1. Pricing Financial Derivatives
Bruno Dupire
Bloomberg L.P/NYU
AIMS Day 1
Cape Town, February 17, 2011

2. Addressing Financial Risks
Over the past 20 years, intense development of Derivatives
in terms of:
volume
underlyings
products
models
users
regions
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3. Vanilla Options European Call:
Gives the right to buy the underlying at a fixed price (the strike) at some future time (the maturity)
European Put:
Gives the right to sell the underlying at a fixed strike at some maturity
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4. Option prices for one maturity
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5. Risk Management Client has risk exposure
Buys a product from a bank to limit its risk
Not Enough
Too Costly
Perfect Hedge
Risk
Exotic Hedge
Vanilla Hedges
Client transfers risk to the bank which has the technology to handle it
Product fits the risk
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6. OUTLINE Theory - Risk neutral pricing - Stochastic calculus
- Pricing methods
B) Volatility
- Definition and estimation
- Volatility modeling
- Volatility arbitrage

7. A) THEORY

8. Risk neutral pricing

9. Warm-up Roulette: A lottery ticket gives:
You can buy it or sell it for $60
Is it cheap or expensive?
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10. 2 approaches Naïve expectation Replication Argument
“as if” priced with other probabilities
instead of
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11. Price as discounted expectation
Option gives uncertain payoff in the future
Premium: known price today
Resolve the uncertainty by computing expectation:
Transfer future into present by discounting
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12. Application to option pricing
Risk Neutral Probability
Physical Probability
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13. Stochastic Calculus

14. Modeling Uncertainty Main ingredients for spot modeling
Many small shocks: Brownian Motion (continuous prices)
A few big shocks: Poisson process (jumps) t S t S
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15. Brownian Motion
From discrete to continuous 10
100
1000
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16. Stochastic Differential Equations
At the limit:
Continuous with independent Gaussian increments a
SDE:
drift noise
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17. Ito’s Dilemma Classical calculus: expand to the first order
Stochastic calculus:
should we expand further?
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18. Ito’s Lemma
At the limit If
for f(x),
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19. Black-Scholes PDE Black-Scholes assumption
Apply Ito’s formula to Call price C(S,t)
Hedged position is riskless, earns interest rate r
Black-Scholes PDE
No drift!
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20. P&L of a delta hedged option
Break-even
points
Option Value
Delta hedge
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21. Black-Scholes Model If instantaneous volatility is constant :
drift:
noise, SD:
Then call prices are given by :
No drift in the formula, only the interest rate r due to the hedging argument.
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22. Pricing methods

23. Pricing methods Analytical formulas Trees/PDE finite difference
Monte Carlo simulations
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24. Formula via PDE The Black-Scholes PDE is Reduces to the Heat Equation
With Fourier methods, Black-Scholes equation:
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25. Formula via discounted expectation
Risk neutral dynamics
Ito to ln S:
Integrating:
Same formula
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26. Finite difference discretization of PDE
Black-Scholes PDE
Partial derivatives discretized as
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27. Option pricing with Monte Carlo methods
An option price is the discounted expectation of its payoff:
Sometimes the expectation cannot be computed analytically:
complex product
complex dynamics
Then the integral has to be computed numerically
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28. Computing expectations basic example
You play with a biased die
You want to compute the likelihood of getting
Throw the die times
Estimate p( ) by the number of over runs
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29. B) VOLATILITY

30. Volatility : some definitions
Historical volatility :
annualized standard deviation of the logreturns; measure of uncertainty/activity
Implied volatility :
measure of the option price given by the market
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31. Historical volatility
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32. Historical Volatility
Measure of realized moves
annualized SD of
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33. Estimates based on High/Low
Commonly available information: open, close, high, low
Captures valuable volatility information
Parkinson estimate:
Garman-Klass estimate:
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34. Move based estimation Leads to alternative historical vol estimation:
= number of crossings of log-price over [0,T]
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35. Black-Scholes Model If instantaneous volatility is constant :
Then call prices are given by :
No drift in the formula, only the interest rate r due to the hedging argument.
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36. Implied volatility
Input of the Black-Scholes formula which makes it fit the market price :
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37. Market Skews Dominating fact since 1987 crash: strong negative skew on
Equity Markets
Not a general phenomenon
Gold: FX:
We focus on Equity Markets K K K
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38. Skews
Volatility Skew: slope of implied volatility as a function of Strike
Link with Skewness (asymmetry) of the Risk Neutral density function ?
Moments
Statistics
Finance 1
Expectation
FWD price 2
Variance
Level of implied vol 3
Skewness
Slope of implied vol 4
Kurtosis
Convexity of implied vol
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39. Why Volatility Skews? Market prices governed by
a) Anticipated dynamics (future behavior of volatility or jumps)
b) Supply and Demand
To “ arbitrage” European options, estimate a) to capture risk premium b)
To “arbitrage” (or correctly price) exotics, find Risk Neutral dynamics calibrated to the market K
Market Skew
Th. Skew
Supply and Demand
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40. Modeling Uncertainty Main ingredients for spot modeling
Many small shocks: Brownian Motion (continuous prices)
A few big shocks: Poisson process (jumps) t S t S
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41. 2 mechanisms to produce Skews (1)
To obtain downward sloping implied volatilities
a) Negative link between prices and volatility
Deterministic dependency (Local Volatility Model)
Or negative correlation (Stochastic volatility Model)
b) Downward jumps K
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42. 2 mechanisms to produce Skews (2)
a) Negative link between prices and volatility
b) Downward jumps
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43. Strike dependency
Fair or Break-Even volatility is an average of squared returns, weighted by the Gammas, which depend on the strike
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44. Strike dependency for multiple paths
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45. A Brief History of Volatility

