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**Unformatted text preview: **P2.T5. Market Risk Measurement & Management
Hull, Options, Futures, and Other Derivatives
Bionic Turtle FRM Study Notes
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HULL, CHAPTER 19: VOLATILITY SMILES.......................................................................... 3
DEFINE VOLATILITY SMILE AND VOLATILITY SKEW. ..............................................................................3
EXPLAIN HOW PUT-CALL PARITY INDICATES THAT THE IMPLIED VOLATILITY USED TO PRICE CALL OPTIONS IS
THE SAME USED TO PRICE PUT OPTIONS. ...........................................................................................6
COMPARE THE SHAPE OF THE VOLATILITY SMILE (OR SKEW) TO THE SHAPE OF THE IMPLIED DISTRIBUTION
OF THE UNDERLYING ASSET PRICE AND TO THE PRICING OF OPTIONS ON THE UNDERLYING ASSET. .............7
EXPLAIN WHY FOREIGN EXCHANGE RATES ARE NOT NECESSARILY LOGNORMALLY DISTRIBUTED AND THE
IMPLICATIONS THIS CAN HAVE ON OPTION PRICES AND IMPLIED VOLATILITY. ..........................................8
DESCRIBE THE VOLATILITY SMILE FOR EQUITY OPTIONS AND GIVE POSSIBLE EXPLANATIONS FOR ITS SHAPE.
..................................................................................................................................................8
DESCRIBE ALTERNATIVE WAYS OF CHARACTERIZING THE VOLATILITY SMILE. ..........................................9
DESCRIBE VOLATILITY TERM STRUCTURES AND VOLATILITY SURFACES AND HOW THEY MAY BE USED TO
PRICE OPTIONS. ............................................................................................................................9
EXPLAIN THE IMPACT OF THE VOLATILITY SMILE ON THE CALCULATION OF THE “GREEKS”. .....................10
EXPLAIN THE IMPACT OF ASSET PRICE JUMPS ON VOLATILITY SMILES. .................................................10
PRACTICE QUESTIONS & ANSWERS: ..............................................................................................11
HULL, CHAPTER 25: EXOTIC OPTIONS ............................................................................ 13
DEFINE AND CONTRAST EXOTIC DERIVATIVES AND PLAIN VANILLA DERIVATIVES. ..................................13
DESCRIBE SOME OF THE FACTORS THAT DRIVE THE DEVELOPMENT OF EXOTIC PRODUCTS. .....................14
EXPLAIN HOW ANY DERIVATIVE CAN BE CONVERTED INTO A ZERO-COST PRODUCT. ...............................14
IDENTIFY AND DESCRIBE HOW VARIOUS OPTION CHARACTERISTICS CAN TRANSFORM STANDARD AMERICAN
OPTIONS INTO NONSTANDARD AMERICAN OPTIONS. .........................................................................14
IDENTIFY AND DESCRIBE THE CHARACTERISTICS AND PAY-OFF STRUCTURE OF: FORWARD START OPTIONS
................................................................................................................................................15
IDENTIFY AND DESCRIBE THE CHARACTERISTICS AND PAY-OFF STRUCTURE OF: COMPOUND OPTIONS ......15
IDENTIFY AND DESCRIBE THE CHARACTERISTICS AND PAY-OFF STRUCTURE OF CHOOSER OPTIONS [AKA, “AS
YOU LIKE IT”] ..............................................................................................................................16
IDENTIFY AND DESCRIBE CHARACTERISTICS AND PAY-OFF STRUCTURE OF BARRIER OPTIONS..................17
IDENTIFY AND DESCRIBE CHARACTERISTICS AND PAY-OFF STRUCTURE OF: BINARY OPTIONS ..................18
IDENTIFY AND DESCRIBE CHARACTERISTICS & PAY-OFF STRUCTURE OF: LOOKBACK OPTIONS .................19
IDENTIFY AND DESCRIBE THE CHARACTERISTICS AND PAY-OFF STRUCTURE OF: SHOUT OPTIONS .............21
IDENTIFY AND DESCRIBE THE CHARACTERISTICS AND PAY-OFF STRUCTURE OF: ASIAN OPTIONS..............22
IDENTIFY AND DESCRIBE THE CHARACTERISTICS AND PAY-OFF STRUCTURE OF: EXCHANGE OPTIONS .......23
IDENTIFY AND DESCRIBE THE CHARACTERISTICS AND PAY-OFF STRUCTURE OF: RAINBOW OPTIONS .........23
IDENTIFY AND DESCRIBE THE CHARACTERISTICS AND PAY-OFF STRUCTURE OF: BASKET OPTIONS ...........23
DESCRIBE AND CONTRAST VOLATILITY AND VARIANCE SWAPS............................................................23
EXPLAIN THE BASIC PREMISE OF STATIC OPTION REPLICATION AND HOW IT CAN BE APPLIED TO HEDGING
EXOTIC OPTIONS. ........................................................................................................................23
PRACTICE QUESTIONS & ANSWERS: ..............................................................................................24 2 Hull, Chapter 19: Volatility smiles
Define volatility smile and volatility skew.
