LO 15.9: Explain the im pact o f a single asset price jum p on a volatility smile.

LO 15.9: Explain the im pact o f a single asset price jum p on a volatility smile.
Price jumps can occur for a number of reasons. One reason may be the expectation of a significant news event that causes the underlying asset to move either up or down by a large amount. This would cause the underlying distribution to become bimodal, but with the same expected return and standard deviation as a unimodal, or standard, price-change distribution.
Implied volatility is affected by price jumps and the probabilities assumed for either a large up or down movement. The usual result, however, is that at-the-money options tend to have a higher implied volatility than either out-of-the-money or in-the-money options. Away-from-the-money options exhibit a lower implied volatility than at-the-money options. Instead of a volatility smile, price jumps would generate a volatility frown, as in Figure 3.
Figure 3: Volatility Smile (Frown) With Price Jump Implied volatility
Strike price
Page 194
2018 Kaplan, Inc.
Topic 15 Cross Reference to GARP Assigned Reading – Hull, Chapter 20
K e y C o n c e p t s
LO 15.1
When option traders allow implied volatility to depend on strike price, patterns of implied volatility resemble volatility smiles.
LO 15.2
Put-call parity indicates that the deviation between market prices and Black-Scholes-Merton prices will be equivalent for calls and puts. Hence, implied volatility will be the same for calls and puts.
LO 15.3
Currency traders believe there is a greater chance of extreme price movements than predicted by a lognormal distribution. Equity traders believe the probability of large down movements in price is greater than large up movements in price, as compared with a lognormal distribution.
LO 15.4
The volatility pattern used by traders to price currency options generates implied volatilities that are higher for deep in-the-money and deep out-of-the-money options, as compared to the implied volatility for at-the-money options.
LO 15.5
The volatility smile exhibited by equity options is more of a smirk, with implied volatility higher for low strike prices. This has been attributed to leverage and crashophobia effects.
LO 15.6
Alternative methods to studying volatility patterns include: replacing strike price with strike price divided by stock price, replacing strike price with strike price divided by the forward price for the underlying asset, and replacing strike price with option delta.
LO 15.7
Volatility term structures and volatility surfaces are used by traders to judge consistency in model-generated option prices.
2018 Kaplan, Inc.
Page 195
Topic 15 Cross Reference to GARP Assigned Reading – Hull, Chapter 20
LO 15.8
Volatility smiles that are not flat require the use of implied volatility functions or trees to correctly calculate option Greeks.
LO 15.9
Price jumps may generate volatility frowns instead of smiles.
Page 196
2018 Kaplan, Inc.
Topic 15 Cross Reference to GARP Assigned Reading – Hull, Chapter 20
C o n c e p t C h e c k e r s
1.
2.
3.
4.
5.
The market price deviations for puts and calls from Black-Scholes-Merton prices indicate: A. equivalent put and call implied volatility. B. equivalent put and call moneyness. C. unequal put and call implied volatility. D. unequal put and call moneyness.
An empirical distribution that exhibits a fatter right tail than that of a lognormal distribution would indicate: A. equal implied volatilities across low and high strike prices. B. greater implied volatilities for low strike prices. C. greater implied volatilities for high strike prices. D. higher implied volatilities for mid-range strike prices.
the same across maturities for given strike prices. the same for short time periods. The sticky strike rule assumes that implied volatility is: A. B. C. the same across strike prices for given maturities. D. different across strike prices for given maturities.
Compared to at-the-money currency options, out-of-the-money currency options exhibit which of the following volatility traits? A. Lower implied volatility. B. A frown. C. A smirk. D. Higher implied volatility.
Which of the following regarding equity option volatility is true? A. There is higher implied price volatility for away-from-the-money equity options. B. Crashophobia suggests actual equity volatility increases when stock prices
decline.
C. Compared to the lognormal distribution, traders believe the probability of large
down movements in price is similar to large up movements.
D. Increasing leverage at lower equity prices suggests increasing volatility.
2018 Kaplan, Inc.
Page 197
Topic 15 Cross Reference to GARP Assigned Reading – Hull, Chapter 20
C o n c e p t C h e c k e r An s w e r s
1. A Put-call parity indicates that the implied volatility of a call and put will be equal for the same
strike price and time to expiration.
