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One of the most significant applications of calculus derivatives is in the domain of fixed income securities. Historically, calculus derivatives have been used to determine the pricing of certain trade functions such as the in bond securities and agriculture. Considering the application of fixed income securities in agriculture, farmers agree to sell their produce at a predetermined price agreed upon before onset of the planting season, but this is subject to go up or lower depending on how weather and demand schedule (a slope variable) will be during the actual sale period (Soo 530). The model of fixed income securities was determined by Nobel prize laureates Fischer Black, Myron Scholes, and Robert Merton, who won the Nobel prize in Economics in 1997 (Soo 530). Their contribution is highly accredited in the economic world of fixed income derivates.
These economists developed what came to be known as the Black-Scholes differential equation and it enabled the extensive development of the derivates market (Soo 530). The fixed income derivate is essentially based on the stochastic calculus principle. A simple demonstration of stochastic calculus is as follows:
dr = b dt + ∂ dx
dr = change in the short rate
b = expected direction of rate change
dt = length of time interval
∂ = standard deviation of changes in the short rate
dx = random process (Fabozzi 62)
How they are used
Fixed income derivatives are extensively applied in the determination of fixed income securities According to Stewart to calculus derivatives enables us to tell whether the function being used is increasing or decreasing by project either a positive slope or negative slope between the respective tangent lines or intervals within which the variables fall (220). In essence, in the case of fixed income securities, there is an element of sensitivity on other variable factors that may affect the final value of the bond. Hence, calculus derivative here is used to determine the convexity, which refers to “the sensitivity of the duration of a bond to changes in the market rate of interest” (Teall and Hasan 271).
Calculus derivative, specifically the stochastic calculus approach has also been applied in the valuation of bonds in the fixed income segment. Here consideration is given to the fact that there will be a possible variation in the future interest rates. Thus, it suggests that there is some level of uncertainty, which necessitates the application of a fixed value on the projected returns, but the maturity value varies depending on the yield.
Application in Investment Portfolios
Calculus derivatives are also applied in the some fixed security investment portfolios where securities are traded by trusted financial institutions with an aim of realizing profit/gain. Under normal circumstances, the trading of securities is a primary of function of institutions such as banks, which actively engage in buying and selling of securities (Sabramani). Here, the market to market process is used to determine the value of the securities going by the market rates during which gains and losses are recorded. These are then transferred into the trading securities of the individual investors (Sabramani). In this regard, the calculus derivate is applied to track the periodic income realized beginning from the date on which the security was deposited. This implies that it can be used in other debt instrument applications to track the value of securities, which do not normally end up with the individuals, but the lending institution.
In line with investment portfolios calculus derivatives can also be used to identify a function that can be used to track the appreciating or depreciating value of fixed assets, especially those which are used as fixed income securities. In this case, calculus derivatives can be used to determine the change on the slope due to the variable interest rate of return. In this case, if a fixed asset is held until maturity, this will depend on the contractual arrangement based on predetermined period of time during which principle payments (fixed value) and interest (variable component) will be satisfactorily addressed.