Temp_store

LO 15.8: Explain the im pact o f the volatility smile on the calculation o f the Greeks.
Option Greeks indicate expected changes in option prices given changes in the underlying factors that affect option prices.
The problem here is that option Greeks, including delta and vega, may be affected by the implied volatility of an option. Remember these guidelines for how implied volatility may affect the Greek calculations of an option:
The first guideline is the sticky strike rule, which makes an assumption that an options
implied volatility is the same over short time periods (e.g., successive days). If this is the case, the Greek calculations of an option are assumed to be unaffected, as long as the implied volatility is unchanged. If implied volatility changes, the option sensitivity calculations may not yield the correct figures.
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The second guideline is the sticky delta rule, which assumes the relationship between an
options price and the ratio of underlying to strike price applies in subsequent periods. The idea here is that the implied volatility reflects the moneyness of the option, so the delta calculation includes an adjustment factor for implied volatility. If the sticky delta rule holds, the options delta will be larger than that given by the Black-Scholes-Merton formula.
Keep in mind, however, that both rules assume the volatility smile is flat for all option maturities. If this is not the case, the rules are not internally consistent and, to correct for a non-flat volatility smile, we would have to rely on an implied volatility function or tree to correctly calculate option Greeks.
P r i c e J u m p s

LO 15.7: Describe volatility term structures and volatility surfaces and how they may be used to price options.
The volatility term structure is a listing of implied volatilities as a function of time to expiration for at-the-money option contracts. When short-dated volatilities are low (from historical perspectives), volatility tends to be an increasing function of maturity. When short-dated volatilities are high, volatility tends to be an inverse function of maturity. This phenomenon is related to, but has a slightly different meaning from, the mean-reverting characteristic often exhibited by implied volatility.
A volatility surface is nothing other than a combination of a volatility term structure with volatility smiles (i.e., those implied volatilities away-from-the-money). The surface provides guidance in pricing options with any strike or maturity structure.
A traders primary objective is to maintain a pricing mechanism that generates option prices consistent with market pricing. Even if the implied volatility or model pricing errors change due to shifting from one pricing model to another (which could occur if traders use an alternative model to Black-Scholes-Merton), the objective is to have consistency in model-generated pricing. The volatility term structure and volatility surfaces can be used to confirm or disprove a models accuracy and consistency in pricing.
T h e O p t i o n G r e e k s

LO 15.6: Describe alternative ways o f characterizing the volatility smile.
The volatility smiles we have characterized thus far have examined the relationship between implied volatility and strike price. Other relationships exist which allow traders to use alternative methods to study these volatility patterns. All alternatives require a replacement of the independent variable, strike price (X).
One alternative method involves replacing the strike price with strike price divided by stock price (X / SQ). This method results in a more stable volatility smile. A second alternative
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approach is to substitute the strike price with strike price divided by the forward price for the underlying asset (X / FQ). The forward price would have the same maturity date as the options being assessed. Traders sometimes view the forward price as a better gauge of at the money option prices since the forward price displays the theoretical expected stock price. A third alternative method involves replacing the strike price with the options delta. With this approach, traders are able to study volatility smiles of options other than European and American options.
V o l a t i l i t y T e r m S t r u c t u r e a n d V o l a t i l i t y S u r f a c e s

