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Diagnosing Zero Black–Scholes Vega from Interest-Rate Units

Article Quant Q&A · Author: Oscar
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Summary

The document describes calculating Black–Scholes delta and vega in R for a sample of executive compensation contracts containing multiple options. The author derives the Black–Scholes call sensitivities and reports that computed vega values are zero, apparently because the normal density term evaluates to zero for the calculated inputs. Delta, by contrast, agrees closely with a comparison dataset.

The response points to a possible unit mismatch in the risk-free rate: a value expressed as 15 percent must be entered as 0.15 in the formula. This highlights how inconsistent rate conventions can distort option pricing inputs and sensitivities. The exchange is brief and does not show corrected calculations or confirm the units used in the example, so it suggests a diagnostic rather than documenting a verified fix.

Key ideas

  • The author computes Black–Scholes delta and vega for a set of options in R.
  • The reported vega values become zero when the normal density term evaluates to zero.
  • Check whether interest rates are represented as decimals rather than whole percentage numbers.
  • The response proposes a rate-unit issue but does not show a corrected computation.

Tags

OptionsDerivatives pricingVolatilityStatistics
Full text
# Compute Vega and Delta in R


# Compute Vega and Delta in R












I am trying to compute greeks for a large sample of CEO compensation contracts in R. However, my vega computations all result in a value of zero.

In doing so, I follow Core and Guay [2002]:

Here is some a snapshot of one contract that contains multiple options:

```
library(tidyverse)
df <- tibble(prccf = rep(36.55, 15),
            Xc = 28:42,
            maturity = seq(0.5, 7.5, by = (1/2)),
            rf = rnorm(15, 4.5, 0.25), 
            d = rep(0.025, 15),
            sigma = rep(0.30, 15))
```

I compute Z:

```
df <- df %>% 
mutate(Zc = (log(prccf / Xc) + maturity * (rf - d + sigma^2 / 2)) / (sigma * sqrt(maturity)))
```

Then I take the first derivative of the Black-Scholes option value, with respect to `prccf` and `sigma`:

```
deriv(~ ((prccf * exp(-d * maturity) * pnorm(Zc)) - (Xc * exp(-rf * maturity) * pnorm(Zc - sigma * sqrt(maturity)))), c("prccf", "sigma"))
```

which gives:

```
delta = exp(-d * maturity) * pnorm(Zc)
vega = Xc * exp(-rf * maturity) * (dnorm(Zc - sigma * sqrt(maturity)) * sqrt(maturity))
```

So:

```
df <- df %>% 
mutate(delta = exp(-d * maturity) * pnorm(Zc),
       vega = Xc * exp(-rf * maturity) * (dnorm(Zc - sigma * sqrt(maturity)) * sqrt(maturity)))
```

But then all values for vega are zero. This happens irrespective of the parameters.

The vega part becomes zero when it gets multiplied with the normal density function: `dnorm(Zc - sigma * sqrt(maturity))`, because `Zc` contains relatively high values for the normal distribution, so it results in 0. Then the whole line gets multiplied by 0, resulting in 0s as final values as well.

I do not really see how I can fix this, since Zc does capture the right value. (The "delta code" is correct, because I can compare it to another dataset (correlation of 0.999).)

This also happens when I rely on packages instead of manually computing it, e.g.:

```
library(derivmkts)

bsopt(s = df$prccf,
      k = df$Xc,
      v = df$sigma,
      r = df$rf,
      tt = df$maturity,
      d = df$d)[['Call']][c('Delta', 'Vega'), ]
```

Can someome help me with this issue? I got stuck, especially because the delta part is correct. Thanks!

## Answer by Enrico Schumann (score 2, accepted)

https://quant.stackexchange.com/a/48820

Are you sure your `rf` values are right? If `15` is to mean 15%, then write it as `0.15`.

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