Topics 13-15

Question 1

Term Structure Model with No Drift (Model 1)

Answer 1

Model 1 is the simplest model for predicting the evolution of short rates, which is used in cases where there is no drift and interest rates are normally distributed.

The probability of up and down movements will be the same from period to period (50% up and 50% down) and the tree will be recombining. Since the tree is recombining, the updown path ends up at the same place as the down-up path in the second time period.

The continuously compounded instantaneous rate, denoted r_t, will change (over time) according to the following relationship:

Question 2

Calculate the short-term rate change and standard deviation of the rate change using a model with normally distributed rates and no drift.

Answer 2

Question 3

Describe methods for addressing the possibility of negative short-term rates in term structure models

Answer 3

• The terminal nodes in the two-period model generate three possible ending rates: r₀ + 2σ(dt)^0.5, r₀, and r₀ - 2σ(dt)^0.5. This discrete, finite set of outcomes does not technically represent a normal distribution. However, our knowledge of probability distributions tells us that as the number of steps increases, the terminal distribution at the nodes will approach a continuous normal distribution.
• One obvious drawback to Model 1 is that there is always a positive probability that interest rates could become negative. On the surface, negative interest rates do not make much economic sense. The negative interest rate problem will be exacerbated as the investment horizon gets longer, since it is more likely that forecasted interest rates will drop below zero.
• There are two reasonable solutions for negative interest rates.
  
  • First, the model could use distributions that are always non-negative, such as lognormal or chi-squared distributions. In this way, the interest rate can never be negative, but this action may introduce other nondesirable characteristics such as skewness or inappropriate volatilities.
  • Second, the interest rate tree can “force” negative interest rates to take a value of zero. In this way, the original interest rate tree is adjusted to constrain the distribution from being below zero. This method may be preferred over the first method because it forces a change in the original distribution only in a very low interest rate environment whereas changing the entire
    distribution will impact a much wider range of rates.
• It is ultimately up to the user to decide on the appropriateness of the model. For example, if the purpose of the term structure model is to price coupon-paying bonds, then the valuation is closely tied to the average interest rate over the life of the bond and the possible effect of negative interest rates (small probability of occurring or staying negative for long) is less important. On the other hand, option valuation models that have asymmetric payoffs will be more affected by the negative interest rate problem.

Question 4

Model 1 Effectiveness

Answer 4

Given the no-drift assumption of Model 1, we can draw several conclusions regarding the effectiveness of this model for predicting the shape of the term structure:

• The no-drift assumption does not give enough flexibility to accurately model basic term structure shapes. The result is a downward-sloping predicted term structure due to a larger convexity effect.
• Model 1 predicts a flat term structure of volatility, whereas the observed volatility term structure is hump-shaped, rising and then falling.
• Model 1 only has one factor, the short-term rate. Other models that incorporate additional factors (e.g., drift, time-dependent volatility) form a richer set of predictions.
• Model 1 implies that any change in the short-term rate would lead to a parallel shift in the yield curve, again, a finding incongruous with observed (non-parallel) yield curve shifts.
• Model 1 is classified as an equilibrium model

Question 5

Term Structure Model with Drift (Model 2)

Answer 5

Question 6

Model 2 Effectiveness

Answer 6

• Model 2 is more effective than Model 1.
• Intuitively, the drift term can accommodate the typically observed upward-sloping nature of the term structure. In practice, a researcher is likely to choose r₀ and λ based on the calibration of observed rates. Hence, the term structure will fit better. The downside of this approach is that the estimated value of drift could be relatively high, especially if considered as a risk premium only.
• On the other hand, if the drift is viewed as a combination of the risk premium and the expected rate change, the model suggests that the expected rates in year 10 will be higher than year 9, for example. This view is more appropriate in the short run, since it is more difficult to justify increases in expected rates in the long run.
• Model 2 is classified as an equilibrium model

Question 7

Construct a short-term rate tree under the Ho-Lee Model with time-dependent drift.

Answer 7

! Ho-Lee model is classified as an arbitrage-free model (due to time-dependent drift which can be used to match the observed prices of securities)

Question 8

Describe uses and benefits of the arbitrage-free models and assess the issue of fitting models to market prices.

Answer 8

• Broadly speaking, there are two types of models: arbitrage-free models and equilibrium models. The key factor in choosing between these two models is based on the need to match market prices.
• Arbitrage models are often used to quote the prices of securities that are illiquid or customized. For example, an arbitrage-free tree is constructed to properly price on-the-run Treasury securities (i.e., the model price must match the market price). Then, the arbitrage-free tree is used to predict off-the-run Treasury securities and is compared to market prices to determine if the bonds are properly valued. These arbitrage models are also commonly used for pricing derivatives based on observable prices of the underlying securities (e.g., options on bonds).
• There are two potential detractors of arbitrage-free models.
  
