Remark on variance swaps pricing
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1. The document discusses the pricing of variance swaps using risk neutral valuation. It defines variance swaps as transactions where the payout is based on the difference between realized variance and a prespecified strike variance. 2. It derives a formula for the strike variance that is equal to the risk-neutral expected value of the average variance over the swap period, where the expectation is calculated using Black-Scholes option pricing. 3. The value of a variance swap is defined as the notional amount multiplied by the difference between realized variance, estimated from historical stock prices, and the strike variance estimated from option prices. The variance swap thus specifies the value of differences between two estimates of the true variance.
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REMARK ON VARIANCE SWAPS PRICING.
Ilya I. Gikhman
6077 Ivy Woods Court,
Mason, OH 45040, USA
ph. 513-573-9348
email: ilyagikhman@mail.ru
JEL : G12, G13
Key words. Variance swap, Black Scholes pricing, risk neutral valuation.
Following [1] we briefly recall scheme of variance pricing. Variance represents a factor that
affects market price of the traded securities. Most popular traded instruments are variance and
volatility swaps. These are netted transactions between two parties based on fixed and floating
rates at maturity. The value of transaction between two counterparties is based on the difference
between realized variance and initially prespecified strike variance times a notional principal.
I. Let S ( t ) denote the prices of a stock at time t 0 and assume
that price of the stock is govern by equation
dS ( t ) = μ ( t ) S ( t ) dt + ( t ) S ( t ) dw ( t ) (1)
where coefficients μ and is a known deterministic continuous function in t. Applying Ito
formula it follows that
d ln S ( t ) = [ μ ( t ) –
2
1
2
( t ) ] dt + σ ( t ) dw ( t )
Subtracting later from former equations we arrive at the formula
)t(S
)t(dS
– d ln S ( t ) =
2
1
2
( t ) dt
This equality can be rewritten in integral form as following](https://image.slidesharecdn.com/remarkonvarianceswapspricing-151108133000-lva1-app6891/85/Remark-on-variance-swaps-pricing-1-320.jpg)
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σˆ 2
]T,0[ =
T
1
T
0
2
( t ) dt =
T
2
[
T
0
)t(S
)t(dS
– ln
)0(S
)T(S
] (2)
Formula (2) defines average variance over [ 0 , T ] interval . It is a parameter which specifies
variance swap. Let us fixed an arbitrary number S *. Then by direct calculation one can easy
verify equality
– ln
*S
)T(S
= –
*S
*S)T(S
+
*SK
( K – S ( T ) ) +
2
K
Kd
+
+
*SK
( S ( T ) – K) +
2
K
Kd
Bearing in mind latter equality we can rewrite (2) in the form
T
1
T
0
2
( t ) dt =
T
2
[
T
0
)t(S
)t(dS
–
*S
*S)T(S
+
(3)
+
*SK
( K – S ( T ) ) +
2
K
Kd
+
*SK
( S ( T ) – K) +
2
K
Kd
]
Now let us define the value of the variance swap. From buyer perspective it is defined by
transaction
V varince = N ( 2
RV – 2
strike ) (4)
that is taking place at expiration date T. Here N is a notional principal and 2
strike , 2
RV are
variance strike and realized variance correspondingly. Buyer of the variance swap receives
amount N 2
RV and pays N 2
strike from swap seller.
