Model for the Term Structure of Interest Rate - Essay Example

Essays on the Model for the term structure of interest rate Table of Contents Essays on the Model for the term structure of interest rate 1 ABSTRACT 3 Introduction 3 The desirable characteristic of Vasicek Model 5 Stochastic duration 7 How Model may be applied to the pricing model of Zero-coupon bond and the interest rate derivatives 7 The visiceck Model 7 The Cox-Ingersoll-Ross model 8 limit Calibration and Estimation 11 Data for Analysis 12 Conclusion 14 Reference list 17 ABSTRACT In this essay, I focus on the question of whether the term structure model of interest rates employed for pricing derivatives instruments are relevant to use in simulation on the basis of context such as an asset liability management research. I take into consideration the study of the Visiceck and Cox-Ingersoll-Ross (cir) model estimates are on the basis of some dataset on zero coupon yields, while estimation of visiceck model is made on using the cross-sectional data. I standardized the Cir model employing the times sequence of the 2 year yields the simulation of visiceck as well as Cir short rate process will be made by their discrete representation. The judgment of the term structure model empirical and dynamics in the research and relaying on the Visiceck depict negative interest which is unrealistic. ,, most models of the term structure of both methodologies as described above have been of dissuasion sorts, even though it is know that simple diffusion models of the term structure like the Gaussian models do to capture entire of the large variations in interest rates. In this regards, the models inclusive of the jumping process might do a better job. Many events might act as a source of jumping in price in bond market. First, the intrusion of the central bank is to be anticipated due to the fact that the target rates are employed as monetary policy, secondly, the demand or supply impetus might lead to [price jump, since anticipated economic news may be a contributing factor Introduction Term structures of interest rates explain the link concerning the rates of interest and bonds with dissimilar maturity period. The paper explains on the two main models that are centered on explaining the term structure of interest rates. The model is visiceck and Cox-Ingersoll-Ross model. The two models are considered as short rate models since they model the dynamics of the instant shot rate r (t) straight. The shot rate is the annualize interest rates for measurable period of time. Nevertheless, the 3 month period is deem the best estimation of the short rate since for instance, instant credits are impacted by aspects that term structure model doesn’t focus to cover. Short rate is therefore define as rt = R(t; 0) = lim T!0 R(t; T) In which t is the moment of time, T is the time of maturity and R (t; T) is the equivalent interest rates. There two key methodologies to modeling the term structure of interest rate. The first approach is to select the number of explanatory stochastic variables or factors which are presumed to explain the attitude of the term structure. The first realistic model of the kind is the model by Vasicek, who employed the short rate as the main factor. Numerous authors have contributed to what is currently the large collections of single as well as multi-factor model, the model by Dothan, Cox et ale, long staff, Black et ale, and Vasicek are the similar framework since they model their diverse factors as Markov procedure that permits the derivation of a partial differential equation also referred as term structure equation in which the term structure might be obtained. The other methodology is to follow the notion rec0ommedn by Heath, Jarrow and Morton, in which it commands the specification of the initial forward rates curve as well as the volatilities of the forward rates. The whole term structure is modeled by constructing these models and then generating the term structure that precisely fits the initially observed yield curve. Usually,, most models of the term structure of both methodologies as described above have been of dissuasion sorts, even though it is know that simple diffusion models of the term structure like the Gaussian models do to capture entire of the large variations in interest rates. In this regards, the models inclusive of the jumping process might do a better job. Many events might act as a source of jumping in price in bond market. First, the intrusion of the central bank is to be anticipated due to the fact that the target rates are employed as monetary policy, secondly, the demand or supply impetus might lead to [price jump, since anticipated economic news may be a contributing factor. The desirable characteristic of Vasicek Model The Vasicek model we signify with P (t, T) the price at time t of a zero coupon bond (ZCB) that pays a unit of currency at time T greater