Convexity adjustment for
volatility swaps
Chrif YOUSSFI
Global Equity Linked Products
Outline
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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:
N . Vol  KVol 
• A variance contract pays at maturity: N . Var  KVar 
• The annualized volatility is defined as the square root of
the variance:
2
Vol  Var 
1   Si 1  
 
ln 


n i 0   Si  
n 1
1  Si 1  Si 



n i 0  Si 
n 1
2
where Si is the closing price of the underlying at the ith
business day and (n+1) is the total number of trade days.
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
80.00%
70.00%
Volatility
60.00%
50.00%
Implied
40.00%
30.00%
20.00%
10.00%
0.00%
0
50
100
150
200
Strik es
250
300
350
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
• p (.) is the spot density at maturity T and  (.) the diffusion
factor which be stochastic:
1
Var  E (
T

T
0
 dt )  
2
t

0
1
E(
T

T
0
 t2 dt | S  K ) p ( K )dK
• A rough estimation of the volatility swap:

2
Var    implied
(T , K ) p( K )dK
0
The weighting is not exact.
Question: What can the moments of the implied volatility
teach us about the value of volatility swap?
MIV: Moment of Implied Volatility
• We define by MIV (n) as the nth moment of the implied
volatility weighted by the risk neutral density.

n
MIV (n)    implied
(T , K ) p( K )dK
0
  2C (T , K ) 

p( K )  
2
 K

• We define the smile convexity by: impvol  MIV (2)  MIV (1)
• The convexity adjustment for the volatility swaps:
vol / var  E(Var)  E(Vol)
• Question: What is the relation between
vol / var and  implied?
Stochastic volatility assumptions
dSt
• The underlying dynamics are:
   t dWt1
St
d
• The volatility itself is log-normal: t   v dWt 2
t
with the initial condition ( S 0 , 0 ) and  dWt1 , dWt 2   dt
• 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))
Forward and backward equations
• The backward equation for the call prices is V (t , S , , T , K ) :


1 2 2 2
Vt    S VSS  2  v S VS ,  v 2 V ,  0
2
The curve
Smile effect
• The transition probability  (t , S0 , 0 , T , K , ) from the state
(t , S0 , 0 ) to (T , S , ) satisfies the forward equation (FPDE) :

1 2
T    2 S 2 
2


2

2

v

S
SS

   0
2
2

v


S ,
 ,
• When integrating the Forward PDE (Tanaka’s formula):
1 2 2 T  2
V (t , S 0 ,  0 , T , K )  ( S 0  K )   K     (t , S 0 ,  0 , T , K , )dT d
t 0
2

Intrinsic
Integral over calendar spreads
Call Price and Implied Volatility
• Define by:
S0
)
K ;   T  t;
z
A( z )  1  2vz   2 v 2 z 2
 0
log(
and
X
• The solution of the system (S) is:
V (t , S ,  0 , T , K )  ( S0  K )  B( z,  0 , v,  ) X 2
2
B ( z ,  0 , v,  ) 
h



 2 h
1
A( z )    vz
log{
}
v
1 
3
2
exp( q)q dq
1
1


  0 | z |  (1   2  02 z 2 ) KS0   O( 4 )
4 
 24

1
{ 02  v 2 (2  3  2 )  6 v 0 } X 2
24
• In the BS case we have a similar formula with v  0 :
 1
2
 BS (T , K )   0 1  vz 
6 v0  v 2 (2  3 2 )   2v 2 z 2 (2  3 2 )
24
 2



  O( 4 )


Volatility swap convexity adjustment
• The expected variance under the model assumptions:
2 
ST 
 1


E (Var )   E log( )    202 1   2v 2  O( 3 ) .
T 
S0 
 2

• The value of the expected volatility:
1


 1 T 2 2 2 
 1

E (Vol )  E    t dt     0 1   2v 2  O( 3 ) 
 
 12

 T 0


It follows that the convexity adjustment is:
Var / Vol
1
  0 v 2 3  O( 4 )
6
Smile convexity
• By considering the value of the log-contract and the square
of the log-profile:
E( z)  
S
1
1
1
E (log(  ))    0 (1   2 v 2 )  O( 4 )
 0
S0
2
2
1 2 2 2 1 2 2 2
E ( z )      0    v   O( 3 )
4
2
2
• The smile convexity of the implied volatility is:
 implied
1 3
   0 v 2  2  O( 4 )
8
Convexity adjustment
• As long as the  2 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:
Var / Vol 
4
3
4


O
(

)
implied
2
• 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.
Numerical Results (4)
Convexity adjustment vs Smile Convexity
4.50%
4.00%
Adjustment
3.50%
3.00%
2.50%
MC
2.00%
Formula
Vol=40%
Rho=-90%
T=1 yr
Epsilon=1
1.50%
1.00%
0.50%
0.00%
0%
10%
20%
30%
40%
voVol
50%
60%
70%
80%
Numerical Results (5)
Convexity adjustment vs Smile Convexity
3.50%
3.00%
Adjustment
2.50%
2.00%
MC
1.50%
Formula
Vol=30%
Rho=-90%
T=1 yr
Epsilon=1
1.00%
0.50%
0.00%
0%
10%
20%
30%
40%
voVol
50%
60%
70%
80%
Numerical Results(6): Heston
Convexity adjustment vs Smile Convexity
3.00%
2.50%
Vol=40%
Rho=-70%
T=1 yr
Lambda=80%
Vbar=16%
2.00%
1.50%
1.00%
MC
Theory
0.50%
0.00%
0%
10%
20%
30%
40%
50%
60%
70%
80%
Numerical Results (7): Heston
Convexity adjustment vs Smile Convexity
3.00%
2.50%
Vol=40%
Rho=-90%
T=1 yr
Lambda=80%
Vbar=16%
2.00%
1.50%
MC
1.00%
Theory
0.50%
0.00%
0%
10%
20%
30%
40%
50%
60%
70%
80%
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.