An Approximation of Caplet Implied Volatilities in Gaussian Models Flavio Angelini, Stefano Herzel Department of Economics University of Perugia Via A. Pascoli, 1 06100 Perugia, Italy. Phone: 0039-75-5855258; Fax: 0039-75-5855221 E-mail:
[email protected],
[email protected]
JEL classification: E43, C13
MS classification: 91B70
Keywords: Interest rate caplets, implied volatility, forward rates, Gaussian HJM models
1
1
Introduction
It is market practice to quote interest rate derivatives traded ”Over the Counter” in terms of their implied volatility. For this reason, the term structure of at-the-money cap volatilities as well as the volatility surface of at the money swaptions are directly observed. This paper analyzes the case of caps. Any analysis of these markets would most likely report two main facts. The first one is that the level of the volatility is inversely related to the level of the interest rates. The second one is that the term structure is either a decreasing or a humped function of maturity. As a reference, one can look at Rebonato (2003) and Brigo and Mercurio (2002)). Rebonato (2003) suggests that the structure of implied volatility is humped in periods of normal market conditions and decreasing when markets are ”excited”. Interpreting and explaining such phenomena is indeed an interesting and important issue. In a HJM model the price of any traded security can (in principle) be determined by the current term structure of the interest rates and the diffusion coefficient that specifies the model. Given the price, one can define the model implied volatility of the security by inverting the appropriate formula used by the market. Therefore, in a HJM context, the facts mentioned above can be explained by the combined action of two factors: the level and the volatility of the interest rate term structure. In this paper we will study an important specification of the HJM framework, the Gaussian HJM models. Such models are quite fashionable because they provide with closed formula for many claims. We shall determine a linear approximation of the model implied volatility, in terms of the level of the forward rate and the standard deviation of the bond yield. With such an approximation, we will be able to disentangle the effect on the implied volatility due to the level of the forward rate and to its stochastic dynamics. We will study the approximation error and give an upper bound of it. The proposed formula allows us to determine which shapes of implied volatility a given model is able to produce and to what extent. We will focus the analysis on some popular Gaussian HJM model, showing for instance under which conditions the Ho-Lee model, whose diffusion coefficient is constant, presents a hump in the implied volatility structure. We will also explore the fitting capabilities of a model and propose a ”rule of thumb” to select a model under particular market conditions. In particular, we show that the Hull-White model is an appropriate choice only under ”very excited” market conditions. We will also analyze the relation between its parameter of mean-reversion and a quantity, inferred from market data, which 1
can be interpreted as implied volatility of the bond. Further applications, with specific references to market data and model calibration, can be found in Angelini and Herzel (2004b). The rest of the paper is structured as follows: Section 2 proves the approximation formula and studies the error; the applications are contained in Section 3.
2
Implied Volatilities in Gaussian Models
In this section we obtain a formula that relates the implied volatility with the level and the volatility of interest rates under the hypothesis that the prices of bonds are log-normal. This is the case of a Gaussian specification of the class of models proposed by Heath, Jarrow and Morton (1992). This class provides with a large variety of models, which are analytically tractable and therefore very suitable for pricing derivatives. Let us consider an at-the-money forward caplet with reset time T − τ and expiration T . Let P (t, T ) be the price, at the current time t, of the zero coupon bonds with maturity T . The simple forward rate with reset T − τ and maturity T is defined as F (t, T − τ, T ) =
P (t, T − τ ) − P (t, T ) . τ P (t, T )
The caplet is at-the-money forward when the strike rate is equal to F (t, T − τ, T ). It is market standard to use Black Formula to price caps and floors; the Black Formula for an at-the-money caplet is ¶ µ v(t, T ) v(t, T ) ) − N (− ) , B(t, T, v(t, T )) = P (t, T )τ F (t, T − τ, T ) N ( 2 2 where N (x) is the distribution function of the standard normal and v(t, T ) is the standard deviation, computed at time t, of the logarithm of F (T − τ, T − τ, T ). The main assumption underlying Black Formula is that the simple forward rate is log-normal. A rigorous derivation under usual noarbitrage conditions was proposed by Brace et al. (1997). The market ) , when B(t, T, v(t, T )) equals implied volatility is defined as σ(t, T ) = √v(t,T T −τ −t the market price of the caplet. Let us consider Gaussian HJM models in which the dynamics of the prices of the zero coupon bonds are dP (u, T ) = (·)du + S(u, T )dWu , P (u, T ) 2
