berlin amamef 2010

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Options on Discrete Realized Variance The Convex Order Conjecture

Martin Keller-ResselETH Zurich

AMaMeF Workshop, TU Berlin

September 28, 2010

Martin Keller-Ressel Options on Variance


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Part I

Realized Variance vs. Quadratic Variation



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Realized Variance

S = S0 exp(X) . . . discounted assetT . . . time horizon in yearsP ...a partition 0 = t0 < < tn = T of [0, T] into n businessdays or other increments.

Realized Variance

The realized variance of X over the partition P is defined as

RV(X, P) =n

k=1

Xtk Xtk12

.

The annualized realized variance is given by 1T

RV(X, P).



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Options on Realized Variance (1)

There exists a considerable number of financial instruments that

are based on realized variance as an underlying

Variance Swap: Pays the difference between a fixed swap rateand the annualized realized variance. The swap rate is chosensuch that todays fair value of the swap is zero.

Volatility Swap: Instead of realized variance its square root(realized volatility) is used.

Call on realized variance: Pays the difference betweenannualized realized variance and the strike value, with a floorat zero.

Puts, Straddles, weighted var swaps, corridor var swaps, . . .

For a good overview see Buhler (2006a) or Gatheral (2006,Chapter 11)



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Options on Realized Variance (2)

If the strike of a call or put option equals the swap rate, theoption is called at-the-money (ATM). These are the most

frequently traded options.

All the mentioned instruments allow hedging of orspeculation on volatility risk, without exposure to othermarket risk factors, e.g. directional (delta) risk.



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Quality of the Approximation

The approximation via quadratic variation works well forclaims with linear payoffs, such as variance swaps, and claimswith close-to-linear payoffs like volatility swaps (cf. Buhler(2006a); Sepp (2008) resp. Broadie and Jain (2008))

However, citing Buhler (2006a):

while the approximation of realized variance via quadraticvariation works very well for variance swaps, it is not sufficient

for non-linear payoffs with short maturities. The effect is

common to all variance curve models (or stochastic volatilitymodels, for that matter).



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Buhlers Examples

Plot from Buhler (2006b).



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Gatherals Example

Plot from Gatheral (2008). Shows prices of ATM Variance

Straddles in a Double CEV (diffusion) model.Martin Keller-Ressel Options on Variance


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Own Example

10 20 30 40 50Days

0.02

0.04

0.06

0.08

Price

QV

RV

Plot from Keller-Ressel and Muhle-Karbe (2010). Shows ATM call

prices in Kous jump-diffusion model.Martin Keller-Ressel Options on Variance


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Empirical Observations

Some observations:

1 Prices of contracts on RV and QV are significantly differentfor short ( 60 days) maturities.1

2 Contracts on RV are more expensive then contracts on QV

3 For short maturities the term structure is decreasing for RV.In diffusion models it is increasing for QV

4 Prices of ATM contracts on QV go to zero for T 0 indiffusion models; prices of contracts on RV do not.

Features seem to be. . .

parameter-independent,

largely model-independent (only diffusion vs. jumps matters)

possibly even payoff independent (for convex payoffs)

1At least if a diffusive component is presentMartin Keller-Ressel Options on Variance


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Goal: Making things rigorous

General Objective: Making these observations mathematicallyrigorous (providing proofs)

Observations 1, 3, 4 have been made rigorous in an asymptoticsense (T 0) in

Keller-Ressel, M. and Muhle-Karbe, J. (2010). Asymptotics and exactpricing of options on variance. arXiv:1003.5514.

In this talk, focus on Observation 2: Are contracts on RV moreexpensive then contracts on QV?



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The convex order conjecture (1)

Convex Order conjecture

The price of a call option on realized variance is bigger than theprice of a call option on quadratic variation

We consider both

Fixed-strike call (x K)+,

(Variable-strike) ATM call (x R)+, where R is the swap rate.

Note that the swap rate is model-dependent! The second case is abit harder, but more relevant in practice.



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We believe that the conjecture is true in a large class ofmodels and for several payoffs; however more numericalevidence is needed.

The fixed-strike convex order conjecture is equivalent to

E [g([X, X]T)] E [g(RV(X, P))]

for all increasing convex g, i.e. RV dominates QV in the

convex order c.


( )


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Centering

Define the centered realized variance (RV) and the centeredquadratic variation (QV) by

RV(X, P) = RV(X, P) E [RV(X, P)] ,

[X, X]T = [X, X]T E [[X, X]T] .

For the variable-strike convex order conjecture it is sufficientthat

E

g

[X, X]T

E

g

RV(X, P)

for all (even non-increasing) convex g, i.e. RV dominates QVin convex order. Note that E

RV

= E

QV

.



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Part II

Showing the Convex Order Conjecture

Assumptions X has conditionally independent increments &satisfies a symmetry condition.

Tools Reverse (Sub)martingales.


A i h l i


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Assumptions on the log-price process

Assumption 1

X is a semi-martingale with H-conditionally independentincrements, for some H F0.

We denote by (B, C, ) the semi-martingale characteristics ofX relative to the truncation function h(x) = 1{|x|1}x.

A semi-martingale has H-cond. indep. increments if and onlyif its characteristics have a H-measurable version.

This assumption includes Levy processes, Sato processes,time-changed Levy processes and stochastic volatility modelswithout leverage.


T l R M ti l


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Tools: Reverse Martingales

Consider a martingale indexed by Z:

. . . , M2, M1, M0, M1, M2, . . .

Going backwards

Yn = Mn, (n Z)

is called a reverse martingale.

Introducing -algebras and generalizing a bit, we get. . .