Explain how put-call parity indicates that the implied volatility used to price call
options is the same used to price put options.
Compare the shape of the volatility smile (or skew) to the shape of the implied
distribution of the underlying asset price and to the pricing of options on the
underlying asset.
Explain why foreign exchange rates are not necessarily lognormally distributed and
the implications this can have on option prices and implied volatility.
Describe the volatility smile for equity options and give possible explanations for its
shape.
Describe alternative ways of characterizing the volatility smile.
Describe volatility term structures and volatility surfaces and how they may be used
to price options.
Explain the impact of the volatility smile on the calculation of the “Greeks”.
Explain the impact of asset price jumps on volatility smiles. Define volatility smile and volatility skew.
The Black-Scholes option pricing model (OPM) solves for an option price given several
inputs, including (and especially) volatility. If we supply the inputs, then output of the
Black-Scholes can be referred to as a model price; i.e., the option model price =
Function of Inputs [price, strike, volatility, term, riskfree rate]. In a textbook
application, volatility is an input (e.g., historical volatility might be used to calibrate
the input) and the output is a model-based price or value.
But implied volatility reverses-engineers to infer volatility from a given price. In
order to iteratively solve for implied volatility, we require an observed (traded) option
price. Then, we infer the implied volatility that is consistent with the traded price. In
this way, the price is the input and, in a manner of speaking, the volatility is the
output. However, the implied volatility, by definition, depends on the option pricing
model.
If we use (historical) volatility as an input and price an option with BSM, we produce
a model-based price; this approach neither utilizes nor requires an observed, traded
market price. Contrast to the implied-volatility that begins with the observed
market price: assuming an option pricing model (OPM) and the other inputs, the
implied volatility matches the OPM output to the observed price.
3 A volatility smile is the result of a plot of option implied volatility (y axis) as a
function of strike price (x axis). Volatility skew simply refers to the smile when it is
skewed: when volatility happens to be a decreasing function of the strike price. Below is an example (somewhat) of actual volatility skew: as the strike price
increases, volatility decreases but higher volatility is found at lower strike prices.
Implied Volatility
GOOGLE options, stock @ $440
120%
100%
80%
60%
40%
20%
0%
340 360 380 400 420 440 460 480 500 520 540 560 580 610
If actual asset returns behaved per the assumptions of the Black-Scholes (i.e., if
asset returns exhibited a lognormal distribution), then implied volatility would be a
straight line. A key point of this reading is that a volatility smile (or skew or grin) is
evidence, at a minimum, that the underlying distributional assumption in the BSM
option pricing model is not correct in practice. 4 Key Questions & Answers (“Define volatility smile and volatility skew”)
Question: In a plot of the volatility smile, what is each axis and how is the function line
plotted?
Answer: The y-axis is implied volatility (not realized historical volatility!). The x-axis is
strike price (K), K/S, K/Forward, or option delta.
The line plots volatility (sigma) that, given an observed market price (c), solves for
option price (c) = BSM[S,K, sigma, T, rho, q].There is not an analytical solution, we must iterate (goal-seek) to solve for the volatility that returns a model price equal to
observed market price.
Question: Identify the axes in a plot of the (i) volatility term structure and (ii) volatility
surface.
Answer: (i) Implied volatility versus option term. (ii) Three dimensions: implied
volatility (Y axis) versus strike or K/S (X axis) versus term (Z axis)
Question: Which is consistent with the lognormal price distribution (GBM) that
underlies classic Black-Scholes Merton: volatility smile or volatility skew?
Answer: Neither. A horizontal line (Y-axis = implied volatility and X-axis = strike price)
is consistent with the lognormal price distribution. A smile (i.e., high implied volatility
for OTM puts and calls) implies heavier tails, while a skew (i.e., high implied volatility
for ITM call/OTM put and low implied volatility for OTM call/ITM put) implies heavy
left tail and light right tail.
Question: True or false: the implied option volatility smile should be the same for all
market participants (traders).
