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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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**Addressing Financial Risks**

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

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**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 Bruno Dupire

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**Option prices for one maturity**

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**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 Bruno Dupire

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**OUTLINE 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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**Warm-up Roulette: A lottery ticket gives:**

You can buy it or sell it for $60 Is it cheap or expensive? Bruno Dupire

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**2 approaches Naïve expectation Replication Argument**

“as if” priced with other probabilities instead of Bruno Dupire

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**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 Bruno Dupire

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**Application to option pricing**

Risk Neutral Probability Physical Probability Bruno Dupire

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Stochastic Calculus

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**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 Bruno Dupire

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Brownian Motion From discrete to continuous 10 100 1000 Bruno Dupire

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**Stochastic Differential Equations**

At the limit: Continuous with independent Gaussian increments a SDE: drift noise Bruno Dupire

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**Ito’s Dilemma Classical calculus: expand to the first order**

Stochastic calculus: should we expand further? Bruno Dupire

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

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**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! Bruno Dupire

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**P&L of a delta hedged option**

Break-even points Option Value Delta hedge Bruno Dupire

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**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. Bruno Dupire

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Pricing methods

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**Pricing methods Analytical formulas Trees/PDE finite difference**

Monte Carlo simulations Bruno Dupire

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**Formula via PDE The Black-Scholes PDE is Reduces to the Heat Equation**

With Fourier methods, Black-Scholes equation: Bruno Dupire

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**Formula via discounted expectation**

Risk neutral dynamics Ito to ln S: Integrating: Same formula Bruno Dupire

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**Finite difference discretization of PDE**

Black-Scholes PDE Partial derivatives discretized as Bruno Dupire

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**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 Bruno Dupire

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**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 Bruno Dupire

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B) VOLATILITY

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**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 Bruno Dupire

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**Historical volatility**

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**Historical Volatility**

Measure of realized moves annualized SD of Bruno Dupire

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**Estimates based on High/Low**

Commonly available information: open, close, high, low Captures valuable volatility information Parkinson estimate: Garman-Klass estimate: Bruno Dupire

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**Move based estimation Leads to alternative historical vol estimation:**

= number of crossings of log-price over [0,T] Bruno Dupire

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**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. Bruno Dupire

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

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**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 Bruno Dupire

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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 Bruno Dupire

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**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 Bruno Dupire

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**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 Bruno Dupire

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**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 Bruno Dupire

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**2 mechanisms to produce Skews (2)**

a) Negative link between prices and volatility b) Downward jumps Bruno Dupire

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Strike dependency Fair or Break-Even volatility is an average of squared returns, weighted by the Gammas, which depend on the strike Bruno Dupire

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**Strike dependency for multiple paths**

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**A Brief History of Volatility**

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Evolution Theory constant deterministic stochastic nD Bruno Dupire

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**A Brief History of Volatility (1)**

: Bachelier 1900 : Black-Scholes 1973 : Merton 1973 : Merton 1976 : Hull&White 1987 Bruno Dupire

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**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 Bruno Dupire

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**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. Bruno Dupire

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**Local Volatility Model**

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**The smile model Black-Scholes: Merton:**

Simplest extension consistent with smile: s(S,t) is called “local volatility” Bruno Dupire

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**From simple to complex European prices Local volatilities**

Exotic prices Bruno Dupire

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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. Bruno Dupire

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**The Risk-Neutral Solution**

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

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**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: Bruno Dupire

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**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) Bruno Dupire

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**Extracting information**

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MEXBOL: Option Prices Bruno Dupire

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**Non parametric fit of implied vols**

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Risk Neutral density Bruno Dupire

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S&P 500: Option Prices Bruno Dupire

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**Non parametric fit of implied vols**

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**Risk Neutral densities**

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Implied Volatilities Bruno Dupire

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Local Volatilities Bruno Dupire

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**Interest rates evolution**

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Fed Funds evolution Bruno Dupire

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**Volatility Arbitrages**

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**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 Bruno Dupire

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**Frequency arbitrage Fair Skew Dynamic arbitrage Volatility derivatives**

Outline Frequency arbitrage Fair Skew Dynamic arbitrage Volatility derivatives

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I. Frequency arbitrage

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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? Bruno Dupire

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**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 : Bruno Dupire

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**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 Bruno Dupire

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II. Fair Skew

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**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 Bruno Dupire

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**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 Bruno Dupire

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**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. Bruno Dupire

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**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. Bruno Dupire

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**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 Bruno Dupire

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**Theoretical Skew from historical prices (4)**

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**Theoretical Skew from historical prices (5)**

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**Theoretical Skew from historical prices (6)**

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**Theoretical Skew from historical prices (7)**

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EUR/$, 2005 Bruno Dupire

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Gold, 2005 Bruno Dupire

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Intel, 2005 Bruno Dupire

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S&P500, 2005 Bruno Dupire

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JPY/$, 2005 Bruno Dupire

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S&P500, 2002 Bruno Dupire

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S&P500, 2003 Bruno Dupire

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MexBol 2007 Bruno Dupire

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III. Dynamic arbitrage

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**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 Bruno Dupire

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**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 Bruno Dupire

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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 Bruno Dupire

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**M arbitrage In a sticky-delta smiley market: K1 K2 S0 S+ S- =(K1+K2)/2**

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**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 Bruno Dupire

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**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 t0 t T S0 K S Bruno Dupire

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**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 Bruno Dupire

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**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 ! Bruno Dupire

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**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 Bruno Dupire

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**IV. Volatility derivatives**

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VIX Future Pricing

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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 : Bruno Dupire

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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) Bruno Dupire

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**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 Bruno Dupire

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**Option prices for one maturity**

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**Perfect Replication of**

We can buy today a PF which gives VIX2T1 at T1: buy T2 options and sell T1 options. Bruno Dupire

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**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). Bruno Dupire

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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 Bruno Dupire

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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 Bruno Dupire

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**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 Bruno Dupire

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The End

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