Stochastic Interest Rates and the Standard Market Model

Jiří Witzany

Chapter 6 in Derivatives, 2020, pp 223-259 from Springer

Abstract: Abstract In Chap. 4 , we have presented the Black-Scholes option valuation model that has become a market standard. However, the model has several limiting assumptions including the one saying that the instantaneous interest rates are constant. But the interest rates are not constant at all in real financial markets. First, there is a term structure of interest rates, 1-year interest rates are usually greater than over-night interest rates, and 5-year interest rates are usually greater than 1-year interest rates. Evaluating a 1-year European stock option, which interest rate should be used? Recall that a European derivative value was obtained as the present value of the expected payoff. Hence, in the Black-Scholes formula, one could propose to use the 1-year interest rate instead of the presumably constant short rate. It turns out that this simple modification, leading to the so-called Standard MarketModel, is correct, but in order to prove it we need to generalize significantly the risk-neutral valuation framework.

Date: 2020
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DOI: 10.1007/978-3-030-51751-9_6

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