Vasicek model

In describing the dynamics of the yield curve, the Vasicek model only captures changes in the short-rate r, and not the long-run average rate b. 

It might be considered to be more reahstic to consider that there are no constant parameters for the drift rate and the standard deviation that would ensure that the price of a zero-coupon bond at any time is exactly the same as that suggested by observed market yields. For this reason, a modified version of the Vasicek model has been described by Hull and White (1990), known as the Hull-White or extended Vasicek model, which we will consider later. 

The Hull-White (1990) model is an extension of the Vasicek model designed to produce a precise fit with the current term structure of rates. It is also known as the extended Vasicek model, with the interest rate following a process described by Equation (3.48) ... 

To recap on the issues involved in fitting the extended Vasicek model or Hull-White model this describes the short-rate process as following the form... 

The Vasicek, Cox-Ingersoll-Ross, Hull-White and other models incorporate mean reversion. As the time to maturity increases and as it approaches infinity, the forward rates converge to a point at the long-run mean reversion level of the current short-rate. This is the limiting level of the forward rate and is a function of the volatility of the current short-rate. As the time to maturity approaches zero, the short-term forward rate converges to the same level as the instantaneous short-rate. In the Merton and Vasicek models, the mean of the short-rate over the maturity period T is assumed to be constant. The same constant for the mean, or the drift of the interest rate, is described in the Ho-Lee model, but not the extended Vasicek or Hull-White model. 

When calcnlating option prices in a one-factor model, a frequently made assnmption is that the process is driven by the short rate often with a mean reversion featnre linked to the short rate. There are several popnlar models which fall into this category, for example, the Vasicek model, and the Cox, Ingersoll, and Ross model both of which will be discussed in more detail later. Calculating option prices in a two-factor model involves both the short- and long-term rates linked by a mean reversion process. 

These models are two more general families of models incorporating Vasicek model and CIR model, respectively. The first one is used more often as it can be calibrated to the observable term structure of interest rates and the volatility term structure of spot or forward rates. However, its implied volatility structures may be unrealistic. Hence, it may be wise to use a constant coefficient P(t) = P and a constant volatility parameter a(t) = a and then calibrate the model using only the term structure of market interest rates. It is still theoretically possible that the short rate r may go negative. The risk-neutral probability for the occurrence of such an event is... 

Dothan model, Black-Karasinski model, and the Exponential Vasicek model given below imply that r is log-normally distributed. While this finding may seem reasonable it is the cause for the explosion of the bank account, that is from a single unit of money one may be able to make, in an infinitesimal interval of time, and infinite amount of money. Sandmann and Sondermann model overcomes this problem by modelling the short rates as above. 

The Extended Exponential Vasicek model (Brigo and Mercurio) is... 

Formulas for bond options were found by Cox, Ingersoll, and Ross using the CIR model (square root process) for short rates, and by Jam-shidian, Rabinovitch, and by Chaplin using the Vasicek model for the short rate process. 

Bonds are traded generally over the counter. Futures contracts on bonds may be more liquid and may remove some of the modelling difficulties generated by the known value at maturity of the bonds. Hedging may be more efficient in this context using the futures contracts on pure discount bonds (provided they are liquid) rather than the bonds themselves. Chen provides closed-form solutions for futures and European futures options on pure discount bonds, under the Vasicek model. 

Initially the first formulas on pricing options on pure discount bonds used the Vasicek model for the term structure of interest rates. Thus, given that r follows equation (18.6), the price of a European call option with maturity Tq with exercise price fC on a discount bond maturing at T(Tq < T) is... 

We will use the Vasicek model for pricing a 3-year European call option on a 10-year zero-coupon bond with face value 1 and exercise price K equal to 0.5. As in Jackson and Staunton, we use for the parameters of this model the values estimated by Chan, Karolyi, Longstaff, and Sanders for US 1-month Treasury bill yield from 1964 to 1989. Thus a = 0.0154, p = 0.1779, and o = 2%. In addition, the value of the short... 

EXHIBIT 18.6 Calculations of Elements for Pricing an European Call Option on a Zero-Coupon Bond when Short Rates are Following the Vasicek Model... 

Taking the same example as that developed to demonstrate the Vasicek model earlier, we now price the 3-year European call option on a 10-year pure discount bond using the CIR model for the short interest rates. Recall that face value is 1 and exercise price K is equal to 0.5. As in the example with the Vasicek model, we consider that o = 2% and tq = 3.75%. The CIR model overcomes the problem of negative interest rates (acknowledged as a problem for the Vasicek model) as long as 2a > o. This is true, for example, if we take a = 0.0189 and P = 0.24. Feeding this information into the above formulae is relatively tedious. A spreadsheet application is provided by Jackson and Staunton, After some work we get that the price of the call is... 

EXHBIT 18.7 Calculations Using Vasicek Model for Separate Zero-Coupon European Call Options the Bond Prices Shown are Calculated with the Estimated... 

We shall repeat the calculation of the coupon-bond call option when the CIR model is employed for the short rates. The procedure is the same as in the case discussed above for the Vasicek model. First we calculate the interest rate such that the present value at the maturity of the option of all later cash flows on the bond equals the strike price. This value is here rjf = 25.05%. Next, we map the strike price into a series of strike prices via equation (18.50) that are then associated with coupon pay-... 

The Vasicek model was the first term-structure model described in the academic literature, in Vasicek (1977). It is a yield-based, one-factor equilibrium model that assumes the short-rate process follows a normal distribution and incorporates mean reversion. The model is popular with many practitioners as well as academics because it is analytically tractable—that is, it is easily implemented to compute yield curves. Although it has a constant volatility element, the mean reversion feature removes the certainty of a negative interest rate over the long term. Nevertheless, some practitioners do not favor the model because it is not necessarily arbitrage-free with respect to the prices of actual bonds in the market. 

The Vasicek model describes the dynamics of the instantaneous short rate as (4.7). 

In Vasiceks model, the short rate r is normally distributed. It therefore has a positive probability of being negative. Model-generated negative rates are an extreme possibility. Their occurrence depends on the initial interest rate and the parameters chosen for the model. They have been generated, for instance, when the initial rate was very low, like those seen in Japan for some time, and volatility was set at market levels. This possibility, which other interest rate models also allow, is inconsistent with a no-arbitrage market as Black (1995) states, investors will hold cash rather than invest at a negative interest rate. For most applications, however, the model is robust, and its tractability makes it popular with practitioners. 

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