Chapter 1 Introduction to Volatility Smiles 1.1 The Black-Scholes Formula and Volatility Smiles Consider the pricing of a call option with payoff C T =(S T - K ) + . Let us work with forward prices. The forward price of the underlying asset is F T t = S t P (t, T ) , where P (t, T ) is the time-t price of discount bond with maturity T . It is well known in finance that F T t is modeled as a lognormal martingale under the T -forward measure, Q T , such that dF T t = F T t σ F dW T t , where σ F is the volatility of the forward price and W T t is a Brownian motion under the T-forward measure Q T . The solution to the above SDE is F T t = F T t exp - 1 2 σ 2 F τ + σ F (W T T - W T t ) = F T t exp - 1 2 σ 2 F τ + σ F √ τ , where τ = T - t is the time to maturity and is a standard normal random variable under Q T . Since all forward prices are martingale under the forward measure, we have, for the forward price of the option, C t P (t, T ) = E Q T h ( F T t - K ) + i = E Q T h F T t 1 F T T >K i - KE Q T h 1 F T T >K i . 1
Transcript
Chapter 1
Introduction to Volatility Smiles
1.1 The Black-Scholes Formula and Volatility Smiles
Consider the pricing of a call option with payoff
CT = (ST −K)+ .
Let us work with forward prices. The forward price of the underlying asset is
F Tt =
StP (t, T )
,
where P (t, T ) is the time-t price of discount bond with maturity T . It is well known infinance that F T
t is modeled as a lognormal martingale under the T -forward measure, QT ,such that
dF Tt = F T
t σFdWTt ,
where σF is the volatility of the forward price and W Tt is a Brownian motion under the
T-forward measure QT . The solution to the above SDE is
F Tt = F T
t exp
(−1
2σ2F τ + σF (W T
T −W Tt )
)= F T
t exp
(−1
2σ2F τ + σF
√τε
),
where τ = T − t is the time to maturity and ε is a standard normal random variable underQT . Since all forward prices are martingale under the forward measure, we have, for theforward price of the option,
CtP (t, T )
= EQT
[(F Tt −K
)+]= EQT
[F Tt 1FT
T >K
]−KEQT
[1FT
T >K
].
1
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The two expectation will be evaluated consecutively. For the second expectation we have
EQT
[1FT
T >K
]= Prob(F T
t > K)
= Prob
(ε > −
lnFT
t
K− 1
2σ2F τ
σF√τ
)= Φ(d2),
(1.1.1)
whereΦ(x) =
1√2π
∫ x
−∞e−
x2
2 dx,
and
d2 =ln
FTt
K− 1
2σ2F τ
σF√τ
.
For the first expectation we have
EQT
[F Tt 1FT
T >K
]=
1√2π
∫ +∞
−d2F Tt exp
(−1
2σ2F τ + σF
√τx
)e−
x2
2 dx
=F Tt√2π
∫ +∞
−d2−σF√τ
e−y2
2 dy = F Tt Φ(d1),
(1.1.2)
whered1 = d2 + σF
√τ .
Combining (1.1.1) and (1.1.2) we arrive at the Black’s formula:
Ct = P (t, T )(F Tt Φ(d1)−KΦ(d2)
)= StΦ(d1)−KP (t, T )Φ(d2).
(1.1.3)
Put options, with terminal payoff of the form
PT = (K − St)+ ,
can be priced by call-put parity:
Pt = Ct − (S − t− P (t, T )K)
= St(Φ(d1)− 1) +KP (t, T )(1− Φ(d2))
= KP (t, T )Φ(−d2)− StΦ(−d1).
(1.1.4)
Note that, due to arbitrage, the put call parity is well observed in the market place.
Given the market prices of call and put options, we can solve for the implied volatil-ities, numerically, and thus obtain {σ(K/St, τ)}, as a function of two variables. Whenplotted, we obtain a surface. If the B-S model is correct, then the surface must be flat, as
1.1. THE BLACK-SCHOLES FORMULA AND VOLATILITY SMILES 3
shown in Figure 1.1. This was indeed the case before the Black Monday of 1987 (October19).
Figure 1.1 A flat implied Black’s volatility surface
Figure 1.2 shows an implied volatility curve for S&P500 index option before 1987.
Figure 1.2 Representative S&P 500 implied volatilities prior to 1987.Data taken from M. Rubinstein, ”Implied Binomial Trees” J. of Finance, 69 (1994) pp. 771-818
The flat volatility curve or surface are consistent with the lognormal assumption of theasset dynamics. However, after the Black Monday, the pattern has changed substantially,making the assumption lognormality flawed.
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1.2 The Reality in Today’s Option Markets
The notion of volatility smiles means the smile-shaped curve of implied Black’s volatilitiesof call or put options against their strikes (In a reasonably liquid market, call and putoptions with identical strike should have identical implied volatility, as is mandated by thecall-put parity). Volatility smiles became a serious existence soon after the Black MondayCrash of 1987. Most likely due to the ”crash ophopia” and the increased demand forout-of-the money put options as portfolio hedging instruments. Today, the smiles havebecome a stylized feature in various derivative markets, including equity, interest-rate,credit, forex, and even the inflation market, although they may ”smile” in some differentway.
Next, we demonstrate, for various markets, the typical patterns of volatility smiles orskews.
1.2.1 Evidences of smiles in various market
Figure 1.3 Representative S&P 500 implied volatilities after 1987.
1.2. THE REALITY IN TODAY’S OPTION MARKETS 5
Figure 1.4 The volatility surface according to S&P options markets.
