RSM332H1 Study Guide - Final Guide: Implied Volatility, Call Option, Capital Asset Pricing Model
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! "#$ %
&! "&' %
.0 2/3/4,34 $ /5
- Risk of the portfolios we obtain this way will
critically depend on the correlation 6&( .
CML: the tangent line from Rf to this M is known as
the capital market line, and is described by
,# [Sharpe ratio]
!"#$ "70 >! "5' "7+/5 and ,#$ /5,5
MVP is the portfolios with the lowest possible risk and
therefore defines the lowest expected return we should be
willing to accept. We would invest in the efficient
portfolios as they offer the highest achievable expected
return for a given level of risk.
!"#$ /5! "50 F ' /5"7
[N O, less risk averse, take
more risk higher portfolio volatility and expected
P ! "Q$ "70NRST "QU "5
XYZ>HWUHI+ [reward-risk ratio]
V N53[\]^ $GH_;HJ
aST "3U "?$aST "3U /3"30 /4"4$ /3,3
.0 /4,34 $
&! "&' %
]$ ! "&' "7U 11,34 $ 6&(,&,(
! "5$ "70 N53[\]^,5
. à N53[\]^ $GHI;HJ
If N f N53[\]^ (You are less risk averse than average
investors), /5g F, ,#g ,5,1! "#g ! "5
If N g N53[\]^, /5f F, ,#f ,5, !"#f ! "5
Why obtain the result (three Sharpe ratio)?
- among all portfolios on the Efficient Frontier, the
market portfolio is the one which has the highest
Sharpe ratio. The Sharpe ratio of the market portfolio
is the higher than that of any other portfolio or any
other individual risky assets.
Why aggregate risk aversion h all the same?
- if they are not equal, assets have the different
reward-risk ratio and there would be an arbitrage
trading which will lead them to be equal eventually.
1. Returns do not follow the normal distribution. 2.
Parameters are not predictable in the long term (e.g., time-
varying correlations). 3. We don’t observe the market
portfolio. S&P500 is a proxy.
CAPM (measures performance of investment
strategies / the required return on a project)
- intuition: an asset commoves closely with the market is a risky
asset that should have a higher return. This is because investors
hold the market portfolio, following the Modern Portfolio Theory.
Because investors hold the market, they are concerned about an
asset that performs poorly when their portfolio (market portfolio)
performs poorly, Therefore, they care about the co-movement
between the return of an asset and the market returns. An asset
that doesn’t give you insurance and instead moves as the market
moves is a highly risky asset.
- the market equity premium is the slope of the SML, during
recessions, if the market equity premium becomes higher, given a
level of beta, an asset should have higher returns. This is because
overall market compensation increases for investors taking risk
even during recession times.
- !"QUX&-? $ "70 N53[\]^RST "QU "5
-"!"QUX&-? $ "7089:;9<
iRST "QU "5"
$ "70 jk>! "5' "7+"
- oQg F,"!"QUX&-? g 1! "5 aggresive
- oQf F, !"QUX&-? f 1! "5 defensive
- oQ$ F,"!"QUX&-? $ 1! "5"
- nature of business [depends on economic condition]
- operating leverage [higher fixed cost, higher beta]
- financial leverage [volatilityp, riskp, higher beta]
- o-$ /o&0 >F ' /+o(
- !"#$ "70 o->! "5' "7+
qQ$ ! "Q' ! "QUX&-? ""
- gives abnormal returns that its riskiness(CAPM)
cannot explain [stock underpriced
v$F 0 ! "Qw rs' ! xv
111111$ >F 0 "70 oQ! "5' "7w rs' !>xv+
Systematic risk = o&
. total risk = ,&
R-Square is 6Q U5 i$ oQ
Criticisms: technically, we
cannot test the CAPM
because we don’t observe the
- getting a right/ having a legal obligation to buy/sell
the underlying asset at a pre-specified price and time.
- 买put: right to sell at strike price. 跌就挣。
- R‚$ƒ„…1)†U ‡‚' ˆ* 'R‚$ 'ƒ„…1)†U ‡‚' ˆ*
- r‚$ƒ„…1)†U ˆ ' ‡‚* 'r‚$ 'ƒ„…1)†U ˆ ' ‡‚*
- riskiness of option
Options in isolation are much riskier
than stocks. That is, with the same
amount of money invested in a
portfolio of options instead of a stock,
one can achieve a much higher risk exposure. Higher
riskiness is achieved via leverage
- Butterfly spreads:
ü buy call option with X1+sell 2 call options with
X2+buy call option with X3
ü I believe stock price will be stable (not volatile),
also I want to protect my downside.
- long straddle:
ü Buy call option with X2+ buy put option X1
ü I believe stock price is extremely volatile
- short straddle:
ü Sell call option with X2+sell put option X1
ü I believe stock price will be extremely stable.
Because it will be stable, I don’t care about my
downside, which can be negative infinite
- bull spread:
Bull: Buy a call and sell a call with a higher strike price.
