Differentiating Black–Scholes Equations for Merton Volatility Estimation
Summary
The document concerns estimating asset value and asset volatility in the Merton structural credit model by solving a nonlinear system with Newton–Raphson. The system relates equity value to a Black–Scholes-style call expression and relates equity volatility to asset volatility through the option’s sensitivity to underlying asset value. To build the Jacobian, the answer clarifies that N denotes the standard normal cumulative distribution function, whose derivative is the standard normal density, then applies the chain rule to differentiate N(d1) with respect to asset value.
The same procedure applies to volatility and the other terms, after which the partial derivatives form the Jacobian used by the iterative solver. The question’s displayed formula for the normal CDF is incorrect; it uses an exponential expression rather than the integral definition. The answer supplies one partial derivative but does not present the complete Jacobian, numerical convergence behavior, or safeguards for a poorly conditioned system.
Key ideas
- The derivative of the standard normal CDF is the standard normal probability density function.
- Derivatives of N(d1) with respect to model parameters require the chain rule through d1.
- The partial derivatives of both equations form the Jacobian for Newton–Raphson estimation.
- The normal CDF expression in the question is incorrect and should not be differentiated as written.
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# Question about calculating asset volatility using Black-Scholes and the Merton Model (Differentiation Question)
# Question about calculating asset volatility using Black-Scholes and the Merton Model (Differentiation Question)
I have a problem where I need to relever equity volatility and take into account debt. I'm trying to solve a system of nonlinear equations for $\sigma_v,V$ using $f(\sigma_v,V) = VN(d_1)-De^{-R_ft}N(d_2)-E$ and $g(\sigma_v,V)=\sigma_e\frac{E}{VN(d_1)}-\sigma_v$. I wanted to do the Newton method using R but I think the first thing I'm going to need to do is a partial differentiation of the two functions with respect to $\sigma_v$ and $V$ to create the Jacobian. I'm worried about differentiating the first equation because of the CDF. As for the equation I will use the quotient rule. $\frac{\partial{f}}{\partial{\sigma_v}}=0N(d_1)+VN'(d_1)-0N(d_2)+De^{-R_ft}N'(d_2)$. My question is about $\frac{\partial{N(d_1)}}{\partial{\sigma_v}}$ and $\frac{\partial{N(d_1)}}{\partial{V}}$. Since $N(d_1)=\frac{1}{2}\left[1+e^{\frac{log\frac{V}{D}+(R_f+.5\sigma_v^{2})t}{\sigma_v\sqrt{t}\sqrt{2}}}\right]$ how would I go about differentiating it w.r.t. $\sigma_v$ and $V$ or is there an easier way to solve this system of equations?
## Answer by user28763 (score 5, accepted)
https://quant.stackexchange.com/a/40974
For one thing, what you have written is incorrect. Black-Scholes uses the standard normal CDF:
$$N(d_1) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{d_1} e^{-x^2/2} \, dx$$
So $N'(d_1) = \frac{e^{-d_1^2/2} }{\sqrt{2 \pi}}$ and using the chain rule we get
$$\frac{\partial N(d_1)}{\partial V} = N'(d_1) \frac{\partial d_1}{\partial V} = \frac{e^{-d_1^2/2} }{\sqrt{2 \pi}}\frac{1}{V\sigma_v\sqrt{t}}$$
Compute the other partial derivatives in the same way and then apply the Newton-Raphson method.Shown 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.