Lecture 4

Transcription

Lecture 4

c 2015 Juliusz Jablecki: Equity and Fixed Income
Equity and Fixed Income
Juliusz Jabłecki
Banking, Finance and Accounting Dept.
Faculty of Economic Sciences
University of Warsaw
[email protected]
and
Head of Monetary Policy Analysis Team
Economic Institute, National Bank of Poland
1
Lecture 4:
VIX and volatility derivatives
Plain vanilla equity options, which we have discussed so far, allow investors to speculate on the movement in the price of the
underlying stock. In this lecture we introduce an important new
class of equity derivatives, in which investors speculate not on
the price of the underlying instrument, but its volatility. In this
lecture we will try to find answers to the following questions:
• what is volatility?
• what’s the rationale for investing in volatility (a priori)?
• how to invest in volatility?
• does it make sense to invest in volatility (ex post)?
2
What is volatility?
We need to start with a philosophical observation. Unlike stock
price, the volatility of stock price is unobservable – what we observe is the movement of prices with the flow of time. Volatility
is merely a statistical metric by which we choose to describe this
movement. And since volatility has to be estimated, this opens a
Pandora’s box of nontrivial statistical dilemmas. Fortunately, in
this lecture we take a market-oriented (rather than econometric)
point of view and will associate volatility with one of two things:
• realized volatility, i.e. annualized standard deviation of
log-returns on a given stock/index in a certain time frame
(calculated under the assumption of a zero mean)
v
u
u
u
u
u
t
252 N
Si+1 2
X 
ln

σ=
N − 1 i=1
Si


• implied volatility, i.e. a parameter Σ(K, T ), which plugged
into the Black-Scholes formula for the value of a European
call option with strike K and expiration time T yields the
market value of that option.
3
The difference between realized and implied volatility
4
Realized volatility is a backward-looking measure (historical) while
implied volatility is forward-looking.Implied volatility is essentially
the price one has to pay for gaining implicit or explicit exposure
to volatility of a given index that will materialize over a certain
future time horizon.
This is clear if we consider a trader who writes a call option C
on an underlying S with implied volatility Σ and hedges away
the underlying (“delta”) risk by going long (∂C/∂S) units of the
underlying instrument (typically a futures contract).
If we strip the option off of its linear componet, we become long
a quadratic payoff:
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To understand this better Taylor-expand C(S, t):
∂C
1 ∂ 2C
∂C
2 +...
(∆S)
C (S + ∆S, t + ∆t) = C(S, t)+ ∆S+ ∆t+
∂S
∂t
2 ∂S 2
Hence the profit-and-loss, P &L, on the delta hedged option is
∂C 
∂C
1 ∂ 2C
2,

P &L = d −C(S, t) +
S  ≈ − ∆t −
(∆S)
∂S
∂t
2 ∂S 2


Since the option is written at Black-Scholes implied volatility Σ,
it also satisfies the Black-Scholes differential equation, i.e.:
∂C
∂C 1 2 2 ∂ 2C
+ rS
+ Σ S
= rC.
2
∂t
∂S 2
∂S
Assuming that the risk free rate r is close to zero:
∂C
1 2 2 ∂ 2C
Σ S
≈− ,
2
∂S 2
∂t
So that

∆S 2

1 2  2
P &L ≈ ΓS Σ ∆t − 
2
S



 