LO 15.5: Describe the volatility smile for equity options and foreign currency options and provide possible explanations for its shape.
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, as shown in Figure 1.
Figure 1: Volatility Smile for Foreign Currency Options Implied volatility
Strike price
The easiest way to see why implied volatilities for away-from-the-money options are greater than at-the-money options is to consider the following call and put examples. For calls, a currency option is going to pay off only if the actual exchange rate is above the strike rate. For puts, on the other hand, a currency option is going to pay off only if the actual exchange rate is below the strike rate. If the implied volatilities for actual currency options are greater for away-from-the-money than at-the-money options, currency traders must think there is a greater chance of extreme price movements than predicted by a lognormal distribution. Empirical evidence indicates that this is the case.
This tendency for exchange rate changes to be more extreme is a function of the fact that exchange rate volatility is not constant and frequently jumps from one level to another, which increases the likelihood of extreme currency rate levels. However, these two effects tend to be mitigated for long-dated options, which tend to exhibit less of a volatility smile pattern than shorter-dated options.
E q u i t y O p t i o n s
The equity option volatility smile is different from the currency option pattern. The smile is more of a smirk, or skew, that shows a higher implied volatility for low strike price options (in-the-money calls and out-of-the-money puts) than for high strike price options
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(in-the-money puts and out-of-the-money calls). As shown in Figure 2, there is essentially an inverse relationship between implied volatility and the strike price of equity options.
Figure 2: Volatility Smile for Equities Implied volatility
Strike price
The volatility smirk (half-smile) exhibited by equity options translates into a left-skewed implied distribution of equity price changes. This left-skewed distribution indicates that 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. Two reasons have been promoted as causing this increased implied volatilityleverage and crashophobia.

Leverage. When a firms equity value decreases, the amount of leverage increases, which essentially increases the riskiness, or volatility, of the underlying asset. When a firms equity increases in value, the amount of leverage decreases, which tends to decrease the riskiness of the firm. This lowers the volatility of the underlying asset. All else held constant, there is an inverse relationship between volatility and the underlying assets valuation.
Crashophobia. The second explanation, used since the 1987 stock market crash, was coined crashophobia by Mark Rubinstein. Market participants are simply afraid of another market crash, so they place a premium on the probability of stock prices falling precipitouslydeep out-of-the-money puts will exhibit high premiums since they provide protection against a substantial drop in equity prices. There is some support for Rubinsteins crashophobia hypothesis, because the volatility skew tends to increase when equity markets decline, but is not as noticeable when equity markets increase in value.
A l t e r n a t i v e M e t h o d s f o r S t u d y i n g V o l a t i l i t y S m i l e s

LO 15.4: Describe characteristics o f foreign exchange rate distributions and their im plications on option prices and im plied volatility.

LO 15.3: Com pare the shape o f the volatility smile (or skew) to the shape o f the im plied distribution o f the underlying asset price and to the pricing o f options on the underlying asset.
Actual option prices, in conjunction with the BSM model, can be used to generate implied volatilities which may differ from historical volatilities. When option traders allow implied volatility to depend on strike price, patterns of implied volatility are generated which resemble volatility smiles. These curves display implied volatility as a function of the options strike (or exercise) price. In this topic, we will examine volatility smiles for both currency and equity options. In the case of equity options, the volatility smile is sometimes referred to as a volatility skew since, as we will see in LO 15.5, the volatility pattern for equity options is essentially an inverse relationship.
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F o r e i g n C u r r e n c y O p t i o n s

LO 15.1: Define volatility smile and volatility skew.

LO 15.2: Explain the im plications o f put-call parity on the im plied volatility o f call and put options.
Recall that put-call parity is a no-arbitrage equilibrium relationship that relates European call and put option prices to the underlying assets price and the present value of the options strike price. In its simplest form, put-call parity can be represented by the following relationship:
c – p = S – PV(X)
where: c p S PV(X) = present value of the strike = price of a call = price of a put = price of a call = price of a put = price of the underlying security
PV(X) can be represented in continuous time by:
PV(X) = Xe-rT
where: r = risk-free rate T =
time left to expiration expressed in years
Since put-call parity is a no-arbitrage relationship, it will hold whether or not the underlying asset price distribution is lognormal, as required by the Black-Scholes-Merton (BSM) option pricing model.
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If we simply rearrange put-call parity and denote subscripts for the option prices to indicate whether they are market or Black-Scholes-Merton option prices, the following two equations are generated:
Pmkt + S0e-V = cmkt + PV(X)
P b SM + S 0e qt = CBSM + ^ ( X )
Subtracting the second equation from the first gives us:
Pmkt PBSM = Cmkt CBSM
This relationship shows that, given the same strike price and time to expiration, option market prices that deviate from those dictated by the Black-Scholes-Merton model are going to deviate in the same amount whether they are for calls or puts. Since any deviation in prices will be the same, the implication is that the implied volatility of a call and put will be equal for the same strike price and time to expiration.
V o l a t i l i t y S m i l e s