  • First, calibrating to market prices is still subject to the suitability of the original pricing model. For example, if the parallel shift assumption is not appropriate, then a better fitting model (by adding drift) will still be faulty.
  • Second, arbitrage models assume the underlying prices are accurate. This will not be the case if there is an external, temporary, exogenous shock (e.g., oversupply of securities from forced liquidation, which temporarily depresses market prices).
• If the purpose of the model is relative analysis (i.e., comparing the value of one security to another), then using arbitrage-free models, which assume both securities are properly priced, is meaningless. Hence, for relative analysis, equilibrium models would be used rather than arbitrage-free models.

Question 9

Describe the process of constructing a simple tree for a short-term rate under the Vasicek Model with mean reversion

Answer 9

Question 10

Describe the process of constructing a recombining tree for a short-term rate under the Vasicek Model with mean reversion

Answer 10

The most interesting observation is that the model is not recombining.

It is possible to modify the methodology so that a recombining tree is the end result. There are several ways to do this, but we will outline one straight-forward method.

• The first step is to take an average of the two middle nodes.
• Next, we remove the assumption of 50% up and 50% down movements by generically replacing them with (p, 1 — p) and (q, 1 — q).
• The final step for recombining the tree is to solve for p and q and r^uu and r^dd
• p and q are the respective probabilities of up movements in the trees in the second period after the up and down movements in the first period. r^uu and r^dd are the respective interest rates from successive (up, up and down, down) movements in the tree.

Question 11

Calculate the Vasicek Model expected rate in T years, and half life

Answer 11

Question 12

Describe the effectiveness of the Vasicek Model

Answer 12

• In development of the mean-reverting model, the parameters r₀ and θ were calibrated to match observed market prices. Hence, the mean reversion parameter not only improves the specification of the term structure, but also produces a specific term structure of volatility. Specifically, the Vasicek model will produce a term structure of volatility that is declining. Therefore, short-term volatility is overstated and long-term volatility is understated. In contrast, Model 1 with no drift generates a flat volatility of interest rates across all maturities.
• Consider an upward shift in the short-term rate. In the mean-reverting model, the short-term rate will be impacted more than long-term rates. Therefore, the Vasicek model does not imply parallel shifts from exogenous liquidity shocks.
• Another interpretation concerns the nature of the shock. If the shock is based on short-term economic news, then the mean reversion model implies the shock dissipates as it approaches the long-run mean. The larger the mean reversion parameter, the quicker the economic news is incorporated. Similarly, the smaller the mean reversion parameter, the longer it takes for the economic news to be assimilated into security prices. In this case, the economic news is long-lived. In contrast, shocks to short-term rates in models without drift affect all rates equally regardless of maturity (i.e., produce a parallel shift).

Question 13

Describe the short-term rate process under a model with time-dependent volatility

Answer 13

The generic continuously compounded instantaneous rate is denoted r_t and will change (over time) according to the following relationship:

Question 14

Calculate the short-term rate change and determine the behavior of the standard deviation of the rate change using a model with time dependent volatility

Answer 14

Consider the following model, which is known as Model 3:

Question 15

Model 3 Effectiveness. Assess the efficacy of time-dependent volatility models

Answer 15

• Time-dependent volatility models add flexibility to models of future short-term rates. This is particularly useful for pricing multi-period derivatives like interest rate caps and floors. Each cap and floor is made up of single period caplets and floorlets (essentially interest rate calls and puts). The payoff to each caplet or floorlet is based on the strike rate and the current short-term rate over the next period. Hence, the pricing of the cap and floor will depend critically on the forecast of σ(t) at several future dates.
• There are some parallels between Model 3 and the mean-reverting drift (Vasicek) model. Specifically, if the initial volatility for both models is equal and the decay rate is the same as the mean reversion rate, then the standard deviations of the terminal distributions are exactly the same. Similarly, if the time-dependent drift in Model 3 is equal to the average interest rate path in the Vasicek model, then the two terminal distributions are identical, an even stronger observation than having the same terminal standard deviation.
• There are still important differences between these models.
  
  • First, Model 3 will experience a parallel shift in the yield curve from a change in the short-term rate.
  • Second, the purpose of the model drives the choice of the model. If the model is needed to price options on fixed income instruments, then volatility dependent models are preferred to interpolate between observed market prices. On the other hand, if the model is needed to value or hedge fixed income securities or options, then there is a rationale for choosing mean reversion models.
• One criticism of time-dependent volatility models is that the market forecasts short-term volatility far out into the future, which is not likely. A compromise is to forecast volatility
  approaching a constant value (in Model 3, the volatility approaches 0). A point in favor of the mean reversion models is the downward-sloping volatility term structure.

