Pricing Equity Options When Interest Rates Are Negative

# Value options when the currency’s risk free rate is negative?


# Value options when the currency’s risk free rate is negative?












How would you handle a negative interest rate in index/equity options valuation?

An example would be negative rates for short term maturities for Swiss Frank (CHF).

## Answer by Matt Wolf (score 4, accepted)

https://quant.stackexchange.com/a/4952

A realistic alternative is to set all CHF rates out till 3 months equal to zero. The differential is so insignificant that for practical pricing valuation purposes it makes zero difference.

Please note that I am in particular talking about the valuation of index/equity options such as you indicated.

But even plugging negative rates in should work fine for BS-models (read a paper on that by Espen Gaarder Haug). However, you need to be more careful when working with interest rate models. Some are not happy with negative rates.

## Answer by AKdemy (score 3)

https://quant.stackexchange.com/a/74180

Nothing new to @Matt Wolf's answer but a few more details. Although this question was something new at the time of writing, it is probably clear now that plugging negative interest rates into Equity / Index options is no problem at all. The reason interest rate options (swaptions, caps, floors etc) switched to pricing with Bachelier (Normal) model is that the underlying itself can turn negative, in which case the logarithm in the Black Scholes model is undefined. That is why interest options are usually priced with the Normal (Bachelier) model now. The difference can be seen below.

Black-76 formula for pricing a call is (in essence Black-Scholes in terms of forward price instead of spot)

$$C_{BS}(K) = F_0 N(d_1) - K N(d_2)$$ where $d_{1,2} = \frac{\log(F_0/K)}{\sigma_{BS}\sqrt{T}}\pm\frac{\sigma_{BS}\sqrt{T}}{2}$.

Bachelier $$C_N(K) = (F_0-K) N(d_N) + \sigma_N\sqrt{T}n(d_N)$$ where $d_N = \frac{F_0-K}{\sigma_N\sqrt{T}}$.

However, the standard Black Scholes model formula works perfectly fine with negative interest rates or dividends. While negative dividends may sound unreasonable, it is no problem mathematically, and also something that is not to be ruled out theoretically for implied dividends or interpreted (used) as representing the borrow cost. There is an interesting paper valled Pricing Vanilla Options with Cash Dividends from Timothy Klassen that addresses dividend and borrow rate assumptions.

Black Scholes looks like this: $$Se^{-q\tau }\Phi (d_{1})-e^{-r\tau }K\Phi (d_{2})$$ where \begin{aligned}d_{1}&={\frac {\ln(S/K)+\left(r-q+{\frac {1}{2}}\sigma ^{2}\right)\tau }{\sigma {\sqrt {\tau }}}}\\d_{2}&={\frac {\ln(S/K)+\left(r-q-{\frac {1}{2}}\sigma ^{2}\right)\tau }{\sigma {\sqrt {\tau }}}}=d_{1}-\sigma {\sqrt {\tau }}\\\phi (x)&={\frac {1}{\sqrt {2\pi }}}e^{-{\frac {1}{2}}x^{2}}\\\Phi (x)&={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{x}e^{-{\frac {1}{2}}y^{2}}\,dy=1-{\frac {1}{\sqrt {2\pi }}}\int _{x}^{\infty }e^{-{\frac {1}{2}}y^{2}}\,dy\end{aligned}

There is no problem with negative rates in the formula. Below is a screenshot from Bloomberg's OVME function where you can see that negative rates are indeed used in pricing as well.

It is also straightforward to verify this claim with some code. I am using Julia in the example below.

```
# load relevant packages 
using Plots, Distributions, PlotThemes

# define the CDF 
N(x) = cdf(Normal(0,1),x)

# define the Black Scholes Merton model 
function BSM(S,K,t,r,q,σ)
  d1 = ( log(S/K) + (r - q + 1/2*σ^2)*t ) / (σ*sqrt(t))
  d2 = d1 - σ*sqrt(t)
  c  = exp(-q*t)S*N(d1) - exp(-r*t)*K*N(d2)
  return c
end

# define inputs
S = 100
K = 100
t = 1
r = -0.1:0.01:0.1 # interest in a range of -10% to 10%
q = 0.0
σ = 0.2

# plot the result
theme(:juno)
plot(r,BSM.(S,K,t,r,q,σ), title = "BSM option value for various interest rates", size = (800,500), legend = false)
xlabel!("interest rate")
ylabel!("Option value")
```

With options on spot (Black Scholes), you have the right to buy / sell spot at some future date. The right to buy means you do not forgo interest (as you still have that money at your disposal). If you have the right to sell (Long put), you could have sold it already, meaning you forgo the potential interest income. That is why the Greek $\rho$, which measures the sensitivity to the interest rate is typically positive for call options. It is also the reason why in the screenshot above, the call value gets smaller and smaller, the lower interest rates.