Download Lecture 7: Quadratic Variation

Lecture 7: Quadratic Variation-based
Payoffs
Jim Gatheral, Merrill Lynch
Case Studies in Financial Modeling Course Notes,
Courant Institute of Mathematical Sciences,
Fall Term, 2004
17
Spanning Generalized European Payoffs
As usual, we assume that European options with all possible strikes and
expirations are traded. In the spirit of the paper by Carr and Madan (1998),
we now show that any twice-differentiable payoff at time T may be statically
hedged using a portfolio of European options expiring at time T .
From Breeden and Litzenberger (1978), we know that we may write the
pdf of the stock price ST at time T as
¯
¯
¯
2
2
∂ C̃(St , K, t, T ) ¯
∂ P̃ (St , K, t, T ) ¯¯
p(ST , T ; St , t) =
=
¯
¯
¯
¯
∂K 2
∂K 2
K=ST
K=ST
where C̃ and P̃ represent undiscounted call and put prices respectively.
Then, the value of a claim with a generalized payoff g(ST ) at time T is
given by
Z ∞
E [ g(ST )| St ] =
dK p(K, T ; St , t) g(K)
0
Z F
Z ∞
∂ 2 P̃
∂ 2 C̃
=
dK
g(K)
+
dK
g(K)
∂K 2
∂K 2
0
F
where F represents the time-T forward price of the stock. Integrating by
parts twice and using the put-call parity relation C̃(K) − P̃ (K) = F − K
gives
¯F
¯∞
Z F
Z ∞
¯
¯
∂ P̃
∂
P̃
∂
C̃
∂ C̃ 0
¯
¯
0
E [ g(ST )| St ] =
g(K)¯ −
dK
g (K) +
g(K)¯ −
dK
g (K)
¯
¯
∂K
∂K
∂K
∂K
0
F
0
F
Z F
Z ∞
∂ P̃ 0
∂ C̃ 0
= g(F ) −
dK
g (K) −
dK
g (K)
∂K
∂K
0
F
Z F
¯F
¯
0
= g(F ) − P̃ (K)g (K)¯ +
dK P̃ (K) g 00 (K)
0
Z ∞ 0
¯∞
¯
dK C̃(K) g 00 (K)
− C̃(K)g 0 (K)¯ +
F
F
Z F
Z ∞
00
= g(F ) +
dK P̃ (K) g (K) +
dK C̃(K) g 00 (K)
(70)
0
F
120
By letting t → T in equation (70), we see that any European-style twicedifferentiable payoff may be replicated using a portfolio of European options
with strikes from 0 to ∞ with the weight of each option equal to the second
derivative of the payoff at the strike price of the option. This portfolio of
European options is a static hedge because the weight of an option with a
particular strike depends only on the strike price and the form of the payoff
function and not on time or the level of the stock price. Note further that
equation (70) is completely model-independent.
Example: European Options
In fact, using Dirac delta-functions, we can extend the above result to payoffs
which are not twice-differentiable. Consider for example the portfolio of
options required to hedge a single call option with payoff (ST − L)+ . In this
case g 00 (K) = δ(K − L) and equation (70) gives
Z F
Z ∞
£
¤
+
+
E (ST − L)
= (F − L) +
dK P̃ (K) δ(K − L) +
dK C̃(K) δ(K − L)
0
F
½
(F − L) + P̃ (L) if L < F
=
C̃(L) if L ≥ F
= C̃(L)
with the last step following from put-call parity as before. In other words,
the replicating portfolio for a European option is just the option itself.
Example: Amortizing Options
A useful variation on the payoff of the standard European option is given
by the amortizing option with strike L with payoff
g(ST ) =
(ST − L)+
ST
Such options look particularly attractive when the volatility of the underlying stock is very high and the price of a standard European option is prohibitive . The payoff is effectively that of a European option whose notional
amount declines as the option goes in-the-money. Then,
¾¯
½
δ(ST − L) ¯¯
2L
00
g (K) = − 3 θ(ST − L) +
¯
ST
ST
ST =K
121
Without loss of generality (but to make things easier), suppose L > F . Then
substituting into equation (70) gives
·
¸
Z ∞
(ST − L)+
E
=
dK C̃(K) g 00 (K)
ST
F
Z ∞
C̃(L)
dK
=
− 2L
C̃(K)
L
K3
L
and we see that an Amortizing call option struck at L is equivalent to a
European call option struck at L minus an infinite strip of European call
options with strikes from L to ∞.