46. Evolution Theory
constant deterministic stochastic nD
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47. A Brief History of Volatility (1)
: Bachelier 1900
: Black-Scholes 1973
: Merton 1973
: Merton 1976
: Hull&White 1987
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48. A Brief History of Volatility (2)
Dupire 1992, arbitrage model
which fits term structure of
volatility given by log contracts.
Dupire 1993, minimal model
to fit current volatility surface
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49. A Brief History of Volatility (3)
Heston 1993,
semi-analytical formulae.
Dupire 1996 (UTV), Derman 1997,
stochastic volatility model
which fits current volatility
surface HJM treatment.
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50. Local Volatility Model

51. The smile model Black-Scholes: Merton:
Simplest extension consistent with smile:
s(S,t) is called “local volatility”
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52. From simple to complex European prices Local volatilities
Exotic prices
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53. One Single Model
We know that a model with dS = s(S,t)dW would generate smiles.
Can we find s(S,t) which fits market smiles?
Are there several solutions?
ANSWER: One and only one way to do it.
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54. The Risk-Neutral Solution
But if drift imposed (by risk-neutrality), uniqueness of the solution
Diffusions
Risk Neutral Processes
Compatible with Smile
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55. Forward Equation BWD Equation: price of one option for different
FWD Equation: price of all options for current
Advantage of FWD equation:
If local volatilities known, fast computation of implied volatility surface,
If current implied volatility surface known, extraction of local volatilities:
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56. Summary of LVM Properties
is the initial volatility surface
compatible with local vol
compatible with
(calibrated SVM are noisy versions of LVM)
deterministic function of (S,t) (if no jumps)
future smile = FWD smile from local vol
Extracts the notion of FWD vol (Conditional Instantaneous Forward Variance)
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57. Extracting information

58. MEXBOL: Option Prices
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59. Non parametric fit of implied vols
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60. Risk Neutral density
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61. S&P 500: Option Prices
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62. Non parametric fit of implied vols
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63. Risk Neutral densities
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64. Implied Volatilities
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65. Local Volatilities
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66. Interest rates evolution
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67. Fed Funds evolution
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68. Volatility Arbitrages

69. Volatility as an Asset Class: A Rich Playfield
Index S
Implied VolatilityFutures
Vanillas
VIX
C(S)
VIX options RV
C(RV) VS
Realized VarianceFutures
Variance Swap
Options on Realized Variance
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70. Frequency arbitrage Fair Skew Dynamic arbitrage Volatility derivatives
Outline
Frequency arbitrage
Fair Skew
Dynamic arbitrage
Volatility derivatives

71. I. Frequency arbitrage

72. Frequencygram
Historical volatility tends to depend on the sampling frequency: SPX historical vols over last 5 (left) and 2 (right) years, averaged over the starting dates
Can we take advantage of this pattern?
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73. Historical Vol / Historical Vol Arbitrage
If weekly historical vol < daily historical vol :
buy strip of T options, Δ-hedge daily
sell strip of T options, Δ-hedge weekly
Adding up :
do not buy nor sell any option;
play intra-week mean reversion until T;
final P&L :
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74. Daily Vol / weekly Vol Arbitrage
On each leg: always keep $a invested in the index and update every Dt
Resulting spot strategy: follow each week a mean reverting strategy
Keep each day the following exposure:
where is the j-th day of the i-th week
It amounts to follow an intra-week mean reversion strategy
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75. II. Fair Skew