Answer: False. Implied volatility, by definition, is model-dependent. It varies with the
model employed. Only if all traders use the same OPM will they all perceive the same
volatility smile. 5 Explain how put-call parity indicates that the implied volatility
used to price call options is the same used to price put options.
Put-call parity applies to both model-based relationship and the market-based
(observed) relationship: cBlack-Scholes Ke rT pBlack-Scholes S0
cmarket Ke rT pmarket S0 Such that the pricing error observed when using the Black-Scholes to price a call
option should be exactly the same as observed when pricing a put option: cBlack-Scholes cmarket pBlack-Scholes pmarket
“This shows that the dollar pricing error when the Black-Scholes model is used to
price a European put option should be exactly the same as the dollar pricing error
when it is used to price a European call option with the same strike price and time to
maturity” – Hull Key Questions & Answers (Put-call parity implied volatility):
Question: Assume a non-dividend paying stock price (S) is $15 with (annualized)
volatility of 30%, and the riskless rate is 4%. What is the BSM call option price for a 1year option (T=1) with strike price of $10?
Answer: d1 = 1.63 and d2 = 1.33.
European call price (c) = 15*N(d1) - 10*EXP(-4%*1 year)*N(1.33) = $5.50
Question: Use parity to find the price of a put option with same strike and maturity.
Answer: Since c + K*EXP(-rT) = p + S,
p = c + K*EXP(-rT) - S = 5.50 + 10*EXP(-4%*1) - 15 = $0.108 or $.11
Question: Assume the traded put price is $0.60 instead (i.e., much higher). What is the
implied volatility of traded put?
Answer: Implied volatility = 50% because BSM[S=15, K=10, sigma = 50%, r=4%,
q=0%, T=1 year] = $0.60. Please note this is almost impossible to do on the exam, so
this is not exactly an exam-type question. 6 Compare the shape of the volatility smile (or skew) to the
shape of the implied distribution of the underlying asset price
and to the pricing of options on the underlying asset.
The volatility smile corresponds to an implied distribution (i.e., the distribution that
would explain the volatility smile). A lognormal asset price distribution is consistent
with a flat volatility smile. If the implied volatility is a smile, the implied distribution
has heavier tails (leptokurtosis; kurtosis > 3 or excess kurtosis > 0) than the lognormal
distribution Sample questions:
Question (Hull 18.01.a.): What volatility smile is likely to be observed when both tails
of the stock price distribution are less heavy than those of the lognormal distribution?
Answer: When both tails of the stock price distribution
are less heavy than those of the lognormal distribution,
Black-Scholes will tend to produce relatively high
prices for options when they are either significantly out
of the money or significantly in the money [i.e., as the
BSM will tend to assume the same constant volatility,
the model price will tend to be higher than the observed
market price]. This leads to an implied volatility pattern
similar to that in the figure above. 7 Question: (Hull 18.01.b): What volatility smile is likely to
be observed when the right tail is heavier, and the left
tail is less heavy, than that of a lognormal distribution?
Answer: When the right tail is heavier and the left tail is
less heavy, Black-Scholes will tend to produce relatively
low prices for out-of-the-money calls and in-the-money
puts. It will tend to produce relatively high prices for out-of-the-money puts and inthe-money calls. This leads to implied volatility being an increasing function of strike
price. Explain why foreign exchange rates are not necessarily
lognormally distributed and the implications this can have on
option prices and implied volatility.
There are two conditions for an asset price to have a lognormal distribution Constant volatility of the asset
Smooth price change without jumps (i.e., a diffusion process) But neither is satisfied for an exchange rate Volatility of FX rate is non-constant
Exchange rates exhibit frequent jumps Describe the volatility smile for equity options and give
possible explanations for its shape.
Hull parrots (i.e., he does not necessarily endorse) two offered explanations for the
observance of a volatility smile for equity options: Capital structure leverage: as a company’s equity value decreases, its leverage
(i.e., debt-to-equity or debt-to-total capital ratio) increases. Higher leverage
implies higher volatility.