17.1
The Log Contract
Now consider
a contract whose payoff at time T is log( SFT ). Then g 00 (K) =
¯
¯
− ST1 2 ¯
and it follows from equation (70) that
ST =K
· µ ¶¸
Z F
Z ∞
dK
dK
ST
= −
P̃ (K) −
C̃(K)
E log
2
F
K
K2
0
F
Rewriting this equation in terms of the log-strike variable k ≡ log
get the promising-looking expression
· µ ¶¸
Z ∞
Z 0
ST
dk c(k)
E log
dk p(k) −
= −
F
0
−∞
y
¡K ¢
F
, we
(71)
y
e )
e )
with c(y) ≡ C̃(F
and p(y) ≡ P̃ (F
representing option prices expressed
F ey
F ey
in terms of percentage of the strike price.
18
Variance and Volatility Swaps
We now revert to our usual assumption of zero interest rates and dividends.
In this case, F = S0 and applying Itô’s Lemma, path-by-path
µ ¶
µ ¶
ST
ST
log
= log
F
S0
Z T
=
d log (St )
0
Z T
Z T
σSt 2
dSt
−
dt
(72)
=
St
2
0
0
122
The second term on the r.h.s of equation (72) is immediately recognizable
as half the total variance WT over the period {0, T }. The first term on the
r.h.s represents the payoff of a hedging strategy which involves maintaining
a constant dollar amount in stock (if the stock price increases, sell stock; if
the stock price decreases, buy stock so as to maintain a constant dollar value
of stock). Since the log payoff on the l.h.s can be hedged using a portfolio
of European options as noted earlier, it follows that the total variance WT
may be replicated in a completely model-independent way so long as the
stock price process is a diffusion. In particular, volatility may be stochastic
or deterministic and equation (72) still applies.
Now taking the risk-neutral expectation of (72) and comparing with
equation (71), we obtain
¾
·Z T
¸
· µ ¶¸
½Z 0
Z ∞
ST
2
dk c(k)
E
σSt dt = −2 E log
=2
dk p(k) +
F
0
0
−∞
(73)
We see explicitly that the fair value of total variance is given by the value
of an infinite strip of European options in a completely model-independent
way so long as the underlying process is a diffusion.
18.1
Variance Swaps
Although variance and volatility swaps are relatively recent innovations,
there is already a significant literature describing these contracts and the
practicalities of hedging them including articles by Chriss and Morokoff
(1999) and Demeterfi, Derman, Kamal, and Zou (1999).
In fact, a variance swap is not really a swap at all but a forward contract
on the realized annualized variance. The payoff at time T is
(
µ
µ ¶¾2 )
¶¾2
½
N ½
1 X
Si
1
SN
N ×A×
log
−
log
− N × Kvar
N i=1
Si−1
N
S0
where N is the notional amount of the swap, A is the annualization factor
and Kvar is the strike price. Annualized variance may or may not be defined
as mean-adjusted in practice so the corresponding drift term in the above
payoff may or may not appear.
From a theoretical perspective, the beauty of a variance swap is that it
may be replicated perfectly assuming a diffusion process for the stock price
123
as shown in the previous section. From a practical perspective, market operators may express views on volatility using variance swaps without having
to delta hedge.
Variance swaps took off as a product in the aftermath of the LTCM
meltdown in late 1998 when implied stock index volatility levels rose to
unprecedented levels. Hedge funds took advantage of this by paying variance
in swaps (selling the realized volatility at high implied levels). The key to
their willingness to pay on a variance swap rather than sell options was that
a variance swap is a pure play on realized volatility – no labor-intensive delta
hedging or other path dependency is involved. Dealers were happy to buy
vega at these high levels because they were structurally short vega (in the
aggregate) through sales of guaranteed equity-linked investments to retail
investors and were getting badly hurt by high implied volatility levels.
18.2
Variance Swaps in the Heston Model
Recall that in the Heston model, instantaneous variance v follows the process:
p
dv(t) = −λ(v(t) − v̄)dt + η v(t) dZ
It follows that the expectation of the total variance WT is given by
¸
·Z T
vt dt
E [WT ] = E
0
Z T
=
v̂t dt
0
1 − e−λT
=
(v − v̄) + v̄T
λ
The expected annualized variance is given by
1
1 − e−λT
E [WT ] =
(v − v̄) + v̄
T
λT
We see that the expected variance in the Heston model depends only on v, v̄
and λ. It does not depend on the volatility of volatility η. Since the value of
a variance swap depends only on the prices of European options, it follows
that a variance swap would be priced identically by both Heston and our
local volatility approximation to Heston.