76. Market Skews Dominating fact since 1987 crash: strong negative skew on
Equity Markets
Not a general phenomenon
Gold: FX:
We focus on Equity Markets K K K
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77. Why Volatility Skews? Market prices governed by
a) Anticipated dynamics (future behavior of volatility or jumps)
b) Supply and Demand
To “ arbitrage” European options, estimate a) to capture risk premium b)
To “arbitrage” (or correctly price) exotics, find Risk Neutral dynamics calibrated to the market K
Market Skew
Th. Skew
Supply and Demand
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78. Theoretical Skew from Prices?
=>
Problem : How to compute option prices on an underlying without options?
For instance : compute 3 month 5% OTM Call from price history only.
Discounted average of the historical Intrinsic Values.
Bad : depends on bull/bear, no call/put parity.
Generate paths by sampling 1 day return recentered histogram.
Problem : CLT => converges quickly to same volatility for all strike/maturity; breaks autocorrelation and vol/spot dependency.
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79. Theoretical Skew from Prices (2)
Discounted average of the Intrinsic Value from recentered 3 month histogram.
Δ-Hedging : compute the implied volatility which makes the Δ-hedging a fair game.
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80. Theoretical Skew from historical prices (3)
How to get a theoretical Skew just from spot price history?
Example:
3 month daily data
1 strike
a) price and delta hedge for a given within Black-Scholes model
b) compute the associated final Profit & Loss:
c) solve for
d) repeat a) b) c) for general time period and average
e) repeat a) b) c) and d) to get the “theoretical Skew” t S K
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81. Theoretical Skew from historical prices (4)
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82. Theoretical Skew from historical prices (5)
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83. Theoretical Skew from historical prices (6)
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84. Theoretical Skew from historical prices (7)
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85. EUR/$, 2005
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86. Gold, 2005
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87. Intel, 2005
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88. S&P500, 2005
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89. JPY/$, 2005
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90. S&P500, 2002
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91. S&P500, 2003
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92. MexBol 2007
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93. III. Dynamic arbitrage

94. Arbitraging parallel shifts
Assume every day normal implied vols are flat
Level vary from day to day
Does it lead to arbitrage?
Implied vol
Strikes
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95. W arbitrage Buy the wings, sell the ATM
Symmetric Straddle and Strangle have no Delta and no Vanna
If same maturity, 0 Gamma => 0 Theta, 0 Vega
The W portfolio: has a free Volga and no other Greek up to the second order
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96. Cashing in on vol moves
The W Portfolio transforms all vol moves into profit
Vols should be convex in strike to prevent this kind of arbitrage
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97. M arbitrage In a sticky-delta smiley market: K1 K2 S0 S+ S- =(K1+K2)/2
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98. Deterministic future smiles
It is not possible to prescribe just any future smile
If deterministic, one must have
Not satisfied in general S0 K T1 T2 t0
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99. Det. Fut. smiles & no jumps => = FWD smileIf
stripped from implied vols at (S,t)
Then, there exists a 2 step arbitrage:
Define
At t0 : Sell
At t:
gives a premium = PLt at t, no loss at T
Conclusion: independent of
from initial smile t0 t T S0 K S
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100. Consequence of det. future smiles
Sticky Strike assumption: Each (K,T) has a fixed independent of (S,t)
Sticky Delta assumption: depends only on moneyness and residual maturity
In the absence of jumps,
Sticky Strike is arbitrageable
Sticky Δ is (even more) arbitrageable
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101. Example of arbitrage with Sticky Strike
Each CK,T lives in its Black-Scholes ( ) world
P&L of Delta hedge position over dt:
If no jump !
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102. Arbitrage with Sticky Delta
In the absence of jumps, Sticky-K is arbitrageable and Sticky-Δ even more so.
However, it seems that quiet trending market (no jumps!) are Sticky-Δ.
In trending markets, buy Calls, sell Puts and Δ-hedge.
Example: S PF
Δ-hedged PF gains from S induced volatility moves.
Vega > Vega S PF
Vega < Vega
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103. IV. Volatility derivatives

104. VIX Future Pricing

105. Vanilla Options
Simple product, but complex mix of underlying and volatility:
Call option has :
Sensitivity to S : Δ
Sensitivity to σ : Vega
These sensitivities vary through time and spot, and vol :
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106. Volatility Games
To play pure volatility games (eg bet that S&P vol goes up, no view on the S&P itself):
Need of constant sensitivity to vol;
Achieved by combining several strikes;
Ideally achieved by a log profile : (variance swaps)
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107. Log Profile Under BS: dS=σS dW, price of For all S,
The log profile is decomposed as:
In practice, finite number of strikes CBOE definition:
Put if Ki<F,
Call otherwise
FWD adjustment
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108. Option prices for one maturity
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109. Perfect Replication of
We can buy today a PF which gives VIX2T1 at T1:
buy T2 options and sell T1 options.
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110. Theoretical Pricing of VIX Futures FVIX before launch
FVIXt: price at t of receiving at T1 .
The difference between both sides depends on the variance of PF (vol vol).
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111. RV/VarS
The pay-off of an OTC Variance Swap can be replicated by a string of Realized Variance Futures:
From 12/02/04 to maturity 09/17/05, bid-ask in vol: 15.03/15.33
Spread=.30% in vol, much tighter than the typical 1% from the OTC market t T T0 T1 T2 T3 T4
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112. RV/VIX
Assume that RV and VIX, with prices RV and F are defined on the same future period [T1 ,T2]
If at T0 , then buy 1 RV Futures and sell 2 F0 VIX Futures
at T1
If sell the PF of options for and Delta hedge in S until maturity to replicate RV.
In practice, maturity differ: conduct the same approach with a string of VIX Futures
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113. Conclusion Volatility is a complex and important field
It is important to
- understand how to trade it
- see the link between products
- have the tools to read the market
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114. The End