Realized variance 2
RV represents a discrete approximation of the value σˆ 2
]T,0[ based on
observed historical data of the equity prices on [0 , T]. On the other hand swap buyer pays its
date-0 estimated value 2
strike represented on the left hand side (3). It is called the strike
variance. The value of realized variance is defined as average of log returns over time period
[ 0 , T ]. It is defined as following](https://image.slidesharecdn.com/remarkonvarianceswapspricing-151108133000-lva1-app6891/85/Remark-on-variance-swaps-pricing-2-320.jpg)
![3
2
RV ( Δt ) =
252
n
n
1j
[ ln
)t(S
)t(S
1-j
j
] 2
(5)
where t i = t i – 1 + Δt , t = t 0 < t 1 < … t n = T. Sometimes in order to get unbiased estimate
one uses factor n – 1 rather than n where n is the size of the sample variance. Let us justify
formula (5). We note that
S ( t i ) = S ( t i – 1 ) +
i
1-i
t
t
μ ( u ) S ( u ) du +
i
1-i
t
t
( u ) S ( u ) dw ( u ) =
= S ( t i – 1 ) exp
i
1-i
t
t
[ μ ( u ) –
2
1
2
( u ) ] du +
i
1-i
t
t
( u ) dw ( u )
Hence
[ ln
)t(S
)t(S
1-i
i
] 2
= [
i
1-i
t
t
[ μ ( u ) –
2
1
2
( u ) ] du +
i
1-i
t
t
( u ) dw ( u ) ] 2
2
( t i – 1 ) ( t i – t i – 1 ) + o ( Δt )
where
0t
l.i.m
o ( Δt ) = 0 . Summing up latter equalities over all time periods lead leads to the
realized variance formula
2
RV ( Δt ) =
252
n
n
1i
2
( t i – 1 ) ( t i – t i – 1 ) + O (Δt) =
=
252
n 2
1-i
i
n
1i
]
)t(S
)t(S
ln[
+ O (Δt)
where
0t
l.i.m
O (Δt) = 0. Taking limit when Δt tends to zero we arrive at the formula
2
RV ( 0 ) =
0t
l.i.m
2
RV ( Δt ) =
T
1
T
0
2
( t ) dt (6)
We can use also other formulas for realized variance presentations. For example bearing in mind
approximation
S ( t i ) S ( t i – 1 ) + μ ( t i – 1 ) S ( t i – 1 ) ( t i – t i – 1 ) +](https://image.slidesharecdn.com/remarkonvarianceswapspricing-151108133000-lva1-app6891/85/Remark-on-variance-swaps-pricing-3-320.jpg)
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+ ( t i – 1 ) S ( t i – 1 ) [ w ( t i ) – w ( t i – 1 ) ]
we note that
[
)t(S
)t(S
1-i
i
– 1 ] = μ ( t i – 1 ) ( t i – t i – 1 ) + ( t i – 1 ) [ w ( t i ) – w ( t i – 1 ) ]
Therefore
2
RV ( Δt ) = [ T
n
1i
2
( t i – 1 ) ( t i – t i – 1 ) ] + O (Δt)
T
n
1i
[
)t(S
)t(S
1-i
i
– 1 ] 2
Following [1] let us perform calculation of the variance strike. Consider first two terms on the
right hand side (3). Taking in equality (3) risk neutral expectation and bearing in mind that risk-
neutral expectation of the real stock is transformed to SDE
dS r ( t ) = r S r ( t ) dt + σ ( t ) S r ( t ) dw ( t ) (7)
on real world we note that
E Q
T
0
)t(S
)t(dS
= E
T
0
)t(S
)t(dS
r
r
= r T
Then next term on the right hand side (3) is
*S
*S)T(S
=
*S
)T(S
– 1
It is interpreted as a date-0 forward contract of the ( S* ) – 1
shares on security S with expiration
date at T. Therefore
E Q
*S
*S)T(S
=
*S
Trexp)0(S
– 1 ,
E Q [
*SK
( K – S ( T ) ) +
2
K
Kd
+
*SK
( S ( T ) – K) +
2
K
Kd
] =](https://image.slidesharecdn.com/remarkonvarianceswapspricing-151108133000-lva1-app6891/85/Remark-on-variance-swaps-pricing-4-320.jpg)
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=
*SK
P ( 0 , S ( 0 ) ; T , K ) 2
K
Kd
+
*SK
C ( 0 , S ( 0 ) ; T , K ) 2
K
Kd
where P ( 0 , S ( 0 ) ; T , K ) and C ( 0 , S ( 0 ) ; T , K ) are put and call options premium at date t
= 0 correspondingly. Thus we arrive at the well known formula
2
strike =
T
2
[ r T – (
*S
Trexp)0(S
– 1 ) – ln
)0(S
*S
+
(8)
+ exp r T
*SK
P ( 0 , S ( 0 ) ; T , K ) 2
K
Kd
+ exp r T
*SK
C ( 0 , S ( 0 ) ; T , K ) 2
K
Kd
In practice there is only a finite set of of options with different strike prices are available on the
market. Therefore approximation formulas for calculation the value 2
strike should be used.
