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For example, for bond options [3] the underlying is a bond, but the source of uncertainty is the annualized interest rate (i.e. the short rate). Here, for each randomly generated yield curve we observe a different resultant bond price on the option's exercise date; this bond price is then the input for the determination of the option's payoff.
For example, for bonds, and bond options, [13] under each possible evolution of interest rates we observe a different yield curve and a different resultant bond price. To determine the bond value, these bond prices are then averaged; to value the bond option, as for equity options, the corresponding exercise values are averaged and present valued.
In finance, the binomial options pricing model (BOPM) provides a generalizable numerical method for the valuation of options.Essentially, the model uses a "discrete-time" (lattice based) model of the varying price over time of the underlying financial instrument, addressing cases where the closed-form Black–Scholes formula is wanting, which in general does not exist for the BOPM.
The binomial correlation approach of equation (5) is a limiting case of the Pearson correlation approach discussed in section 1. As a consequence, the significant shortcomings of the Pearson correlation approach for financial modeling apply also to the binomial correlation model. [citation needed]
Binomial Lattice for equity, with CRR formulae Tree for an bond option returning the OAS (black vs red): the short rate is the top value; the development of the bond value shows pull-to-par clearly . In quantitative finance, a lattice model [1] is a numerical approach to the valuation of derivatives in situations requiring a discrete time model.
Because interest rate caps/floors are equivalent to bond puts and calls respectively, the above analysis shows that caps and floors can be priced analytically in the Hull–White model. Jamshidian's trick applies to Hull–White (as today's value of a swaption in the Hull–White model is a monotonic function of today's short rate).
The model was introduced by Fischer Black, Emanuel Derman, and Bill Toy. It was first developed for in-house use by Goldman Sachs in the 1980s and was published in the Financial Analysts Journal in 1990. A personal account of the development of the model is provided in Emanuel Derman's memoir My Life as a Quant. [4]
The CIR model uses a special case of a basic affine jump diffusion, which still permits a closed-form expression for bond prices. Time varying functions replacing coefficients can be introduced in the model in order to make it consistent with a pre-assigned term structure of interest rates and possibly volatilities.