where W is a d-dimensional Brownian motion and the bond price volatility S(u, T ) is a d-dimensional, deterministic vector. The drift (not explicitly written) depends on the probability measure under which the dynamics are written. In particular, when the martingale measure is adopted, it is equal to the short rate. An equivalent representation in terms of the dynamics of the instantaP (t,T ) is (see, for instance, Bj¨ork (1998), neous forward rate f (t, T ) = − ∂ log∂T Prop. 15.5) df (u, T ) = (·)du + β(u, T )dWu , where the instantaneous volatility of P (t, T ) is related to that of the instantaneous forward rate as Z T S(u, T ) = − β(u, x)dx. u
From Ito’s lemma one can recover the dynamics of the simple forward rate dF (u, T − τ, T ) = (·)du +
P (u, T − τ ) (S(u, T − τ ) − S(u, T )) dWu . P (u, T )
From a standard argument (see for instance Brigo and Mercurio (2001) §6.3) it follows that, under the ”forward measure” QT associated to the numeraire P (u, T ), F (u, T − τ, T ) is a martingale satisfying F (u, T − τ, T ) + τ1 dF (u, T − τ, T ) = s(u, T − τ, T )dWuT , F (u, T − τ, T ) F (u, T − τ, T )
(2.1)
where W T is a Brownian motion under QT and s(u, T − τ, T ) = S(u, T − τ ) − S(u, T ). The solution of Equation (2.1) is ¡ ¢ F (u,³T − τ, T ) = F (t, T − τ, TR ) + τ1 · ´ u Ru ||s(y,T −τ,T )||2 dy exp t s(y, T − τ, T )dW T − t − τ1 . 2 We observe that the simple forward rate is not log-normal, hence the model price of a caplet will not be given by Black Formula. Instead, computing the expected value with respect to the forward measure leads to µ ¶µ ¶ 1 Σ(t, T ) Σ(t, T ) M(t, T, Σ(t, T )) = P (t, T )τ F (t, T − τ, T ) + N( ) − N (− ) , τ 2 2 3
where
Z
T −τ
Σ(t, T )2 =
||s(y, T − τ, T )||2 dy.
t
Recall that Σ(t, T ) is the standard deviation, computed at time t, of the Gaussian random variable log(P (T − τ, T )). We observe that, as Σ(t, T ) increases, the function M returns values that can be higher than the upper bound of the corresponding Black Formula. This is due to the fact that, in a Gaussian model, interest rates can become negative with positive probability that increases with Σ(t, T ). (t,T ) The model implied volatility is defined as σM (t, T ) = √vM , where T −τ −t vM (t, T ) is such that B(t, T, vM (t, T )) = M(t, T, Σ(t, T )). Thus vM (t, T ) is defined implicitly as a function of Σ(t, T ). In the case of at-the-money options, one gets (here and in the sequel, when not needed, we drop the variables t and T to simplify notations) vM (Σ) = 2N −1 ((H(t, T )[2N (Σ/2) − 1] + 1)/2) , where H(t, T ) =
τ F (t, T − τ, T ) + 1 . τ F (t, T − τ, T )
(2.2)
(2.3)
Formula (2.2) permits to recover the implied volatility of a model when the function Σ is specified. Our goal is to determine an approximation of it. As noted above, Σ has to be sufficiently small in order to limit the possibility of negative interest rates. For this reason, we will study the function vM (Σ) for small values of Σ, proposing an expansion around Σ = 0. Let us observe that 2N (vM (Σ)/2) − 1 = H(2N (Σ/2) − 1),
(2.4)
vM (Σ) > Σ,
(2.5)
hence that is, the Black implied volatility of a caplet is always greater than the standard deviation of the logarithm of the price of the underlying bond. Note that vM (0) = 0. Differentiating Formula (2.4) with respect to Σ, we get n(Σ/2) (1) vM (Σ) = H , n(vM /2) 4
where n(x) is the probability density of the standard normal distribution. (1) Hence vM (Σ) is an increasing function and vM (0) = H. Differentiating again we get (2)
vM (Σ) =
H n(Σ/2) (Hn(Σ/2)vM − n(vM /2)Σ) , 4 n(vM /2)2
which is zero when Σ = 0 and positive when Σ is positive, because of (2.5). We do not report the third and fourth derivative and we just observe that (3)
vM (0) = H(H 2 − 1)/4; (4)
vM (0) = 0. Note that, being vM (Σ) an odd smooth function, all the derivatives of even order vanish in zero. Summarizing, we get the following Proposition 2.1 Let the forward rates evolve according to (2.1). Then σ1 (t, T ) = H(t, T )Z(t, T ), where H(t, T ) is defined by (2.3) and Σ(t, T ) Z(t, T ) = √ , T −τ −t
(2.6)
is a first order approximation for the model implied volatility σM (t, T ) of the at-the-money forward caplet with maturity T . The error of the approximation is ¡ ¢ 1 √ H(t, T ) H(t, T )2 − 1 Σ(t, T )3 σM (t, T ) − σ1 (t, T ) = 24 T − τ − t + o(Σ(t, T )5 ), Σ(t, T ) → 0. The approximating formula for the implied volatility can be written as σM (t, T ) ≈ H(t, T )Z(t, T ).
