Revisiting Black–Scholes: A Smooth Wiener Approach to Derivation and a Self-Contained Solution

Abstract

This study presents a self-contained derivation and solution of the Black and Scholes partial differential equation (PDE), replacing the standard Wiener process with a smoothed Wiener process, which is a differentiable stochastic process constructed via normal kernel smoothing. By presenting a self-contained, Itô-free derivation, this study bridges the gap between heuristic financial reasoning and rigorous mathematics, bringing forth fresh insights into one of the most influential models in quantitative finance. The smoothed Wiener process does not merely simplify the technical machinery but further reaffirms the robustness of the Black and Scholes framework under alternative mathematical formulations. This approach is particularly valuable for instructors, apprentices, and practitioners who may seek a deeper understanding of derivative pricing without relying on the full machinery of stochastic calculus. The derivation underscores the universality of the Black and Scholes PDE, irrespective of the specific stochastic process adopted, under the condition that the essential properties of stochasticity, volatility, and of no arbitrage may be preserved.

1. Introduction

The Black and Scholes model is a cornerstone of contemporary financial mathematics which provides a theoretical framework for pricing options and other financial derivatives. The derivation of the Black and Scholes PDE model traditionally relies on Itô’s Lemma, a fundamental tool in stochastic calculus which allows for the differentiation of functions of stochastic processes, handling the non-differentiability of the standard Wiener process (Brownian motion) in particular; Itô’s Lemma is albeit mathematically complex, obscuring the underlying economic principles of no arbitrage and of risk-neutral pricing, besides mathematical elegance for readers unaccustomed with stochastic differential equations.

In this study, we explore an alternative derivation of the Black and Scholes PDE which avoids the use of Itô’s Lemma by employing a smoothed Wiener process constructed using a normal or Gaussian kernel. Our approach preserves the essential properties of stochasticity and volatility while introducing differentiability, allowing for a more intuitive and economically grounded derivation; by replacing the standard Wiener process with a smoothed version, we maintain the mathematical consistency of the model while simplifying its derivation. The advantages of our alternative approach are (1) pedagogical clarity, (2) uncompromising mathematical rigor, and (3) enhanced interpretability, further described below.

1. Pedagogical clarity. By avoiding Itô’s Lemma, the derivation becomes accessible to a broader audience, including apprentices and practitioners with a background in differential equations but limited exposure to stochastic calculus. The smoothed Wiener process preserves the essential properties of stochasticity and volatility while allowing for classical differentiation, simplifying the analysis of stochastic terms.

2. Uncompromising mathematical rigor. The smoothed Wiener process retains the structure of stochasticity and volatility proper to the original model, ensuring that the resulting PDE may be identical to the classical Black and Scholes PDE. The solution method, drawing from the heat equation and Fourier analysis, is presented in full detail, evidencing the deep connection between financial mathematics and classical physics.

3. Enhanced interpretability. The use of a differentiable process clarifies the role of delta hedging and gamma hedging by making the stochastic terms tractable without advanced tools. The derivation explicitly links the volatility parameter to the smoothing kernel, reinforcing the fact that market randomness is encoded in the model.

2. Literature Review and Contribution

The Black and Scholes PDE was first introduced by Fischer Black and Myron Scholes (1973) [1] in their seminal paper by the title of ‘The pricing of options and corporate liabilities’. The model revolutionized the field of financial mathematics by providing an analytical solution for the price of a European call option. The traditional derivation of the Black and Scholes PDE relies on Itô’s Lemma, which was developed by Kiyoshi Itô (1944) [2] as part of stochastic calculus.

Alternative derivations of the Black and Scholes PDE have been explored. Foremost, Merton (1973) [3] extends the Black and Scholes PDE to include continuous dividends and provides a more general framework for option pricing, incorporating dividend yields.

As part of a comprehensive mathematical discussion on derivative pricing, Wilmott et al. (1995) [4] provide a detailed mathematical treatment of the Black and Scholes PDE and explore various methods for deriving it, entailing those which do not rely on Itô’s Lemma but emphasize PDE approaches and risk-neutral valuation.

Shreve (2004) [5] sets forth a rigorous treatment of stochastic calculus and of its applications in financial modeling, entailing alternative derivations of the Black and Scholes PDE, emphasizing measure-theoretic rigor. Hull (2018) [6] finally discusses alternative derivations, extensions, and generalizations of the Black and Scholes PDE.

In addition, Föllmer (1981) [7] develops stochastic calculus by focusing on paths with finite quadratic variation and deriving an Itô formula applicable to individual trajectories rather than stochastic processes in the probabilistic sense. The use of such smoothed stochastic processes as a smoothed Wiener process has been comparably explored in the context of financial modeling to simplify derivations and improve interpretability (see Kloeden and Platen, 1992 [8], Bichteler, 2002 [9], Fouque et al., 2000 [10], and Øksendal, 2003) [11]).

In other words, the Black and Scholes PDE has been derived and solved through a variety of mathematical approaches. The Itô calculus method (Black and Scholes, 1973 [1], Merton, 1973 [3]) employs stochastic differential equations, while integrated Wiener processes (Rogers and Williams, 2000 [12], Øksendal, 2003 [11]) and wavelet-based smoothing (Meyer, 1992 [13], Mallat, 1999 [14]) or kernel-based smoothing (Silverman, 1986 [15], Wand and Jones, 1995 [16]) provide differentiable alternatives to Brownian motions, akin to our approach; pathwise hedging arguments (Wilmott et al. 1995 [4], Hull, 2018 [6]) bypass stochastic calculus entirely, instead relying on no arbitrage principles.

For solving the Black and Scholes PDE reduction to the heat equation (Kac, 1951 [17], Evans, 2010 [18]), the Feynman and Kac formula (Kac, 1949 [19], Karatzas and Shreve, 1998 [20]) offer probabilistic solutions, whereas Mellin transforms (Panini and Srivastav, 2004 [21]), Adomian decomposition (Adomian, 1994 [22]), and homotopy methods (Liao, 2003 [23]) provide analytical and perturbative techniques. Such diverse methods underscore the Black and Scholes PDE’s adaptability, bridging stochastic analysis, PDEs, and computational mathematics to address option pricing under varying assumptions and constraints.

By convolving the standard Wiener process with a normal kernel, we obtain a differentiable process which retains the essential properties of stochasticity and volatility, making it suitable for deriving the Black and Scholes PDE without Itô’s Lemma. In other words, the smoothed Wiener process is a modification of the standard Wiener process which introduces differentiability while still capturing the essential properties of stochasticity and volatility, being achieved by convolving the standard Wiener process with a normal kernel, smoothing out the rough, fractal paths of the standard Wiener process.

In the next section, we explain the meaning of the smoothed Wiener process, the way in which it preserves stochasticity and volatility, and the reason for which it is suitable for the task of deriving the Black and Scholes PDE. More generally, the upcoming sections fully derive the Black and Scholes PDE for the price of an option or financial derivative together with its solution, beginning from first principles, employing the theory of PDEs and of Fourier analysis in the process.

3. Smoothed Wiener Process

Let time $t\in [0,T]\subseteq {\mathbb{R}}_{+};$ then

$$\partial W\left(t\right)=\underset{\Delta \to 0}{lim}\Delta W\left(t\right)=\underset{\Delta \to 0}{lim}\epsilon \left(t\right)\sqrt{\Delta t}=\epsilon \left(t\right)\sqrt{\partial t}$$

is the infinitesimal difference of a Wiener process

$$W\left(t\right)=W(t+\Delta t)-\epsilon \left(t\right)\sqrt{\Delta t}\sim \mathcal{N}(0,t)$$

(1)

for the standard white noise $\epsilon \left(t\right)\sim \mathcal{N}(0,1)$ and scaled white noise $W(t+\Delta t)-W\left(t\right)=\epsilon \left(t\right)\sqrt{\Delta t}\sim \mathcal{N}(0,\Delta t)$ such that Wiener process $W(t+\Delta t)\sim \mathcal{N}(0,t+\Delta t)$; notice that limit

$$\underset{\Delta \to 0}{lim}W\left(t\right)=\underset{\Delta \to 0}{lim}[W(t+\Delta t)-\epsilon \left(t\right)\sqrt{\Delta t}]=W(t+\partial t)-\epsilon \left(t\right)\sqrt{\partial t}=W(t+\partial t)-\partial W\left(t\right).$$

Wiener process $W\left(t\right)$ is non-differentiable, as the rate of change

$$W_t\left(t\right)=\underset{\Delta t\to 0}{lim}{\displaystyle \frac{W(t+\Delta t)-W\left(t\right)}{\Delta t}}$$