Bear: Buy a put and sell a put with a lower strike price.
- ‰ $ Š 0 ‹ ' Œ
- call: strike price 20, 30: 20 will be more likely that
this option will pay off. Therefore, there is a higher
intrinsic value and option price should be higher than
the one with the
strike price of
If someone believes
that stock will be
around $15, the person would create this security. To
illustrate this point, suppose you buy only put option with the
strike price of $30, and the stock price turns out to be $15.
Your net payoff is $30−$15−$19.7 =−$4.7. Your net payoff
is negative even though the stock price goes down below the
strike price. However, if you construct the above security
(Bear spread), then your net payoff is$30−$15−$14.69 =
- Black-Scholes-Merton Model
Assume: 1. stocks are uncorrelated over timeàEMH
2. stock moves for infinitesimal small time (z;[^+
3. stock returns follow normal distribution
4. constant interest rate and volatility 5. No friction
‰ $ ‡sŽ•F ' ˆz;[‚Ž •2
A‚ and •2 $ •F ' , –
a) Implied volatility:
1) Volatility for the BSM that makes the BSM price equal
the market call option price. It reflects investors expected
future volatility, whereas historical volatility captures past
2) Smile and Smirk: Patterns that arise from the gap
between the normal distribution which the BSM is based on
and the investors’ expected distribution which is reflected in
the market option prices.
3) VIX: Implied volatility for S&P500 which aggregates all
option implied volatilities.
- —˜™š›œ $ •
sŽ•F ' ˆz;[‚ Ž>•2+
where •F $
A ‚ and •2 $ •F ' , –
- Debt value: •
- Debt yield: Ÿ ¡ $ 1 v
- Credit spread: Ÿ ¡ ' ¨
- Probability of default: F ' Ž>•2+
- the higher asset volatility, the higher equity value, the lower
debt value and the higher credit risk [call option prices
increase with the volatility and sell put option (part of debt
payoff) decrease with the volatility]/ the higher face value,
higher credit risk.
-higher volatility increases the expected equity payoff and
decreases the expected debt payoff. Since it leads to a higher
possibility that the equity holders make a lot of money while
their downside is limited. For debt holders, a higher
volatility translates in to a higher chance of not getting
money back. Therefore, they have to use a high discount
rate, resulting in a lower debt value
-The Firm A has a higher credit risk represented by the
higher probability of default, higher credit spread. The reason
for this is firm A total asset is more volatile than firm B,
therefore, there is a higher chance of not generating enough
cash flow to repay the debt.
Reason for market value < debt value?
You can think of this case as an out-of-the money call which
doesn’t have an intrinsic value. The reason for a positive equity
value is time value together with volatility. Even though the asset
value is lower than the debt claim, after two years, it is possible
that this firm will generate enough cash flow to cover the debt.
Beta different reason: We can think about the cyclicality of
these companies. We can imagine Starbucks profit is more
cyclical than McDonald as Starbucks have a premium price
Alpha different reason: This is because while their simple
average returns are similar, Starbucks have a higher beta. After
we consider the riskiness of Starbucks, Starbucks' stock didn't
perform well compared to McDonald. That is why alpha for
Starbucks is not statistically significant. This result implies that
Starbucks returns can be explained by its exposure to the market
risk and there is no additional compensation beyond the required
rate of returns implied by the CAPM.
Alpha CAPM: The sign is positive, which is consistent with the
CAPM because the slope of security market line is the market
equity premium which is positive.
How to use CAPM to measure mutual fund performance?
- By running a regression. Firstly, we compute the excess returns
of a mutual fund and the excess returns of the market for a certain
time period. Secondly, we run a regression of the excess returns
of a mutual fund on the excess returns of the market. Then the
intercept from the regression (alpha) captures the risk-adjusted
average returns in excess of the CAPM-implied expected returns.
Implied volatility and strike price: The pattern shows the
volatility smirk. That is, for options with low strike prices (in-the-
money call), the implied volatilities are high. For options with
high strike prices (out-the-money call), the implied volatilities are
higher than at-the-money call, but lower than in-the-money call
Reason: You can see this pattern because stock returns
distribution that investors expect is negatively skewed while the
BSM model assumes the normal distribution of stock returns.
When investors believe that an extreme negative event can
happen, they want to buy an option which can pay o during
extreme negative times. When investors buy those options, the
option prices increase. In order to match the observed high call
option price, the implied volatility must be high, which will drive
up the BSM model option price.
VIX related to the S&P500 returns: When overall stock prices
go down, firms have a higher debt-to-equity or debt-to-asset ratio.
When a debt accounts for a greater portion of firm asset, a credit
risk increases, which is factored into the volatility (leverage
effect). Another reason is risk aversion and uncertainty. When
stock prices go down, investors become more risk averse. A less
risk-taking behavior can lead to a further drop in the stock prices
and therefore uncertainty of the market increases, which is
factored into the volatility.
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