124
18.3
Dependence on Skew and Curvature
We know that the implied volatility of an at-the-money forward option in
the Heston model is lower than the square root of the expected variance
(just think of the shape of the implied distribution of the final stock price
in Heston). In practice, we start with a strip of European options of a given
expiration and we would like to know how we should expect the price of a
variance swap to relate to the at-the-money-forward implied volatility, the
volatility skew and the volatility curvature (smile).
It turns out that there is a very elegant exact expression for the fair value
of variance; the proof is given in Appendix A. Define
√
k
σBS (k) T
√ +
z(k) = d2 = −
2
σBS (k) T
Intuitively, z measures the log-moneyness of an option in implied standard
deviations. Then,
Z∞
2
E[WT ] =
dz N 0 (z) σBS
(z)T
(74)
−∞
To see this formula is plausible, it is obviously correct when there is no
volatility skew.
Now consider the following simple parameterization of the BS implied
variance skew:
2
σBS
(z) = σ02 + α z + β z 2
Substituting into equation (74) and integrating, we obtain
E[WT ] = σ02 T + βT
We see that skew makes no contribution to this expression, only the curvature contributes. The intuition for this is simply that increasing the skew
does not change the average level of volatility but increasing the curvature
β increases the prices of puts and calls in equation (71) and always increases
the fair value of variance.
18.4
The Effect of Jumps
Suppose we have some continuous process and add a jump of size ² that
occurs between times ti and ti+1 . Total variance is computed as a sum of
125
terms of the form
µ
µ
¶¶2
Si+1
log
Si
The jump will change this particular term in the sum to
¶¶2
µ µ
Si+1 e²
log
Si
The net impact of the jump on the total variance for this realization of the
process is then given by
µ
¶
Si+1
2
² + 2² log
Si
which in expectation is just E[²2 ]. So, assuming that jumps are independent
of the underlying continuous process, to get the effect of jumps on the expected total variance, we need only figure out the expected number of jumps
in the interval (0, T ] and their average squared magnitude.
In the Merton and SVJ models, jumps are generated by a Poisson process
with parameter λ and the size of the jump is lognormally distributed with
mean α and standard deviation δ. In this case, the expected number of
jumps is just λT and the average squared magnitude is given by α2 + δ 2 .
The expected total variance E[WT ] is then given by
¡
¢
E[WT ] = E[WT ]c + λT α2 + δ 2
where E[WTc ] denotes the expected variance of the same process without
jumps. This formula tells us in absolute terms how the value of a variance
swap would change but not in terms of the European option hedge whose
value obviously must also change when we add jumps to the underlying
process.
To see how the value of the variance swap relates to the value of the
log-contract (the infinite strip of European options), we note that from the
definition of the characteristic function
¯
· µ ¶¸
¯
∂
ST
= −i
φT (u)¯¯
E log
F
∂u
u=0
Also, recall that if jumps are independent of the continuous process as they
are in both the Merton and SVJ models, the characteristic function may be
written as the product of a continuous part and a jump part
φT (u) = φcT (u) φjT (u)
126
Then
¯
¯
· µ ¶¸
¯
¯
ST
∂ c
∂
j
E log
= −i
φT (u)¯¯
−i
φT (u)¯¯
F
∂u
∂u
u=0
u=0
From equation (72), the first term on the r.h.s is just − 12 E[WTc ]. To get the
second term, we use the explicit form of φjT (u) given by
n
³
´
³
´o
2
2 2
φjT (u) = exp −iuλT eα+δ /2 − 1 + λT eiuα−u δ /2 − 1
Then
So
¯
³
´
¯
∂ j
2
= λT 1 + α − eα+δ /2
−i
φT (u)¯¯
∂u
u=0
· µ ¶¸
³
´
ST
1
c
α+δ 2 /2
E log
= − E[WT ] + λT 1 + α − e
F
2
and the difference in value between the log-contract hedge (2 log contracts)
and the variance swap is given by
· µ ¶¸
³
´
¡
¢
ST
2
= λT α2 + δ 2 + 2λT 1 + α − eα+δ /2
E[WT ] + 2E log
F
¡
¢
1
= − λT α α2 + 3δ 2 + higher order terms
3
This shows that two log contracts is a good average hedge for a variance
swap even with jumps. Putting λ = 0.6, α = −0.15 and δ = 0.1, we get
an error of only 0.0016 on a one-year variance swap which corresponds to
around 0.25 vol points if the volatility level is 30%.