(2.7)
It separates, in a simple and natural way, the effects of the level of the forward rates (the term H) and of the volatility of bond prices (the term Z) on the implied volatility of caplets. Proposition 2.1 gives the order of magnitude of the error as Σ goes to zero. It is also possible to determine an upper bound on bounded intervals. In fact, we observe that e(Σ) := σM (t, T ) − σ1 (t, T ) 5
is a convex, increasing function when Σ is positive. Since e(0) = 0, we ¯ with the segment can bound the function e(Σ) for Σ in the interval (0, Σ) ¯ e(Σ)), ¯ that is connecting the points (0, 0) and (Σ, e(Σ) ≤
¯ e(Σ) ¯ Σ, Σ
¯ 0 ≤ Σ ≤ Σ.
To study the dependence of the error from the level of the forward rate, we compute its derivative with respect to H ∂e 2N (Σ/2) − 1 − n(vM /2)Σ √ (Σ, H) = . ∂H T − τ − t n(vM /2) The numerator is positive being vM > Σ > 0, therefore the error increases with H, hence decreases with the level of the forward rate. Figure (1) exhibits the error function e(Σ) for a 10-year caplet. The thicker line corresponds to an underlying forward rate of 5%, the thinner to a forward rate of 10%. Values of Σ are in the interval [0,0.04], corresponding to model implied volatilities ranging from 0 to 61% in the case of a forward rate of 5% and from 0 to 28% in the case of a forward rate of 10%. We remark that the ranges for implied volatility are in line with real market data and with the typical inverse relation between implied volatility and underlying rate. Note that the errors are higher when the forward rate is smaller. We end the section with an intuitive explanation of the approximating formula: Black formula for caplet is theoretically justified by the following dynamics of the simple forward rate, under the measure QT (see Brace et al. (1997)), dF (u, T − τ, T ) = σ(u, T )dWuT ; (2.8) F (u, T − τ, T ) the implied volatility of the caplet is then the annualized version of the RT square root of t σ(u, T )2 du. A possible way to approximate the model implied volatility is by ”freezing” the coefficient of s(u, T − τ, T ) in Equation 2.1 to its value at time t to obtain à ! 1 2 Z T −τ F (t, T − τ, T ) + τ (T −τ −t)σM (t, T )2 ≈ s(u, T −τ, T )2 du. (2.9) F (t, T − τ, T ) t This would lead, in a more straightforward, but less formal way (and without an estimate of the error) to Formula (2.7). 6
3
Applications
Formula (2.7) relates three term structures: the annualized volatility of bond prices Z(t, T ) which depends on the model selected, the function H(t, T ) which depends on the current level of the forward rates and can be extracted from market data and the implied volatility of caplets σ(t, T ) which are produced by the model. A possible way to measure the goodness of a model is to look at the distance between model and market implied volatilities. In particular, the capability of reproducing the shape of the volatility term structure is considered a nice feature of a model. We start the section by showing some typical shapes of market volatilities, then we briefly describe three popular Gaussian HJM models and their functions Z(t, T ). We use Formula (2.7) to analyze the structures of model implied volatility, answering to questions like what kind of shapes a given model is able to produce, under what conditions and to what extent. We will conclude by suggesting a ”rule of thumb” to select a model in the Gaussian class according to market conditions, with the objective of minimizing the number of parameters and offering a good fit to the market volatilities. Usually, caplet volatilities are not directly observed on the market. Instead, commonly quoted are implied volatilities of at-the-money-forward caps with maturities 1, 2, 3, 4, 5, 7, 10 years. Caplet volatilities can be obtained through a bootstrap procedure from cap volatilities. We denote by σ ¯ (t, T ) the caplet bootstrapped volatilities at time t, with T = t + 2τ, t + 3τ, . . . , t + 20τ and τ = 0.5. We show market implied volatility structures of caps together with the results of such a procedure in Figure 2, where four typical shapes are shown. Namely, humped at medium-long maturity, humped (not pronounced) at short maturity, decreasing and again humped (pronounced) at short maturity. The type of structure of caplet volatilities is similar to that of caps, but it exhibits a more irregular behaviour. The cap volatility is essentially an average of the volatilities of the caplets involved in the cap. To illustrate the use of the formula we consider three Gaussian HJM models. The continuous time version of the Ho-Lee model (Ho-Lee (1986)), denoted by HL, whose discrete version set the path to the work of Heath, Jarrow and Morton. The extension of the Vasicek model (Vasicek (1977)) proposed by Hull and White (1990), denoted by HW, a widely used model which incorporates mean-reversion and has a short rate representation. The model proposed by Mercurio and Moraleda (1996) and Ritchen and Chuang (1999), henceforth MM, introduced with the specific purpose of fitting the hump of the volatility structure. Although we restrict our analysis to these three 7