(2)

does not exist on account of variance

$$\underset{\Delta t\to 0}{lim}Var\left[{\displaystyle \frac{W(t+\Delta t)-W\left(t\right)}{\Delta t}}\right]={\displaystyle \frac{\Delta t}{\Delta t^2}}={\displaystyle \frac{1}{\Delta t}}=\infty ,$$

itself due to the random fluctuations of numerator $W(t+\Delta t)-W\left(t\right),$ which is fractal and scales non-linearly by $\sqrt{\Delta t}.$ As a consequence, let the smoothed Wiener process

$$\tilde{W}\left(t\right)={\int }_{\mathbb{R}}W\left(s\right)\varphi (t-s)ds$$

(3)

such that the normal kernel is

$$\varphi (t-s)={\displaystyle \frac{1}{\varsigma \sqrt{2\pi }}}{e}^{-{\displaystyle \frac{{(t-s)}^2}{2{\varsigma }^2}}}$$

(4)

for the smoothing parameter (bandwidth) $\varsigma \in {\mathbb{R}}_{+},$ whereby the rate of change is

$${\tilde{W}}_t\left(t\right)={\int }_{\mathbb{R}}W\left(s\right){\varphi }_t(t-s)ds.$$

(5)

The smoothed Wiener process $\tilde{W}\left(t\right)$ convolves the Wiener process $W\left(s\right)$ with the normal kernel $\varphi (t-s)$: it is a convolution or weighted average of the Wiener process $W\left(s\right)$ over time t such that the normal kernel $\varphi (t-s)$ assigns greater weights to the Wiener process $W\left(s\right)$ near time $t,$ thereby preserving stochasticity (randomness) as spread out over time t by the smoothing effect of the normal kernel $\varphi (t-s).$ Indeed, the smoothed Wiener process $\tilde{W}\left(t\right)$ preserves security $S\left(t\right)$ volatility owing to the fact that the normal kernel $\varphi (t-s)$ does not alter the variance of the Wiener process $W\left(t\right)$:

$$Var\left[\tilde{W}\left(t\right)\right]={\int }_{\mathbb{R}}Var\left[W\left(s\right)\right]{\varphi }^2(t-s)ds$$

(6)

for $Var\left[W\right(s\left)\right]=s,$ whereby the variance of $\tilde{W}\left(t\right)$ depends on $\varsigma .$ Moreover, in the limit, as smoothing parameter $\varsigma$ tends to zero and the normal kernel $\varphi (t-s)$ thereby approaches Dirac delta function $\delta (t-s)$, the smoothed Wiener process $\tilde{W}\left(t\right)$ recovers the Wiener process $W\left(t\right),$ preserving its stochastic properties:

$$W\left(t\right)=\underset{\varsigma \to 0}{lim}\tilde{W}\left(t\right)=\underset{\varsigma \to 0}{lim}{\int }_{\mathbb{R}}W\left(s\right)\varphi (t-s)ds.$$

In summary, the smoothed Wiener process, constructed using a normal kernel, provides a differentiable alternative to the standard Wiener process while still capturing the stochasticity and volatility necessary for the Black and Scholes framework. This allows us to derive the Black and Scholes PDE without relying on Itô’s Lemma while maintaining the mathematical and economic consistency of the model.

The rest of this study proceeds as follows: in Section 4, we lay the foundations for the derivation of the Black and Scholes PDE from first principles without relying on Itô’s Lemma but by means of the smoothed Wiener process developed in the previous section; we achieve such a derivation in Section 5.

In Section 6, we especially introduce the initial condition and the two boundary conditions whereby the Black and Scholes PDE is to become a well-posed problem. Section 7, Section 8, Section 9, Section 10, Section 11 and Section 12 are devoted to the computation of the solution of the Black and Scholes PDE.

In Section 7, we introduce time and space dummies for computational purposes. In Section 8, we especially compute partial derivatives $V_t,V_S$, and $V_{SS}$ in accordance with our change of variables, and we substitute them into the Black and Scholes PDE.

In Section 9, we further our computation of partial derivatives in terms of our changed variables in order to reduce our problem to that of the heat equation, which we are able to solve analytically. In Section 10, we thoroughly solve the resulting heat equation by means of Fourier analysis.

In Section 11, we thoroughly extend our solution to the heat equation by considering the initial condition and in Section 12, we perform a backwards substitution to finally obtain the Black and Scholes PDE solution, which we present and discuss in terms of the core logic of its applications in Section 13. Section 14 presents the conclusion.

4. Changes in Security and Option Prices

In this section, we lay the foundations for the derivation of the Black and Scholes PDE from first principles without relying on Itô’s Lemma but by means of the smoothed Wiener process developed in the previous section. We specifically define an equation for the rate of change $S_t\left(t\right)$ of security price $S\left(t\right);$ we then proceed to the analysis of the local behavior of option price $V(S,t)$ in terms of small changes $\partial S$ in security price S by means of a second-order Taylor series expansion, which we rearrange for a no-arbitrage condition towards the derivation of the Black and Scholes PDE.

Let the price of a security $S\left(t\right)\in \mathbb{R}$ for function $S:[0,T]\subseteq {\mathbb{R}}_{+}\to \mathbb{R},$ its expected growth rate $\alpha \in \mathbb{R},$ its dividend yield $\delta \in \mathbb{R}$, and its volatility $\sigma \in {\mathbb{R}}_{+}.$ The rate of change in the price of the security $S_t\left(t\right)$ equals the price of the security $S\left(t\right)$ weighted by the sum of (1) the expected capital growth rate (expected growth rate minus dividend yield) $\alpha -\delta$ and (2) the fluctuations augmented for volatility (volatility times fluctuations) $\sigma {\tilde{W}}_t\left(t\right)$:

$$\begin{array}{cc}\hfill & S_t\left(t\right)=[(\alpha -\delta )+\sigma {\tilde{W}}_t\left(t\right)]S\left(t\right)=\hfill \\ \hfill & =(\alpha -\delta )S\left(t\right)+\sigma S\left(t\right){\tilde{W}}_t\left(t\right)⟶\hfill \\ \hfill & \partial S\left(t\right)=(\alpha -\delta )S\left(t\right)\partial t+\sigma S\left(t\right)\partial \tilde{W}\left(t\right).\hfill \end{array}$$

(7)

The price of an option $V(S,t)$ is such that $V_S$ and $\partial S$, respectively, model the sensitivity in the price of the security S (delta hedging parameter) and its rate of change (expected deterministic growth) so that $V_S\partial S$ may model the expected change in the option’s price V proportionally due to the drift of the security price S (deterministic), quantifying the impact of its trend.

The price of an option $V(S,t)$ is also such that $V_{SS}$ and $\sigma$, respectively, model the curvature in the price of the security S (gamma hedging parameter) and its volatility (randomness in price changes), so that ${\displaystyle \frac{{\sigma }^2{S}^2}{2}}{V}_{SS}$ may model the expected change in the option’s price proportionally due to the volatility of the security price (stochastic), quantifying the impact of its fluctuations.

In detail, coefficient ${\displaystyle \frac{{\sigma }^2{S}^2}{2}}$ scales volatility $\sigma$ by security price S (higher fluctuations for higher price or volatility) and symmetrically normalizes their multiplied square ${\sigma }^2{S}^2$ through division by $2,$ in which variance ${\sigma }^2$ is necessary to quantify volatility $\sigma .$

Otherwise put, focus is on the local behavior of option price $V(S,t)$ in response to small changes $\partial S$ in security price $S,$ which can be approximated by means of a second-order Taylor series expansion for security price $S:\forall f:{\mathbb{R}}^n\to \mathbb{R},$ $f\in C^{\infty },$ $n\in {\mathbb{N}}_{+},\mathbf{a}={[a_1\dots a_n]}^{\top }\in {\mathbb{R}}^n,$ $f\left(\mathbf{x}\right)=f(x_1,\dots ,x_n)$ = $f\left(\mathbf{a}\right)+{\sum }_{j=1}^{\infty }{\displaystyle \frac{D^{\left[j\right]}f\left(\mathbf{a}\right)}{j!}}{(\mathbf{x}-\mathbf{a})}^j$ = $f(a_1,\dots ,a_n)+{\sum }_{j=1}^{\infty }{\sum }_{i=1}^n{\displaystyle \frac{{\partial }^{j}f(a_1,\dots ,a_n)}{\partial a_i\partial a_{\neg i}}}(x_i-a_i)(x_{\neg i}-a_{\neg i}),$

$$\begin{array}{cc}\hfill & V=V+{\displaystyle \frac{V_S}{1!}}\partial S+{\displaystyle \frac{V_{SS}}{2!}}\partial S^2=V+V_S\partial S+{\displaystyle \frac{\partial S^2}{2}}{V}_{SS}⟶\hfill \\ \hfill & 0=V_S[(\alpha -\delta )S\partial t+\sigma S\partial \tilde{W}]+{\displaystyle \frac{{\sigma }^2{S}^2{(\partial {\tilde{W}}_t)}^2}{2}}{V}_{SS}=\hfill \\ \hfill & =V_S[(\alpha -\delta )S\partial t+\sigma S\partial \tilde{W}]+{\displaystyle \frac{{\sigma }^2{S}^2\partial t}{2}}{V}_{SS}⟶\hfill \\ \hfill & 0=V_S[(\alpha -\delta )S+\sigma S{\tilde{W}}_t]+{\displaystyle \frac{{\sigma }^2{S}^2}{2}}{V}_{SS}=V_S\partial S+{\displaystyle \frac{{\sigma }^2{S}^2}{2}}{V}_{SS},\hfill \end{array}$$