18.5
Volatility Swaps
Realized volatility ΣT is the square root of realized variance VT and we know
that the expectation of the square root of a random variable is less than√(or
equal to) the square root of the expectation. The difference between VT
and ΣT is known as the convexity adjustment.
Figure 1 shows how the payoff of a variance swap compares with the
payoff of a volatility swap.
Intuitively, the magnitude of the convexity adjustment must depend on
the volatility of realized volatility. Note that volatility does not have to be
127
Figure 1: Payoff of a variance swap (dashed line) and volatility swap (solid
line) as a function of realized volatility. Both swaps are stuck at 30% volatility. ΣT
Payoff
0.1
0.05
0.15
0.2
0.25
0.3
0.35
0.4
ST
-0.05
-0.1
-0.15
-0.2
stochastic for realized volatility to be volatile; realized volatility ΣT varies
according to the path of the stock price even in a local volatility model.
In general, there is no replicating portfolio for a volatility swap and the
magnitude of the convexity adjustment is highly model-dependent. As a
consequence, market makers’ prices for volatility swaps are both wide (in
terms of bid-offer) and widely dispersed. As in the case of live-out options,
price takers such as hedge funds may occasionally have the luxury of being
able to cross the bid-offer – that is, buy on one dealer’s offer and sell on the
other dealer’s bid.
Assuming no jumps however (Matytsin (1999) discusses the impact of
jumps), as we will see in Section 18.7, the convexity adjustment is model
independent. We will now compute it for the Heston model.
18.6
Convexity Adjustment in the Heston Model
To compute the expectation of volatility in the Heston model we use the
following trick:
£
¤
Z ∞
hp i
1 − E e−ψWT
1
dψ
(75)
E
WT = √
ψ 3/2
2 π 0
128
From Cox, Ingersoll, and Ross (1985), the Laplace transform of the total
RT
variance WT = 0 vt dt is given by
£
¤
E e−ψWT = A e−ψvB
where
½
2φ e(φ+λ)T /2
A =
(φ + λ)(eφT − 1) + 2φ
2 (eφT − 1)
B =
(φ + λ)(eφT − 1) + 2φ
¾2λv̄/η2
p
with φ = λ2 + 2ψη 2 .
With some tedious algebra, we may verify that
¯
∂ £ −ψWT ¤¯¯
E [WT ] = − E e
¯
∂ψ
ψ=0
=
1 − e−λT
(v − v̄) + v̄T
λ
as we found earlier in Section 18.2.
Computing the integral in equation (75) numerically using the usual
parameters from Homework 2 (v = 0.04, v̄ = 0.04, λ = 10.0, η = 1.0), we get
the graph of the convexity adjustment as a function of time to expiration
shown in Figure 2.
Using Bakshi, Cao and Chen parameters (v = 0.04, v̄ = 0.04, λ =
1.15, η = 0.39), we get the graph of the convexity adjustment as a function of time to expiration shown in Figure 3.
To get intuition for what is going on here, compute the limit of the
variance of VT as T → ∞ with v = v̄ using
£
¤
var [WT ] = E WT 2 − {E [WT ]}2
(
¯
¯ )2
∂ 2 £ −ψWT ¤¯¯
∂ £ −ψWT ¤¯¯
−
=
E e
E e
¯
¯
∂ψ 2
∂ψ
ψ=0
ψ=0
= v̄T
η2
+ O(T 0 )
λ2
Then, as T → ∞, the q
standard deviation of annualized variance has the
v̄ η
leading order behavior
. The convexity adjustment should be of the
T λ
129
Figure 2: Annualized Heston convexity adjustment as a function of T with
parameters from Homework 2.
Convexity Adjustment
0.01
0.008
0.006
0.004
0.002
1
2
3
4
5
T
Figure 3: Annualized Heston convexity adjustment as a function of T with
Bakshi, Cao and Chen parameters.