In the next remark we specified the essence of the risk neutral valuation that was used in
derivation of the 2
strike in formula (8).
II. Let us briefly recall the essence of the Risk Neutral World notion. Stock price
is given by equation (1).Solution of the equation (1) is a random process which is defined on
original probability space ( Ω , F , P ). This probability space is also referred to as to real world.
The risk neutral world can be introduced as following. Denote ν μ measure corresponding to the
solution of the equation (1) and let ν r be a measure corresponding to the solution of the
equation (7). Then density measure ν r with respect to measure ν μ can be represented in the
form
μ
r
νd
νd
( S μ ( * ) ) = exp {
T
0
λ ( t , S μ ( t )) d w ( t ) –
2
1
T
0
λ 2
( t , S μ ( t )) d t }
Here λ ( t , S ) =
)t(σ
)t(μ)t(r
is a constant that does not depend on S. For an arbitrary
Borel bounded function f ( x ) , x [ 0 , + ) we note that
E f ( S r ( * ) ) = ),0[C
f ( u ) d ν r ( u ) = ),0[C
f ( u ) q ( 0 , T ; u ) d ν μ ( u )
where C [ 0 , T ] denote the space of continuous functions on [ 0 , T ] and
q ( 0 , T ) =
μ
r
νd
νd
( S μ ( * ) )](https://image.slidesharecdn.com/remarkonvarianceswapspricing-151108133000-lva1-app6891/85/Remark-on-variance-swaps-pricing-5-320.jpg)
![6
i.e. d ν r = q ( 0 , T ) d ν μ . Therefore integration with respect to measure corresponding to the
solution of the equation (7) is equivalent to integration with respect to probabilistic measure
q ( 0 , T ) d ν μ . Then latter equality can be written in the form
E f ( S r ( * ) ) = E q ( 0 , T ) f ( S μ ( * ) ) = E Q f ( S μ ( * ) )
This formula shows that calculations of the Q-expectation of a functional of the process S μ
leads us to the P-expectation of the functional with respect to S r . Thus the process S μ on
‘risk neutral world ( Ω , F , Q ) is equivalent in terms of distributions to S r on the real world
( Ω , F , P ).
Risk neutral technique is used to highlight the relationship between underlying stock price and
option values. Note that real world underlying of the Black Scholes option price is solution of the
equation (7) and therefore option is taking its value based on values risk neutral heuristic process
S r . The real stock is does not directly effect on option prices. Formula (3) is used for strike
calculations. Multiply both sides of the equality (3) by q ( 0 , T ) and taking expectation leads to
risk neutral expectation which then justifies formula (8). Equalities (3) and (5) represent the
essence of the variance swap pricing. Formula (5) shows that the price N 2
RV is assigned to the
variance parameter 2
RV . In theory the diffusion coefficient ( t ) is known. In this case we
know deterministic function which we assign to ( t ) as its price. In these case derivatives such
as for example a variance swap does not make sense to define. Indeed, in practice the function
( t ) is unknown and we use two different approximations for estimation the value V 2
price
defined by (4). One estimate is based on observations on stock prices. This estimate leads to
realized volatility 2
RV . Other estimate is based on call - put Black-Scholes options prices and
lead to the value 2
strike . Variance swap is defined by the value of the transaction (4) at
expiration date. Following interest rate swap terminology variance price is associated with the
value 2
strike . The variance swaps specify the value of the difference of two estimations of the