Figure 1.5 The implied volatility surface for interest-rate caps
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Figure 1.6 Implied volatility smile for USD/EUR on 17Dec01
Figure 1.7 Implied volatility smile for USD/JPY on 17Dec01
1.2. THE REALITY IN TODAY’S OPTION MARKETS 7
Figure 1.8 Implied volatility smile for inflation caps/floors on 4Sep08
1.2.2 Characteristic of Equity Implied Volatility Smiles
The characteristics of volatility smiles in equity derivatives is summarized in Derman(2009).
1. Volatilities are steepest for small expiration as a function of strike, shallower forlonger expirations.
2. The minimal volatility as a function of strike occurs near ATM strike or ATM strikefor the forward price.
3. The volatility of implied volatility is greatest for short maturities, as with Treasuryrates.
4. There is a negative correlation between implied ATM volatility and the underlyingasset ( for DAX).
5. Implied volatility appears to be mean reverting with a life of about 60 days.
6. Implied volatility tends to rise fast and decline slowly.
7. Shocks across the implied vol surface are highly correlated.
The fourth characteristic is labeled as ”Leverage effect” by Fisher Black (1976). Sup-pose that the firm value follows a lognormal diffusion process, Black observed that the
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volatility of the firm’s equity price will be a decreasing function of the equity price. To seethis, we start from the relationship
Firm value= Stock price + Liability.
Let A, B and S denote the firm value, the liability and equity price of the firm, respectively,then there are
S = A−B,dA
A= σAdWt,
dS
S=dA
S=AσAdWt
S=
(1 +
B
S
)σAdWt,
=⇒ σS = σA
(1 +
B
S
).
So, under the Black’s model for firm value, equity volatility decreases when equity priceincreases.
1.2.3 Why Do Volatility Smiles Arise?
• Idealized assumptions of the Black-Scholes (BS) Model.
• The fair price of deep OTM or deep ITM options is more expensive than would beassumed by the BS formulation.
• By increasing prices for such options, volatility smile could be the markets’ indi-rect way of achieving such higher prices within the imperfect framework of the BSmodel.
• In equity markets, volatility skew could reflect investors’ fear of market crasheswhich would cause them to bid up the prices of options at strikes below currentmarket levels.
• Volatility smiles also reflect supply and demand as well as risk premium paid to theoptions.
As we have already seen, volatility smiles have become a stylized feature in derivativesmarkets of various assets. If a production model cannot accommodation the smiles, therewill be consequences in pricing and hedging of other derivatives.
1. Pricing of Exotic Derivatives
1.2. THE REALITY IN TODAY’S OPTION MARKETS 9
• Using the BS model effectively means that for each option, a new IV must beused, and therefore each option is priced using a different model (i.e., differentunderlying distribution).
• This is not an issue for the vanilla market options, like caps/floors and swap-tions, because this is just a model to allow price quoting through volatility.
• However, a problem arises when pricing structured derivatives because it isno longer clear which of these models should be used, and in general, none ofthese are suitable.
• Suppose that we need to price an exotic option, say down-and-out call option,with strike and knock-out barrier at . Which IV should we use, the IV at the callstrike or the IV at the barrier ?
• Clearly, this option cannot be priced without a single, self-consistent, modelthat works for all strikes without ”adjustment”.
2. Hedging large book of options
• Traders aren’t interested only in static volatility surfaces. They also want toknow how smile will respond to the time-to-maturity and changes in the un-derlying’s value.
• Since different models are being used for different strikes, it is not clear thatdelta and vega risks calculated at one strike are consistent with the same riskscalculated at other strikes.
For example, suppose that an option book is long strike options with a total delta risk of$1M, and is long how strike options with a delta risk of -$1M. Is our book delta neutral?Not really as it may leave residual delta risk that needs to be hedged due to using differentmodel for each strike.
1.2.4 The Mechanism for Volatility Smiles
There are strong evidences that the ”Black-Scholes world” is too much an idealized de-scription of the financial markets, and the geometric Brownian motion (GBM) is a tooidealized description of asset-price dynamics. Let us use the S&P 500 time series data to
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make the points.
Figure 1.9 SPX daily log return from 31Dec1984 to 31Dec2004
Figure 1.10 Frequency distribution of (77 years of )SPX daily log return comparedwith normal distributions, where the -23% return on October 19, 1987 is not visible.
1.2. THE REALITY IN TODAY’S OPTION MARKETS 11
Figure 1.11 Q-Q plot of the SPX log daily return compared with normal distribution.Note the extreme tails.
Figure 1.12 The three-month ATM IV of DAX index option.
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Figure 1.13 SPX index v.s. three-month ATM IV of SPX options
Based on the previous plots we can make a few points.
1. The normal distribution underestimates the risks of big moves, practically the riskof big drawdown.
2. Returns are not only always continuous, jumps in prices are possible.
3. The return distribution has the ”leptokurtic feature”, the peak is higher and the tailsare fatter.
4. The implied volatilities can also jump.
If we add additional sources of risks, namely, stochastic volatilities and jumps, will theprice of the options become more accurate qualitatively or quantitatively? The answer ispositive. This has motivated a wave of researches.
Nonetheless, adding either stochastic volatilities or jumps make the market incom-plete, in the sense that the additional risks cannot be hedged off, and risk premiums enterpricing. In the early days, traders and quarts, who have been living in a fantasy world,disdain that, and want to stay in a complete market. Dupire provided a desired solution—- the local volatility model. The local volatility models allow consistent pricing of exoticderivatives, and it has a close relationship with the stochastic volatility models.