Convexity Adjustment
0.012
0.01
0.008
0.006
0.004
0.002
1
2
130
3
4
5
T
order of the standard deviation of annualized volatility over the life of the
contract. From the last result, we expect this to scale as λη . Comparing
Bakshi, Cao and Chen (BCC) parameters with Homework 2 parameters, we
deduce that the convexity adjustment should be roughly 3.39 times greater
for BCC parameters and that’s what we see in the graphs.
18.7
Replication of Generic Quadratic Variation-Based
Claims in a Diffusion Context
Ground-breaking work by Peter Carr and Roger Lee (Carr and Lee 2003)
extends the well-known replication results for variance-swaps presented in
Section 17 to arbitrary functions of realized variance assuming a diffusion
process for the underlying and zero correlation between moves in the underlying and moves in volatility. We proceed to sketch out their results.
From equation (70), we know that generalized payoffs may be spanned
according to
Z F
Z ∞
00
E [ g(ST )] = g(F ) +
dK P̃ (K, T ) g (K) +
dK C̃(K, T ) g 00 (K) (76)
0
F
where C̃ and P̃ represent undiscounted call and put prices respectively and
F represents the time-T forward price of the stock. With the substitution
g(ST ) = STp , we obtain the replicating portfolio for a power payoff:
Z F
Z ∞
p
p
p−2
E [ ST ] = F +p (p−1)
dK P̃ (K, T ) K +p (p−1)
dK C̃(K, T ) K p−2
0
F
Define the log-strike k ≡ log(K/F ). Then
½
Z
Z 0
p
pk
p
1 + p (p − 1)
dk p̂(k) e + p (p − 1)
E [ ST ] = F
−∞
¾
∞
dk ĉ(k) e
0
pk
(77)
where p̂(k) and ĉ(k) denote the prices of puts and calls respectively in terms
of percentage of strike.
If there is zero correlation between moves in the underlying and volatility
moves, put-call symmetry holds, so that p̂(k) = e−k ĉ(−k). We can then
rewrite equation (77) as
½
¾
Z ∞
p
p
k/2
E [ ST ] = F
1 + 2 p (p − 1)
dk e ĉ(k) cosh (p − 1/2) k
0
131
18.7.1
The Laplace transform of quadratic variation
The main contribution of Carr and Lee (2003) under their assumptions of
diffusion and zero correlation between spot moves and volatility moves is to
have derived an expression for the Laplace transform (moment generating
function) of the quadratic variation WT ≡ hxiT in terms of the fair value of
the power payoff just derived in equation (77). Since we know the replicating
strip for the power payoff, it follows that we know the replicating strip for
quadratic variation. Explicitly, Carr and Lee show that
£
¤
£
¤
E eλhxiT = E ep(λ)xT
p
where p(λ) = 1/2 ± 1/4 + 2 λ.
From equation (77), we then have that
½Z 0
Z
£ λhxi ¤
p(λ) k
T
dk p̂(k) e
+
E e
= 1 + p(λ) (p(λ) − 1)
¾
∞
dk ĉ(k) e
0
−∞
p(λ) k
(78)
or alternatively
£
λhxiT
E e
¤
Z
∞
= 1 + 2 p(λ) (p(λ) − 1)
dk ek/2 ĉ(k) cosh (p(λ) − 1/2) k
0
Noting that p(λ) (p(λ) − 1) = 2 λ, we may simplify further to obtain
Z ∞
£ λhxi ¤
T
dk ek/2 ĉ(k) cosh (p(λ) − 1/2) k
E e
= 1 + 4λ
(79)
0
The fair value of an exponential quadratic-variation (realized volatility) payoff follows immediately from equation (79). By taking derivatives, the fair
value of any positive integral power of quadratic variation also follows. As
a check, consider the fair value of the first power of quadratic variation –
132
expected realized total variance. We have
¯
∂ £ λhxiT ¤¯¯
E e
E [ hxiT ] =
¯
∂λ
½
¾¯
Zλ=0
∞
¯
∂
k/2
=
1 + 4λ
dk e ĉ(k) cosh (p(λ) − 1/2) k ¯¯
∂λ
0
λ=0
Z ∞
= 4
dk ek/2 ĉ(k) cosh (p(0) − 1/2) k
Z0 ∞
= 4
dk ek/2 ĉ(k) cosh k/2
Z0 ∞
©
ª
= 2
dk ĉ(k) 1 + ek
½0Z ∞
¾
Z 0
= 2
dk ĉ(k) +
dk p̂(k)
0
−∞
which agrees exactly with our earlier result (73).