unknown (6) and it does not specify itself the value of the asset variance. Taking limit in (4)
when Δt tends to zero we note that
V varince = N [
T
1
T
0
2
( t ) dt – 2
strike ] (4′)
where 2
strike is defined by (8). Formula (4′) implies that coefficient ( t ) is known. On the other
hand formula (8) implies that Black Scholes pricing model is used for variance pricing. In Black
Scholes model we assume that S ( t ) follows equation (1) with known deterministic continuous
functions μ ( t ) , ( t ). Hence the value V varince defined by the right hand side of the formula
(4′) is assumed to be known. Thus
N 2
strike = N
T
1
T
0
2
( t ) dt (9)
Equality (9) is implied by formulas (3) and (8).](https://image.slidesharecdn.com/remarkonvarianceswapspricing-151108133000-lva1-app6891/85/Remark-on-variance-swaps-pricing-6-320.jpg)
Remark on variance swaps pricing
- 1. 1 REMARK ON VARIANCE SWAPS PRICING. Ilya I. Gikhman 6077 Ivy Woods Court, Mason, OH 45040, USA ph. 513-573-9348 email: ilyagikhman@mail.ru JEL : G12, G13 Key words. Variance swap, Black Scholes pricing, risk neutral valuation. Following [1] we briefly recall scheme of variance pricing. Variance represents a factor that affects market price of the traded securities. Most popular traded instruments are variance and volatility swaps. These are netted transactions between two parties based on fixed and floating rates at maturity. The value of transaction between two counterparties is based on the difference between realized variance and initially prespecified strike variance times a notional principal. I. Let S ( t ) denote the prices of a stock at time t 0 and assume that price of the stock is govern by equation dS ( t ) = μ ( t ) S ( t ) dt + ( t ) S ( t ) dw ( t ) (1) where coefficients μ and is a known deterministic continuous function in t. Applying Ito formula it follows that d ln S ( t ) = [ μ ( t ) – 2 1 2 ( t ) ] dt + σ ( t ) dw ( t ) Subtracting later from former equations we arrive at the formula )t(S )t(dS – d ln S ( t ) = 2 1 2 ( t ) dt This equality can be rewritten in integral form as following
- 2. 2 σˆ 2 ]T,0[ = T 1 T 0 2 ( t ) dt = T 2 [ T 0 )t(S )t(dS – ln )0(S )T(S ] (2) Formula (2) defines average variance over [ 0 , T ] interval . It is a parameter which specifies variance swap. Let us fixed an arbitrary number S *. Then by direct calculation one can easy verify equality – ln *S )T(S = – *S *S)T(S + *SK ( K – S ( T ) ) + 2 K Kd + + *SK ( S ( T ) – K) + 2 K Kd Bearing in mind latter equality we can rewrite (2) in the form T 1 T 0 2 ( t ) dt = T 2 [ T 0 )t(S )t(dS – *S *S)T(S + (3) + *SK ( K – S ( T ) ) + 2 K Kd + *SK ( S ( T ) – K) + 2 K Kd ] Now let us define the value of the variance swap. From buyer perspective it is defined by transaction V varince = N ( 2 RV – 2 strike ) (4) that is taking place at expiration date T. Here N is a notional principal and 2 strike , 2 RV are variance strike and realized variance correspondingly. Buyer of the variance swap receives amount N 2 RV and pays N 2 strike from swap seller. Realized variance 2 RV represents a discrete approximation of the value σˆ 2 ]T,0[ based on observed historical data of the equity prices on [0 , T]. On the other hand swap buyer pays its date-0 estimated value 2 strike represented on the left hand side (3). It is called the strike variance. The value of realized variance is defined as average of log returns over time period [ 0 , T ]. It is defined as following
