Convexity adjustment for volatility swaps

Chrif YOUSSFI

Global Equity Linked Products

Outline

• Generalities about volatility/variance swaps.

• Intuition and motivation

• The framework of stochastic volatility.

• Convexity adjustment under stochastic volatility.

• Convexity adjustment and current smile.

• Numerical results.

• Conclusions.

Volatility and Variance Swaps• A volatility swap is a forward contract on the annualized

volatility that delivers at maturity:

• A variance contract pays at maturity:

• The annualized volatility is defined as the square root of the variance:

where is the closing price of the underlying at the ith business day and (n+1) is the total number of trade days.

VolKVolN .

1

0

2

11

0

2

1 1ln

1 n

i i

iin

i i

i

S

SS

nS

S

nVarVol

iS

VarKVarN .

Hedge and Valuation

• When there no jumps, the variance swaps valuation and hedging are model independent.

• The vega-hedge portfolio for variance swaps is static and the value is directly calculated from the current smile.

• The valuation of a volatility swap is model dependent and

the pricing requires model calibration and simulations.

• The vega hedge portfolio is not static.

Motivation

Smile of volatility

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%

0 50 100 150 200 250 300 350

Strikes

Vola

tili

ty

Implied

Smile of volatility generated by a stochastic volatility model where:spot is 100,maturity 1y and correlation is estimated at -70%

• What is the price of ATM option?

By considering the linearity of the option price w.r.t. volatility, the price is approximately 14.11%

• What is the value of the variance swap? 16.17%.

• What is the value of the volatility Swap? More difficult.

Intuition

• is the spot density at maturity T and the diffusion factor which be stochastic:

• A rough estimation of the volatility swap:

Question: What can the moments of the implied volatility teach us about the value of volatility swap?

0 0

2

0

2 )()|1

()1

( dKKpKSdtT

EdtT

EVarT

t

T

t

(.)p (.)

0

2 )(),( dKKpKTVar implied

The weighting is not exact.

MIV: Moment of Implied Volatility • We define by MIV (n) as the nth moment of the implied

volatility weighted by the risk neutral density.

• We define the smile convexity by:

• The convexity adjustment for the volatility swaps:

• Question: What is the relation between and ?

)1()2( MIVMIVimpvol

0

)(),()( dKKpKTnMIV nimplied

2

2 ),()(

K

KTCKp

)()(var/ VolEVarEvol

var/vol implied

Stochastic volatility assumptions

• The underlying dynamics are:

• The volatility itself is log-normal:

with the initial condition and

• The dynamics correspond to the short time analysis and the factor can be considered proportional to the square root of time to maturity.

(Patrick Hagan Model (1999))

1 ttt

t dWS

dS

2 tt

t dWvd

),( 00 S dtdWdW tt , 21

Forward and backward equations• The backward equation for the call prices is :

• The transition probability from the state

to satisfies the forward equation (FPDE) :

• When integrating the Forward PDE (Tanaka’s formula):

0 2 2

1,

2,

222 VvVSvVSV SSSt

),,,,( KTStV

T

tdTdKTStKKSKTStV

0 00222

000 ),,,,,(2

1)(),,,,(

),,,,,( 00 KTSt

),,( 00 St ),,( ST

Smile effect The curve

0 2 2

1,

22,

2222 vSvS SSST

Integral over calendar spreadsIntrinsic

• Define by:

and

• The solution of the system (S) is:

• In the BS case we have a similar formula with :

; ;

)log(

0

0

tTK

S

z

}

1

)(log{

1

vzzA

vX22221)( zvvzzA

h

X dqqqvzBKSKTStV2

2

2

2

3

000 )exp(),,,()(),,,,(

)() 24

11(||

4

1),,,( 4

022

02

00

OKSzzvzB

)()32(2)32(6242

11 ),( 422222

0

2

0 OzvvvvzKTBS

0v

20

2220 }6)32({

24

1Xvvh

Call Price and Implied Volatility

Volatility swap convexity adjustment

• The expected variance under the model assumptions:

• The value of the expected volatility:

It follows that the convexity adjustment is:

.)(2

11)log(

2)( 3222

02

0

Ov

S

SE

TVarE T

)(

12

11

1 )( 322

0

2

1

0

22 Ovdt T

EVolET

t

)(6

1 4320/ OvVolVar

Smile convexity

• By considering the value of the log-contract and the square of the log-profile:

• The smile convexity of the implied volatility is:

)()2

11(

2

1))(log(

1)( 422

000

OvS

SEzE

)(2

1

4

1)( 322222

022 OvzE

)(8

1 4220

3 Ovimplied

Convexity adjustment• As long as the is large enough (which is satisfied in the

equity markets), to the leading orders show that the relation between the two convexities is very simple:

• There no dependencies on maturity and volatility of the volatility.

• The value of the volatility swap does not depend on the correlation, however the implied volatility depends on and therefore intuitively we need to strip off this dependency.

)(3

4 42/

OimpliedVolVar

2

Numerical Results (4)

Convexity adjustment vs Smile Convexity

0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

3.00%

3.50%

4.00%

4.50%

0% 10% 20% 30% 40% 50% 60% 70% 80%

voVol

Adj

ustm

ent

MC

FormulaVol=40%Rho=-90%T=1 yrEpsilon=1

Numerical Results (5)

Convexity adjustment vs Smile Convexity

0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

3.00%

3.50%

0% 10% 20% 30% 40% 50% 60% 70% 80%

voVol

Adj

ustm

ent

MC

FormulaVol=30%Rho=-90%T=1 yrEpsilon=1

Numerical Results(6): Heston

Convexity adjustment vs Smile Convexity

0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

3.00%

0% 10% 20% 30% 40% 50% 60% 70% 80%

MC

Theory

Vol=40%Rho=-70%T=1 yrLambda=80%Vbar=16%

Numerical Results (7): Heston

Convexity adjustment vs Smile Convexity

0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

3.00%

0% 10% 20% 30% 40% 50% 60% 70% 80%

MC

Theory

Vol=40%Rho=-90%T=1 yrLambda=80%Vbar=16%

Conclusion

• This analysis shows that option prices can be very insightful to estimate the convexity adjustment.

• Even though the results are derived in the case of Hagan model, they can be extended to other models of stochastic volatility (Heston) as long as the correlation is in an appropriate range.

• It sheds some light on the importance of the curve factors to decide the value of a volatility swap.