In principle, since we have an explicit expression for the moment generating function (mgf ) of realized variance, we know its entire terminal (time
T ) pseudo-probability distribution and we may compute the fair value of
any European-style claim on realized variance – knowing only market prices
of European options!
18.7.2
Replicating volatility
It turns out√that by far the dominant contribution to the fair value of realized
volatility ( WT ) is the at-the-money forward European option. Indeed we
have
p
√
E[ WT ] ≈ 2 π ĉ(0)
This should be no surprise as we are already very familiar with the extremely
accurate approximation
√
σBS T
ĉ(0) ≈ √
2π
for at-the-money forward European options.
We see however that the volatility replication strategy differs fundamentally from the variance replication strategy. In the variance case, we trade
a strip of options at inception and thereafter daily rebalancing in the underlying only and that only to maintain a constant dollar amount of the
133
underlying in the hedge portfolio. In contrast, in the volatility case we have
to continuously maintain a position in the at-the-money option: each day,
we sell the entire position (which is no longer at-the-money) and buy a new
one. Unlike the variance strategy, this strategy is clearly not practical – the
option bid-offer would kill the hedger.
Similar comments apply to the hedging in practice of any generic claim
using this strategy: it involves daily rebalancing in options.
Nevertheless, Carr and Lee’s result is powerful because it shows that
the fair value of any claim on quadratic variation (under their assumptions)
depends only on the prices of European options and is otherwise completely
model-independent. We can suppose that a way will be found to extend their
results further to the case of non-zero correlation between volatility moves
and underlying moves although not to the general non-diffusive case (see
Friz and Gatheral (2004) for example). Despite this, given our observations
in Section 18.4 on the effect of jumps in practice, it seems likely that any
such extension will be valuable in practice as well as in theory.
19
The VIX futures contract
Earlier this year, the CBOE listed futures on the VIX - an implied volatility
index. Originally, the VIX computation was designed to mimic the implied
volatility of an at-the-money 1 month option on the OEX index. It did
this by averaging volatilities from 8 options (puts and calls from the closest
to ATM strikes in the nearest and next to nearest months). To facilitate
trading in the VIX, the definition of the index was changed. Here is an
excerpt from the FAQ section on the CBOE website:
“How is VIX being changed? Three important changes are being
made to update and improve VIX: 1. The New VIX is calculated using a wide
range of strike prices in order to incorporate information from the volatility
skew. The original VIX used only at-the-money options. 2. The New VIX
uses a newly developed formula to derive expected volatility directly from
the prices of a weighted strip of options. The original VIX extracted implied
volatility from an option-pricing model. 3. The New VIX uses options on
the S&P 500 Index, which is the primary U.S. stock market benchmark. The
original VIX was based on S&P 100 Index (OEX) option prices.
Why is the CBOE making changes to the VIX? CBOE is changing VIX to provide a more precise and robust measure of expected market
134
volatility and to create a viable underlying index for tradable volatility products.
The New VIX calculation reflects the way financial theorists, risk managers and volatility traders think about - and trade - volatility. As such, the
New VIX calculation more closely conforms to industry practice than the
original VIX methodology. It is simpler, yet it yields a more robust measure of expected volatility. The New VIX is more robust because it pools
information from option prices over a wide range of strike prices thereby
capturing the whole volatility skew, rather than just the volatility implied
by at-the-money options. The New VIX is simpler because it uses a formula
that derives the market expectation of volatility directly from index option
prices rather than an algorithm that involves backing implied volatilities out
of an option-pricing model. The changes also increase the practical appeal
of VIX. As noted previously, the New VIX is calculated using options on
the S&P 500 index, the widely recognized benchmark for U.S. equities, and
the reference point for the performance of many stock funds, with over $800
billion in indexed assets. In addition, the S&P 500 is the domestic index
most often used in over-the-counter volatility trading. This powerful calculation supplies a script for replicating the New VIX with a static portfolio of
S&P 500 options. This critical fact lays the foundation for tradable products
based on the New VIX, critical because it facilitates hedging and arbitrage
of VIX derivatives. CBOE has announced plans to list VIX futures and
options in Q4 2003, pending regulatory approval. These will be the first of
an entire family of volatility products.”
This explanation is intriguing. To see what the CBOE means, consider
the VIX definition (converted to our notation) as specified in the CBOE
white paper (CBOE 2003).
