Deriving the Black–Scholes Vanna Formula
Summary
Vanna measures the sensitivity of an option’s delta to volatility, equivalently the sensitivity of its vega to the underlying price. The document frames its derivation by differentiating the Black–Scholes option value with respect to spot to obtain delta, then differentiating delta with respect to volatility. It asks why a term involving d2 divided by volatility can be rewritten using one minus d1 in a displayed formula.
The identity follows from the standard definitions: d2 equals d1 minus volatility multiplied by the square root of time to expiry. Rearranging the definition of d1 also gives d1 as a volatility-scaled log-moneyness and rate term plus half the volatility-time term; combining these relations yields the stated substitution under the formula’s conventions. The source provides the question and relevant formulas but no complete derivation or worked numerical check. Its notation contains apparent inconsistencies, so the relationship should be verified against the precise definitions and time convention being used.
Key ideas
- Vanna is the cross-sensitivity of option value to spot and volatility.
- It can be obtained by differentiating delta with respect to volatility.
- The relationship between d1 and d2 follows from their Black–Scholes definitions.
- Formula conventions and notation should be checked before applying the identity.
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# How does $(d_2/\sigma) = (1-d_1)$ while deriving the Vanna Formula from BSM?
# How does $(d_2/\sigma) = (1-d_1)$ while deriving the Vanna Formula from BSM?
Just realized there was a quant finance board, so I figured I'd post it here instead.
I'm trying to derive Vanna from the Black-Scholes Model (BSM) equation, but had a hook up on one of the manipulations in the formula.
Vanna, also referred to as DvegaDspot and DdeltaDvol, is a second order derivative of the option value, once to the underlying spot price and once to volatility.
Essentially it tells you how much Vega (sensitivity to volatility) will change if price changes; it's also equal to how much delta will change if volatility changes. It can be useful for hedging options against changing volatility for various strategies.
You can get it by taking the BSM equation, deriving it with respect to price (which leaves you with delta - the rate of change with respect to the underlying price), then you take the derivative of delta with respect to volatility, which leaves you with this next formula:
In the below PDF, on step 5 (page 4):
$$e^{-qt}\sqrt{T-t} N'(d_1)(d_2/\sigma)$$
Step 6, they adjust it into (with the assumption of no dividend, q=0)
$$\sqrt{T-t} N'(d_1)(1-d_1)$$
by making the substitution (from step 5):
$$d_2/\sigma = 1 - d_1$$
I can't seem to figure out how they did it. I'd be grateful if anyone had a little bit of knowledge they'd want to share. (I included a link to the actual PDF below as well.) I think I can figure out everything else, but that one adjustment is confusing me. Makes me wonder if I'm forgetting a basic rule or something.
Various Formulas:
Black-Scholes(-Merton) Model: $$C = S_tN(d_1)-Ke^{-rt}N(d_2)$$
Where:
C = Call Option Price
S = Current Stock Price
K - Strike Price
r - risk free interest rate
t - time to maturity
N = Normal CDF
$$d_1 = ((\ln(S_0/X) +t(r-q+\sigma^2/2))/\sigma\sqrt{\pi})$$
$$d_2 = D_1 -\sigma \sqrt{t}$$
N = Normal cumulative distribution function
$N^\prime$ = (Normal?) Probability Density Function
$N^{\prime \prime}$ = $$-(x*e^{-x^2/2})/(\sqrt{2}*\sqrt{pi})$$
$N^{\prime \prime}$ is my best guess, I checked with a calc, but could be wrong.
The paper I referenced:
Kun Huang: Vanna Volga and Smile-consistent Implied Volatility Surface of Equity Index Option (2019?)
Cheers,
NNShown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.