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FRM Part II · FRM Exam Part II · The Vasicek and Gauss+ Models

A risk analyst uses the Vasicek model dr = k(θ − r)dt + σ dw to price a European option on a zero-coupon bond. Which feature of the Vasicek model most directly allows the option price to be obtained in closed form?

The Vasicek short rate is normally distributed because its volatility is constant. Zero-coupon bond prices are exponentials of affine functions of the rate, hence lognormal, which lets a Black-Scholes-type formula price bond options in closed form.

  1. AShort rate is normally distributed, so bond prices are lognormally distributedCorrect
  2. BShort rate is lognormally distributed, so bond prices are normal
  3. CVolatility of the short rate is proportional to the rate level
  4. DThe model has no mean reversion, so the drift is constant

Explanation

In Vasicek, volatility is constant, so the short rate is normal. Zero-coupon bond prices are exponentials of affine functions of r, so they are lognormal, and a Black-Scholes-style formula applies. Lognormal rates or proportional volatility describe other models (e.g., Black-Karasinski, CIR), and Vasicek does have mean reversion.

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