen ,  $ / Yen )  0
Kernal (t , xt ; T , xT ; r $ , r Yen ,  $ / Yen )
 
1 $ / Yen 2

$
Yen
ik
x

x

(
r

r


)(
T

t
)


t
T

 dk
2


exp  
 2
 1 2 $ / Yen 2


k

(
T

t
)
 2

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Black-Scholes-Merton (1973) European Vanilla
Option Pricing Equation = Heat (Probability)
Equation:
xt  ln S t$ / Yen ;
xT  ln ST$ / Yen
FXCallOption(t , S t$ / Yen ;T , K $ / Yen ; r $ , r Yen , $ / Yen )  Martingale
FXCallOption(t , S t$ / Yen ;T , K $ / Yen ; r $ , r Yen , $ / Yen )  Martingale
FXCallOption(T , ST$ / Yen ;T , K $ / Yen ; r $ , r Yen , $ / Yen )
 Max ( ST$ / Yen  K $ / Yen ,0)
FXCallOption(t , S t$ / Yen  0;T , K $ / Yen ; r $ , r Yen , $ / Yen )  0
FXCallOption(t , S t$ / Yen  ;T , K $ / Yen ; r $ , r Yen , $ / Yen )  S t$ / Yen
BlackScholesMertonCall (t , S t$ / Yen ;T , K $ / Yen ; r $ , r Yen , $ / Yen )
 FXCallOption(t , S t$ / Yen ;T , K $ / Yen ; r $ , r Yen , $ / Yen )

  dxT Kernal (t , xt ;T , xT ; r $ , r Yen , $ / Yen )

 Max ( ST$ / Yen  K $ / Yen ,0)
 Vanilla Options with European Exercize
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Derivatives are Traded on Wall Street, City of
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London, Hong Kong, Tokyo ~ $10 Notional
 Asset Classes in all G11 Currencies
 Foreign Exchange
 Interest Rates (e.g. Swaps)
 Commodities
 Equities
 Credit Derivatives
 Mortgage-backed
 Asset Backed (e.g. Automobiles)
 All Asset Classes in Emerging Markets and
Local Currencies
 Derivative Classes
 Exchange-traded Vanilla
 Over-the-counter (OTC) Vanilla
 OTC Vanilla Exotics (e.g. Barriers)
 OTC Exotics
 Exercize Style
 American
 Bermudan
 European
 Asian
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Why does probability mathematics have
explanatory and predictive power? Financial
Contracts and Derivatives Agreements (i.e.
analogy with odds-based Poker)
 Financial Derivatives Markets
 Financial contracts are (very) binding
agreements
 Industry-wide agreements:
 No-arbitrage self-consistency: certain
new contracts are super-positions of
existing contracts
 Industry-wide agreement to use
probability to drive a large part of
(vanilla and exotic) derivatives
pricing
 Black Scholes pricing of vanilla
European options
 Price
 Risk (Greeks)
 Industry-wide agreement to quote
“Implied Volatility” severely constrains
possible methods of option pricing
 (Credit-based) Implied Volatility Skew:
human behaviour generates independent
markets (like poker)
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What special skills do physicists have for
Financial Derivatives?
 Partial differential equations
 Very complicated boundary conditions
specified to absorb proprietary and customer
risk
 Analytic (especially Interest Rates)
 Lattice methods
 Monte-Carlo methods
 Choice of Monte-Carlo vs. lattice integration
techniques is governed by the usual criteria for
PDEs
 Remember:
~ $1013 Notional
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Implied Volatility (i.e. analogy with human skill
in Poker): European exercize
VanillaFXC allOption (t , St$ / Yen ;T , K $ / Yen ; r $ , r Yen , $ / Yen )
 Max ( ST$ / Yen  K $ / Yen ,0)
VanillaFXP utOption(t , St$ / Yen ;T , K $ / Yen ; r $ , r Yen , $ / Yen )
 Max ( K $ / Yen  ST$ / Yen ,0)
BlackScholesMerton(t , S t$ / Yen ;T , K $ / Yen ; r $ , r Yen , $ / Yen )
is not quite the right price when 
volatility taken from historical data) is used
 Calls are less valuable than
BlackScholesMerton formula
 Puts are more valuable than
BlackScholesMerton formula
 Supply and demand
 Credit-based
$ / Yen
historical (i.e.
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Implied Volatility (i.e. analogy with human skill
in Poker)
 Price-quoting convention:
$ / Yen
BlackScholesMerton(t , S t$ / Yen ; T , K $ / Yen ; r $ , r Yen ,  Im
plied )
 with Implied Volatility

$ / Yen
Im plied
(T , K
$ / Yen
)
 This price-quoting convention
 Imposes a severe constraint (i.e. by
agreement!) on options markets
 Restores mathematical applicability,
explanatory power and predictive
power
 Allows market-making experts to
harvest $
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Conclusions
 Comparison of Applicability, Explanatory
Power and Predictive Power of
Mathematics of Probability
 Finance Industry Standard Practice:
 Hedging of Derivatives Risk
 Implied Volatility
 Physicists special skill: solve partial
differential equations
17
Post Script:
Financial Industry Standard Practice:
Diversification of Risk for Cash Portfolios
 Portfolios of Cash Secutities
 Industry Agreement: trade securities by
Markowitz’ “factor decomposition”
 Value at Risk (VaR): 1-day portfolio loss with
95% probability
 Delta Risk is “Diversified” (i.e. statistical
fluctuations  0 but smaller, due to
correlations)
 Barra, BlackRock, Wilshire, Morgan Stanley,
ML, etc. build large covariance matrices
 Factors are used to buy securities: e.g.
 Industries
 Credit Quality
 Country
 Interest Rates
 Foreign Exchange
 Mortgage pre-payment
 Factor Covariance ~ 800 X 800
6
6
 Asset Covariance ~ 10 X 10
 Large covariance matrix calculations are
possible only because
 Analytic result:
 Normal Distribution + Normal
Distribution = Normal Distribution
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Financial Industry Innovation: Diversification
of Non-linear Derivatives risk
 VaR for portfolios of cash and derivatives
 Quadratic Approximation
 Gamma Risk
 Normal Distribution + Chi-Square
Distribution = Analytic Integral
Distribution
 VaR for portfolios of cash and derivatives:
Full Revaluation
 Monte-Carlo event generation in very high
dimensions
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An Unsolved Monte Carlo Financial
Mathematical Problem
Finance Industry Innovation: Diversification of
Nonlinear Derivatives Risk
 Beyond Quadratic Approximation
 Unsolved Monte Carlo problem in
mathematical finance
 Monte Carlos in very high dimensions
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