than t. In conventional, deterministic financial calculation we characteristically have P (t, T) = exp(−r(T − t) in which r is a constant force of interest or constantly compounded interest rate. The interest rates are variable hence, possible extension is to take into consideration the deterministic r(s). In this regards, the earlier formula come to be P(t, T ) = exp(− T t r(s)ds) Where r(t) is considered as stochastic quantity, hence the ZCBs turns to be risk asset since, their price might be worked out while applying the risk neutrality pricing model, which is a computation of the discounted value of the anticipated value of the payoff under the risk neutrality distribution which provides P(t, T ) = E[exp(− T t r(s)ds) In which the anticipation is calculated with regards to the risk distribution of r(t) The primary notion of short rate is to model r(t) as stochastic and afterward to clearly calculate the ZCB price from the equation 3 above. Form a financial point of view, r(t) might be considered as an interest rate equivalent to short maturity such as the three months periods. It might be depicted through the Black Scholes partial differential equation lined to the Vasicek model that the bond pricing for Vasicek, certainly , has just 3 limit ’s which are the ,  and µ . The diverse limit s that yield the bond price are;, µ, ,  and , µ  , , 0 . Denotes workings ,hence  = 0. In this case µ  might be a risk0adjusted depiction of the long term average short rate Vasicek depict that the yield curvature such as the term structure is a function of some fixed as well as short rate, the representation of this with zero coupon bond depict function of short rate.. Stochastic duration In conventional immunization model, on the basis of the outcome of Fisher-Weil and Redington, the assumption is that the dynamics of the interest rates curve is made of parallel shits. The assumptions provides that the main duty of creating immunized portfolio is played by the idea of duration , which is connected to the semi0elasticity of bond price with regards to parallel variation in interest rates. The consideration is the one factor short rate model that generates ZCB prices as a function of short rate r(t), form the expression P(τ ) = A(τ ) exp(−rtB(τ ) we may calculate the semi-elasticity of the ZCB prices with regards to short rate r(t) which we gate 1 p ∂ ∂/rP(τ ) = ∂ /∂r lnP(τ) = −B(τ ). It can be observed that the duration shall be B(τ ) rather than τ . When k → 0, we have that B(τ) ∼ τ, improving the conventional case. How Model may be applied to the pricing model of Zero-coupon bond and the interest rate derivatives The visiceck Model Analytic bond price formula Assuming that the market price of risk λ to be fixed, independent of r and t, then it are likely to develop analytic bond price equation. The Vasicek mean reversion model matches to u = α(γ − r) and w = ρ thus the equation for the bond price becomes ∂B/∂t+ ρ2∂2B/∂r2 + [α(γ − r) − λρ] ∂B∂r− rB = 0 Assuming that the result to be of the form B(r, t; T) = a(t,T)e−b(t,T)r e^−b(t,T)r} The long term internal rate of returns presumed from the present model is depicted to be fixed. Consider that the R(t, T) and in B(r, t; T) are linear function of r(t) and because r(t) depicts a normal distribution implying that r(t) depict a normal distribution and B(t;tlT) is log normally distributed. Supposing that we set T = T1 and T = T2 in and afterward remove r(t), we get a link between R(t,T2) which place dependence on the limit values. The implication is that under the Vasicek model , the instant returns on bonds of diverse maturities are correctly linked. Nevertheless, in real life,. Bond returns over a known period are not linked correctly. The Cox-Ingersoll-Ross model In the Vasicek model, it might happen that the interest rates turns to be negative. To correct this, the Cox, Ingersoll and Ross (1985) suggested the resulting square root diffusion procedure for the short rate: dr = α(γ − r) dt + ρ√r dZ, α,γ >0. With an originally non-negative interest rate, r(t) will not be negative. This is ascribed to the mean returning drift rate with tendency of pulling r(t) in the direction of the long run average γ hence the weakening instability as r(t) reduces to nil.