Download Pricing Volatility Derivatives with General Risk Functions Alejandro Balbás University Carlos III

Pricing Volatility Derivatives
with General Risk Functions
Alejandro Balbás
University Carlos III of Madrid
[email protected]
Content
•
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Introduction.
Describing volatility derivatives.
Pricing and hedging methods.
Using infinitely many options.
Synthetic variance swap.
Synthetic volatility swap.
Synthetic variance and volatility swap options.
Dealing with the available options and risk measures.
Applications in Nordpool
Conclusions.
2
Introduction
• The underlying variable is the realized
volatility
financial
contract,
asset.
or variance of a
asset over the life
rather than the price
traded
of the
of this
3
Introduction
• They are traded because:
–They easily provide diversification. They don’t
depend on prices evolution, but on how fast prices
are changing.
–As the empirical evidence points out, they provide
adequate hedging when facing market turmoil,
which usually implies correlations quite close to
one.
–Hedge funds and other risk seeking investors
usually sell these products due to their high risk
premium.
• The last two reasons imply very asymmetric returns,
with heavy tails.
• Sellers usually earns money very often, but the profits
are really small. They rarely loose money, but looses are
very high when they arise.
4
Introduction
• The are usually traded in OTC markets,
so estimating the total trading volume
is problematic.
• But recent estimates for daily trading
volume on indices are in the region of
$30 to $35 millions of notional.
(Broadie, M. and A. Jain, 2008. “Pricing and
hedging volatility derivatives”. Journal of
Derivatives. 15, 3, 3-24.)
5
Introduction
• Approximately one year and a half ago the
CBOE (Chicago Board Option Exchange) has
been trading volatility derivatives. They have
several derivatives:
–VIX
–VXN, etc.
www.CBOE.com
• Besides, there are volatility funds, such us
the
Euroption
Strategic
Fund
(www.europtionfund.com), that only deals
with volatility derivatives, mainly on
European indexes.
6
Describing volatility derivatives
• Variance and Volatility Swap:
dSt
= µ ( St , t )dt + σ ( St , t )dz
St
is a Brownian Motion reflecting the evolution of the
underlying security. The realized variance in [0,T] is
defined by the stochastic integral
1 T 2
W = ∫ σ ( St , t )dt
T 0
T
and the realized volatility is:
T
V = W
T
7
Describing volatility derivatives
• Variance and Volatility Swap:
–It may be proved that
T
T
ˆ
ˆ
Lim( ∆t →0 )V (n) = V
and
T
T
ˆ
ˆ
Lim( ∆t →0 )W (n) = W
with convergence in probability at least.
(Demeterfi, K., E. Derman, M. Karmal, and J. Zou. “A
Guide to Volatility and Variance Swaps”. Journal of
Derivatives, 4 (1999), pp. 9 - 32.)
8
Describing volatility derivatives
• Variance and Volatility Swap:
– Consider a finite set of trading dates:
– The estimated variance in [0, T] is given by:
n
1
2
Wˆ T (n) =
r
∑
i
n(∆t ) i =1
 S ti
where ri = L
 St
 i−1




and ∆t = ti − ti −1 , i = 1,2,..., n . Similarly, the estimated
volatility is given by the square root of the previous
n
expression:
1
T
2
Vˆ (n) =
r
∑
n(∆t )
i
i =1
9
Describing volatility derivatives
• Variance and Volatility Swap:
– A long position in a Variance TSwap implies the
payment of a (numerical) price Wˆ 0 at t = 0, so as to
T
receive the random amount ΦWˆ ( n) at t = T,
Φ > 0 being a notional value known by the agents.
T
– The buyer of a Volatility Swap must pay Vˆ0
T
at t = 0 so as to receive the random pay-off ΦVˆ ( n)
at t = T.
10
Describing volatility derivatives
• Variance Swap Future:
– A long position in a Variance Swap Future with
maturities T1 < T2 implies to accept at t = 0 a
commitment so as to purchase a Variance Swap at
T1 with maturity at T2 .Consider the trading
dates
0 = t0 < t1 < ... < t k = T1 < t k +1 < ... < t k + h = T2
with
∆t = ti − ti −1 , i = 1,2,..., k + h
11
Describing volatility derivatives
• Variance Swap Future:
– Then:
k +h
1
2
Wˆ (k + h) =
r
∑i
(k + h)(∆t ) i =1
T2
k
1
2
Wˆ T1 (k ) =
r
∑
i
k (∆t ) i =1
k +h
1
(T ,T )
2
ˆ
W
( h) =
ri
∑
(h)(∆t ) i = k +1
1
2
manipulating we obtain:
(T1 ,T2 )
ˆ
W
( h) =
T2 ˆ T2
T1 ˆ T1
W ( k + h) −
W (k )
T2 − T1
T2 − T1
12
Describing volatility derivatives
• Variance Swap Future:
–Theorem 1: The Variance Swap Future with maturities at
T1 and T2 is replicated with:
the purchase of
T2
T2 − T1
Variance Swaps with maturity at
T2
T1
− r (T2 −T1 )
e
the sale of
T2 − T1
Variance Swaps with maturity at
and borrowing the price of the strategy above
T1
T2 ˆ T2
T1 ˆ T1
W0 −
W0
T2 − T1
T2 − T1
during the time interval [0, T ]
1
–Therefore, the price of the Variance Swap Future is given by
 T2 ˆ T2
T1 ˆ T1 − r (T2 −T1 )  rT1
(T1 ,T2 )
ˆ
e
W0
= 
W0 −
W0 e
T2 − T1
 T2 − T1