(8)

as

$$\partial S^2={[(\alpha -\delta )S\partial t+\sigma S\partial \tilde{W}]}^2={(\alpha -\delta )}^2{S}^2{(\partial t)}^2+2(\alpha -\delta )S\partial t\sigma S\partial \tilde{W}+{\sigma }^2{S}^2{(\partial \tilde{W})}^2,$$

(9)

for which

$$\underset{\partial t\to 0}{lim}\partial S^2=\mathbb{E}\left[{(\partial S)}^2\right]=\mathbb{E}\left[{\sigma }^2{S}^2{(\partial \tilde{W})}^2\right]={\sigma }^2{S}^2\partial t$$

(10)

and

$$\begin{array}{cc}\hfill & \mathbb{E}\left\{{[\partial \tilde{W}\left(t\right)]}^2\right\}=\mathbb{E}\left\{{\int }_{\mathbb{R}}{\int }_{\mathbb{R}}{[\partial W\left(s\right)]}^2{\varphi }^2(t-s){\left(ds\right)}^2\right\}=\hfill \\ \hfill & =\mathbb{E}\left\{{\int }_{\mathbb{R}}{\varphi }^2(t-s)ds{\int }_{\mathbb{R}}{[\partial W\left(s\right)]}^{2}ds\right\}=\mathbb{E}[{\epsilon }^2\left(t\right)\partial t]=\partial t.\hfill \end{array}$$

(11)

No arbitrage is such that the sum $V_S\partial S+{\displaystyle \frac{{\sigma }^2{S}^2}{2}}{V}_{SS},$ which is the expected total change in the option’s price, equals the risk-free interest rate $r\in \mathbb{R}$ at zero; the rate of change in the price of the option over time $V_t$ is then added to the equation in order to also account for exogenous time changes in option price V:

$$\begin{array}{cc}\hfill & V_S\partial S+{\displaystyle \frac{{\sigma }^2{S}^2}{2}}{V}_{SS}=[(\alpha -\delta )+\sigma {\tilde{W}}_t]S{V}_S+{\displaystyle \frac{{\sigma }^2{S}^2}{2}}{V}_{SS}=\hfill \\ \hfill & =(\alpha -\delta )S{V}_S+{\displaystyle \frac{{\sigma }^2{S}^2}{2}}{V}_{SS}+\sigma S{V}_S{\tilde{W}}_t=r=0⟶\hfill \\ \hfill & V_t=(\alpha -\delta )S{V}_S+{\displaystyle \frac{{\sigma }^2{S}^2}{2}}{V}_{SS}+V_t+\sigma S{V}_S{\tilde{W}}_t⟶\hfill \\ \hfill & \partial V=\left[(\alpha -\delta )S{V}_S+{\displaystyle \frac{{\sigma }^2{S}^2}{2}}{V}_{SS}+V_t\right]\partial t+\sigma S{V}_S\partial \tilde{W},\hfill \end{array}$$

(12)

which would follow from Itô’s Lemma if $W\left(t\right)$ were used instead. Indeed, by having replaced the Wiener process with such a differentiable process as a smoothed Wiener process, we have been able to maintain the validity of the Black and Scholes PDE derivation, still capturing the essential properties of stochasticity and volatility while allowing for differentiability.

The final Black and Scholes PDE is to remain unchanged, as the economic principle of no arbitrage and the dynamics of the asset price have been preserved. In detail, the no-arbitrage principle ensures that the expected change in the option price due to drift and volatility may balance the risk-free rate, leading to the Black and Scholes PDE.

5. Investor Portfolio

In this section, we achieve the derivation of the Black and Scholes PDE from first principles without relying on Itô’s Lemma but by means of the smoothed Wiener process developed in the previous section. We specifically define the investor’s portfolio P in order to derive the Black and Scholes PDE. For quantity of securities $q\in {\mathbb{R}}_{++}$ and borrowing $B\in \mathbb{R}$ the investor’s portfolio P is such that

$$\begin{array}{cc}\hfill & V+B=qS⟶\hfill \\ \hfill & P=V-qS+B=0⟶\hfill \\ \hfill & B=qS-V⟶\hfill \\ \hfill & B_S=q-V_S=0\hfill \end{array}$$

(13)

at borrowing efficiency, whence the quantity of securities $q=V_S⟶B=S{V}_S-V.$ The rate of change in the investor’s portfolio P is such that for ${\displaystyle \frac{\partial B}{\partial t}}=rB⟶\partial B=rB\partial t$ and ${\displaystyle \frac{\partial S}{\partial t}}=S_t+\delta S⟶\delta S=0$

$$\begin{array}{cc}\hfill & P_t\left(t\right)=V_t-q{S}_t+rB=V_t-q(S_t+\delta S)+rB=0⟶\hfill \\ \hfill & \partial P=\partial V-q(\partial S+\delta S\partial t)+rB\partial t=\partial (V-qS+B)=\hfill \\ \hfill & =\left[(\alpha -\delta )S{V}_S+{\displaystyle \frac{{\sigma }^2{S}^2}{2}}{V}_{SS}+V_t\right]\partial t+\sigma S{V}_S\partial \tilde{W}-q(\partial S+\delta S\partial t)+rB\partial t=\hfill \\ \hfill & =(\alpha -\delta )S{V}_S\partial t+\sigma S{V}_S\partial \tilde{W}+\left({\displaystyle \frac{{\sigma }^2{S}^2}{2}}{V}_{SS}+V_t\right)\partial t-q(\partial S+\delta S\partial t)+rB\partial t=\hfill \\ \hfill & =[(\alpha -\delta )S\partial t+\sigma S{V}_S\partial \tilde{W}]V_S+\left({\displaystyle \frac{{\sigma }^2{S}^2}{2}}{V}_{SS}+V_t\right)\partial t-q(\partial S+\delta S\partial t)+rB\partial t=\hfill \\ \hfill & =V_S\partial S+\left(V_t+{\displaystyle \frac{{\sigma }^2{S}^2}{2}}{V}_{SS}\right)\partial t-q(\partial S+\delta S\partial t)+rB\partial t=\hfill \\ \hfill & =(V_S-q)\partial S+\left(V_t+{\displaystyle \frac{{\sigma }^2{S}^2}{2}}{V}_{SS}-\delta qS+rB\right)\partial t=\hfill \\ \hfill & =0·\partial S+\left[V_t+{\displaystyle \frac{{\sigma }^2{S}^2}{2}}{V}_{SS}-\delta S{V}_S+r(S{V}_S-V)\right]\partial t=\hfill \\ \hfill & =\left[V_t+{\displaystyle \frac{{\sigma }^2{S}^2}{2}}{V}_{SS}+(r-\delta )S{V}_S-rV\right]\partial t=0,\hfill \end{array}$$

(14)

as investor portfolio $P=0⟶{\displaystyle \frac{\partial P}{\partial t}}=0⟶\partial P=0,$ whence the Black and Scholes PDE

$$V_t+{\displaystyle \frac{{\sigma }^2{S}^2}{2}}{V}_{SS}+(r-\delta )S{V}_S=rV⟶V_t+{\displaystyle \frac{{\sigma }^2{S}^2}{2}}{V}_{SS}+rS{V}_S=rV,$$

(15)

for dividend yield $\delta =0.$

6. Well-Posed Problem

In this section, we introduce the initial condition and the two boundary conditions whereby the Black and Scholes PDE is to become a well-posed problem. We also cover the sufficient rudiments of PDEs in order for the reader to contextualize the endeavor towards its solution. The Black and Scholes PDE features one initial condition and two boundary conditions:

$$\begin{array}{cc}\hfill & V(S,T)=max(S-K,0);\hfill \\ \hfill & V(0,t)=0,\forall t\in [0,T)\subset {\mathbb{R}}_{+};\hfill \\ \hfill & V(S,t)=S{e}^{-\delta (T-t)}-K{e}^{-r(T-t)},\forall S\to \infty ,t\in [0,T)\subset {\mathbb{R}}_{+},\hfill \end{array}$$

