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

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Bruno Dupire 2 Addressing Financial Risks volume underlyings products models users regions Over the past 20 years, intense development of Derivatives in terms of:

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Bruno Dupire 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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Bruno Dupire 4 Option prices for one maturity

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Bruno Dupire 5 Risk Management Client has risk exposure Buys a product from a bank to limit its risk Risk Not EnoughToo CostlyPerfect Hedge Vanilla Hedges Exotic Hedge Client transfers risk to the bank which has the technology to handle it Product fits the risk

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OUTLINE A)Theory - Risk neutral pricing - Stochastic calculus - Pricing methods B) Volatility - Definition and estimation - Volatility modeling - Volatility arbitrage

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A) THEORY

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Risk neutral pricing

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Bruno Dupire 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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Bruno Dupire 10 2 approaches Naïve expectation Replication Argument “as if” priced with other probabilities instead of

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Bruno Dupire 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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Bruno Dupire 12 Application to option pricing Risk Neutral Probability Physical Probability

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Stochastic Calculus

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Bruno Dupire 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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Bruno Dupire 15 Brownian Motion From discrete to continuous

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Bruno Dupire 16 Stochastic Differential Equations At the limit: Continuous with independent Gaussian increments SDE: drift noise a

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Bruno Dupire 17 Ito’s Dilemma Classical calculus: expand to the first order Stochastic calculus: should we expand further?

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Bruno Dupire 18 Ito’s Lemma At the limit for f(x), If

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Bruno Dupire 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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Bruno Dupire 20 P&L Break-even points Option Value Delta hedge P&L of a delta hedged option

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Bruno Dupire 21 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. drift: noise, SD:

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Pricing methods

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Bruno Dupire 23 Pricing methods Analytical formulas Trees/PDE finite difference Monte Carlo simulations

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Bruno Dupire 24 Formula via PDE The Black-Scholes PDE is Reduces to the Heat Equation With Fourier methods, Black-Scholes equation:

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Bruno Dupire 25 Formula via discounted expectation Risk neutral dynamics Ito to ln S: Integrating: Same formula

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Bruno Dupire 26 Finite difference discretization of PDE Black-Scholes PDE Partial derivatives discretized as

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Bruno Dupire 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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Bruno Dupire 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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B) VOLATILITY

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Bruno Dupire 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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Bruno Dupire 31 Historical volatility

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Bruno Dupire 32 Historical Volatility Measure of realized moves annualized SD of

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Bruno Dupire 33 Commonly available information: open, close, high, low Captures valuable volatility information Parkinson estimate: Garman-Klass estimate: Estimates based on High/Low

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Bruno Dupire 34 Move based estimation Leads to alternative historical vol estimation: = number of crossings of log-price over [0,T]

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Bruno Dupire 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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Bruno Dupire 36 Implied volatility Input of the Black-Scholes formula which makes it fit the market price :

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Bruno Dupire 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 KK

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Bruno Dupire 38 Skews Volatility Skew: slope of implied volatility as a function of Strike Link with Skewness (asymmetry) of the Risk Neutral density function ? MomentsStatisticsFinance 1ExpectationFWD price 2VarianceLevel of implied vol 3SkewnessSlope of implied vol 4KurtosisConvexity of implied vol

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Bruno Dupire 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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Bruno Dupire 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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Bruno Dupire 41 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 2 mechanisms to produce Skews (1)

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Bruno Dupire 42 2 mechanisms to produce Skews (2) –a) Negative link between prices and volatility –b) Downward jumps

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Bruno Dupire 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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Bruno Dupire 44 Strike dependency for multiple paths

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A Brief History of Volatility

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Bruno Dupire 46 Evolution Theory constant deterministic stochastic nD

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Bruno Dupire 47 A Brief History of Volatility (1) – : Bachelier 1900 – : Black-Scholes 1973 – : Merton 1973 – : Merton 1976 – : Hull&White 1987

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Bruno Dupire 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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Bruno Dupire 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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Local Volatility Model

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Bruno Dupire 51 Black-Scholes: Merton: Simplest extension consistent with smile: (S,t) is called “local volatility” The smile model

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Bruno Dupire 52 European prices Local volatilities Exotic prices From simple to complex

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Bruno Dupire 53 One Single Model We know that a model with dS = (S,t)dW would generate smiles. –Can we find (S,t) which fits market smiles? –Are there several solutions? ANSWER: One and only one way to do it.