13
Describing volatility derivatives
• Variance Swap Option:
– We will consider that the buyer of a Variance Swap
European Call (Put) with maturities T1 < T2 has the
right (no obligation) to buy (sell) a Variance Swap at
T1 with maturity at T2 , and he will pay (receive) the
strike E.
– The price of the option is paid at t = 0, whereas the
strike is paid at T1
14
Describing volatility derivatives
• Covariance Swap:
– Consider a finite set of trading dates:
0 = t0 < t1 < ... < t n = T
– Consider also two underlying assets whose price
processes will be denoted by S t and S t . Their
realized covariance in [0,T] is given by
where
n
1
Cˆ T (n) =
ri ri
∑
n(∆t ) i =1
 St 

ri = L
 St 


i
i −1
15
Describing volatility derivatives
• Covariance Swap:
– The buyer of a Covariance Swap with
maturity at T will pay at t = 0 the
numerical price Cˆ 0T, and he will receive
T
ˆ
the random pay-off ΦC ( n) at t = T.
– Where Φ is the notional price of one
covariance point.
16
Pricing and hedging methods
• Mainly there are two methods:
–Method 1: using infinitely many
options.
–Method 2: using stochastic complete
pricing models.
17
Pricing and hedging methods
• Method 1:
–It was introduced by:
Neuberger, A.J. “The Log Contract”. Journal of Portfolio
Management, Vol. 20, No. 2 (1994), pp. 74 – 80.
–And further developments were given by Demeterfi, K.
et at.
–The first article considered an extension of the BlackScholes model and proved that the realized volatility
equals:
Φ
g vas ( ST ) =
T
 ST
ST

