Negative interest rates are a new and strange phenomenon. This concept causes a lot of head scratching, particularly regarding option valuation and VAR calculation, since the lognormal paradigm, applied for the last thirty years, breaks down. In a lognormal world, the next period’s rate is calculated as this period’s rate multiplied by e (2.718…) raised to the sum of the drift (a certain number) plus the diffusion (volatility multiplied by a normal random number).
Lognormal Model: Rate Next Period = Rate Today * e Drift+Diffusion
No matter what the diffusion is, e is positive and therefore the rate next period cannot cross zero. If this period’s rate is positive then the next period’s rate must be positive and vice-versa.
For example, consider the valuation of the right to lend to the US Government at 0% interest for the whole of 2018 (an interest rate floor). Under a lognormal model, the loan rate cannot be negative and therefore the option has no value. If this were the case, you would always choose to let the option expire and lend to the US government at the prevailing rate rather than exercise the option. However, if you believe it is possible (no matter how unlikely) for rates to become negative then the option to lend the US Government at 0% is valuable, and your model must reflect this. Under the normal model, however, the next period’s rate is determined as this period’s rate plus a drift and diffusion coefficient.
Normal Model: Rate Next Period = Rate Today + Drift + Diffusion
This means that the rate can go negative and the above option would still have value.
Imagine Fair Valuation of the above T-Bill Floor under the normal and lognormal models.
A further point of interest is the volatility (the diffusion multiplier) between the two models is not equivalent. For example, suppose the rate in question is 1%, and we assume no drift, and an annualized volatility of 5% in a lognormal setting. This implies the two standard deviations range of rates with two standard deviations is approximately 0.91% and 1.1%. However, in a normal model a volatility of 5% would imply this range to be approximately -9% to 11%. Clearly, it is very important to adjust the volatility between the two.
At Imagine, we let our users choose to value their options using a lognormal or normal model for both option valuation and VAR calculation. In addition, for our risk calculation, we handle the conversion and provide the lognormal equivalent volatility so the earlier-mentioned issues do not occur. Please contact Imagine if you would like to know more about how we can help.
About the Author
Jack Goss has been a Consultant at Imagine for six months. Previously, over the course of five years, he gained financial experience as both an investment manager at a large British pension fund and analyst for a hedge fund at Henderson Global Investors. During this time he earned both his CFA and CAIA. Jack received his bachelor’s in economics from Cambridge University and is currently studying for a master’s degree in mathematical trading and finance from CASS business school. Outside of finance he enjoys bridge and cycle touring. He can be reached by email and phone: +44 (0) 20 7440 0713.