Question 16

Describe the short-term rate process under the Cox-Ingersoll-Ross (CIR) model

Answer 16

Another issue with the Models 1-3 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.

Question 17

Describe the short-term rate process under the lognormal models.

Answer 17

Question 18

Calculate the short-term rate change and describe the basis point
volatility using the CIR and lognormal models.

Answer 18

• 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 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.
• 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.

Question 19

Lognormal Model with Deterministic Drift

Answer 19

The lognormal model with drift is shown as follows:

Question 20

Lognormal Model with Mean Reversion

Answer 20

Question 21

Term structure models. Summary

Answer 21

Question 22

Put-call parity

Answer 22

Put-call parity is a no-arbitrage equilibrium relationship that relates European call and put option prices to the underlying asset’s price and the present value of the option’s strike price. In its simplest form, put-call parity can be represented by the following relationship:

Question 23

Explain the implications of put-call parity on the implied volatility of call and put options

Answer 23

Question 24

Define volatility smile and volatility skew. 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.

Answer 24

• 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 the case of equity options, the volatility smile is sometimes referred to as a volatility skew, the volatility pattern for equity options is essentially an inverse relationship.

Question 25

Describe characteristics of foreign exchange rate distributions and their implications on option prices and implied volatility

Answer 25

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.

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.

According to Hull, we should expect foreign currency options to exhibit a volatility smile because exchange rates are not lognormally distributed (per the underlying BSM assumption).

• One “violation” of the lognormal assumption is that exchange rates do not exhibit constant volatility.
• Another “violation” is that exchange rates do not change smoothly per a diffusion model; i.e., they jump.

Question 26

Describe the volatility smile for equity options and provide possible explanations for its shape.

Answer 26

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 (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.

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 volatility— leverage and “crashophobia.”

• Leverage. When a firm’s equity value decreases, the amount of leverage increases, which essentially increases the riskiness, or “volatility,” of the underlying asset. When a firm’s 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 asset’s 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 precipitously—deep 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 Rubinstein’s crashophobia hypothesis, because the volatility skew tends to increase when equity markets decline, but is not as noticeable when equity markets increase in value.

! Volatility smile conveys information about the implied distribution of the asset price. Below is the so-called implied volatility smile (skew) for equities; i.e., decreasing as a function of stock price. Such a skew implies a heavier-than left tail and lighter-than right tail, in comparison to a lognormal distribution. This implied distribution has a heavier left tail than the lognormal, not necessarily lighter than left tail of normal.

Volatility Skew appears for options on Equity. What has been observed by traders is that, as the Strike price of an option rises, the implied volatility of an option falls. The volatility is the same as that used in either Put or Call option, it is only their moneyness that would change.

So, if you have a larger strike price for an option, you have a ITM Put, as the volatility is less, the price of the option would also be less; on the other hand, if the strike price is less, you have an OTM Put and according to the skew, you have a higher volatility and thus a higher price for the OTM Put. This is in direct contrast to the Lognormal distribution that we use for pricing options under BSM, this assumes the value of the OTM Puts as well as the probability of occurrence of the extreme events like the OTM Put to be significantly lower than what would happen if we vary the volatility w.r.t the Strike price. In short, in the case of Put options, as the value of the underlying asset falls, the OTM Puts based on lower strike prices would have higher values and probability based on the volatility skew.
Therefore, if we plot the probability distribution, the curve is slightly skewed to the left, with fatter tails on the left and thus, higher probability of occurrence of OTM puts.

Question 27

Describe alternative ways of characterizing the volatility smile

Answer 27

• One alternative method involves replacing the strike price with strike price divided by stock price (X / S₀). This method results in a more stable volatility smile.
• A second alternative approach is to substitute the strike price with strike price divided by the forward price for the underlying asset (X / F₀). 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 prkices since the forward price displays the theoretical expected stock price.
• A third alternative method involves replacing the strike price with the option’s delta. With this approach, traders are able to study volatility smiles of options other than European and American options.

Question 28

Describe volatility term structures and volatility surfaces and how they may be used to price options

Answer 28

• 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 model’s accuracy and consistency in pricing.

Question 29

Explain the impact of the volatility smile on the calculation of the
“Greeks.”

Answer 29

The 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 option’s 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.
• 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.

Question 30

Explain the impact of a single asset price jump on a volatility smile

Answer 30

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.