−
1
−
L
 FT
F0T
 0



18
Pricing and hedging methods
• Method 1:
• Later, the second extended the analysis so as to
•
involve some stochastic volatility models.
The authors also showed that minor
modifications of the formula allow us to accept
the existence of jumps in the underlying asset
price process.
19
Pricing and hedging methods
• Method 1:
• From the formula above and the equality
h
g ( S ) = ( g (h) − hg ' (h)) + Sg ' (h) + ∫ g ' ' (k )(k − S )dk
S
for a tow times differentiable function those papers
stated that the variance swap may be replicated:
–Buying
–Buying
Φ
T
Φ
T
1
dk
2
k
1
dk
2
k
European Puts with strike k, 0 <
European Calls with strike k,
k < F0T
F0T < k < ∞
where the maturity of the options equals the maturity of the
variance swap.
20
Pricing and hedging methods
• Proof:
• Case 1:
0
0
•
T
0 makes the remaining terms vanish.
h=F
21
Pricing and hedging methods
• Case 2:
0
0
22
Pricing and hedging methods
• Method 2:
–Javaheri, Wilmott, and Haug (2002) discussed the
valuation of volatility swaps in the GARCH(1, 1)
stochastic volatility model.
(Javaheri, A., P. Wilmott, and E. Haug. “Garch and
Volatility
Swaps”.
Working
paper,
2002,
http://www.wilmott.com).
–Little and Pant (2001) developed a finite difference
method for the valuation of variance swaps in the case
of discrete sampling in an extended Black-Scholes
framework.
(Little, T., and V. Pant. “A Finite Difference Method for
the Valuation of variance Swaps”. Journal of
Computational Finance, Vol. 5, No. 1 (2001), pp. 74 80).
23
Pricing and hedging methods
• Method 2:
–Carr et al. (2005) priced options on realized variance
by directly modeling the quadratic variation of the
underlying process using a Lévy process.
(Carr, P., H. German, D. Madan, and M. Yor. “Pricing
Options on Realized Variance” . Finance and
Stochastics, Vol. 9, No. 4 (2005), pp. 453 - 475).
24
Pricing and hedging methods
–Therefore, the variance swap contract has been priced
by using infinitely many options, whereas the remaining
contracts have been priced by using an stochastic
volatility pricing model.
–The first method has advantage since it doesn’t
depend on the theoretical price we use.
–Option prices are given by the market.
–The drawback is that an infinite number of options
can’t be traded.
25
Pricing and hedging methods
–In a recent paper, Broadie and Jain (2008) use the
Heston model and propose to price and hedge volatility
swaps and variance swap options by using variance
swaps.
–Thus, they hedge a position in volatility by trading
variance swaps and consequently by trading infinitely
many options.
–They use the pay-off variance as a risk measure to be
minimized when approximating the variance swap
contract by a finite combination of the available options.
26
Pricing and hedging methods
–In this paper we propose to use infinitely many options
to replicate every volatility derivative without using the
variance swap to connect the options and the asset to
be priced and hedged.
–This makes our approach be independent on any
pricing model.
–We will use both European and digital options since
digital options may be easily priced in practice from the
combination provided by the European options.
27
Pricing and hedging methods
–Furthermore, the committed error due to the lack of infinitely
many available options will be computed by using a general risk
function, (an expectation bounded risk measure or a deviation
measure).
(Rockafellar, R.T., S. Uryasev and M. Zabarankin, 2006.
“Generalized Deviations In Risk Analysis”. Finance & Stochastics,
10, 51 - 74).
–Since heavy tails and asymmetries are usual when dealing with
volatility the use of the variance could be a shortcoming.
(Ogryczak, W. and A. Ruszczynsky, 2002. “Dual Stochastic
Dominance and Related Mean Risk Models”. SIAM Journal on
Optimization, 13, 60 - 78).
28
Using infinitely many options
• From the formulas
S
g ( S ) = g (h) + ∫ g ' (k )dk
a
for a differentiable function, and
h
g ( S ) = ( g (h) − hg ' (h)) + Sg ' (h) + ∫ g ' ' (k )(k − S )dk
S
for a 2 times differentiable function, we can
prove the following results:
29
Using infinitely many options
• Theorem 2: Let be a, b ∈ ℜ ∪ {−∞, ∞} and g : (a, b) → ℜ
an arbitrary function such that g and its first derivative
exist and are continuous out of a finite set
D = {d1 < d 2 < ... < d m } ⊂ (a, b) . Suppose that g may
be extended and become continuous on (a, d1 ] , and the
same property holds if (a, d1 ] is replaced by
[d i , d i +1 ], i = 1,..., m − 1 , or by [d m , b) . Denote by
J i the jump of g at d i , i = 1,..., m . Then, if
h ∈ (a, b), h ≤ d1 and ST is a (random) price at T
such that
µ ( ST ∈ (a, b)) = 1, µ ( ST ∈ D ∪ {h}) = 0
the final pay-off
g ( ST )
may be replicated by:
30
Using infinitely many options
−
−rf T
• Investing g (h) e euros in the riskless asset.
• Buying J i Digital Calls with strike d i , i = 1,2,..., m .
• Buying g ' (k )dk Digital Calls for every strike
k ∈ ( a , b ) \ D, k > h .
• Selling g ' (k )dk Digital Puts for every strike
k ∈ ( a , b ) \ D, k < h .
31
Using infinitely many options
• Proof:
1
0
32
Using infinitely many options
• Theorem 3: Let be a, b ∈ ℜ ∪ {−∞, ∞}and g : (a, b) → ℜ
an arbitrary function such that g and its first and
second derivatives g’ and g’’ exist and are continuous
out of a single element d ∈ (a, b) . Suppose that g may
be extended and become continuous on ( a, d ], and the
same property holds if ( a, d ] is replaced by [ d , b) .
Denote by J d and J 'd the jumps of g and g’ at d.
Then, if ST is a (random) price at T such that
µ ( ST ∈ (a, b)) = 1, µ ( ST = d ) = 0
the final pay-off
g ( ST ) may be replicated by:
33
Using infinitely many options
• Investing (g (d ) − hg ' (d ) )e
−
•
•
•
•
•
−
−rf T
euros in the riskless
asset.
−
g
'
(
d
)
Buying
units of the underlying asset.
Buying g ' ' ( k ) dk European Puts with strike k for
every k ∈ ( a, d ) .
Buying g ' ' ( k ) dk European Calls with strike k for
every k ∈ ( d , b) .
Buying J 'd European Calls with strike d.
Buying J d Digital Calls with strike d.
34
Using infinitely many options
• These theorems apply to replicate every pay-off
g regular enough. In particular, the variance
•
and volatility swaps.
For volatility swaps notice that the pay-off
function
Φ
g vas ( ST ) =
T
 ST
ST