.
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Reason #1: Volatility is negatively correlated with equity returns.
Why invest in volatility?
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Reason #2:Volatility tends to increase with market uncertainty.
8
Reason #3: Equity volatility is negatively correlated with credit returns (recall the Merton model!)
9
Hence, there are two typical reasons for investing in volatility
• Expressing views on the most likely future path of volatility.
This can be either a relatively simple strategy of betting on
the future value of (expected) volatility by going outright
long or short volatility, or a more sophisticated trade based
on the whole term structure of volatility (e.g. going short
near-term volatility and long medium-term volatility), as well
as a relative value trade between implied/realized vol.
• Risk mitigation. Many categories of investors hold positions
that are explicitly or implicitly short volatility (e.g. bond and
equity investors are short volatility because of the negative
correlation between equity and credit returns and volatility).
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How to invest in volatility?
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Instrument 1: Variance swaps
Variance swaps are bets on future realized volatility. The payoff
of a variance swap at maturity can be expressed as:
V = N otional × (RV 2 − Σ2)
s
2
n ln (S /S
)
is the realized volatility, i.e.
where RV = N
t
t−1
t=1
n
standard deviation of log-returns (with an assumed zero mean)
during the n days until maturity, N is the number of trading
days, and Σ is the strike value of the swap. What is the fair
strike Σ of a variance swap?
P
V = 0 ⇐⇒ Σ2 = E RV 2
!
It turns out that:
Σ2 ≈
2erT
T


M
C(T, Ki)
X P (T, Ki )
X
 N



∆K
+
∆K

i
i ,
2
2
Ki
Ki
i=1
i=N +1
where K1 < ... < KN < ... < KM are the successive strikes of
N puts worth P (T, Ki) and M calls worth C(T, Ki).
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Instrument 2: Implied volatility (VIX) futures
Implied volatility futures allow betting on future implied volatility
of a stock or, more commonly, an index. But implied volatility
of which option?
CBOE has made investors’ lives easier by introducing an implied
volatility index called VIX which is defined as the strike of a one
month variance swap:
V IX =
v
u
u
u
u
u
u
t
2erT
T


N P (T, Ki )
M
C(T, Ki)
X

 + ∆K
+
∆K

i
i
2
2
Ki
Ki
i=1
i=N +1
 X


13
VIX represents the market’s expectation of volatility over the next
month. VIX futures are futures contracts on the VIX.
E.g. on 17 April 2015 the price of May 2015 VIX futures expiring
on May 20, 2015 was 15.685. This means that the market was
pricing the volatility over the period May 20 to June 17, 2015 at
15.685%.
14
Instrument 3: VIX options
Two years after the launch of VIX futures, in February 2006, plain
vanilla call and put options on the VIX index began trading on
the CBOE. Options are European style – i.e. can be exercised
only on the expiration date – and settle to the T -day opening
quotation of the VIX. On the settlement date T the payoffs are:
call: $100×max(0, V IXT −K), put: $100×max(0, K−V IXT )
where K is the strike price quoted in volatility points. The contracts are structured so that the value of one volatility point is
$100. Although there are currently only 6 listed expiries for VIX
options, corresponding to six following months (versus 8 expiries
for VIX futures), VIX options are the second most liquid group
of option contracts listed on CBOE, slowly approaching SPX options in terms of open interest.
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The benefits of investing in volatility
To analyze the benefits of investing in volatility consider the effects of adding volatility exposure to a benchmark portfolio of
stocks and bonds. Volatility exposure is defined in three different
ways: (i) a long position in implied volatility; (ii) short position
in realized volatility; and (iii) a combination of a short position
in realized volatility and a long position in implied volatility. The
analysis uses daily data for the period 2003-2013.
Benchmark portfolio (40% equity + 60% bonds)
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Long volatility strategy: adding VIX futures to benchmark
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Short realized volatility strategy: adding variance swap to benchmark
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Return
Std. dev
Sharpe ratio
99% VaR
1. Benchmark
0.068
0.080
0.602
-0.307
2. Long vol
0.092
0.073
0.993
-0.169
3. Short vol
0.100
0.107
0.748
-0.440
4. Long/short vol
0.147
0.080
1.586
-0.116
Adding any kind of volaility exposure improves the performance of the benchmark portfolio on a risk-adjusted basis. However, the most beneficial strategy – in the sense of
maximizing return and minizming value-at-risk – combines exposure to both implied
volatility and realized volatility.
Performance of the strategies:
19
Exam-like problems
1. Please explain the difference between implied and realized
volatility.
2. Please list and briefly describe the main volatility derivatives.
3. Please list the main reasons for investing in volatility.
4. Explain why a delta-hedged option contains exposure to
volatility.
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Lecture 4

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