2. C An empirical distribution with a fat right tail generates a higher implied volatility for higher
strike prices due to the increased probability of observing high underlying asset prices. The pricing indication is that in-the-money calls and out-of-the-money puts would be expensive.
3. B The sticky strike rule, when applied to calculating option sensitivity measures, assumes
implied volatility is the same over short time periods.
4. D Away-from-the-money currency options have greater implied volatility than at-the-money
options. This pattern results in a volatility smile.
5. D There is higher implied price volatility for low strike price equity options. Crashophobia is based on the idea that large price declines are more likely than assumed in Black-Scholes- Merton prices, not that volatility increases when prices decline. Compared to the lognormal distribution, traders believe the probability of large down movements in price is higher than large up movements. Increasing leverage at lower equity prices suggests increasing volatility.
Page 198
2018 Kaplan, Inc.
S e l f -Te s t : M a r k e t R i s k M e a s u r e m e n t a n d M a n a g e m e n t
10 Q u e stio n s: 3 0 M in u te s
1.
2.
3.
4.
An analyst for Z Corporation is determining the value at risk (VaR) for the corporations profit/loss distribution that is assumed to be normally distributed. The profit/loss distribution has an annual mean of $3 million and a standard deviation of$3.3 million. Using a parametric approach, what is the VaR with a 99% confidence level? A. $0,775 million. B.$3,155 million. C. $5,775 million. D.$8,155 million.
The Basel Committee requires backtesting of actual losses to VaR calculations. How many exceptions would need to occur in a 250-day trading period for the capital multiplier to increase from three to four? two to five. A. B. five to seven. C. seven to nine. D. ten or more.
The top-down approach to risk aggregation assumes that a banks portfolio can be cleanly subdivided according to market, credit, and operational risk measures. In contrast, a bottom-up approach attempts to account for interactions among various risk factors. In order to assess which approach is more appropriate, academic studies evaluate the ratio of integrated risks to separate risks. Regarding studies of top-down and bottom-up approaches, which of the following statements is incorrect? A. Top-down studies suggest that risk diversification is present. B. Bottom-up studies sometimes calculate the ratio of integrated risks to separate
risks to be less than one.
C. Bottom-up studies suggest that risk diversification should be questioned. D. Top-down studies calculate the ratio of integrated risks to separate risks to be
greater than one.
Commercial Bank Z has a $3 million loan to company A and a$3 million loan to company B. Companies A and B each have a 5% and 4% default probability, respectively. The default correlation between companies A and B is 0.6. What is the expected loss (EL) for the commercial bank under the worst case scenario? a. b. c. d.
$83,700.$133,900. $165,600.$233,800.
2018 Kaplan, Inc.
Page 199
Book 1 Self-Test: Market Risk Measurement and Management
5.
6.
7.
8.
A risk manager should always pay careful attention to the limitations and advantages of applying financial models such as the value at risk (VaR) and Black-Scholes- Merton (BSM) option pricing model. Which of the following statements regarding financial models is correct? a. Financial models should always be calibrated using most recent market data
because it is more likely to be accurate in extrapolating trends.
b. When applying the VaR model, empirical studies imply asset returns closely
follow the normal distribution.
c. The Black-Scholes-Merton option pricing model is a good example of
the advantage of using financial models because the model eliminates all mathematical inconsistences that can occur with human judgment.
d. A good example of a limitation of a financial model is the assumption of
constant volatility when applying the Black-Scholes-Merton (BSM) option pricing model.
Assume that a trader wishes to set up a hedge such that he sells $100,000 of a Treasury bond and buys TIPS as a hedge. Using a historical yield regression framework, assume the DV01 on the T-bond is 0.072, the DV01 on the TIPS is 0.051, and the hedge adjustment factor (regression beta coefficient) is 1.2. What is the face value of the offsetting TIPS position needed to carry out this regression hedge? A.$138,462. B. $169,412. C.$268,499. D. $280,067. A constant maturity Treasury (CMT) swap pays ($1,000,000 / 2) x (yCMT 9%) every six months. There is a 70% probability of an increase in the 6-month spot rate and a 60% probability of an increase in the 1 -year spot rate. The rate change in all cases is 0.50% per period, and the initial yCMT is 9%. What is the value of this CMT swap? A. $2,325. B.$2,229. C. $2,429. D.$905.