(16)

in which $K\in \mathbb{R}$ is the strike price of the security; notice that $V(S,t)=S-K{e}^{-r(T-t)},\forall S\to \infty ,t\in [0,T)\subset {\mathbb{R}}_{+}$ for $\delta =0.$ In detail, the initial condition $V(S,T)=max(S-K,0)$ is such that the price of the option $V(S,T)$ at the time of exercise $T>t$ excludes a loss; the boundary condition $V(0,t)=0$ before terminal time period T is such that the price of the option $V=0$ for the price of the security $S=0;$ the boundary condition $V(S,t)=S{e}^{-\delta (T-t)}-K{e}^{-r(T-t)}$ as the price of the security S increases without bound is such that the price of the option V is the discounted difference between the security price S and the strike price $K.$ The Black and Scholes equation is a PDE of order $2,$ which assumes the general form

$$a{u}_{xx}+2b{u}_{xy}+c{u}_{yy}+d{u}_x+e{u}_y+fu=g,$$

for ordered pair $(x,y)\in \mathrm{\Omega }\subseteq {\mathbb{R}}^2,$ in which $\mathrm{\Omega }$ is an open domain for function $u:\mathrm{\Omega }\to \mathbb{R}$ such that $u\in C^2,$ and for images $a(x,y),b(x,y)$ and $c(x,y)$ open and continuous in hyperplane ${\mathbb{R}}^2.$ A PDE is classified as follows:

$$b^2-ac\left\{\begin{array}{c}>0,\mathrm{hyperbolic}\\ =0,\mathrm{parabolic}\\ <0,\mathrm{elliptic}\end{array}\right..$$

The Black and Scholes equation is thus a parabolic PDE of order $2,$ as parameter $b=c=0$ for function $u=V,$ variables $x=S$ and $y=t$, and domain $\mathrm{\Omega }=\{(S,t):S\in {\mathbb{R}}_{+},t\in [0,T]\subseteq {\mathbb{R}}_{+}\}.$

7. Time and Strike Price

For computational scopes, in this section, we introduce the time dummy $\tau$ and space dummy x for security price $S=K{e}^x.$ Let time period $t=T-{\displaystyle \frac{\tau }{\frac{{\sigma }^2}{2}}},$ in which $T>t$ is the terminal time period at which the option is exercised and subtrahend numerator $\tau$ is found by rearrangement:

$$t=T-{\displaystyle \frac{\tau }{\frac{{\sigma }^2}{2}}}⟶{\displaystyle \frac{\tau }{\frac{{\sigma }^2}{2}}}=T-t⟶\tau ={\displaystyle \frac{(T-t){\sigma }^2}{2}}.$$

(17)

Let the security price S be expressed in terms of the discounted strike price $K{e}^x$ such that exponent x is found by rearrangement:

$$\begin{array}{cc}\hfill & S=K{e}^x⟶lnS=lnKln{e}^x=lnKxlne=lnKx⟶\hfill \\ \hfill & x={\displaystyle \frac{lnS}{lnK}}=ln\left({\displaystyle \frac{S}{K}}\right)=lnS-lnK.\hfill \end{array}$$

(18)

8. Change of Variables

In this section, we compute partial derivatives $V_t,V_S$, and $V_{SS}$ for option price $V(S,t)=Kv(x,\tau )$ and we substitute them into the Black and Scholes PDE; we additionally discern the initial condition $V(S,T)=max(S-K,0)$ as being $Kv(x,0)=Kmax(e^x-1,0).$ Let the price of the option $V(S,t)=Kv(x,\tau )$ and compute the following partial derivatives:

$$\begin{array}{cc}\hfill & V_t=K{v}_{\tau }{\tau }_t=K{v}_{\tau }\left[(-1){\displaystyle \frac{{\sigma }^2}{2}}\right];\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill & V_S=K{v}_x{x}_S=K{v}_x\left({\displaystyle \frac{1}{S}}\right);\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill & V_{SS}=K{v}_{xx}{x}_S\left({\displaystyle \frac{1}{S}}\right)+K{v}_x\left(-{\displaystyle \frac{1}{S^2}}\right)=\hfill \\ \hfill & =K{v}_{xx}\left({\displaystyle \frac{1}{S^2}}\right)-K{v}_x\left({\displaystyle \frac{1}{S^2}}\right)=\hfill \\ \hfill & =K\left({\displaystyle \frac{1}{S^2}}\right)(v_{xx}-v_x).\hfill \end{array}$$

(21)

The initial condition is accordingly written such that $V(S,T)=max(S-K,0)=max(K{e}^x-K,0)=Kmax(e^x-1,0)$ and $V(S,T)=Kv(x,0),$ whence $Kv(x,0)=Kmax(e^x-1,0).$ Take the Black and Scholes PDE and substitute the changed variables into it:

$$\begin{array}{cc}\hfill & V_t+{\displaystyle \frac{{\sigma }^2{S}^2}{2}}{V}_{SS}+rS{V}_S=rV⟷\hfill \\ \hfill & \left(-K{v}_{\tau }{\displaystyle \frac{{\sigma }^2}{2}}\right)+{\displaystyle \frac{{\sigma }^2{S}^2}{2}}\left[K\left({\displaystyle \frac{1}{S^2}}\right)(v_{xx}-v_x)\right]+rS\left[K{v}_x\left({\displaystyle \frac{1}{S}}\right)\right]=rKv⟶\hfill \\ \hfill & -{\displaystyle \frac{{\sigma }^2}{2}}{v}_{\tau }+{\displaystyle \frac{{\sigma }^2}{2}}(v_{xx}-v_x)+r{v}_x=rv⟶{\displaystyle \frac{{\sigma }^2}{2}}{v}_{\tau }={\displaystyle \frac{{\sigma }^2}{2}}(v_{xx}-v_x)+r{v}_x-rv⟶\hfill \\ \hfill & v_{\tau }=v_{xx}-v_x+{\displaystyle \frac{2r}{{\sigma }^2}}{v}_x-{\displaystyle \frac{2r}{{\sigma }^2}}v=v_{xx}-v_x+k{v}_x-kv⟶\hfill \\ \hfill & v_{\tau }=v_{xx}+(k-1)v_x-kv.\hfill \end{array}$$

(22)

9. Heat Equation

In order to reduce our problem to that of the heat equation, which we are able to solve analytically, in this section, we compute partial derivatives $v_{\tau },v_x$, and $v_{xx}$ for function $v(x,\tau )=e^{\alpha x+\beta \tau }u(x,\tau );$ we additionally discern the initial condition $V(S,T)=u(x,0)=max\left[e^{{\displaystyle \frac{(k+1)x}{2}}}-e^{{\displaystyle \frac{(k-1)x}{2}}},0\right].$

Let function $v(x,\tau )=e^{\alpha x+\beta \tau }u(x,\tau )⟶lnv=(\alpha x+\beta \tau )lne+lnu(x,\tau )=(\alpha x+\beta \tau )+lnu(x,\tau )$ such that parameters $\alpha$ and $\beta$ are found by rearrangement through the computation of the following partial derivatives:

$$\begin{array}{l}{v}_{\tau }=\beta e^{\alpha x+\beta \tau }u+e^{\alpha x+\beta \tau }{u}_{\tau };\hfill \end{array}$$

(23)

$$\begin{array}{l}{v}_x=\alpha e^{\alpha x+\beta \tau }u+e^{\alpha x+\beta \tau }{u}_x;\end{array}$$

(24)

$$\begin{array}{l}{v}_{xx}=({\alpha }^2{e}^{\alpha x+\beta \tau }u+\alpha e^{\alpha x+\beta \tau }{u}_x)+(\alpha \alpha e^{\alpha x+\beta \tau }{u}_x+e^{\alpha x+\beta \tau }{u}_{xx})=\\ ={\alpha }^2{e}^{\alpha x+\beta \tau }u+2\alpha e^{\alpha x+\beta \tau }{u}_x+e^{\alpha x+\beta \tau }{u}_{xx},\end{array}$$