−
1
−
L
T
T

F0
 F0



is continuous, but its first derivative presents a
jump at ST = F0T whose value is
2Φ .
T
0
2T F
35
Using infinitely many options
36
Synthetic Variance Swap
• If µ ( ST > 0) = 1 holds, the following strategies
•
replicate the variance swap:
Strategy 1: (Digital options)
 ST  − r f T euros.
– Lending Φ  h
 T − 1 − L T  e
T  F0
 F0 
– Buying Φ  1 1  Digital Calls for every strike k,
 T −  dk
T  F0 k 
h<k <∞.
– Selling Φ  1 − 1  dk Digital Puts for every strike k,


T  F0T k 
0<k <h.
37
Synthetic Variance Swap
• Strategy 2: (European options)
– Borrowing Φ  L h  e − r f T euros.


T   F0T 
– Buying Φ  1 − 1  units of the underlying asset.

T  F0T

h
Φ 1
dk
2
T k
0<k <h.
– Selling Φ 1
dk
2
T k
h<k <∞.
– Buying
European Puts with strike k,
European Calls with strike k,
38
Synthetic Volatility Swap
T
µ
(
S
=
F
• If µ ( ST > 0) = 1 and
T
0 ) = 0 , then:
• Strategy 1: (Digital options)
1
1
Φ
k
−
– Buying
dk Digital Calls for every strike
T
F0 T g vas (k )
(
T
0
k ∈ F ,∞
).
1 Φ k −1
dk Digital Puts for every strike
– Selling T
F0 T g vas (k )
(
k ∈ 0, F0T
).
39
Synthetic Volatility Swap
• Strategy 2: (European
options)
T
Φ F0 − 1 − r f T euros.
e
T
2T F0
Φ F0T − 1
– Borrowing
– Selling
– Buying
( )
T 2
0
2T F
4 F0T
units of the underlying asset.
Φ g vas (k ) 4 − (k − F0T ) 2
dk
2k g vas (k )
T
(1)
T
European Puts with strike k, for every k ∈ (0, F0 ) .
T
– Buying (1) European Calls with strike k for every k ∈ ( F0 , ∞) .
T
– Buying 2 / 2 European Calls with strike F0 .
40
Synthetic Variance and Volatility
Swap Options
• For the sake of simplicity let us assume that the option maturity
equals the variance or volatility swap maturity. In the general case,
under very weak assumptions we can also prove that the volatility
swap final pay-off is a function depending on ST, underlying price at 41
the option maturity.
Synthetic Variance and Volatility
Swap Options
• The Variance Swap Option can be replicated with
•
2 strategies.
Strategy 1: (Digital options):
– Buying
Φ 1 1
 T − dk
T  F0 k 
Digital Calls for every strike
k > S2 .
– Selling
Φ 1 1
 T − dk
T  F0 k 
Digital Puts for every strike
k < S1 .
42
Synthetic Variance and Volatility
Swap Options
• Strategy 2: (European options):
S1 
Φ
– Investing
1 − T  euros in the riskless asset.
T  F0 
Φ 1
1
 T −  units of the underlying asset.
– Buying
T  F0 S1 
Φ 1
– Buying
dk European Puts for every strike k < S1 .
2
T k
Φ 1
– Buying
dk European Calls for every strike k > S 2.
2
T k
Φ 1
1 
 − T  European Calls with strike
T  S1 F0 
1 
Φ 1
 European Calls with strike
– Buying  T −
T  F0 S 2 
– Buying
S1
.
S2 .
43
Synthetic Variance and Volatility
Swap Options
• Volatility Swap Options:
– Analogously, from theorems 2 and 3, we can see that
a volatility swap option may be replicated by using
infinitely many Digital or European options.
44
Dealing with the available options
and risk measures
• As already said Demeterfi et al. proposed an
•
•
heuristic procedure to approximate the variance
swap whereas Broadie and Jain solve this
problem by minimizing the variance of the
committed error.
This authors don’t consider any transaction
costs.
We
will
provide
a
general
method
simultaneously dealing with the lack of infinitely
many options, transaction costs and a general
risk functions.
45
Dealing with the available options
and risk measures
• rf is the risk free rate.
• ST will be the underlying final pay-off.
• Y1 ,...Yn will be the final pay-offs provided by the
•
available options.
There are 4 cases:
0, ST < E j
– For Digital Calls: Y j = 1, S > E
j
 T
1, ST < E j
– For Digital Puts: Y j = 
0, ST > E j
+
(
)
S
−
E
– For European Calls: T
j
(
– For European Puts: E j − ST
)
+
46
Dealing with the available options
and risk measures
•
•
•
0
S 0 ≤ Si :bid/ask of the underlying asset.
yi ≤ y :bid/ask of Y .
ρ denotes
i
an expectation bounded risk
p
measure, defined over
, where p satisfies
p
p
ST ∈ L . Thus, Y1 ,...Yn ∈ L too.
• Usually p = 1.
0
1
n
( x f , x0 , x , x1 , x ,..., xn , x ) will be the hedging
strategy.
L
47
Dealing with the available options
and risk measures
rf T
• P( x) = x f e + ( x 0 − x0 ) ST + ( x1 − x1 )Y1 + ... + ( x n − xn )Yn
will be the pay-off.
• p( x) = x f + x 0 S 0 − x0 S0 + x1 y1 − x1 y1 + ... + x n y n − xn yn
will be the price. min ρ ( x
• We may prove
fe
rf T
n
+ ∑ ( x i − xi )Yi − g ( ST ))

