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Quantitative Finance · Glosarium

Apa itu Bachelier model, normal volatility?

Dikenal juga sebagai: Bachelier model · normal volatility

Definition 13.4 Markets II: Rates, FX and Credit · Bab 13 — The Rates Options Market

The Bachelier model, published by Louis Bachelier in 1900, assumes that the forward rate follows dFt=σN dWtdF_t = \sigma_N\,dW_t: it moves by normally distributed amounts, independent of its level, and can go negative. Its parameter σN\sigma_N, in basis points a year, is the normal volatility; the implied normal volatility of an option is the σN\sigma_N at which the model reproduces its price.

The Black volatility that gives the same one-year at-the-money price as a normal volatility of 90 basis points, by forward rate, on a logarithmic scale. The same option looks calm at 5% and wild at 0.5%; below the dashed line no Black volatility exists. Data: the chapter’s tutorial.
Figure 13.3. The Black volatility that gives the same one-year at-the-money price as a normal volatility of 90 basis points, by forward rate, on a logarithmic scale. The same option looks calm at 5% and wild at 0.5%; below the dashed line no Black volatility exists. Data: the chapter’s tutorial.

Contoh

Example 13.6 (A straddle)

On a flat 4% curve the 1y×\times10y forward annuity is 7.80. At 95 basis points of normal volatility, a straddle on USD 100 million costs 7.80×0.0095×2/π=5.91%7.80 \times 0.0095 \times \sqrt{2/\pi} = 5.91\% of notional, USD 5.91 million. Divided by the annuity it is 75.8 basis points: the ten-year rate must end more than about 76 basis points away from 4% for the straddle held to expiry to pay for itself.

Example 13.2 (A cap and a zero-cost collar)

On a flat 4% curve, a five-year cap at 4.5% on USD 100 million of an annual rate, with the first period already fixed, has four caplets fixing in one to four years. At a normal volatility of 100 basis points a year they are worth USD 182 874, 310 339, 401 400 and 470 709: USD 1.37 million in all, 137 basis points of notional. A floor at 3.5% is worth exactly the same, since the forward sits halfway between the strikes and the normal model is symmetric: the borrower who buys the cap and sells the floor pays nothing and keeps its rate between 3.5% and 4.5%.

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