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January 09, ; Accepted Date: February 22, ; Published Date: J Appl Computat Math 3: This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Three contents for the pricing of bond options on the arbitrage-free model with jump are included in this paper.
There is a substantial difference between bond option prices which are obtained by the HW model with jump and the HJM model based on jump. Through the empirical simulation of our method suggested, we obtain a better accurate estimation for the pricing of bond options.
In pricing and hedging with financial derivatives, term structure models with jump are particularly important [ 1 ], since ignoring jumps in financial prices may cause inaccurate pricing and hedging rates [ 2 ]. Solutions of term structure model under jump-diffusion processes are justified because of movements in interest rates displaying both continuous and discontinuous behaviors [ 3 ]. Moreover, to explain term structure movements used in the latent factor models, it means how macro variables affect bond prices and the dynamics of the yield curves [ 4 ].
Current research using jump-diffusion processes relies mostly on two classes of models: In this paper, we show the actual proof analysis of the HJM model based on jump easily under the extended restrictive condition of Ritchken and Sankarasubramanian RS [ 7 ]. By beginning with certain forward rate volatility processes, it is possible to obtain classes of interest models under HJM model based on jump that closely resembles the traditional models [ 8 ].
Finally, we confirm that there is a substantial difference between bond option prices which are obtained by HW model with jump and HJM model based on jump through the empirical computer simulation which used MCS, which is used by many financial engineers to place a value on financial derivatives.
For this, we use the well-known MSE. We make sure that lower value of PCS in the proposed models corresponds to sharper estimates [ 9 ].
These results mean an accurate estimate in the empirical computer. The structure of the remainder of this paper is as follows. In section 4, investigate the pricing of bond on arbitrage-free models with jump. In section 4, the pricing of bond option on arbitrage-free models with jump are presented.
Section 6, explains the simulation procedure of the proposed models using MCS. Finally, Section 8 concludes this paper. All our models will be set up in a given complete probability space and an argument filtration generated by an Winear process and N t represents a Poisson process with intensity rate h and the total number of extreme shocks that occur in a financial market until time t [ 10 ]. In the same way that a model for the asset price is proposed as a lognormal random walk, let us suppose that the interest rate r and the forward rate f are governed by a SDE of the form.
When interest rates follow the SDE 1 , a bond has a price of the form V t; T ; the dependence on T will only be made explicit when necessary. To get the bond pricingequation with jump, we set up a riskless portfolio containing two bonds with different maturities T1 and T2. Hence, we derive the partial differential equation PDE for bond pricing. If r satisfies SDE 1 , then the zero-coupon bondpricing equation with jumps is.
Boundary conditions depend on the form of u r, t and w r, t. We now consider a quite different type of random environment. In this paper, we extend jump-diffusion version of equilibrium single factor model to reflect this time dependence. This leads to the following model for r t:.
Under the process specified in equation 5 , r t is defined as:. It can be shown that the probability density function for r t. Therefore, the conditional expectation and variance of jump-diffusion process given the current level are.
Thus, we get the partial differential difference bond pricing equation:. Bond price derivatives can be calculated from 4 , and then the substitution of these derivatives into 9. Thus, equating powers of r t yields the following equations for A and B. Then the price at time t of a discount bond with maturity T, is defined as.
Where is a standard Wiener process generated by the risk-neutral measure Q, and dN t is the Poisson process with intensity rate h. In similar way as before, therefore, the conditional expectation and variance of the SDE 15 given the current level are. In the study, we use the relation between short rate and forward rate process to obtain the formula of bond price under the extended restrictive condition of RS. Let be the jump-diffusion process in short rate r t is the equation 5.
Let be the volatility form is. We know the SDE for forward rate Then we obtain theequivalent model is. By the theorem 4, we derive the relation between short rate and forward rate. Using the equation 19 , we obtain bond pricing equation as follows;. Then discount bond price V t, T for forward rate is given by. We derive a CFS for bond options when the prices of the underlying instantaneous interest and forward rate evolve as discontinuous processes. We now consider the value of European options on discount bond equations 4 and Thus, the equation 21 becomes.
In similar way, we obtain the price of a put option on the discount bond. In this section, we explain about the simulation procedure ofthe pricing of bond options on the arbitrage-free models withjump. The MCS is actually a very general tool and its applicationsare by no means restricted to numerical integration. For small time steps, wecan obtain the bond price by sampling n short and forwardrates paths under the discrete version of the risk-adjusted SDEs 5 and The bond price estimate is given by:.
The option price is evaluated using formula 21 , using the Euler-Maruyama scheme for the integration, as.
Lower values of PCS in equation 25 correspond to sharper estimates. In this section, we investigate the pricing of bond options on the arbitrage-free models with jump. Tables 1 and 2 represent the pricing of bond options on arbitrage-free models using the MCS.
We nowinvestigating the pricing of bond options on the HJM model based on jump which is shown in Figure 2 and Table 2. For this experiment, the parameter values are assumed as before. In empirical computer simulation Tables 1 and 2 , we show that the lower values of PCS in the proposed models correspond to sharper estimates using the Mathematica [ 11 ]. After investigating the models which allow the short term interest and the forward rate following a jump-diffusion process, we obtained the closed-form solutions on jump models, which are more useful to evaluate the accurate estimate for the values of bond options in the financial market.
Through the MCS simulation of these solutions with jump, the price of the expected stable figure like right-downward flow as maturity increases while the graph of bond options on the HW-Jump model with the short term interest rate is humped. We need further investigation on this difference which can be caused by performing jump term simulation of different interest rate cases.
Also, we obtained the more accurate estimate in empirical computing by showing the fact that the PCS for the HJM based on jump is lower than that for the HW model withjump. There are still problems remained for further research. Some of them, for instance, are i using the MCS to simulate more complicated two factors of the proposed models; ii considering a dynamic algorithm to predict the bond option prices using actual data set of bond. Home Publications Conferences Register Contact.
Research Article Open Access. February 25, Citation: The pricing of bond options on the HW model with jump. The pricing of bond options on the HJM model based on jump. Select your language of interest to view the total content in your interested language.
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