(25)

whence

$$\begin{array}{cc}\hfill & v_{\tau }=v_{xx}+(k-1)v_x-kv⟷\hfill \\ \hfill & \beta e^{\alpha x+\beta \tau }u+e^{\alpha x+\beta \tau }{u}_{\tau }=({\alpha }^2{e}^{\alpha x+\beta \tau }u+2\alpha e^{\alpha x+\beta \tau }{u}_x+e^{\alpha x+\beta \tau }{u}_{xx})+\hfill \\ \hfill & +(k-1)(\alpha e^{\alpha x+\beta \tau }u+e^{\alpha x+\beta \tau }{u}_x)-k\left(e^{\alpha x+\beta \tau }u\right)⟶\hfill \\ \hfill & \beta u+u_{\tau }={\alpha }^{2}u+2\alpha u_x+u_{xx}+(k-1)(\alpha u+u_x)-ku⟶\hfill \\ \hfill & u_{\tau }=u_{xx}+(2\alpha +k-1)u_x+[{\alpha }^2+\alpha (k-1)-k]u-\beta u=\hfill \\ \hfill & =u_{xx}+(2\alpha +k-1)u_x+[{\alpha }^2+k(\alpha -1)-\alpha -\beta ]u,\hfill \end{array}$$

(26)

which is the heat equation $u_{\tau }=u_{xx}$ for $2\alpha +k-1=0$ and ${\alpha }^2+k(\alpha -1)-\alpha -\beta =0.$ Thus:

$$\begin{array}{c}2\alpha +k-1=0⟶\alpha ={\displaystyle \frac{1-k}{2}}={\displaystyle \frac{-(k-1)}{2}};\hfill \end{array}$$

(27)

$$\begin{array}{c}{\alpha }^2+k(\alpha -1)-\alpha -\beta =0⟶\beta ={\alpha }^2+k(\alpha -1)-\alpha =\hfill \\ ={\left[{\displaystyle \frac{-(k-1)}{2}}\right]}^2+k\left[{\displaystyle \frac{-(k-1)}{2}}-1\right]-{\displaystyle \frac{-(k-1)}{2}}=\hfill \\ ={\displaystyle \frac{{(k-1)}^2}{4}}+\left[{\displaystyle \frac{-k(k-1)}{2}}-k\right]+{\displaystyle \frac{k-1}{2}}=\hfill \\ ={\displaystyle \frac{{(k-1)}^2-2k(k-1)-4k+2(k-1)}{4}}=\hfill \\ ={\displaystyle \frac{k^2-2k+1-2k^2+2k-4k+2k-2}{4}}=\hfill \\ ={\displaystyle \frac{-k^2-2k-1}{4}}={\displaystyle \frac{-(k^2+2k+1)}{4}}={\displaystyle \frac{-{(k+1)}^2}{4}}.\hfill \end{array}$$

(28)

Consequently, the weighted initial condition $v(x,0)=K^{-1}V(S,T)$ becomes

$$\begin{array}{cc}\hfill & v(x,0)=e^{\alpha x}u(x,0)=K^{-1}V(S,T)⟶V(S,T)=u(x,0)=\hfill \\ \hfill & =e^{-\alpha x}v(x,0)=e^{-\alpha x}max(e^x-1,0)=\hfill \\ \hfill & =max\left[e^{(1-\alpha )x}-e^{-\alpha x},0\right]=\hfill \\ \hfill & =max\left\{e^{\left\{1-\left[{\displaystyle \frac{-(k-1)}{2}}\right]\right\}x}-e^{-\left[{\displaystyle \frac{-(k-1)}{2}}\right]x},0\right\}=\hfill \\ \hfill & =max\left\{e^{\left[1+{\displaystyle \frac{(k-1)}{2}}\right]x}-e^{{\displaystyle \frac{(k-1)x}{2}}},0\right\}=\hfill \\ \hfill & =max\left[e^{{\displaystyle \frac{(k+1)x}{2}}}-e^{{\displaystyle \frac{(k-1)x}{2}}},0\right].\hfill \end{array}$$

(29)

10. Heat Equation Solution

In this section, we thoroughly solve the resulting heat equation by means of Fourier analysis; we obtain the heat kernel $u(x,\tau )={\displaystyle \frac{1}{2\sqrt{\pi \tau }}}{\int }_{\mathbb{R}}f\left(y\right)e^{-{\displaystyle \frac{{(x-y)}^2}{4\tau }}}dy.$ Consider the heat equation $u_{xx}=u_{\tau }$ with the initial condition $u(x,0)=f\left(x\right)$ or, more generally, ${lim}_{\tau \to 0_{+}}u(x,\tau )=f\left(x\right).$ The Fourier transform of function $u(x,\tau )$ is $\widehat{u}(s,\tau )={\int }_{\mathbb{R}}u(x,\tau )e^{-isx}dx;$ compute its partial derivatives with respect to x:

$$\begin{array}{cc}\hfill & {\widehat{u}}_x=-is{\int }_{\mathbb{R}}u{e}^{-isx}dx=-is\widehat{u};\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill & {\widehat{u}}_{xx}={(-is)}^2{\int }_{\mathbb{R}}u{e}^{-isx}dx=-s^2\widehat{u},\hfill \end{array}$$

(31)

whence for $\widehat{u}(s,\tau )=e^{z\tau },$

$$\begin{array}{cc}\hfill & {\widehat{u}}_{xx}={\widehat{u}}_{\tau }⟷-s^2\widehat{u}={\widehat{u}}_{\tau }⟷-s^2{e}^{z\tau }={\left(e^{z\tau }\right)}_{\tau }⟶\hfill \\ \hfill & -s^2{e}^{z\tau }=z{e}^{z\tau }⟶z=-s^2⟶\widehat{u}(s,\tau )=\widehat{u}(s,0)e^{-s^2\tau };\hfill \end{array}$$

(32)

indeed,

$${\widehat{u}}_{\tau }=-s^2\widehat{u}(s,0)e^{-s^2\tau }=-s^2\widehat{u}=-s^2{\int }_{\mathbb{R}}u{e}^{-isx}dx.$$

(33)

Notice that the initial condition $\widehat{u}(s,0)$ is the Fourier transform of the initial condition $u(x,0):\widehat{u}(s,0)=\widehat{f}\left(s\right)={\int }_{\mathbb{R}}f\left(x\right)e^{-isx}dx$ such that $\widehat{u}(s,\tau )=\widehat{u}(s,0)e^{-s^2\tau }=\widehat{f}\left(s\right)e^{-s^2\tau }.$ It follows that function $u(x,\tau )$ is the inverse Fourier transform of its Fourier transform and is called the heat kernel:

$$\begin{array}{cc}\hfill & u(x,\tau )={\displaystyle \frac{1}{2\pi }}{\int }_{\mathbb{R}}\widehat{u}(s,\tau )e^{isx}ds={\displaystyle \frac{1}{2\pi }}{\int }_{\mathbb{R}}\widehat{f}\left(s\right)e^{-s^2\tau }{e}^{isx}ds=\hfill \\ \hfill & ={\displaystyle \frac{1}{2\pi }}{\int }_{\mathbb{R}}\left[{\int }_{\mathbb{R}}f\left(y\right)e^{-isy}dy\right]e^{-s^2\tau }{e}^{isx}ds=\hfill \\ \hfill & ={\int }_{\mathbb{R}}f\left(y\right)\left[{\displaystyle \frac{1}{2\pi }}{\int }_{\mathbb{R}}{e}^{-s^2\tau }{e}^{is(x-y)}ds\right]dy=\hfill \\ \hfill & ={\displaystyle \frac{1}{2\sqrt{\pi \tau }}}{\int }_{\mathbb{R}}f\left(y\right)e^{-{\displaystyle \frac{{(x-y)}^2}{4\tau }}}dy,\hfill \end{array}$$

(34)

in which ${\displaystyle \frac{1}{2\pi }}{\int }_{\mathbb{R}}{e}^{-s^2\tau }{e}^{is(x-y)}ds={\displaystyle \frac{1}{\sqrt{4\pi \tau }}}{e}^{-{\displaystyle \frac{{(x-y)}^2}{4\tau }}}$ is a standard normal inverse Fourier transform of $e^{-s^2\tau }.$ In detail:

$$\begin{array}{cc}\hfill & -s^2\tau +is(x-y)=-\tau \left[s^2-{\displaystyle \frac{is(x-y)}{\tau }}\right]=\hfill \\ \hfill & =-\tau \left\{{\left[s-{\displaystyle \frac{i(x-y)}{2\tau }}\right]}^2-{\left[{\displaystyle \frac{i(x-y)}{2\tau }}\right]}^2\right\}=\hfill \\ \hfill & =-\tau {\left[s-{\displaystyle \frac{i(x-y)}{2\tau }}\right]}^2+\tau \left[{\displaystyle \frac{-{(x-y)}^2}{4{\tau }^2}}\right]=\hfill \\ \hfill & =-\tau u^2-{\displaystyle \frac{{(x-y)}^2}{4\tau }}\hfill \end{array}$$

(35)

such that for $du=ds$ and the normal integral ${\int }_{\mathbb{R}}{e}^{-w^2}dw=\sqrt{\pi }$ it follows that