·
¸2
2 X ∆Ki
1 F
V IX =
Qi (Ki ) −
−1
T i Ki2
T K0
2
where Qi is the price of the out-of-the-money option with strike Ki and K0
is the highest strike below the forward price F . We recognize this formula
as a simple discretization of equation (73) and makes clear the reason why
the CBOE implies that the new index permits replication of volatility.
135
19.1
How did they perform?
Volume on VIX futures (strictly speaking VXB futures where the VXB is
defined to be 10 times the VIX) has been negligible. For example, at the
time of writing, the Jan-05 VXB future had been trading an average of 155
contracts per day. At the current price of 143.90, that corresponds to roughly
$2.25 notional per day. In contrast, Dec04 SPX futures trade around 50,000
contracts per day which corresponds to around $15bn notional per day.
As a practical example of how the replicating strip (73) is associated with
expected variance, settlement is on the Wednesday before the third Friday
based on the opening prices of options expiring in the following month.
19.2
A subtlety
VXB futures settle based on the square root of the value of the replicating
strip (i.e. on volatility rather than variance) so there must be a convexity
adjustment. However, this isn’t exactly the same as the convexity adjustment
p
p
E[WT ] − E[ WT ]
that we computed earlier.
As of the valuation date, the convexity adjustment relevant to the VXB
futures contract is given by
·q
¸
q
E[Wt,T ] − E
Et [Wt,T ]
where t is the settlement date, T is the expiration date of options in the
strip and the expectation is computed as of some valuation date prior to t.
As before, we can easily compute E[Wt,T ] from strips of options expiring
hp at times
i t and T respectively but we need a model to compute
E
Et [Wt,T ] .
One obvious question is where the market prices this convexity adjustment. As of December 8th, 2004 the spot VXB was at 132.30. Computing
the expected forward variances E[Wt,T ] from market prices of options we
obtain the empirical convexity adjustments shown in Table 1.
The next question is what the VXB futures are worth theoretically.
136
Table 1: Empirical VXB convexity adjustments as of 08-Dec-2004.
Expiry
Spot
Jan-05
Feb-05
19.3
VXB Price Fwd Variance Convexity adjustment
132.30
132.30
0.00
144.20
149.09
4.89
153.30
159.63
6.33
VXB convexity adjustment in the Heston Model
To compute the VXB convexity adjustment in the Heston model, we first
note that expected one-month expected total variance Et [Wt,T ] given the
instantaneous variance vt is
f (vt ) = Et [Wt,T ] = (vt − v̄)
with τ = T − t.
Then
·q
¸
Z
E
Et [Wt,T ] =
1 − e−λ τ
+ v̄ τ
λ
∞
q(vt |v0 )
p
f (vt ) dv
(80)
0
where q(v|v0 ) is the pdf of instantaneous variance v given the initial variance
v0 .
Computing the integral in equation (80) numerically using Heston parameters from a fit to December 8, 2004 data, we get the graph of the
convexity adjustment as a function of time to expiration shown in Figure 4.
We see that for the two futures maturities listed in Table 1, the Heston
convexity adjustments were respectively around 7.4 and 8.6 points - not so
different from the empirically-observed numbers.
19.4
Futures on realized variance
The CBOE also lists futures on 3-month realized variance with symbol VT.
These futures really hardly trade at all. Their payoff is given by
N
1 X 2
Realized Variance = 252
r
N −1 i i
where N is the number of observation days in the 3-month trading period.
The VT future becomes decreasingly volatile as settings are made; settlement
137
Figure 4: Annualized Heston VXB convexity adjustment as a function of t
with Heston parameters from 08-Dec-2004 SPX fit.
VXB Adjustment
14
12
10
8
6
4
2
0.25
0.5
0.75
1
1.25
1.5
t
is effectively based on average realized daily variance. It would be nice to
compare the price of the VT future with theory but as an example of its
illiquidity, there wasn’t a single trade on December 8, 2004.
19.5
Spin-off benefits
Although these futures contracts weren’t very successful by usual futures
standards, they did stimulate the OTC market to very substantially tighten
spreads. Now SPX variance swaps can be traded from 6 months to 5 years
with bid-offer spreads of as little as 0.5 vol points. Single-stock variance
swaps are also now actively quoted; for the top 50 US single stock names,
bid-offer spreads are around 2.5 vol points.