- 3. 3 2 RV ( Δt ) = 252 n n 1j [ ln )t(S )t(S 1-j j ] 2 (5) where t i = t i – 1 + Δt , t = t 0 < t 1 < … t n = T. Sometimes in order to get unbiased estimate one uses factor n – 1 rather than n where n is the size of the sample variance. Let us justify formula (5). We note that S ( t i ) = S ( t i – 1 ) + i 1-i t t μ ( u ) S ( u ) du + i 1-i t t ( u ) S ( u ) dw ( u ) = = S ( t i – 1 ) exp i 1-i t t [ μ ( u ) – 2 1 2 ( u ) ] du + i 1-i t t ( u ) dw ( u ) Hence [ ln )t(S )t(S 1-i i ] 2 = [ i 1-i t t [ μ ( u ) – 2 1 2 ( u ) ] du + i 1-i t t ( u ) dw ( u ) ] 2 2 ( t i – 1 ) ( t i – t i – 1 ) + o ( Δt ) where 0t l.i.m o ( Δt ) = 0 . Summing up latter equalities over all time periods lead leads to the realized variance formula 2 RV ( Δt ) = 252 n n 1i 2 ( t i – 1 ) ( t i – t i – 1 ) + O (Δt) = = 252 n 2 1-i i n 1i ] )t(S )t(S ln[ + O (Δt) where 0t l.i.m O (Δt) = 0. Taking limit when Δt tends to zero we arrive at the formula 2 RV ( 0 ) = 0t l.i.m 2 RV ( Δt ) = T 1 T 0 2 ( t ) dt (6) We can use also other formulas for realized variance presentations. For example bearing in mind approximation S ( t i ) S ( t i – 1 ) + μ ( t i – 1 ) S ( t i – 1 ) ( t i – t i – 1 ) +
- 4. 4 + ( t i – 1 ) S ( t i – 1 ) [ w ( t i ) – w ( t i – 1 ) ] we note that [ )t(S )t(S 1-i i – 1 ] = μ ( t i – 1 ) ( t i – t i – 1 ) + ( t i – 1 ) [ w ( t i ) – w ( t i – 1 ) ] Therefore 2 RV ( Δt ) = [ T n 1i 2 ( t i – 1 ) ( t i – t i – 1 ) ] + O (Δt) T n 1i [ )t(S )t(S 1-i i – 1 ] 2 Following [1] let us perform calculation of the variance strike. Consider first two terms on the right hand side (3). Taking in equality (3) risk neutral expectation and bearing in mind that risk- neutral expectation of the real stock is transformed to SDE dS r ( t ) = r S r ( t ) dt + σ ( t ) S r ( t ) dw ( t ) (7) on real world we note that E Q T 0 )t(S )t(dS = E T 0 )t(S )t(dS r r = r T Then next term on the right hand side (3) is *S *S)T(S = *S )T(S – 1 It is interpreted as a date-0 forward contract of the ( S* ) – 1 shares on security S with expiration date at T. Therefore E Q *S *S)T(S = *S Trexp)0(S – 1 , E Q [ *SK ( K – S ( T ) ) + 2 K Kd + *SK ( S ( T ) – K) + 2 K Kd ] =
- 5. 5 = *SK P ( 0 , S ( 0 ) ; T , K ) 2 K Kd + *SK C ( 0 , S ( 0 ) ; T , K ) 2 K Kd where P ( 0 , S ( 0 ) ; T , K ) and C ( 0 , S ( 0 ) ; T , K ) are put and call options premium at date t = 0 correspondingly. Thus we arrive at the well known formula 2 strike = T 2 [ r T – ( *S Trexp)0(S – 1 ) – ln )0(S *S + (8) + exp r T *SK P ( 0 , S ( 0 ) ; T , K ) 2 K Kd + exp r T *SK C ( 0 , S ( 0 ) ; T , K ) 2 K Kd In practice there is only a finite set of of options with different strike prices are available on the market. Therefore approximation formulas for calculation the value 2 strike should be used. In the next remark we specified the essence of the risk neutral valuation that was used in derivation of the 2 strike in formula (8). II. Let us briefly recall the essence of the Risk Neutral World notion. Stock price is given by equation (1).Solution of the equation (1) is a random process which is defined on original probability space ( Ω , F , P ). This probability space is also referred to as to real world. The risk neutral world can be introduced as following. Denote ν μ measure corresponding to the solution of the equation (1) and let ν r be a measure corresponding to the solution of the equation (7). Then density measure ν r with respect to measure ν μ can be represented in the form μ r νd νd ( S μ ( * ) ) = exp { T 0 λ ( t , S μ ( t )) d w ( t ) – 2 1 T 0 λ 2 ( t , S μ ( t )) d t } Here λ ( t , S ) = )t(σ )t(μ)t(r is a constant that does not depend on S. For an arbitrary Borel bounded function f ( x ) , x [ 0 , + ) we note that E f ( S r ( * ) ) = ),0[C f ( u ) d ν r ( u ) = ),0[C f ( u ) q ( 0 , T ; u ) d ν μ ( u ) where C [ 0 , T ] denote the space of continuous functions on [ 0 , T ] and q ( 0 , T ) = μ r νd νd ( S μ ( * ) )