(remember that unpredictability is fixed in the Vasicek model). It might be depicted that r(t) might go to zero simply if ρ2 > 2αγ; although the rising drift is adequately robust to make r(t) = 0 unbearable where 2αγ ≥ ρ2. Every time r(t) is zero, it jumps up into the positive region promptly. The likelihood concentration of the interest rate at time T, provisional on its worth at the present time t, shall be; g(r(T); r(t)) = ce^{u−u) (v/ vu)q/2/q(2(uv)1/2) The supply of future interest depicts the attributes depicted below; (i) as α→∞, the mean tends to γ and the variation to zero, (ii) as α → 0+, the mean tends to r(t) and the variance to ρ2(T − t)r(t). In finding the price of zero coupon bonds short rate on the basis of present square root diffusion procedure, it is assumed that similar outcome as provide in the equation above shall be relevant. The conforming new duo of differential equations for a(t, T) and b(t, T) shall be b(t, T) ={2/eθ(T−t) – 1/(θ + ψ)_eθ(T−t) – 1/+ 2θ} From the equation above, it is apparent that the market price of risk λ is depicted throughout the sum ψ. The limitation of the Model as well as how the issues might be addressed The main limitation of Vasicek Model is that, it is a hypothetically likely for the interest rate to be negative, an adverse feature under the pre-crisis hypothesis. The inadequacy was fixed in the cox-ingersoll-Ross model, exponential Vasicek model among many others. The Vasicek model is well a recognized model of the affine term structure model, along with the cox-ingersoll-Ross model. The fixed rate of interest is that the model is in a position to rebuild diverse shapes either up or downward curve or a numbed molded curvature which are calculated by the diverse limit values divide by the figure one below depicts the three shapes with table one corresponding to limit s values. Table 1; limit s value for the Visiceck Model limit Down slope Ascending slope Humped Shaped rt 0.09 0.02 0.06 0.08 0.08 0.08 0.05 0.05 0.05 0.2 0.2 0.2 0 0 0 The disadvantage of the Visiceck Model is that it might generate negative interest rates. Where the real interest rate is model, this doesn’t have to big issues since real interest rates might be negative in real life situation. The normal rates, will never be negative in reality. Example 1. limit Values rt 0.02 0.03 0.15 0.2 0 P r0.01 0 ={0.02 0.2(0.03 0.02)0.01 0.15 0.01 }   1.335  0.091. Figure 2: Replication of the short rate under the Vasicek model. From figure 2 above, it can be observed that the short rates under the Vasicek Model are negative. The replication with 251 times stages of the short rate is conducted. On many occasion in time, the short rate turns to be adverse. It is realized that adverse interest rates just form the problems in the short turn. In understanding at the conditional anticipation of the Omstein-Unlenbeck procedure, it Can be realized that T, the exponential above as well as the comprehensive 2nd term turns to be zero which is an indication that the long term anticipation to be same to µ: Et rT ={µ + rt µ e(T)} limit Calibration and Estimation In making a comparison of the fit of term structure models, it is imperious that one uses the similar dataset to entire models. Regrettably, other limit s does not evaluate the arrangement of the models. Because the three dimension; yield, time to maturity and time must be considered, diverse estimation approach might be employed. I might decide to perform the cross-sectional approximation which is an approximation of limits whilst taking into consideration the diverse maturity time at a consent moment in time. For the Vasicek method, it has its advantages and disadvantages. Approximation of the mean reversion limit will be hard with time series since, numerous observations ranging many years are needed. In cross-sectional approach, just a day should be observed so long as adequate bonds are traded. The cross-sectional estimation may yield unreasoned estimates. For instance, estimates of the long time averaging 21%. The dissimilarity between the time series and cross-sectional estimates must be small where one factor model of the term structure of interest is correct. Term structure models just on the basis of short rate which may not be accurate in entire situations, but still may better in trying to fit the term structure. Data for Analysis The data for analysis are the monthly yield curvatures for zero coupon bonds, retrieved with the use of the data of Australian bonds and the treasury bills. Consider that the zero coupon bonds for maturities with a horizontal is more than a year are not directly discernable which implies that they require some computation. The methodology employed to derive the Australian Zero coupon yield is provided by Jonson and Metzler (2004).In trying to work out the yield by extrapolating the cross-sectional estimations to the period , it will be undependable . As a result, I just center on yield with maturity time of less than 24 years. Time to maturity has an interlude of 0.24 years, 3 months, commencing at 0.24 years and thus I take into consideration the 100 diverse maturity times. Even though the Australian government doesn’t issues bonds with maturities just like any other government, an explanation of the yield with such maturities is provided by interpolating. Figure 4 depict the monthly yield curve for the zero coupon bond, with declining trend in interest rate for the entire maturity time. The policy central bank is to lower the shot rates to almost 0% in an effort of boosting the economy after the credit crisis