20
Epilog
I hope that this series of lectures has given students an insight into how
financial mathematics is used in the derivatives industry. It should be apparent that modeling is an art in the true sense of the word – not a science,
although when it comes to implementing the chosen solution approach, science becomes necessary. We have seen several examples of claims which may
be priced differently under different modeling assumptions even though the
138
models generate identical prices for European options. The importance of
lateral thinking outside the framework of a given model cannot be overemphasized. For example, what is the impact of jumps? of stochastic
volatility? of skew? and so on. Finally, intuition together with the ability
to express this intuition clearly is ultimately what counts; without intuition
and the ability to communicate it clearly to non-mathematicians, financial
mathematics becomes no more than a theoretical exercise divorced from the
real business of making money.
139
References
Breeden, D., and R. Litzenberger, 1978, Prices of state-contingent claims
implicit in option prices, Journal of Business 51, 621–651.
Carr, Peter, and Roger Lee, 2003, The robust replication of volatility
derivatives,
http://www.math.nyu.edu/research/carrp/papers/pdf/Voltradingoh1.pdf.
Carr, Peter, and Dilip Madan, 1998, Towards a theory of volatility trading, in
Robert A. Jarrow, ed.: Volatility: New Estimation Techniques for Pricing
Derivatives . chap. 29, pp. 417–427 (Risk Books: London).
CBOE, 2003, VIX: CBOE volatility index,
http://www.cboe.com/micro/vix/vixwhite.pdf.
Chriss, Neil, and William Morokoff, 1999, Market risk for variance swaps,
Risk 12, 55–59.
Cox, John C., Jonathan E. Ingersoll, and Steven A. Ross, 1985, A theory of
the term structure of interest rates, Econometrica 53, 385–407.
Demeterfi, Kresimir, Emanuel Derman, Michael Kamal, and Joseph Zou,
1999, A guide to volatility and variance swaps, Journal of Derivatives 6,
9–32.
Friz, Peter, and Jim Gatheral, 2004, Valuing volatility derivatives as an
inverse problem, Discussion paper Courant Institute of Mathematical Sciences, NYU.
Matytsin, Andrew, 1999, Modeling volatility and volatility derivatives,
Columbia Practitioners Conference on the Mathematics of Finance.
140
A
Proof of Equation (74)1
As usual, the undiscounted European call option price is given by
½ µ
µ
√ ¶
√ ¶¾
k
w
k
w
k
C = F N −√ +
− e N −√ −
2
2
w
w
where k := ln(K/F ) is the log-strike.
Differentiating wrt the strike K we obtain
µ
µ
√ ¶
√ ¶ √
k
1 ∂C
k
∂C
w
w ∂ w
0
+ N −√ −
=
= −N − √ −
∂K
K ∂k
2
2
∂k
w
w
Then with the notation
√
√
k
w
k
w
d1 = − √ +
; d2 = − √ −
2
2
w
w
and differentiating again wrt K, we obtain
½
µ
µ
√ ¶
√ ¶ √ ¾
∂ 2C
1 ∂
k
w
k
w ∂ w
0
=
−N − √ −
+ N −√ −
2
∂K
K ∂k
2
2
∂k
w
w
½
µ
√ ¶
√ ¾
0
2
N (d2 )
∂d2
∂ w
∂ w
=
−
1 + d2
+
K
∂k
∂k
∂k 2
Ã
(
!
µ
¶
√
√
√ )
2
0
2
∂ w
N (d2 )
1
2k ∂ w
∂ w
√
d1 d2
=
−√
+1 +
K
∂k
∂k 2
w
w ∂k
From Section 18, the fair value of a variance swap under diffusion may be
obtained by valuing a contract that pays 2 ln (ST /F ) at maturity T . Bruteforce calculation leads to
·
¸
µ ¶ 2
Z ∞
ST
K ∂ C
2E ln
= 2
dK ln
F
F ∂K 2
µ
½
√ ¶
√ ¾
Z0 ∞
∂d2
∂ w
∂2 w
0
= 2
dk k N (d2 ) −
1 + d2
+
∂k
∂k
∂k 2
−∞
½
√ ¾
Z ∞
∂d2 ∂ w
−
= 2
dk N 0 (d2 ) −k
∂k
∂k
−∞
Z ∞
∂d2
=
dk N 0 (d2 )
w
∂k
−∞
which recovers equation (74) as required.
1
This particularly neat proof is due to Chiyan Luo.
141