models, we would like to stress that the approximating formula could be used to analyze any Gaussian HJM model. Other specifications of Gaussian HJM models can be found in the literature, like for instance that with affine volatility function studied by Amin and Morton (1994), not very flexible, or the non-parametric model proposed by Brace and Musiela (1994) to exactly match the market volatilities. We chose those three models because they are quite popular and their volatility term structures present different characteristics. The three models are specified by the volatilities of the instantaneous forward rates which can be nested as β(u, T ) = (σ + γ(T − u)) exp(−a(T − u)). The HL model is obtained by setting γ = a = 0, the HW model by γ = 0. Therefore the HL model has constant volatility, the HW model a monotone volatility function, increasing (decreasing) when a is negative (positive), the MM model a volatility function which can be (when T > u) humped, decreasing or increasing. A simple computation gives the function Z(t, T ) as defined in (2.6). For convenience of the reader, we write Z(t, T ) in terms of β(t, T ): 1 Z (t, T ) = T −τ −t
Z
2
T −τ
µZ
t
T
¶2 β(u, x)dx du.
T −τ
The shape of Z(t, T ) is similar to that of the volatility function β(t, T ), as we have ZHL (t, T ) = στ ; ZHW (t, T ) =
(3.10)
s
σ (1 − e−aτ ) a
1 − e−2a(T −τ −t) . 2a(T − τ − t)
(3.11)
The HL case is particularly simple because ZHL (t, T ) is constant. ZHW (t, T ) is a decreasing function in T when a is positive and increasing when a is negative, starting in both cases from the positive value σa (1 − e−aτ ) (which is the limit as T goes to t + τ ). In both cases, the higher is the absolute value of a the steeper is the function. We do not report the function Z(t, T ) for the MM model because it is quite involved; the interested reader can find it in Mercurio and Moraleda (1996). Typically, when calibrating the MM model on market data, one finds either humped or decreasing shapes. An example of the functions Z’s for typical parameter values can be found in the bottom left box of Figure 4. 8
Every day, from the observed term structure of interest rates one can get, using various techniques, a forward rate curve. For instance, one can obtain, by log-linear interpolation, the discount factors for maturities with six-month lags, T = t + τ, t + 2τ, . . . , t + 20τ years and τ = 0.5 and then compute the simple forward rates with reset T − τ and maturity T . Then one can calculate the function H(t, T ), as defined by (2.3). The shape of this function is important to our analysis and can be easily recovered from the term structure of the forward rates F (t, T − τ, T ). In general, since the forward rate curve can assume any form, so can H(t, T ). Some typical observed forms for the forward curve (apart from some irregularity) could be, for instance: approximately constant, increasing and with a throat, corresponding to approximately constant, decreasing and humped shapes of the function H(t, T ). Some of these shapes are shown in Figure 3, where we chose to pick the same days as in Figure 2. By comparing Figure 2 and 3 we note a relation between the behaviour of the function H(t, T ) with that of the term structure of caplet implied volatilities. In our experience 1 , we found that this is often the case. An analysis of such a relation is indeed an important issue, but it is beyond the scope of the present paper. Here we give just some graphical evidence of this fact: in the top right boxes, where H(t, T ) is approximately constant, the implied volatility is humped at medium maturities, but assumes values in a more narrow range than in the other cases. In the top left, the shapes of the two are quite similar, but differ because the caplet structure presents a small hump at short maturities. In the bottom ones the two structures are very similar to each other. Notice that different interpolation techniques could be used to determine the structure of forward rates, for instance by fitting a NelsonSiegel family, a consistent family or others (see for instance Angelini and Herzel (2002)). This may give different shapes of the