Figure 4. Australian zero-coupon yields Table 9 below depicts the equivalent descriptive statistics in which the term structure is upward sloping. Furthermore, the short rates are somehow more variable unlike the long maturity interest rates, while the long rate seems to be steadier unlike the shot rates. Table 9: Descriptive Data of the Australian zero-coupon yields. Maturity 3 months 5 Months 10 years 20 years 25 yeas Mean 0.045 0.046 0.064 0.047 0.472 Stand.Dev. 0.441 0.049 0.444 0.0422 0.422 ˆ(1) r 0.94 0.984 0.944 0.943 0.94 ˆ(12) r 0.744 0.764 0.844 0.841 0.874 ˆ(24) r 0.544 0.572 0.474 0.74 0.744 A comparison conventional fact of the data with the conventional facts generated by the modeling the Vasicek model on the basis of time series estimation, it can be observed that there are numerous incongruity .for instances, the average term structure of the experiential nominal yield is upward-sloping while the simulation generates the download sloping yield curve. Furthermore, skewedness co-efficient Kurtosis as well as the autocorrelation are same for entire maturity time as per the simulation, whilst the difference in the experiential data are apparent. The simulation reproduces a greater instability for short rates as compared for the long rates, which is in conformity with the real data. Nevertheless, the volatility is underestimated conventional of December 2015 depicts incongruity with job since the values for averages as well as volatility are somehow more representative. The simulation on Cir model generates same outcome to the Vasicek simulation on the basis of time series estimates. The skewedness co-efficient, kurtosis as well as the autocorrelations are same for the entire maturity times. The average curve somehow is 2.1 %, with a little bulge in the medium run. Modeled short rates are very unpredictable unlike the long rates but it is undervalued Conclusion It can therefore be concluded that the instability of the term structure is a significant feature in controlling the value dependent claims more specifically in an actuarial science. In using the 3 months treasury bills yields, it can be realized that the models with γ ≥ 1 might capture the dynamics of the short term interest rate much better unlike those of the model with γ < 1. The association concerning the interest rates volatility as well as the level of r is significant, as in relation to the mean reversion features, in description of interest rate model. The integration of term structure normally cause intricacies in the evaluation of the term structure and because main reversion is important, the extra generalization of the calculation of the mean reversion in a model might not be well reasonable. The visiceck model is normally criticized for permitting the undesirable interests. Nevertheless, their stern deficit is the hypothesis of γ = 0 in the model. The hypothesis means that the conditional instability of variation in interest rates is fixed, independent on the level of r. Another stern concern is that the term structure derived from the model creates a scarce family that cannot precisely value many linked bonds. These intricacies from the inherent inadequacy derived from the model price interest rates derivatives with regards to hypothetical yield curvature instead of essentially observing the curve. When the process for r is completely defined, everything concerning the initial term structure as well as how it might emerge at future time is afterwards completely defined. The initial term structure is an output from the model instead of input to the model. ,, most models of the term structure of both methodologies as described above have been of dissuasion sorts, even though it is know that simple diffusion models of the term structure like the Gaussian models do to capture entire of the large variations in interest rates. In this regards, the models inclusive of the jumping processes might do a better job. Many events might act as a source of jumping in price in bond market. First, the intrusion of the central bank is to be anticipated due to the fact that the target rates are employed as monetary policy, secondly, the demand or supply impetus might lead to [price jump, since anticipated economic news may be a contributing factor It might be interesting to research on the exact estimation process on the replication. This might be interpreted into attuning the model into a subgroup of the experiential data such that form instance the low yields because of credit crisis are ignored. This might as well be interpreted into employing a totally diverse approximation process. The cross-sectional approximation process omits the interest rates since it takes into consideration the yield at a constant moment in time. As a result, one may focus on an approximation process that takes into consideration entire data such as the panel data estimation. Reference list Read More