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Bruno Dupire 54 Diffusions Risk Neutral Processes Compatible with Smile The Risk-Neutral Solution But if drift imposed (by risk-neutrality), uniqueness of the solution

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Bruno Dupire 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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Bruno Dupire 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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Extracting information

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Bruno Dupire 58 MEXBOL: Option Prices

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Bruno Dupire 59 Non parametric fit of implied vols

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Bruno Dupire 60 Risk Neutral density

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Bruno Dupire 61 S&P 500: Option Prices

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Bruno Dupire 62 Non parametric fit of implied vols

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Bruno Dupire 63 Risk Neutral densities

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Bruno Dupire 64 Implied Volatilities

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Bruno Dupire 65 Local Volatilities

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Bruno Dupire 66 Interest rates evolution

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Bruno Dupire 67 Fed Funds evolution

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Volatility Arbitrages

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Bruno Dupire 69 Volatility as an Asset Class: A Rich Playfield S C(S) VIX RV C(RV) VS Index Vanillas Variance Swap Options on Realized Variance Realized Variance Futures Implied Volatility Futures VIX options

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Outline I.Frequency arbitrage II.Fair Skew III.Dynamic arbitrage IV.Volatility derivatives

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I. Frequency arbitrage

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Bruno Dupire 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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Bruno Dupire 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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Bruno Dupire 74 Daily Vol / weekly Vol Arbitrage -On each leg: always keep $ invested in the index and update every t -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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II. Fair Skew

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Bruno Dupire 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 KK

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Bruno Dupire 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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Bruno Dupire 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. 1)Discounted average of the historical Intrinsic Values. Bad : depends on bull/bear, no call/put parity. 2)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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Bruno Dupire 79 Theoretical Skew from Prices (2) 3)Discounted average of the Intrinsic Value from recentered 3 month histogram. 4) Δ-Hedging : compute the implied volatility which makes the Δ- hedging a fair game.

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Bruno Dupire 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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Bruno Dupire 81 Theoretical Skew from historical prices (4)

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Bruno Dupire 82 Theoretical Skew from historical prices (5)

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Bruno Dupire 83 Theoretical Skew from historical prices (6)

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Bruno Dupire 84 Theoretical Skew from historical prices (7)

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Bruno Dupire 85 EUR/$, 2005

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Bruno Dupire 86 Gold, 2005

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Bruno Dupire 87 Intel, 2005

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Bruno Dupire 88 S&P500, 2005

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Bruno Dupire 89 JPY/$, 2005

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Bruno Dupire 90 S&P500, 2002

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Bruno Dupire 91 S&P500, 2003

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Bruno Dupire 92 MexBol 2007

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III. Dynamic arbitrage

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Bruno Dupire 94 Arbitraging parallel shifts Assume every day normal implied vols are flat Level vary from day to day Does it lead to arbitrage? Strikes Implied vol

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Bruno Dupire 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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Bruno Dupire 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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Bruno Dupire 97 M arbitrage K1K2 S0S0 S+S+ S-S- =(K1+K2)/2 In a sticky-delta smiley market:

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Bruno Dupire 98 S0S0 K T1T1 T2T2 t0 Deterministic future smiles It is not possible to prescribe just any future smile If deterministic, one must have Not satisfied in general

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Bruno Dupire 99 Det. Fut. smiles & no jumps => = FWD smile If 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 S t0t0 tT S0S0 K

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Bruno Dupire 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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Bruno Dupire 101 Each C K,T lives in its Black-Scholes ( ) world P&L of Delta hedge position over dt: If no jump Example of arbitrage with Sticky Strike !

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Bruno Dupire 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 Vega > Vega PF Vega < Vega SPF Δ-hedged PF gains from S induced volatility moves.

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IV. Volatility derivatives

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VIX Future Pricing

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Bruno Dupire 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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Bruno Dupire 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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Bruno Dupire 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 K i

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Bruno Dupire 108 Option prices for one maturity

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Bruno Dupire 109 Perfect Replication of We can buy today a PF which gives VIX 2 T1 at T 1 : buy T 2 options and sell T 1 options.

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Bruno Dupire 110 Theoretical Pricing of VIX Futures F VIX before launch F VIX t : price at t of receiving at T 1. The difference between both sides depends on the variance of PF (vol vol).

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Bruno Dupire 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 tT T1T1 T2T2 T3T3 T0T0 T4T4

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Bruno Dupire 112 RV/VIX Assume that RV and VIX, with prices RV and F are defined on the same future period [T 1,T 2 ] If at T 0, then buy 1 RV Futures and sell 2 F 0 VIX Futures at T 1 Ifsell 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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Bruno Dupire 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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The End

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