function H(t, T ). In Figure 3 the log-linear method was used to interpolate discount factors because it is the most broadly used. We are now ready to apply Formula (2.7) to study the three models selected. First we show a toy case that illustrates the relevance of H(t, T ) in the determination of implied volatilities. Its results are reported in Figure 4. We chose a particular forward curve (top left) to which corresponds a strongly humped H(t, T ) (top right). The functions Z(t, T ) of the three models (bottom left) are so chosen to have the same order of magnitude and with reasonable parameter values. The parameter values are as follows: we 1 With this expression, here and in the sequel, we mean to refer to Angelini and Herzel (2004b)
9
set σ equal to 0.008, 0.01 and 0.005 respectively for the HL, the HW and the MM model , a equal to 0.2 and 0.5 respectively for the HW and the MM model and γ equal to 0.01 for the MM model. The resulting implied volatilities (bottom right) are humped, as predicted by the approximation formula, even for the HL and the HW model. In this case the function ZHW (t, T ) is decreasing and therefore the hump produced by the HW model is less pronounced than that of the HL model. The MM model produces the most pronounced hump. Finally, the approximated values are very close to the exact ones. From (2.7) and (3.10) it is clear that, for the HL model, the term structure of caplet implied volatility has basically the same shape as H(t, T ). Since the latter may in general assume any form, so can the implied volatilities produced by the model. As noted above the shapes of the market caplet volatilities and that of the H(t, T ) are often, but not always, related. Hence, as it is confirmed by our experience, the performance of the HL model depends on how similar the shapes of H(t, T ) and of market implied volatilities are. As for the HW model, when a is positive, which is the condition required for the model to be mean-reverting, only a very humped or initially rapidly increasing H(t, T ) (compared to the steepness of Z(t, T )) may force the model implied volatility to have a hump. In our experience, this happens rarely, but it is still possible (see for instance the toy case of Figure 4). A more realistic ”humped” case is when the value of parameter a is small (say less than 0.01) and the function H(t, T ) is sufficiently humped. However, when a is very small, the model function Z(t, T ) (as well as the instantaneous volatility of instantaneous forward rates) is almost constant. In such a case one should consider adopting a more parsimonious model like the HL, whose fitting capabilities in these cases should be comparable. Higher values of parameter a give a more decreasing function Z(t, T ). In these cases it would be quite hard for the caplet implied volatilities to present a hump at short maturities. It is usually reported (Amin and Morton (1994), Mercurio and Moraleda (1996)) that, on real data, the estimates of a are often negative. When a is negative, the model is no longer mean-reverting, but rather mean-diverging, and its implied volatilities result to be increasing for long maturities. Since this is not observed in the market, it may lead to serious errors at long maturities. A suitable H(t, T ) (like that of Figure 4) may, in principle, force the model volatility structure to be humped or decreasing. However, for long maturities, the model volatilities would tend to increase again, with the result of a throat in the structure. 10
As it is well known, the MM model is the most flexible of the three and this may be explained in part because it has one parameter more. Also it is the most capable to reproduce the humped shape of volatilities. Formula (2.7) shows that this is because of the flexibility of the corresponding function Z(t, T ). Formula (2.7) may be also used as a simple tool to select the model in order to have a satisfactory reproduction of the shape of market volatilities. The calibration of a model may be performed by minimizing the relative errors between model and market prices of caplets. This turns out to be similar to minimizing the relative errors in caplet volatilities (see Angelini and Herzel (2004a) for details). Let us consider the approximation of the annualized standard deviation of the logarithm of model bond prices implied by the market volatility σ ¯ (t, T ) given by ¯ (t, T ) ¯ T) = σ . Z(t, H(t, T ) ¯ T ) may be seen as a short cut of the method proposed Looking at Z(t, by Brace and Musiela (1994). Given that the function H(t, T ) may be considered independent of the model and can be extracted from