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{2\pi }}{\int }_{\mathbb{R}}{e}^{-s^2\tau }{e}^{is(x-y)}ds={\displaystyle \frac{1}{2\pi }}{e}^{-{\displaystyle \frac{{(x-y)}^2}{4\tau }}}{\int }_{\mathbb{R}}{e}^{-\tau u^2}du=\hfill \\ \hfill & ={\displaystyle \frac{1}{2\pi }}{e}^{-{\displaystyle \frac{{(x-y)}^2}{4\tau }}}\sqrt{{\displaystyle \frac{\pi }{\tau }}}={\displaystyle \frac{1}{2}}{\pi }^{-1}{\pi }^{0.5}{\tau }^{-0.5}{e}^{-{\displaystyle \frac{{(x-y)}^2}{4\tau }}}=\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}{\pi }^{-0.5}{\tau }^{-0.5}{e}^{-{\displaystyle \frac{{(x-y)}^2}{4\tau }}}={\displaystyle \frac{1}{2\sqrt{\pi \tau }}}{e}^{-{\displaystyle \frac{{(x-y)}^2}{4\tau }}},\hfill \end{array}$$

(36)

as $w^2=\tau u^2⟶w=\sqrt{\tau }u⟶u={\displaystyle \frac{w}{\sqrt{\tau }}}$ for ${\int }_{\mathbb{R}}{e}^{-u^2}du={\int }_{\mathbb{R}}{e}^{-{\left({\displaystyle \frac{w}{\sqrt{\tau }}}\right)}^2}dw=\sqrt{{\displaystyle \frac{\pi }{\tau }}}.$ Notice that the normal integral is computed as follows: for $\left|\right|x\left|\right|=\sqrt{x_1^2+x_2^2}$ and $x_1=rcos\phi$ and $x_2=rsin\phi$, it arises that $dx=d{x}_{1}d{x}_2=det\left(J\right)drd\phi =rdrd\phi ,$ as

$$\begin{array}{c}J=\left[{\displaystyle \frac{\partial x_1}{\partial r}}{\displaystyle \frac{\partial x_1}{\partial \phi }}{\displaystyle \frac{\partial x_2}{\partial r}}{\displaystyle \frac{\partial x_2}{\partial \phi }}\right]=\left[\begin{array}{cc}cos\phi & -rsin\phi \\ sin\phi & rcos\phi \end{array}\right]⟶\hfill \\ det\left(J\right)=cos\phi rcos\phi -(-rsin\phi )\left(sin\phi \right)=r(co{s}^2\phi +si{n}^2\phi )=r\hfill \end{array}$$

(37)

and $co{s}^2\phi +si{n}^2\phi =1;$ consequently,

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{R}}^2}{e}^{-{\left|\right|x\left|\right|}^2}dx={\int }_{{\mathbb{R}}^2}{e}^{-\sqrt[2]{x_1^2+x_2^2}}d{x}_{1}d{x}_2=\hfill \\ \hfill & ={\int }_0^{2\pi }{\int }_0^{\infty }{e}^{-r^2}rdrd\phi ={\int }_0^{2\pi }d\phi {\int }_0^{\infty }{e}^{-r^2}dr=\hfill \\ \hfill & ={\left[\phi \right]}_0^{2\pi }{\int }_0^{\infty }{e}^{-r^2}dr=2\pi {\int }_0^{\infty }{e}^{-r^2}dr=2\pi {\int }_0^{\infty }\left(-{\displaystyle \frac{1}{2}}\right){\displaystyle \frac{d{e}^{-r^2}}{dr}}dr=\hfill \\ \hfill & =-\pi {\int }_0^{\infty }{\displaystyle \frac{d{e}^{-r^2}}{dr}}dr=-\pi {\left[e^{-r^2}\right]}_0^{\infty }=-\pi (e^{-\infty }-e^{-0})=\pi \hfill \end{array}$$

(38)

such that

$$\begin{array}{cc}\hfill & \pi ={\int }_{{\mathbb{R}}^2}{e}^{-{\left|\right|x\left|\right|}^2}dx={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }{e}^{-(x_1^2+x_2^2)}d{x}_{1}d{x}_2=\hfill \\ \hfill & ={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }{e}^{-x_1^2}{e}^{-x_2^2}d{x}_{1}d{x}_2=\hfill \\ \hfill & ={\int }_{-\infty }^{\infty }{e}^{-x_1^2}d{x}_1{\int }_{-\infty }^{\infty }{e}^{-x_2^2}d{x}_2=\hfill \\ \hfill & ={\left({\int }_{-\infty }^{\infty }{e}^{-w^2}dw\right)}^2⟶{\int }_{\mathbb{R}}{e}^{-w^2}dw=\sqrt{\pi }.\hfill \end{array}$$

(39)

11. Heat Kernel with Initial Condition

In this section, we thoroughly extend our solution to the heat equation by considering the initial condition; we ultimately obtain function $u(x,\tau )=e^{{\displaystyle \frac{\tau {(k+1)}^2}{4}}+{\displaystyle \frac{(k+1)x}{2}}}N\left(d_1\right)-e^{{\displaystyle \frac{\tau {(k-1)}^2}{4}}+{\displaystyle \frac{(k-1)x}{2}}}N\left(d_2\right).$ Consider the heat kernel $u(x,\tau )={\displaystyle \frac{1}{2\sqrt{\pi \tau }}}{\int }_{\mathbb{R}}f\left(y\right)e^{-{\displaystyle \frac{{(x-y)}^2}{4\tau }}}dy$ such that variable

$$\begin{array}{cc}\hfill & y=x-z\sqrt{2\tau }⟶x-y=z\sqrt{2\tau }⟶\hfill \\ \hfill & z={\displaystyle \frac{x-y}{\sqrt{2\tau }}}⟶z^2={\displaystyle \frac{{(x-y)}^2}{2\tau }},\hfill \end{array}$$

(40)

whence the derivatives are

$$\begin{array}{cc}\hfill & {\displaystyle \frac{dy}{dx}}=1⟶dy=dx\mathrm{and}\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{dz}{dx}}={\displaystyle \frac{1}{\sqrt{2\tau }}}⟶{\displaystyle \frac{dz}{dy}}={\displaystyle \frac{1}{\sqrt{2\tau }}}⟶\hfill \\ \hfill & dy=\sqrt{2\tau }dz.\hfill \end{array}$$

(42)

Consequently, the heat kernel becomes

$$\begin{array}{cc}\hfill & u(x,\tau )={\displaystyle \frac{1}{2\sqrt{\pi \tau }}}{\int }_{\mathbb{R}}f\left(y\right)e^{-{\displaystyle \frac{{(x-y)}^2}{4\tau }}}dy=\hfill \\ \hfill & ={\displaystyle \frac{1}{2\sqrt{\pi \tau }}}{\int }_{\mathbb{R}}f(x-z\sqrt{2\tau })e^{-{\displaystyle \frac{z^2}{2}}}\sqrt{2\tau }dz=\hfill \\ \hfill & ={\displaystyle \frac{1}{\sqrt{2\pi }}}{\int }_{\mathbb{R}}f(x-z\sqrt{2\tau })e^{-{\displaystyle \frac{z^2}{2}}}dz\hfill \end{array}$$

(43)

with the initial condition $u(x,0)=f\left(y\right)=e^{{\displaystyle \frac{(k+1)y}{2}}}-e^{{\displaystyle \frac{(k-1)y}{2}}}=f(x-z\sqrt{2\tau })=e^{{\displaystyle \frac{(k+1)(x-z\sqrt{2\tau })}{2}}}-e^{{\displaystyle \frac{(k-1)(x-z\sqrt{2\tau })}{2}}}>0,$ as time dummy $\tau >0.$ It follows that the initial condition $f\left(y\right)>0⟶y=x-z\sqrt{2\tau }>0⟶{\displaystyle \frac{x}{\sqrt{2\tau }}}>z,$ whence the heat kernel is

$$\begin{array}{cc}\hfill & u(x,\tau )={\displaystyle \frac{1}{\sqrt{2\pi }}}{\int }_{-\infty }^{{\displaystyle \frac{x}{\sqrt{2\tau }}}}\left[e^{{\displaystyle \frac{(k+1)(x-z\sqrt{2\tau })}{2}}}-e^{{\displaystyle \frac{(k-1)(x-z\sqrt{2\tau })}{2}}}\right]e^{-{\displaystyle \frac{z^2}{2}}}dz=\hfill \\ \hfill & ={\displaystyle \frac{1}{\sqrt{2\pi }}}{\int }_{-\infty }^{{\displaystyle \frac{x}{\sqrt{2\tau }}}}{e}^{{\displaystyle \frac{(k+1)(x-z\sqrt{2\tau })}{2}}-{\displaystyle \frac{z^2}{2}}}dz-{\displaystyle \frac{1}{\sqrt{2\pi }}}{\int }_{-\infty }^{{\displaystyle \frac{x}{\sqrt{2\tau }}}}{e}^{{\displaystyle \frac{(k-1)(x-z\sqrt{2\tau })}{2}}-{\displaystyle \frac{z^2}{2}}}dz\hfill \end{array}$$