- 6. 6 i.e. d ν r = q ( 0 , T ) d ν μ . Therefore integration with respect to measure corresponding to the solution of the equation (7) is equivalent to integration with respect to probabilistic measure q ( 0 , T ) d ν μ . Then latter equality can be written in the form E f ( S r ( * ) ) = E q ( 0 , T ) f ( S μ ( * ) ) = E Q f ( S μ ( * ) ) This formula shows that calculations of the Q-expectation of a functional of the process S μ leads us to the P-expectation of the functional with respect to S r . Thus the process S μ on ‘risk neutral world ( Ω , F , Q ) is equivalent in terms of distributions to S r on the real world ( Ω , F , P ). Risk neutral technique is used to highlight the relationship between underlying stock price and option values. Note that real world underlying of the Black Scholes option price is solution of the equation (7) and therefore option is taking its value based on values risk neutral heuristic process S r . The real stock is does not directly effect on option prices. Formula (3) is used for strike calculations. Multiply both sides of the equality (3) by q ( 0 , T ) and taking expectation leads to risk neutral expectation which then justifies formula (8). Equalities (3) and (5) represent the essence of the variance swap pricing. Formula (5) shows that the price N 2 RV is assigned to the variance parameter 2 RV . In theory the diffusion coefficient ( t ) is known. In this case we know deterministic function which we assign to ( t ) as its price. In these case derivatives such as for example a variance swap does not make sense to define. Indeed, in practice the function ( t ) is unknown and we use two different approximations for estimation the value V 2 price defined by (4). One estimate is based on observations on stock prices. This estimate leads to realized volatility 2 RV . Other estimate is based on call - put Black-Scholes options prices and lead to the value 2 strike . Variance swap is defined by the value of the transaction (4) at expiration date. Following interest rate swap terminology variance price is associated with the value 2 strike . The variance swaps specify the value of the difference of two estimations of the unknown (6) and it does not specify itself the value of the asset variance. Taking limit in (4) when Δt tends to zero we note that V varince = N [ T 1 T 0 2 ( t ) dt – 2 strike ] (4′) where 2 strike is defined by (8). Formula (4′) implies that coefficient ( t ) is known. On the other hand formula (8) implies that Black Scholes pricing model is used for variance pricing. In Black Scholes model we assume that S ( t ) follows equation (1) with known deterministic continuous functions μ ( t ) , ( t ). Hence the value V varince defined by the right hand side of the formula (4′) is assumed to be known. Thus N 2 strike = N T 1 T 0 2 ( t ) dt (9) Equality (9) is implied by formulas (3) and (8).
- 7. 7 References. 1. K. Demeterfi, E. Derman, M. Kamal, J. Zou. More Than You Ever Wanted To Know About Volatility Swaps. Goldman Sachs Quantitative Strategies Research Notes, 1999, p.52.