the current ¯ T ) represents a term structure of interest rate, looking at the shape of Z(t, ”rule of thumb” for the choice of the model. This function is usually quite irregular but its overall shape is often humped. In this case one should adopt the MM model. However, this is not always true, as, in our experience, there are also approximately constant or decreasing cases. In the constant case, the HL model should be able to fit the market implied volatility, almost as well as the other two. In the decreasing case, the HW model would give a satisfactory performance similar to that of the MM model, with a positive value of parameter a. One may expect parameter estimates with the following characteristics: in the constant case, the parameter σ for the three models very close to each other, while parameters γ and a (for both the HW and the MM model) all close to zero. In the decreasing case, σ of the HW and the MM model very close to each other, while that of the HL model significantly different, a positive a of the HW model and γ again close to zero. This is indeed what we found in our experience. One should also expect a strict correspondence between a positive estimate of parameter a ¯ T ), for reasons that should be clear of the HW model and a decreasing Z(t, from the discussion on the HW model. In conclusion, if one wants a model as parsimonious as possible to explain the volatility term structure, one should choose the HL model in the constant case and the HW model in the decreasing one. 11
We conclude with an observation related to the negative estimates of the parameter a of the HW model, often reported in the literature. A common explanation relates such fact to the presence of a hump in the volatility structure. The idea is that the HW model follows the initial growth of the structure with an increasing Z(t, T ). However, such interpretation is not always correct, as negative estimates of a do sometimes occur also when the volatility structure is decreasing. In fact, it should be clear now that ¯ T ). As stated above, the model a negative a is a sign of a humped 2 Z(t, Z(t, T ) and consequently the caplet volatilities would be too high for long maturities, hence far from market ones, leading to serious mis-pricing of ¯ T ) is humped and a correct long maturity derivatives. Therefore, when Z(t, pricing of long maturity derivatives is important, a model like MM should be preferred to the HW one.
References [1] K.I. Amin, A.J. Morton (1994), Implied Volatility Functions in Arbitrage-Free Term Structure Models, Journal of Financial Economics, 35, 141-180. [2] F. Angelini, S. Herzel, Consistent initial curves for interest rate models, Journal of Derivatives, 9, (2002), 8-18. [3] F. Angelini, S. Herzel, Consistent Calibration of HJM Models to Cap Implied Volatilities, preprint, http://www.unipg.it/angelini (2004a). [4] F. Angelini, S. Herzel, An Analysis of Market Volatilities of Caps, preprint, http://www.unipg.it/angelini (2004b). [5] T. Bj¨ork, Arbitrage Theory in Continuous Time, Oxford University Press, 1998. [6] A. Brace, M. Musiela, A Multifactor Gauss Markov Implementation of Heath, Jarrow, and Morton, Mathematical Finance, 4, No. 1 (1994), 259281. [7] A. Brace, D. Gatarek, M. Musiela, The market model of interest rate dynamics Mathematical Finance, 7, No. 7 (1997), 127-147. ¯ T ), which is not In principle, a negative a could also be due to a decreasing Z(t, consistent with market behaviour. 2
12
[8] D. Brigo, F. Mercurio, Interest Rate Models: Theory and Practice, Springer-Verlag, 2001. [9] D. Heath, R. Jarrow, A. Morton, Bond Pricing and the Term Structure of Interest Rates Econometrica, 60, No. 1 (1992), 77-106. [10] T. Ho, S. Lee, Term Structure Movements and Pricing Interest Rate Contingent Claims. The Journal of Finance, 41 (1986), 1011-1029. [11] J. Hull, A. White, Pricing Interest Rate Derivative Securities. The Review of Financial Studies, 3 (1990), 573-592. [12] Mercurio F. and J.M. Moraleda, An Analytically Tractable Interest Rate Model with Humped Volatility, discussion paper TI 96-116-2, (1996), Tinbergen Institute, Erasmus University Rotterdam, The Netherlands. [13] R. Rebonato, Interest Rate Option Models, 2nd edition, Wiley, Chicherster, 1998. [14] R. Rebonato, Volatility and Correlation, Wiley, Chicherster, 1999. [15] R. Rebonato, Term structure models: a review, preprint (2003) [16] P. Ritchken, I. Chuang, Interest Rate Option Pricing with Volatility Humps, Review of Derivatives Research, 3, (1999), 237-262. [17] O. Vasicek, An Equilibrium Characterization of the Term Structure. Journal of Financial Economics, 5 (1977), 177-188.
