*The Business Professor*, updated February 24, 2020, last accessed October 28, 2020, https://thebusinessprofessor.com/lesson/blacks-model-definition/.

### Black’s Model Definition

The Black’s model is a mathematical model for pricing derivative instruments such as options contract, swaptions, bond options, and other interest-rate derivatives. The Black’s model is otherwise called Black-76, it was first presented in 1976 by Fischer Black in one of his papers. This model is a variant of the Black-Scholes options pricing model, as a revised version, the Black’s model can be applied to interest rate cap loans and other derivatives.

### A Little More on What is Black’s Model

The Black model is an option pricing model that can be applied to derivatives and capped variable rate loans. The Black model was developed in 1976 by the collaborative efforts of Fischer Black, an American economist and other developers such as Robert Merton and Myron Scholes who established the Black-Scholes model. This model was developed to enhance how commodities and options contracts were valued in the market. The pricing of commodity options is important, and this model offers an adjustment to the Black-Scholes model on how commodity options are priced.

According to Fischer Black, the price at which traders agree to buy or sell a security at a future time is the futures price. Black 76 has several positions on the option pricing model different from that of the Black-Scholes model. The major difference between these two models that that Black’s model uses forward prices to value futures option while the Black-Scholes model uses spot prices.

### Reference for “Black’s Model”

https://www.investopedia.com/terms/b/blacksmodel.asp

https://en.wikipedia.org/wiki/Black_model

https://www.analystforum.com/forums/cfa-forums/cfa-level-ii-forum/91342378

https://www.mathworks.com › … › Price Derivative Instruments

### Academics research on “Black’s Model”

International **investment **restrictions and closed‐end country fund prices, **Bonser-Neal, C., Brauer, G., Neal, R., & Wheatley, S. (1990). International investment restrictions and closed****‐****end country fund prices. ***The Journal of Finance***, ***45***(2), 523-547. **Some closed‐end country funds trade at large premiums relative to their net asset values. This paper examines whether international investment restrictions raise country fund price‐net asset value ratios by segmenting international capital markets. We test whether a relation exists between announcements of changes in investment restrictions and changes in these ratios using weekly data from May 1981 to January 1989. The results provide evidence that some foreign markets are at least partially segmented from the U.S. capital market.

The capital asset pricing model: A fundamental critique, **Dayala, R. (2012). The capital asset pricing model: A fundamental critique. ***Business Valuation Review***, ***31***(1), 23-34. **The Capital Asset Pricing Model (CAPM) derives an ex post equilibrium relationship for the price of non-diversifiable risk based on investors utilizing two criteria only when making investment decisions: expected value and standard deviation. This article investigates the ex ante and ex post state of the CAPM in a hypothetical three-asset universe: either the CAPM irrationally indicates identical respective discount rates for different amounts of risks (i.e., total risk versus non-diversifiable risk) or the CAPM circularly indicates the ex ante price of total risk (read: standard deviation) depending on the ex post price of non-diversifiable risk.

**Black’s model **of interest rates as options, eigenfunction expansions and Japanese interest rates, **Gorovoi, V., & Linetsky, V. (2004). Black’s model of interest rates as options, eigenfunction expansions and Japanese interest rates. ***Mathematical Finance: An International Journal of Mathematics, Statistics and Financial Economics***, ***14***(1), 49-78. ****Black’s (1995)** model of interest rates as options assumes that there is a shadow instantaneous interest rate that can become negative, while the nominal instantaneous interest rate is a positive part of the shadow rate due to the option to convert to currency. As a result of this currency option, all term rates are strictly positive. A similar model was independently discussed by **Rogers (1995)**. When the shadow rate is modeled as a diffusion, we interpret the zero‐coupon bond as a Laplace transform of the *area functional* of the underlying shadow rate diffusion (evaluated at the unit value of the transform parameter). Using the method of eigenfunction expansions, we derive analytical solutions for zero‐coupon bonds and bond options under the Vasicek and shifted CIR processes for the shadow rate. This class of models can be used to model low interest rate regimes. As an illustration, we calibrate the model with the Vasicek shadow rate to the Japanese Government Bond data and show that the model provides an excellent fit to the Japanese term structure. The current implied value of the instantaneous shadow rate in Japan is negative.

Pricing interest rate futures options with futures-style margining, **Ren-Raw, C., & Scott, L. (1993). Pricing interest rate futures options with futures-style margining. ***The Journal of Futures Markets (1986-1998)***, ***13***(1), 15.**

**Investment **barriers and international asset pricing, **Padmanabhan, P. (1992). Investment barriers and international asset pricing. ***Review of Quantitative Finance and Accounting***, ***2***(3), 299-319. **In this article a multicountry model of international asset pricing is developed. This model incorporates a more general representation of the degree of segmentation in the international capital market. Specifically,*N* types of investors and*N* classes of securities are postulated. In general, the*n*th (*n*=1, 2, 3, …*N*) type of investor has access to all security markets up to and including the*n*th class. Using the standard mean-variance framework, closed form equilibrium risk return relationships are obtained for all classes of securities. It is also shown that class 1 securities are priced as if markets are integrated, class*n* (*n*=2, 3 …*N*) securities command*n*different risk premia. Finally, the nature of the model specification allows us to investigate the effects of partial integration on investor welfare. It is shown that, in general, all investors prefer full integration to any form of partial integration.