LO 14.6: Describe lognorm al models with deterministic drift and mean reversion.
Lognorm al M odel with Determ inistic D rift
For this LO, we detail two lognormal models, one with deterministic drift and one with mean reversion. The lognormal model with drift is shown as follows:
d[ln(r)] = a(t)dt + crdw
The natural log of the short-term rate follows a normal distribution and can be used to construct an interest rate tree based on the natural logarithm of the short-term rate. In the spirit of the Flo-Lee model, where drift can vary from period to period, the interest rate tree in Figure 2 is generated using the lognormal model with deterministic drift.
Figure 2: Interest Rate Tree with Lognormal Model (Drift)
0.5
If each node in Figure 2 is exponentiated, the tree will display the interest rates at each node. For example, the adjusted period 1 upper node would be calculated as:
exp(ln r0 + ajdt + aVdt) = roe(ai(lt+ CJ^ )
Figure 3: Lognormal Model Rates at Each Node
In contrast to the Ho-Lee model, where the drift terms are additive, the drift terms in the lognormal model are multiplicative. The implication is that all rates in the tree will always be positive since ex > 0 for all x. Furthermore, since ex 1+ x, and if we assume a1 = 0 and
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dt = 1, then: r0eCT r0(l + a). Hence, volatility is a percentage of the rate. For example, if a = 20%, then the rate in the upper node will be 20% above the current short-term rate.
Lognorm al M odel with M ean Reversion
The lognormal distribution combined with a mean-reverting process is known as the Black- Karasinski model. This model is very flexible, allowing for time-varying volatility and mean reversion. In logarithmic terms, the model will appear as:
d[ln(r)] = k(t)[ln0(t) – ln(r)]dt + a(t)dw
Thus, the natural log of the short-term rate follows a normal distribution and will revert to the long-run mean of ln[0(t)] based on adjustment parameter k(t). In addition, volatility is time-dependent, transforming the Vasicek model into a time-varying one. The interest rate tree based on this model is a bit more complicated, but it exhibits the same basic structure as previous models.
Figure 4: Interest Rate Tree with Lognormal Model (Mean Revision)
k(l)(ln0(l) – lnr0)dt +a(l)Vdt
r e k(l)(ln0(l) – lnr0)dt -a(l)Vdt
The notation jq is used to condense the exposition. Therefore, the In (upper node) = lmq + cr(l)Vdt and ln(lower node) = lmq – cr(l) V dt. Following the intuition of the mean- reverting model, the tree will recombine in the second period only if:
q(l) – g(2)
CJr(l)dt
Recall from the previous topic that in the mean-reverting model, the nodes can be forced to recombine by changing the probabilities in the second period to properly value the weighted average of paths in the next period. A similar adjustment can be made in this model. However, this adjustment varies the length of time between periods (i.e., by manipulating the ^variable). After choosing dt^, dt2 is determined with the following equation:
g(2)7dt2 o'(l)N/dtT
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K e y C o n c e p t s
LO 14.1
The generic continuously compounded instantaneous rate with time-dependent drift and volatility will evolve over time according to dr = \(t)dt + a(t)dw. Special cases of this model include Model 1 (dr = odw) and the Ho-Lee model (dr = \(t)dt + crdw).
LO 14.2
The relationships between volatility in each period could take on an almost limitless number of combinations. To analyze this factor, it is necessary to assign a specific parameterization of time-dependent volatility such that: dr = \(t)dt + ae_atdw, where a is volatility at t = 0, which decreases exponentially to 0. This model is referred to as Model 3.
LO 14.3
Time-dependent volatility is very useful for pricing interest rate caps and floors that depend critically on the forecast of a(t) on multiple future dates. Under reasonable conditions, Model 3 and the mean-reverting drift (Vasicek) model will have the same standard deviation of the terminal distributions. One criticism of time-dependent volatility models is that the market forecasts short-term volatility far out into the future. A point in favor of the mean- reversion models is the downward-sloping volatility term structure.
LO 14.4
Two common models that avoid negative interest rates are the Cox-Ingersoll-Ross (CIR) model and lognormal model. Although avoiding negative interest rates is attractive, the non-normality of the distributions can lead to derivative mispricings.
LO 14.3
The CIR mean-reverting model has constant that increases at a decreasing rate:
volatility (a) and basis-point volatility (cr Vr)
dr = k(0 – r)dt + a Vr dw
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LO 14.6
There are two lognormal models of importance: (1) lognormal with deterministic drift and (2) lognormal with mean reversion.