13
0.08
F=0.05 F=0.10
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
0
0.005
0.01
0.015
0.02 Σ
0.025
0.03
0.035
0.04
Figure 1 Error of approximation Formula (2.7) on a 10-year caplet as a function of Σ, when the simple forward rate F is 5% or 10%. The interval for Σ is bounded by 0.04, hence the model implied volatility ranges from 0 to 61% (when F=5%) or from 0 to 28% (when F=10%).
14
0.2
0.23 caplet cap
caplet cap
0.19
0.22
0.18 0.21 0.17 0.2
0.16
0.15
0.19
0.14
0.18
0.13 0.17 0.12 0.16
0.11
0.1
1
2
3
4
5
6
7
8
9
10
0.5
0.15
1
2
3
4
5
6
7
8
9
0.8 caplet cap
caplet cap
0.45
0.7
0.4
0.6
0.35
0.5
0.3
0.4
0.25
0.3
0.2
0.2
0.15
10
1
2
3
4
5
6
7
8
9
10
0.1
1
2
3
4
5
6
7
8
9
10
Figure 2 US market cap implied volatilities and results of the bootstrap procedure from cap to caplet volatilities on four days (24-1-2000, 27-3-2001, 2412-2001, 3-3-2003). The cap structure are respectively humped at mediumlong maturity (top left), humped (not pronounced) at short maturity (top right), decreasing (bottom left) and again humped (pronounced) at short maturity (bottom right).
15
110
110
100
100
90
90
80
80
70
70
60
60
50
50
40
40
30
30 1
2
3
4
5
6
7
8
9
10
110
110
100
100
90
90
80
80
70
70
60
60
50
50
40
40
30
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
30 1
2
3
4
5
6
7
8
9
10
Figure 3 The function H(t, T ) on the same days as Figure 2, 24-1-2000, 27-3-2001, 24-12-2001, 3-3-2003. Notice the relation among these figures and the corresponding ones of Figure 2.
16
0.1
70
65 0.09 60 0.08
55
50 0.07 45 0.06 40
35
0.05
30 0.04 25
0.03
1
2
3
4
5
6
7
8
9
10
−3
5
20
1
2
3
4
5
6
7
8
9
10
0.4
x 10
HL HW MM
HL exact HL approximated HW exact HW approximated MM exact MM approximated
0.35
4.5
0.3 4
0.25 3.5
0.2
3
0.15
2.5
2
0.1
1
2
3
4
5
6
7
8
9
10
0.05
1
2
3
4
5
6
7
8
9
10
Figure 4 An example where all the models produce a humped implied volatility. The forward curve (top left) induces a very humped function H(t, T ) (top right). The functions ZHL (t, T ), ZHW (t, T ) and ZM M (t, T ) (bottom left) are obtained by setting σ equal to 0.008, 0.01 and 0.005 respectively for the HL, the HW and the MM model, a equal to 0.2 and 0.5 respectively for the HW and the MM model and γ equal to 0.01 for the MM model. As predicted by the approximating formula, the implied volatility structures are all humped, the least pronounced that of the HW model, the most pronounced that of the MM model (bottom right). Both exact and approximated values of the implied volatilities are shown.
17