Suppose the market expects that the current 1-year rate for zero-coupon bonds with a face value of $1 will remain at 5%, but the 1-year rate in one year will be 3%. What is the 2-year spot rate for zero-coupon bonds? A. 3.995%. B. 4.088%. C. 4.005%. D. 4.115%. Page 200 2018 Kaplan, Inc. Book 1 Self-Test: Market Risk Measurement and Management 9. An analyst is modeling spot rate changes using short rate term structure models. The current short-term interest rate is 5% with a volatility of 80bps. After one month passes the realization of dw, a normally distributed random variable with mean 0 and standard deviation Vdt, is -0.5. Assume a constant interest rate drift, \ , of 0.36%. What should the analyst compute as the new spot rate? A. 5.37%. B. 4.63%. C. 5.76%. D. 4.24%. 10. Which of the following statements is incorrect regarding volatility smiles? A. Currency options exhibit volatility smiles because the at-the-money options have higher implied volatility than away-from-the-money options. B. Volatility frowns result when jumps occur in asset prices. C. Equity options exhibit a volatility smirk because low strike price options have greater implied volatility. D. Relative to currency traders, it appears that equity traders expectations of extreme price movements are more asymmetric. 2018 Kaplan, Inc. Page 201 S e l f -Te s t A n s w e r s : M a r k e t R i s k M e a s u r e m e n t a n d M a n a g e m e n t 1. B The population mean and standard deviations are unknown; therefore, the standard normal z-value of 2.33 is used for a 99% confidence level. VaR(l%) = -5.0 million + ($3.5 million)(2.33) = -5.0 million + 8.155 million = 3.155 million (See Topic 1)
2. D Ten or more backtesting violations require the institution to use a capital multiplier of four.
(See Topic 3)
3. D Top-down studies calculate this ratio to be less than one, which suggests that risk
diversification is present and ignored by the separate approach. Bottom-up studies also often calculate this ratio to be less than one; however, this research has not been conclusive, and has recently found evidence of risk compounding, which produces a ratio greater than one. Thus, bottom-up studies suggests that risk diversification should be questioned. (See Topic 5)
4. C The default probability of company A is 5%. Thus, the standard deviation for company A is:
^0.05(1 0.05) = 0.2179
Company B has a default probability of 4% and, therefore, will have a standard deviation of 0.1960. We can now calculate the expected loss under the worst case scenario where both companies A and B are in default. Assuming that the default correlation between A and B is 0.6, the joint probability of default is:
P(AB) = 0.6^0.05(0.95) x 0.04(0.96) + 0.05 x 0.04 = 0.6V0.001824 + 0.002 = 0.0276
Thus, the expected loss for the commercial bank is $165,600 (= 0.0276 x$6,000,000). (See Topic 6)
5. D The Black-Scholes-Merton (BSM) option pricing model assumes strike prices have a
constant volatility. However, numerous empirical studies find higher volatility for out-of- the-money options and a volatility skew in equity markets. Thus, this is a good example of a limitation of financial models. The choice of time period used to calibrate the parameter inputs for the model can have a big impact on the results. Risk managers used volatility and correlation estimates from pre-crisis periods during the recent financial crisis, and this resulted in significantly underestimating the risk for financial models. All financial models should be stress tested using scenarios of extreme economic conditions. VaR models often assume asset returns have a normal distribution. However, empirical studies find higher kurtosis in return distributions. High kurtosis implies a distribution with fatter tails than the normal distribution. Thus, the normal distribution is not the best assumption for the underlying distribution. Financial models contain mathematical inconsistencies. For example, in applying the BSM option pricing model for up-and-out calls and puts and down-and-out calls and puts, there are rare cases where the inputs make the model insensitive to changes in implied volatility and option maturity. (See Topic 8)
Page 202
2018 Kaplan, Inc.
Book 1 Self-Test Answers: Market Risk Measurement and Management
6. B Defining
and P * as the face amounts of the real and nominal bonds, respectively, and
their corresponding DVOls as DV01R and DV01N, a DV01 hedge is adjusted by the hedge adjustment factor, or beta, as follows:
RF = 100,000 x
xl.2 = 169,412
‘ d v o i n ‘ X 0 R DV01 \ / 0.072 .051 J
/ V0
FR = FN x
(See Topic 10)
7. A The payoff in each period is ($1,000,000 / 2) x (yCMT – 9%). For example, the 1-year payoff of$5,000 in the figure below is calculated as ($1,000,000 / 2) x (10% – 9%) =$5,000. The other numbers in the year one cells are calculated similarly.