The lognormal model with drift is:
d [In (r) ] = a(t)dt + adw
This model is very similar in spirit to the Ho-Lee Model with additive drift. The interest rate tree is expressed in rates, as opposed to the natural log of rates, which results in a multiplicative effect for the lognormal model with drift.
The lognormal model with mean reversion is:
d [In (r) ] = k(t)[ln0(t)-ln(r)]dt + cr(t)dw
This model does not produce a naturally recombining interest rate tree. In order to force the tree to recombine, the time steps, dt, must be recalibrated.
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C o n c e p t C h e c k e r s
1.
Regarding the validity of time-dependent drift models, which of the following statements is (are) correct? I. Time-dependent drift models are flexible since volatility from period to period
can change. However, volatility must be an increasing function of short-term rate volatilities.
II. Time-dependent volatility functions are useful for pricing interest rate caps and
floors. A. I only. B. II only. C. Both I and II. D. Neither I nor II.
2.
Which of the following choices correctly characterizes basis-point volatility and yield volatility as a function of the level of the rate within the lognormal model?
Basis-point volatilitv A. increases B. increases C. decreases D. decreases
Yield volatilitv constant decreases constant decreases
3.
4.
3.
Which of the following statements is most likely a disadvantage of the CIR model? A. Interest rates are always non-negative. B. Option prices from the CIR distribution may differ significantly from
lognormal or normal distributions.
C. Basis-point volatility increases during periods of high inflation. D. Long-run interest rates hover around a mean-reverting level.
Which of the following statements best characterizes the differences between the Ho-Lee model with drift and the lognormal model with drift? A. In the Ho-Lee model and the lognormal model the drift terms are
multiplicative. In the Ho-Lee model and the lognormal model the drift terms are additive.
B. C. In the Ho-Lee model the drift terms are multiplicative, but in the lognormal
model the drift terms are additive.
D. In the Ho-Lee model the drift terms are additive, but in the lognormal model
the drift terms are multiplicative.
Which of the following statements is true regarding the Black-Karasinski model? A. The model produces an interest rate tree that is recombining by definition. B. The model produces an interest rate tree that is recombining when the dt
variable is manipulated.
C. The model is time-varying and mean-reverting with a slow speed of adjustment. D. The model is time-varying and mean-reverting with a fast speed of adjustment.
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C o n c e p t C h e c k e r An s w e r s
1. B Time-dependent volatility models are very flexible and can incorporate increasing,
decreasing, and constant short-term rate volatilities between periods. This flexibility is useful for valuing interest rate caps and floors because there is a potential payout each period, so the flexibility of changing interest rates is more appropriate than applying a constant volatility model.
2. A Choices B and D can be eliminated because yield volatility is constant. Basis-point volatility under the CIR model increases at a decreasing rate, whereas basis-point volatility under the lognormal model increases linearly. Therefore, basis-point volatility is an increasing function for both models.
3. B Choices A and C are advantages of the CIR model. Out-of-the money option prices may
differ with the use of normal or lognormal distributions.
4. D The Ho-Lee model with drift is very flexible, allowing the drift terms each period to vary.
Hence, the cumulative effect is additive. In contrast, the lognormal model with drift allows the drift terms to vary, but the cumulative effect is multiplicative.
5. B A feature of the time-varying, mean-reverting lognormal model is that it will not recombine
naturally. Rather, the time intervals between interest rate changes are recalibrated to force the nodes to recombine. The generic model makes no prediction on the speed of the mean reversion adjustment.
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The following is a review of the Market Risk Measurement and Management principles designed to address the learning objectives set forth by GARP. This topic is also covered in:
Vo l a t il it y S m i l e s
E x a m F o c u s
This topic discusses some of the reasons for the existence of volatility smiles, and how volatility affects option pricing as well as other option characteristics. Focus on the explanation of why volatility smiles exist in currency and equity options. Also, understand how volatility smiles impact the Greeks and how to interpret price jumps.
Topic 15
P u t -C a l l Pa r i t y