“Crashophobia:” Mark Rubinstein offered this theory that market participants
fear a market crash and price options accordingly. Crashophobia is a theory that, subsequent to the October 1987 stock market crash , a
constant volatility assumption (i.e., a lognormal price distribution) does not give
sufficient weight to tail events. The theory suggests that out-of-money (OTM) puts
become more expensive because either:
1. The probability of a rapid drop in equities is greater than had been anticipated
(i.e., a fundamental explanation related to very heavy-tails), or
2. OTM puts are simply more demanded, as an insurance instrument 8 In Lecturing Birds on Flying, Pablo Triana writes about the birth of the crashophobia
phenomenon: “But at the same time, something funny happened as a direct result of
the crash [i.e, the Crash of October 1987]. The currently ubiquitous volatility smile
was born, a reflection of freshly developed crashophobia on the part of traders. After
witnessing the massacre, it became clear to options pros that markets cannot be
assumed to behave “normally” as the mathematics behind Black-Scholes (another
big-time casualty of Black Monday) assumes, and that rare events do happen and
can be truly criminal. In essence, traders realized that they had been hopelessly
underestimating the value of crash protection. From then on, crash-protecting tools
(such as out-of-the-money puts) would have to be priced upwards.” Describe alternative ways of characterizing the volatility
smile.
The typical volatility smile plots volatility against strike / spot price (K/S0). But
alternatives include: Implied volatility versus Strike / Forward (K / F0). In this approach, “at the
money” corresponds to K = F0
Implied volatility versus delta: “At the money” call with delta = 0.5 or put
with delta = -0.5 (so-called 50-delta options) Implied
Vol () Implied
Vol ()
Strike (K) / Forward Delta Describe volatility term structures and volatility surfaces and
how they may be used to price options.
The volatility term structure is the relationship between volatility and maturity. The
volatility term structure tends to be an increasing function for a call option (volatility
increases with a longer maturity)
The volatility surface is a combination of volatility smile and volatility term structure. 9 Whereas the volatility smile plots implied volatility (Y-axis) vs. stock/strike price,
volatility term structure plots implied volatility (Y-axis) vs. time to maturity (X-axis). ? Volatility ()
? Time/Maturity (T) Explain the impact of the volatility smile on the calculation of
the “Greeks”.
Volatility smile complicates calculation of the “Greeks.” Sticky Strike rule: implied volatility of an option remains constant from one
day to the next
Sticky Delta rule: relationship between an option price and the ratio stock-tostrike (S/K) is constant Explain the impact of asset price jumps on volatility smiles. This concerns a situation where a large jump—either up or down—is
anticipated. In such a case, the actual distribution is not lognormal but rather
bimodal; i.e., it has two camel-like humps. Under such a distribution, an at-themoney option has a higher volatility than both an out-of-the-money option and
an in-the-money option. The implication of this inverted situation is an upside-down volatility smile
that peaks in the middle, where the strike price equals the stock price; this is
also known as a “volatility frown.” 10 Practice Questions & Answers:
1. In a plot of the volatility smile, what are each of the axes and how is the function
line plotted?
2. Assume a non-dividend paying stock price (S) is $15 with (annualized) volatility of
30%, and the riskless rate is 4%. What is the BSM call option price for a 1-year
option (T=1) with strike price of $10?
3. Question: what is the SHAPE of the implied volatility smile if the equity log
return distribution is platykurtotic (i.e., kurtosis < 3)?
4. What is the advantage of using delta on the x-axis (i.e., instead of strike price) to
plot the volatility smile?
5. Which would be the most plausible explanation for a downward sloping volatility
term structure?
a)
b)
c)
d) Mean reversion
Platykurtosis in returns (relative to lognormal)
Serial correlation in returns
Independence 11 Answers:
1. Y axis is implied volatility. X axis is strike price (K), K/S, K/Forward, or option
delta. The line plots volatility (sigma) that, given an observed market price (c), solves
for option price (c) = BSM[S, K, sigma, T, rho, (q)]. There is not an analytical solution
for implied volatility, we must iterate (goal-seek) to solve for the volatility that returns
a model price equal to observed market price.
2. d1 = 1.63 and d2 = 1.33
c = 15*N(d1) - 10*EXP(-4%*1 year)*N(1.33) = $5.50
See spreadsheet:
3. Inverted smile or FROWN; i.e., light tails imply lower implied volatility for
ITM/OTM
4. The use of delta incorporates (indirectly) time to expiration as delta incorporates
option term.
Further, as Hull says, “This approach [i.e., plotting implied volatility against delta]
sometimes makes it possible to apply volatility smiles to options other than European
and American calls and puts.”
5. A. mean reversion: If the short-dated volatility is high, then mean reversion of
volatility would imply a downward sloping term structure.
Platykurtosis in returns would impact the smile; i.e., light tails (relative to lognormal)
imply an inverted smile (frown). Serial correlation might imply greater long-run
volatility (upward sloping) 12 Hull, Chapter 25: Exotic Options
Define and contrast exotic derivatives and plain vanilla derivatives.
Describe some of the factors that drive the development of exotic products.
Explain how any derivative can be converted into a zero-cost product.
Identify and describe how various option characteristics c...

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