In six months, the payoff if interest rates increase to 9.50% is ($1,000,000 / 2 ) x (9.5% – 9.0%) =$2,500. Note that the price in this cell equals the present value of the probability weighted 1 -year values plus the 6-month payoff:
months, U
($5,000×0.6)+ ($0x0.4)
+ 0.095 1
+ $2,500 =$5,363.96
The other cell value in six months is calculated similarly and results in a loss of $4,418.47. The value of the CMT swap today is the present value of the probability weighted 6-month values: ($5,363.96 x 0.7) + (-$4,418.47 x 0.3) + 0.09 1$2,324.62
yCMT=10% Price = $5,000 yCMT = 9 0 /0 Price =$0
yCMT = 8% Price = -$5,000 yCMT= 8*5% Price = -$4,418
T od ay
6 m on th s
1 year
Thus the correct response is A. The other answers are incorrect because they do not correctly discount the future values or omit the 6-month payoff from the 6-month values.
(See Topic 11)
2018 Kaplan, Inc.
Page 203
Book 1 Self-Test Answers: Market Risk Measurement and Management
8. A The 2-year spot rate is computed as follows:
f (2) = 2/(1.05) (1.03) – 1 = 3.995%
(See Topic 12)
9. B This short rate process has an annualized drift of 0.36%, so it requires the use of Model 2
(with constant drift). The change in the spot rate is computed as:
dr = Xdt + adw
dr = (0.36% / 12) + (0.8% x -0.5) = -0.37% = -37 basis points
Since the initial short-term rate was 5% and dr is -0.37%, the new spot rate in one month
5% – 0.37% = 4.63% (See Topic 13)
10. A Currency options exhibit volatility smiles because the at-the-money options have lower
implied volatility than away-from-the-money options.
Equity traders believe that the probability of large price decreases is greater than the probability of large price increases. Currency traders beliefs about volatility are more symmetric as there is no large skew in the distribution of expected currency values (i.e., there is a greater chance of large price movements in either direction).
(See Topic 15)
Page 204
2018 Kaplan, Inc.
delta-normal VaR: VaR(a%) = (p,r + a r x za ) x Pt_1
lognormal VaR: VaR(a%) = Pt_1 x ^1 e^R aRXZa j
standard error of a quantile: se (q)
V p ( l- p ) /n
f(q)
Topic 2
age-weighted historical simulation: w(i)
x ^ q – x )
l – X ”
Topic 3
model accuracy test: z
x – p T
V p (l- p )T
unconditional coverage test statistic:
LR = 2ln[(1 – p)TNpN] + 2ln{ [1 – (N/T)]t -n (N/T)nJ
Topic 4
V(Rp) is variance of portfolio return: V(Rp) = (3p x V(Rjyj) +
N
i= l
x CT,i
General market risk: (3p x V (Rm)
Specific risk:
N
i=l
wf x
2018 Kaplan, Inc.
Page 205
\ /
/
pt + p t v
pt-i
geometric return: R t = In
i
. i arithmetic return: rt = —————- = ———— 1
P t+ ^ t
Pt-1
p t- i
p t- i
F o r m u l a s
Topic 1
profit/loss data: P/Lt = P + D t Pt l
M arket R isk M easurem ent and M anagem ent
Book 1 Formulas
Undiversified VaR =
N
i=l
x Vj
Diversified VaR
a J x ‘ ^ x = ^ (x x V )’R (x x V )
Topic 6
portfolio mean return: pp = wxpx + wYPy
V 2 2
W
XCTX T Wytty + 2wyW y C O V y y
2 2
covariance: cov^y
n E ( X t – f e ) ( Y t – ^ Y) t=l__________________
n 1
correlation: PxY
CQVXY CTxCTy
realized correlation: Prealized
ZX
2 n – n i>] correlation swap payoff: notional amount x (preaBzej Pfixecj)
joint probability of default: P(AB) = pAB ^/PDA(1 PDa ) x PDb(1 PDB) + PDA x PDB
Topic 7
mean reversion rate: St S
j = a([i S
j)
autocorrelation: AC(pt,pt_i)
cov(pt,pt- 1) ff(pt)xof(Pt-i)
Topic 8
correlation with expectation values: PxY
E(XY) – E(X)E( Y)
E(X2)-(E (X ))2
E(Y2)-(E (Y ))2
n
i=l
n(n2 1)