LO 14.3: Calculate the short-term rate change and describe the basis point volatility using the C IR and lognormal models.
.Another issue with the aforementioned models is that the basis-point volatility of the short- term rate is determined independently of the level of the short-term rate. This is questionable on two fronts. First, imagine a period of extremely high inflation (or even hyperinflation). The associated change in rates over the next period is likely to be larger than when rates are closer to their normal level. Second, if the economy is operating in an extremely low interest rate environment, then it seems natural that the volatility of rates will become smaller, as rates should be bounded below by zero or should be at most small, negative rates. In effect, interest rates of zero provide a downside barrier which dampens volatility.
A common solution to this problem is to apply a model where the basis-point volatility increases with the short-term rate. Whether the basis-point volatility will increase linearly or non-linearly is based on the particular functional form chosen. A popular model where the basis-point volatility (i.e., annualized volatility of dr) increases proportional to the square root of the rate (i.e., (Wr) is the Cox-Ingersoll-Ross (CIR) model where dr increases at a decreasing rate and a is constant. The CIR model is shown as follows:
dr = k(0 r)dt + a Vr dw
As an illustration, lets continue with the example from LO 14.2, given the application of the CIR model. Assume a current short-term rate of 3%, a long-run value of the short-term rate, 6, of 24%, speed of the mean revision adjustment, k, of 0.04, and a volatility, a, of 1.30%. As before, also assume the dw realization drawn from a normal distribution is 0.2. Using the CIR model, the change in the short-term rate after one month is calculated as:
dr = 0.04(24% – 5%)(1/12) + 1.5%
S %x 0.2
dr = 0.063% + 0.067% = 0.13%
Therefore, the expected short-term rate of 5% plus the rate change (0.13%) equals 5.13%.
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A second common specification of a model where basis-point volatility increases with the short-term rate is the lognormal model (Model 4). An important property of the lognormal model is that the yield volatility, O’, is constant, but basis-point volatility increases with the level of the short-term rate. Specifically, basis-point volatility is equal to or and the functional form of the model, where a is constant and dr increases at err, is:
dr = ardt + crrdw
For both the CIR and the lognormal models, as long as the short-term rate is not negative then a positive drift implies that the short-term rate cannot become negative. As discussed previously, this is certainly a positive feature of the models, but it actually may not be that important. For example, if a market maker feels that interest rates will be fairly flat and the possibility of negative rates would have only a marginal impact on the price, then the market maker may opt for the simpler constant volatility model rather than the more complex CIR.
The differences between the distributions of the short-term rate for the normal, CIR, and lognormal models are also important to analyze. Figure 1 compares the distributions after ten years, assuming equal means and standard deviations for all three models. As mentioned in Topic 13, the normal distribution will always imply a positive probability of negative interest rates. In addition, the longer the forecast horizon, the greater the likelihood of negative rates occurring. This can be seen directly by the left tail lying above the x-axis for rates below 0%. This is clearly a drawback to assuming a normal distribution.
Figure 1: Terminal Distributions
-5%
0%
5% Rate
10%
15%
——–CIR ——– N o r m a l——-Lognormal
In contrast to the normal distribution, the lognormal and CIR terminal distributions are always non-negative and skewed right. This has important pricing implications particularly for out-of-the money options where the mass of the distributions can differ dramatically.
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