R S b ti l


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Reverse Submartingales

Reverse G-(Sub)martingale

(Gn) . . . decreasing sequence of -algebrasYn. . . sequence of integrable random variables such that Yn Gn.

(Yn) is called a reverse (sub)martingale with respect to Gn, if

E [ Yn1| Gn] = () Yn (1)

for all n N.

Note that (Gn) is decreasing, and thus not a filtration.

Doobs martingale convergence theorem yields that a reversesubmartingale converges a.s. to a limit Y. (For forwardsubmartingales further conditions are needed!)


Re e se S b a ti gales a d Co e O de


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Reverse Submartingales and Convex Order

By Jensens inequality, a reverse submartingale is decreasing inconvex order:

E [g(Yn)] E [g(Yn+1)] . . .E [g(Y)] ,

for all increasing and convex g.

Conversely, any two random variables B c A in convex order, canbe connected by a reverse submartingale with Y0 = A, Y = B, bya result of Strassen (1965).

Can we find a reverse submartingale connecting realized varianceand quadratic variation?


Reverse Martingales from Realized Variance (1)


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TheoremLet X be a semi-martingale with H-conditionally independent,symmetric increments, and let (Pn) be a sequence of nestedpartitions of [0, T]. Then the realized variance of X over thepartitions (Pn) is a reverse martingale, i.e. it satisfies

E

RV(X, Pn1)Gn

= RV(X, Pn), (n N)

whereGn = H RV(X, P

n), RV(X, Pn+1), . . ..Moreover, if the mesh size of Pn goes to zero, thenRV(X, Pn) [X, X]T a.s.




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Used implicitly by Levy (1940) to show that the realizedvariance of Brownian motion converges to t almost surelywhen partitions are nested.

Shown by Cogburn and Tucker (1961) for processes withindependent symmetric increments.

Here extended to semi-martingales with conditionallyindependent symmetric increments.

Proof relies on a reflection principle based on the symmetry ofincrements.


Convex Order Relations (1)


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Denote by G =

nN Gn the tail -algebra of Gn.

Convex Order RelationIf X is a semi-martingale with H-conditionally independent,symmetric increments, then

E [ RV(X, P)| G] = [X, X]T.

This equation implies the convex order relation

RV(X, P) c [X, X]T.

However, even in the Black-Scholes model, the log-price X doesnot have symmetric increments, because of the exponentialmartingale drift:

Xt = Bt t

2



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Theorem

Let X be a semi-martingale with conditionally independentincrements and symmetric jump measure. Suppose that the drift Bof X is continuous. Then there exists a -algebra G, such that

E [ RV(X, P)| G] = [X, X]T + RV(B, P).

If B is deterministic, it also holds that

E

RV(X, P)

G

= [X, X]T,



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Consequences for Option Pricing (1)


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Models with deterministic drift B:

Exponential Levy models with symmetric jumps

Symmetric Sato Processes A Sato process is a process with indep.increments and the self-similarilty property

Xtd= Xt, for a fixed > 0.

Has been used for variance options e.g. in (Carret al. 2010).

In these models QV underprices RV for the following payoffs:

Fixed Strike Calls (x K)+

Relative Strike Calls (x kR)+ where k 1 and R is theswap rate

ATM Puts and ATM Straddles with payoffs (R x)+ and|R x| respectively.




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Models with continuous random drift B:Time-changed Levy processes with symmetric jumps. The time

change is arbitrary, but we assume it to beindependent from the Levy process.

Stochastic Volatility models with jumps We assume that the

volatility process is independent from the Brownianmotion and jump measure driving the stock price (noleverage effect).

In these models QV underprices RV for:

Fixed Strike Calls (x K)+

i.e. the convex order conjecture holds for RV and QV but notnecessarily for the centered quantities RV and QV.


To Do


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To Do

Remove the symmetry condition on jumps, or provide acounterexample to show that it is necessary.

Incorporate cash dividends and consider options on weighted

realized variance.

On a more general note. . .

Bring together the risk management perspective and theeconometric perspective on Realized Variance.



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Thank you for your attention!



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M. Broadie and A. Jain. The effect of jumps and discrete sampling on volatilityand variance swaps. International Journal of Theoretical and AppliedFinance, 11:761797, 2008.

Hans Buhler. Volatility Markets Consistent modeling, hedging and practicalimplementation. PhD thesis, TU Berlin, 2006a.

Hans Buhler. Options on variance: Pricing & hedging. Presentation at theIQPC Volatility Trading Conference, London, 2006b.

Peter Carr, Helyette Geman, Dilip B. Madan, and Marc Yor. Options on

realized variance and convex orders. Quantitative Finance, iFirst:110,2010.

Robert Cogburn and Howard G. Tucker. A limit theorem for a function of theincrements of a decomposable process. Transactions of the AmericanMathematical Society, 99:278284, 1961.

Jim Gatheral. The Volatilty Surface. Wiley Finance, 2006.

Jim Gatheral. Consistent modeling of spx and vix options. Presentation at the5th Bachelier Congress, London, 2008.

Martin Keller-Ressel and Johannes Muhle-Karbe. Asymptotic behaviour andpricing of options on variance. Preprint, 2010.

Paul Levy. Le mouvement brownien plan. American Journal of Mathematics,

62(1):487 550, 1940.Martin Keller-Ressel Options on Variance


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A. Neuberger. Volatility trading. London Business School Working Paper, 1992.

A. Sepp. Analytical pricing of double-barrier options under a

double-exponential jump-diffusion process.International Journal of

Theoretical and Applied Finance, 7:151175, 2008.

Volker Strassen. The existence of probability measures with given marginals.The Annals of Mathematical Statistics, 36(2):423439, 1965.


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