(44)

and exponent

$$\begin{array}{cc}\hfill & {\displaystyle \frac{(k\pm 1)(x-z\sqrt{2\tau })}{2}}-{\displaystyle \frac{z^2}{2}}=-{\displaystyle \frac{1}{2}}\left[z^2+z(k\pm 1)\sqrt{2\tau }\right]+{\displaystyle \frac{(k\pm 1)x}{2}}=\hfill \\ \hfill & =-{\displaystyle \frac{1}{2}}\left\{{\left[z+{\displaystyle \frac{(k\pm 1)}{2}}\sqrt{2\tau }\right]}^2-{\displaystyle \frac{2\tau {(k\pm 1)}^2}{4}}\right\}+{\displaystyle \frac{(k\pm 1)x}{2}}=\hfill \\ \hfill & =-{\displaystyle \frac{1}{2}}{\left[z+(k\pm 1)\sqrt{{\displaystyle \frac{\tau }{2}}}\right]}^2+{\displaystyle \frac{\tau {(k\pm 1)}^2}{4}}+{\displaystyle \frac{(k\pm 1)x}{2}}.\hfill \end{array}$$

(45)

The heat kernel can therefore be rewritten as follows:

$$\begin{array}{cc}\hfill & u(x,\tau )={\displaystyle \frac{1}{\sqrt{2\pi }}}{\int }_{-\infty }^{{\displaystyle \frac{x}{\sqrt{2\tau }}}}{e}^{-{\displaystyle \frac{1}{2}}{\left[z+(k+1)\sqrt{{\displaystyle \frac{\tau }{2}}}\right]}^2+{\displaystyle \frac{\tau {(k+1)}^2}{4}}+{\displaystyle \frac{(k+1)x}{2}}}dz-{\displaystyle \frac{1}{\sqrt{2\pi }}}{\int }_{-\infty }^{{\displaystyle \frac{x}{\sqrt{2\tau }}}}{e}^{-{\displaystyle \frac{1}{2}}{\left[z+(k-1)\sqrt{{\displaystyle \frac{\tau }{2}}}\right]}^2+{\displaystyle \frac{\tau {(k-1)}^2}{4}}+{\displaystyle \frac{(k-1)x}{2}}}dz=\hfill \\ \hfill & ={\displaystyle \frac{1}{\sqrt{2\pi }}}{e}^{{\displaystyle \frac{\tau {(k+1)}^2}{4}}+{\displaystyle \frac{(k+1)x}{2}}}{\int }_{-\infty }^{{\displaystyle \frac{x}{\sqrt{2\tau }}}}{e}^{-{\displaystyle \frac{1}{2}}{\left[z+(k+1)\sqrt{{\displaystyle \frac{\tau }{2}}}\right]}^2}dz-{\displaystyle \frac{1}{\sqrt{2\pi }}}{e}^{{\displaystyle \frac{\tau {(k-1)}^2}{4}}+{\displaystyle \frac{(k-1)x}{2}}}{\int }_{-\infty }^{{\displaystyle \frac{x}{\sqrt{2\tau }}}}{e}^{-{\displaystyle \frac{1}{2}}{\left[z+(k-1)\sqrt{{\displaystyle \frac{\tau }{2}}}\right]}^2}dz=\hfill \\ \hfill & ={\displaystyle \frac{1}{\sqrt{2\pi }}}{e}^{{\displaystyle \frac{\tau {(k+1)}^2}{4}}+{\displaystyle \frac{(k+1)x}{2}}}{\int }_{-\infty }^{{\displaystyle \frac{x}{\sqrt{2\tau }}}+(k+1)\sqrt{{\displaystyle \frac{\tau }{2}}}}{e}^{-{\displaystyle \frac{y^2}{2}}}dy-{\displaystyle \frac{1}{\sqrt{2\pi }}}{e}^{{\displaystyle \frac{\tau {(k-1)}^2}{4}}+{\displaystyle \frac{(k-1)x}{2}}}{\int }_{-\infty }^{{\displaystyle \frac{x}{\sqrt{2\tau }}}+(k-1)\sqrt{{\displaystyle \frac{\tau }{2}}}}{e}^{-{\displaystyle \frac{y^2}{2}}}dy=\hfill \\ \hfill & =e^{{\displaystyle \frac{\tau {(k+1)}^2}{4}}+{\displaystyle \frac{(k+1)x}{2}}}N\left(d_1\right)-e^{{\displaystyle \frac{\tau {(k-1)}^2}{4}}+{\displaystyle \frac{(k-1)x}{2}}}N\left(d_2\right),\hfill \end{array}$$

(46)

as $z={\displaystyle \frac{x}{\sqrt{2\tau }}}$ and $y=z+(k\pm 1)\sqrt{{\displaystyle \frac{\tau }{2}}}⟶{\displaystyle \frac{dy}{dz}}=1⟶dy=dz.$ Notice that, for $d_{1,2}={\displaystyle \frac{x}{\sqrt{2\tau }}}+(k\pm 1)\sqrt{{\displaystyle \frac{\tau }{2}}},$

$$N\left(d_{1,2}\right)={\displaystyle \frac{1}{\sqrt{2\pi }}}{\int }_{-\infty }^{d_{1,2}}{e}^{-{\displaystyle \frac{y^2}{2}}}dy$$

(47)

is the standard normal integral for log-normal probabilities; indeed, the equation of the standard normal distribution for random variable $x\in \mathbb{R},$ mean $\mu \in \mathbb{R}$, and standard deviation $\sigma \in {\mathbb{R}}_{+}$ is $f\left(x\right)={\displaystyle \frac{1}{\sigma \sqrt{2\pi }}}{e}^{-{\displaystyle \frac{{(x-\mu )}^2}{2{\sigma }^2}}}.$

12. Reverse Change of Variables

In this section, we perform a backwards substitution on function $v(x,\tau )=e^{\alpha x+\beta \tau }u(x,\tau )$ by means of function $u(x,\tau )=e^{{\displaystyle \frac{\tau {(k+1)}^2}{4}}+{\displaystyle \frac{(k+1)x}{2}}}N\left(d_1\right)-e^{{\displaystyle \frac{\tau {(k-1)}^2}{4}}+{\displaystyle \frac{(k-1)x}{2}}}N\left(d_2\right)$ and parameters $\alpha =-{\displaystyle \frac{(k-1)}{2}}$ and $\beta =-{\displaystyle \frac{{(k+1)}^2}{4}};$ we finally obtain the Black and Scholes PDE solution $V(S,t)=SN\left(d_1\right)-K{e}^{-r(T-t)}N\left(d_2\right)$ for dividend yield $\delta =0.$ Consider function $v(x,\tau )=e^{\alpha x+\beta \tau }u(x,\tau )$ such that parameters $\alpha =-{\displaystyle \frac{(k-1)}{2}}$ and $\beta =-{\displaystyle \frac{{(k+1)}^2}{4}}$:

$$\begin{array}{cc}\hfill & v(x,\tau )=e^{-{\displaystyle \frac{(k-1)x}{2}}-{\displaystyle \frac{{(k+1)}^2\tau }{4}}}\left[e^{{\displaystyle \frac{\tau {(k+1)}^2}{4}}+{\displaystyle \frac{(k+1)x}{2}}}N\left(d_1\right)-e^{{\displaystyle \frac{\tau {(k-1)}^2}{4}}+{\displaystyle \frac{(k-1)x}{2}}}N\left(d_2\right)\right]=\hfill \\ \hfill & =e^{x}N\left(d_1\right)-e^{-k\tau }N\left(d_2\right),\hfill \end{array}$$

(48)

as

$$\begin{array}{cc}\hfill & -{\displaystyle \frac{(k-1)x}{2}}+{\displaystyle \frac{(k+1)x}{2}}={\displaystyle \frac{2x}{2}}=x,\hfill \end{array}$$

(49)

$$\begin{array}{cc}\hfill & -{\displaystyle \frac{{(k+1)}^2\tau }{4}}+{\displaystyle \frac{\tau {(k+1)}^2}{4}}=0,\hfill \end{array}$$

(50)

$$\begin{array}{cc}\hfill & -{\displaystyle \frac{(k-1)x}{2}}+{\displaystyle \frac{(k-1)x}{2}}=0\mathrm{and}\hfill \end{array}$$