i =1

n
n

i i
x f + ∑ x y − ∑ xi yi ≤ A

i =1
i =1

x≥0


−rf T
− rf T

F
(
A
)
e
+
A
=
F
(
0
)
e
that
remains
constant and doesn’t depend on A. And it will be
call the ask price of g ( ST ) .
48
Dealing with the available options
and risk measures
• The solution x of the problem doesn’t depend on
A, except for the risk-free asset.
• The previous problem may be solved in practice
•
•
by duality methods. So, suppose that q is the
1 1
conjugate of p
+ =1
Usually p = 1 and q = ∞ . p q
∆ ρ ⊂ Lq is the sub gradient of ρ (Rockafellar et
at. (2006)). For example if ρ = CVaR95% then:
{
}
∆ ρ = z ( ST ) ∈ L∞ ; E ( z ( ST )) = 1,0 ≤ z ( ST ) ≤ 20
49
Dealing with the available options
and risk measures
• The dual problem is:
max E ( g ( ST ) z ( ST ))


E ( z ( ST )) = 1

rf T

0 rf T
S 0 e ≤ E ( z ( ST ) ST ) ≤ S e

Y e r f T ≤ E ( z ( S )Y ) ≤ Y j e r f T , j = 1,..., n
T
j
 j
• The dual problem is always linear.
• Both problems are analyzed and solved in Balbas, A., R.
Balbas and S. Mayoral (2009), Portfolio Choice Problems
and Optimal Hedging with General Risk Functions,
European Journal of Operational research, 192, 2, 603620.
50
Conclusions
• Volatility derivatives are becoming very useful
•
because they make it easy to compose
diversified portfolios and also offer protection for
market turmoil.
We have proved that all of them , including
volatility swaps and variance or volatility futures
and call and put options may be replicated by a
combination of infinitely many European options.
51
Conclusions
• They can be also replicated with digital options.
•
It seems to be an interesting result since digital
options are easily priced and replicated by using
the underlying.
Moreover, to overcome the lack of infinitely
many options in the market, we have provided a
pricing and hedging method related to
Expectation Bounded Risk Measures. The
method also incorporates the usual market
imperfections.
52
Thank you so much
for your attention
Alejandro Balbás
University Carlos III of Madrid, Spain