Relating Option Delta to Strike Sensitivity Under Volatility Skew

# Mathematical equation relating $\frac{dV}{dS}$ to $\frac{dV}{dK}$


# Mathematical equation relating $\frac{dV}{dS}$ to $\frac{dV}{dK}$












Please help me figure out what is the mathematical relationship between $\frac{dV}{dS}$ (Delta) and $\frac{dV}{dK}$ ($K$=strike), taking into account vol skew.

I ask this because I want to figure out the value of a digital at a certain strike, given that I know the vanilla option delta there, and also I have the volatility smile.

I know that there is the equation: $$ \frac{d V}{d S} = \frac{\partial V}{\partial S} + \frac{\partial V}{\partial \sigma} \frac{\partial \sigma}{\partial S} , $$ so I suppose there is something similar for what I need.

As I recall , $dV/dK = N(d2)$ and $dV/dS = N(d1)$ (ignoring dividends and risk-free rates), so I just need some simple approximation linking $\frac{dV}{dS}$ and $\frac{dV}{dK}$ (approximation if an exact relationship is too complicated).

## Answer by Quantuple (score 11, accepted)

https://quant.stackexchange.com/a/35310

If your working modelling assumptions are such that the dynamics of the log price process $\ln(S_t)$ is space homogeneous, you have that the price of a European vanilla option is itself a space-homogeneous function of degree one. You can then appeal to Euler theorem to get the relationship you need.

More specifically, define the price at time $t$ of the option expiring at $T$ and struck at $K$ as

$$ V = DF(t,T)\, \Bbb{E}_t^\Bbb{Q} \left[ (w(S_T - K))^+ \right] := V(S_t, K, T-t, \theta) $$ where $\theta$ figures the relevant model parameters and $w=\pm1$ the call/put factor. Now under the space homogeneity assumption we've just mentioned, you can write that $$ V(xS_t,xK,T-t,\theta) = x V(S_t,K,T-t,\theta), \forall x \geq 0$$

Taking the derivative with respect to $x$ on both sides and then setting $x=1$ gives:

$$ \frac{\partial V}{\partial S} S + \frac{\partial V}{\partial K} K = V $$ hence $$ \frac{\partial V}{\partial K} = \frac{1}{K} \left( V - \frac{\partial V}{\partial S} S \right) $$

which is what you are looking for.

And indeed if you are pricing a digital call ($D$ below) for instance, using the notation $C$ to denote the European call price \begin{align} D &= -\frac{dC}{dK} \\ &= -\left[ \frac{\partial C}{\partial K} + \frac{\partial C}{\partial \Sigma} \frac{\partial \Sigma}{\partial K} \right] \\ &= -\left[ \frac{1}{K}\left( C - \Delta S\right) + \nu \frac{\partial \Sigma}{\partial K} \right] \end{align} where for a maturity $T$ and strike level $K$, $C$ is the corresponding European call price, $\Delta$ its BS Delta, $\nu$ its BS Vega and $\partial \Sigma/\partial K$ the IV skew. We have moved from the second line to the third using the result which we just derived.