Spearmans rank correlation: Ps
1 –
Kendalls t : t
n c ~ n d n(n 1) / 2
Page 206
2018 Kaplan, Inc.
Book 1 Formulas
Topic 12
2- year spot rate: r(2) = ^ (l + r1)(l +
~ 1
3- year spot rate: r (3) = ^ (l + q ) (l + ^ ) (l + ) 1
Jensens inequality: E
1 (i+0
1
> E[l + r
Topic 13
Model 1:
dr = crdw = annual basis-point volatility of rate changes where: dr = change in interest rates over small time interval, dt dt = small time interval (measured in years) o r = annual basis-point volatility of rate changes dw = normally distributed random variable with mean 0 and standard deviation Vdt
Model 2: dr = \d t + crdw
Vasicek model:
dr = k(0 – r)dt + crdw
where: k 0 r
= a parameter that measures the speed of reversion adjustment = long-run value of the short-term rate assuming risk neutrality = current interest rate level
long-run value of short-term rate:
X A
0 rj H k
where: ri = the long-run true rate of interest
2018 Kaplan, Inc.
Page 207
Book 1 Formulas
Topic 14
Model 3:
dr = \(t)dt + cre-atdw
where: a = volatility at t = 0, which decreases exponentially to 0 for a > 0
CIR model: dr = k(0 r)dt + a Vr dw
Model 4: dr = ardt + crrdw
Topic 13
put-call parity: c p = S PV(X)
Page 208
2018 Kaplan, Inc.
U s in g t h e C u m u l a t iv e Z-Ta b l e
Probability Example
Assume that the annual earnings per share (EPS) for a large sample of firms is normally distributed with a mean of $5.00 and a standard deviation of$1.50. What is the approximate probability of an observed EPS value falling between $3.00 and$7.25?
If EPS = x = $7.25, then z = (x – p)/a = ($7.25 – $5.00)/$1.50 = +1.50
If EPS = x = $3.00, then z = (x – p)/a = ($3.00 – $5.00)/$1.50 = -1.33
For z-value of 1.50: Use the row headed 1.5 and the column headed 0 to find the value 0.9332. This represents the area under the curve to the left of the critical value 1.50.
For z-value of1.33: Use the row headed 1.3 and the column headed 3 to find the value 0.9082. This represents the area under the curve to the left of the critical value +1.33. The area to the left o f1.33 is 1 0.9082 = 0.0918.
The area between these critical values is 0.9332 0.0918 = 0.8414, or 84.14%.
Hypothesis Testing One-Tailed Test Example
A sample of a stocks returns on 36 non-consecutive days results in a mean return of 2.0%. Assume the population standard deviation is 20.0%. Can we say with 95% confidence that the mean return is greater than 0%?
H q: p < 0.0%, Ha : p > 0.0%. The test statistic = ^-statistic = = (2.0 – 0.0) / (20.0 / 6) = 0.60.
x-po
The significance level = 1.0 0.95 = 0.05, or 5%.
Since this is a one-tailed test with an alpha of 0.05, we need to find the value 0.95 in the cumulative stable. The closest value is 0.9505, with a corresponding critical .z-value of 1.65. Since the test statistic is less than the critical value, we fail to reject H Q.
Hypothesis Testing Two-Tailed Test Example
Using the same assumptions as before, suppose that the analyst now wants to determine if he can say with 99% confidence that the stocks return is not equal to 0.0%.
H q: p = 0.0%, Ha : p ^ 0.0%. The test statistic (z-value) = (2.0 0.0) / (20.0 / 6) = 0.60. The significance level = 1.0 0.99 = 0.01, or 1%.
Since this is a two-tailed test with an alpha of 0.01, there is a 0.005 rejection region in both tails. Thus, we need to find the value 0.995 (1.0 0.005) in the table. The closest value is 0.9951, which corresponds to a critical .z-value of 2.58. Since the test statistic is less than the critical value, we fail to reject H Q and conclude that the stocks return equals 0.0%.
2018 Kaplan, Inc.
Page 209