(51)

$$\begin{array}{cc}\hfill & -{\displaystyle \frac{{(k+1)}^2\tau }{4}}+{\displaystyle \frac{\tau {(k-1)}^2}{4}}=\hfill \\ \hfill & ={\displaystyle \frac{-(k^2+2k+1)\tau +(k^2-2k+1)\tau }{4}}={\displaystyle \frac{-4k\tau }{4}}=-k\tau .\hfill \end{array}$$

(52)

Accordingly, space dummy $x=ln\left({\displaystyle \frac{S}{K}}\right)$ and time dummy $\tau ={\displaystyle \frac{{\sigma }^2(T-t)}{2}}$:

$$\begin{array}{cc}\hfill & v(x,\tau )=e^{ln\left({\displaystyle \frac{S}{K}}\right)}N\left(d_1\right)-e^{-k\left[{\displaystyle \frac{{\sigma }^2(T-t)}{2}}\right]}N\left(d_2\right)=\hfill \\ \hfill & ={\displaystyle \frac{S}{K}}N\left(d_1\right)-e^{-{\displaystyle \frac{k{\sigma }^2(T-t)}{2}}}N\left(d_2\right)\hfill \end{array}$$

(53)

and

$$\begin{array}{cc}\hfill & d_{1,2}={\displaystyle \frac{x}{\sqrt{2\tau }}}+(k\pm 1)\sqrt{{\displaystyle \frac{\tau }{2}}}={\displaystyle \frac{ln\left(\frac{S}{K}\right)}{\sqrt{2\left[\frac{{\sigma }^2(T-t)}{2}\right]}}}+(k\pm 1)\sqrt{{\displaystyle \frac{\left[\frac{{\sigma }^2(T-t)}{2}\right]}{2}}}=\hfill \\ \hfill & ={\displaystyle \frac{ln\left(\frac{S}{K}\right)}{\sigma \sqrt{T-t}}}+(k\pm 1)\sigma \sqrt{{\displaystyle \frac{T-t}{4}}}={\displaystyle \frac{ln\left(\frac{S}{K}\right)}{\sigma \sqrt{T-t}}}+(k\pm 1){\displaystyle \frac{\sigma }{2}}\sqrt{T-t}=\hfill \\ \hfill & ={\displaystyle \frac{ln\left(\frac{S}{K}\right)+(k\pm 1)\frac{{\sigma }^2}{2}(T-t)}{\sigma \sqrt{T-t}}}={\displaystyle \frac{ln\left(\frac{S}{K}\right)+\left(\frac{{\sigma }^{2}k}{2}\pm \frac{{\sigma }^2}{2}\right)(T-t)}{\sigma \sqrt{T-t}}}=\hfill \\ \hfill & ={\displaystyle \frac{ln\left(\frac{S}{K}\right)+\left(r\pm \frac{{\sigma }^2}{2}\right)(T-t)}{\sigma \sqrt{T-t}}},\hfill \end{array}$$

(54)

as $k={\displaystyle \frac{2r}{{\sigma }^2}}⟶r={\displaystyle \frac{{\sigma }^{2}k}{2}}.$ Thus,

$$\begin{array}{cc}\hfill & v(x,\tau )={\displaystyle \frac{S}{K}}N\left[{\displaystyle \frac{ln\left(\frac{S}{K}\right)+\left(r+\frac{{\sigma }^2}{2}\right)(T-t)}{\sigma \sqrt{T-t}}}\right]-e^{-r(T-t)}N\left[{\displaystyle \frac{ln\left(\frac{S}{K}\right)+\left(r-\frac{{\sigma }^2}{2}\right)(T-t)}{\sigma \sqrt{T-t}}}\right]⟶\hfill \\ \hfill & V(S,t)=Kv(x,\tau )=\hfill \\ \hfill & =K\left\{{\displaystyle \frac{S}{K}}N\left[{\displaystyle \frac{ln\left(\frac{S}{K}\right)+\left(r+\frac{{\sigma }^2}{2}\right)(T-t)}{\sigma \sqrt{T-t}}}\right]-e^{-r(T-t)}N\left[{\displaystyle \frac{ln\left(\frac{S}{K}\right)+\left(r-\frac{{\sigma }^2}{2}\right)(T-t)}{\sigma \sqrt{T-t}}}\right]\right\}=\hfill \\ \hfill & =SN\left[{\displaystyle \frac{ln\left(\frac{S}{K}\right)+\left(r+\frac{{\sigma }^2}{2}\right)(T-t)}{\sigma \sqrt{T-t}}}\right]-K{e}^{-r(T-t)}N\left[{\displaystyle \frac{ln\left(\frac{S}{K}\right)+\left(r-\frac{{\sigma }^2}{2}\right)(T-t)}{\sigma \sqrt{T-t}}}\right]=\hfill \\ \hfill & =SN\left(d_1\right)-K{e}^{-r(T-t)}N\left(d_2\right),\hfill \end{array}$$

(55)

which for $\delta \ne 0$ becomes $V(S,t)=SN\left(d_1\right)-K{e}^{-(r-\delta )(T-t)}N\left(d_2\right).$

13. Black and Scholes PDE Solution

In this section, we present the solution to the Black and Scholes PDE for call options and put options and discuss the core logic of its applications. Finally, the solution to the Black and Scholes PDE is twofold, one being the call option solution (buy) and the other being the put option solution (sell):

$$\begin{array}{cc}\hfill & V(S,t)\equiv C(S,t)=SN\left(d_1\right)-K{e}^{-(r-\delta )(T-t)}N\left(d_2\right);\hfill \end{array}$$

(56)

$$\begin{array}{cc}\hfill & P(S,t)=K{e}^{-(r-\delta )(T-t)}N(-d_2)-SN(-d_1).\hfill \end{array}$$

(57)

The Black and Scholes PDE solution formulates the fairest price V for the share of a capital stock or security S according to the European system whereby the option is exercised at the end of the time interval $T-t.$ Let us analyze the call option solution $V(S,t).$

Option price V is always paid by the buyer to the seller at the beginning of the time interval $T-t$ such that (1) if $S\ge K+V$ at the end of the time interval $T-t$ then the buyer enjoys a non-loss and (2) if $S\le K+V$ at the end of the time interval $T-t$, then the buyer incurs a non-profit.

Specifically, while the price of the security $S,$ its volatility $\sigma ,$ the risk-free interest rate r, and the dividend yield $\delta$ may be exogenous, the buyer and the seller agree on the strike price K and on the time interval $T-t.$ However, the log-normal probabilities $N\left(d_{1,2}\right)$, whereby the price of the security S is to change at the end of the time interval $T-t$ and thereby determine the non-profit or non-loss of both parties, are proportional to the volatility $\sigma$ of the security, which is exogenous.

For example, suppose the share price were $S=250$ USD, its strike price were $K=300$ USD, and the time interval were $T-t=100$ days. For given volatility $\sigma ,$ risk-free interest rate r, and dividend yield $\delta ,$ further suppose the Black and Scholes PDE solution returned option price $V=5$ USD per share and also suppose the buyer wished to consider buying 100 shares.

The buyer would thus pay the seller 5 USD $\times 100$ shares $=500$ USD upfront and at the end of the time interval $T-t=100$ days the buyer could exercise the option of buying the stock of 100 shares at the strike price of $K=300$ USD per share.

If the stock price $S\le 300+5=305$ USD then the buyer would make a non-profit, as the market value of the stock would not exceed the sum of the strike price and the option price, having the option of not proceeding with the purchase. If the stock price $S\ge 300+5=305$ USD then the buyer would make a non-loss, as the market value of the stock price would not be exceeded by the sum of the strike price and the option price, having the option of proceeding with the purchase.

14. Conclusions

By having replaced the standard Wiener process with a smoothed Wiener process, we have derived the Black and Scholes PDE without relying on Itô’s Lemma. Such an approach has preserved the essential properties of stochasticity and volatility while having introduced differentiability, having made the derivation more intuitive and financially grounded.

We have additionally provided a self-contained solution to the Black and Scholes PDE with a stress on pedagogical clarity and uncompromising mathematical rigor, simultaneously reaching out to a broader audience and highlighting the contiguity between financial mathematics and classical physics.

In summary, our notional contributions have been (1) a differentiable stochastic process which replaces the standard Wiener process while preserving stochasticity and volatility and (2) a pedagogical alternative to Itô-based derivations, emphasizing economic intuition; accordingly, our methodological contributions have been (1) a complete derivation of the Black and Scholes PDE using classical calculus and PDE techniques and (2) a transparent solution method via Fourier transforms and the heat equation.

Our study is finally endowed with flexibility for extensions, as our framework naturally accommodates such generalizations as stochastic volatility or jump processes, which are to be achieved by modifying the smoothing kernel while maintaining differentiability.