Deriving the Interest-Rate Sensitivity PDE for a Black–Scholes Value

Summary

The document considers the Black–Scholes partial differential equation for a derivative value with a terminal payoff, then defines the sensitivity of that value to the interest rate. Differentiating the original equation gives a second PDE for this sensitivity, with a source term involving the value and its underlying-price delta, and a zero terminal condition.

The question asks whether the sensitivity equals time to maturity multiplied by delta exposure minus value. It assumes the original value and the delta-scaled underlying price have specified growth behavior near maturity, and invokes uniqueness of the PDE solution as a possible proof strategy. However, the document contains no accepted derivation or answer; it is an open mathematical question. It therefore offers a setup for studying PDE sensitivity and uniqueness, but does not establish the proposed identity or explain its validity conditions.

Key ideas

• Differentiating the Black–Scholes equation with respect to the interest rate yields a PDE for rate sensitivity.
• The sensitivity has a zero terminal condition when the payoff does not depend on the interest rate.
• The derived PDE includes a source term involving the option value and its delta exposure.
• The proposed closed-form identity is posed as a question and is not proved in the document.

Tags

# PDE and Black Scholes problem


# PDE and Black Scholes problem












Consider Black Scholes problem $\frac{\partial V}{\partial t} + \frac{\sigma^2 S^2}{2}\frac{\partial^2V}{\partial S^2} + rS\frac{\partial V}{\partial S} -rV = 0$ with boundary condition $V(S,T)=f(S)$, where $V(S,t)$ is the solution to the PDE above.

Let $ρ(S,t)=\frac{∂V}{∂r}(S,t)$, then it can be shown that $\rho$ satisfies the following PDE: $\frac{∂ρ}{∂t} + \frac{σ^2 S^2}{2}\frac{∂^2ρ}{∂S^2} + rS\frac{∂ρ}{∂S} −rρ=V−S\frac{∂V}{∂S}$ with boundary condition $\rho(S,T)=0$ .

${\it QUESTION}$: given that the PDE above uniquely determines $ρ$ , and assuming that both $V$ and $S\frac{∂V}{∂S}$ are of the order $o(1/(T−t))$ as $t→T^−$ , show that $ρ(S,t)=(T−t)\Big(S\frac{∂V}{∂S}(S,t) −V(s,t)\Big) $,

I have tried to tackle the problem by following this:

Under sufficient regularity/growth/etc. conditions, an equation $Lu=v$ with specified initial and boundary conditions has a unique solution. So to verify that two solutions $f$ and $g$ of the same equation $Lu=v$ with identical initial and boundary conditions are the same, it is sufficient to verify that their difference satisfies $L(f−g)=0 $ with zero initial and boundary conditions. So it's enough to verify regularity conditions and check that the l.h.s. satisfies exactly the same equation and conditions as the r.h.s.

However I couldnt produce the required result.

I would be grateful if anyone could suggest how to approach this and similar problems.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.