Stochastic Calculus

Abstract

Would be including the notes and topics from Shreve’s and Grezelak’s book as well topics that would fall into this category.Including code implementation in python based of the book.

DISCLAIMER THERE IS AT TIMES LOT OF HAND WAVING ,THIS IS NOT AS MATHEMATICALLY RIGOROUS AS YOU WOULD EXPECT LIKE IN A PURE MATH TEXTBOOK.

Resources

These are list of books , videos and courses that I am currently referencing for content on this page , for more detailed list please check out the resource section above.

• Mathematical modelling and computation in finance by Cornerlis Ooseterlee and Lech Grzelak (Main Text)

• Stochastic CalcuIus for Finance II Continuous-Time Models by Steven Shreve (Main Reference text)

• Stochastic Differential Equations By Bernt Oksendal

Preliminaries

Definition 1 (\sigma algebra) : Let \Omega be a nonempty set , and let \mathcal{F} be a set of collections of subsets of \Omega. We say that \mathcal{F} is a \sigma algebra provided that

1. The empty set \emptyset belongs to \mathcal{F}
2. A and A^{c} belong to \mathcal{F} whenever A \in \mathcal{F}
3. whenever A_1 ,A_2 ,A_3 ..... \in \mathcal{F} then \bigcup_{i=1}^\infty A_{i} \in \mathcal{F}

Definition 2 (Probability measure) : Let \Omega be a nonempty set , and let \mathbb{F} be \sigma algebra of subsets of \Omega. A probability measure \mathbb{P} is a function that, to every set A \in \mathcal{F} , assigns a number in [0,1] called the probability of A written as \mathbb{P}(A) .We require

1. \mathbb{P}(\Omega) = 1 and

2. (Countable Additivity) Whenever A_1,A_2,A_3.... is a sequence of disjoint sets in \mathcal{F} , then \mathbb{P}(\bigcup_{n=1}^\infty A_n) = \sum_{n=1}^{\infty} \mathbb{P}(A_n)

The Triple (\Omega, \mathcal{F},\mathbb{}P) is called probability measure

From the above it follows that for finitely many disjoint sets A_1, A_2,A_3....A_n \mathbb{P}(\bigcup_{n=1}^N) = \sum_{n=1}^N \mathbb{P}(A_n)

and \mathbb{P}(A^c) = 1 - \mathbb{P}(A)

Example 1 (Uniform(Lebesgue) measure on [0,1])
\Omega = [0,1]
\mathbb{P}[a,b] = b-a ,0 \leq a \leq b \leq1
single points have zero probability and
(a,b) can be written as \bigcup_{n=1}^\infty[a+\frac{1}{n} , b -\frac{1}{n}]

The \sigma-algebra beginning with closed intervals and adding everything else necessary in order to have a \sigma- algebra is called a Borel \sigma-algebra and denoted by \mathcal{B}[0,1]. Borel \sigma-algebras are \sigma-algebra generated by open sets and equivalently closed sets over any topological space.

Definition 3 let (\Omega,\mathcal{F},\mathbb{P}) be a probability space. If a set A \in \mathcal{F} satisfies \mathbb{P}(A) = 1, we say the event A occurs almost surely

Example 2 Let \mathbb{P} be the uniform measure as defined in example 1. Define X(\omega) = \omega and Y(\omega) = 1 - \omega for all \omega \in [0,1] , then the distribution measure of X is uniform \mu_{X}[a,b] = \mathbb{P}\{\omega;a\leq X(w)\leq b\} = \mathbb{P}[a,b] = b-a, 0\leq a \leq b \leq 1 by definition of \mathbb{P}.
Its easy to see to see that X and Y have the same distribution. But under the probability measure\mathbb{\widetilde{P}} on [0,1] defined by \mathbb{\widetilde{P}[a,b]} = \int_a^b 2\omega \, d\omega = b^2-a^2 , 0 \leq a\leq b \leq1 X and Y have different distributions.

Definition 4 (cumulative distribution function and density function) F(x) = \mathbb{P}\{X \leq x\} , x \in \mathbb{R} \mu_X[a,b] = \mathbb{P}\{a \leq X \leq b \} = \int_a^b f(x) \, dx , -\infty < a \leq b < \infty f(x) is called the density funtion and F(x) is called the cummulative distribution function.

Example 3 (Standard normal random variable) Let \phi(x) = \frac {a}{\sqrt{2\pi}}e^{\frac{-x^2}{2}} be the standard normal density and definiing the cummulative normal distribution function as N(x) = \int_{-\infty}^x \phi(\xi)\,d\xi The function N(x) is strictly increasing function which is surjective onto (0,1) from \mathbb{R}, so it has a strictly increasing inverse function N^{-1}(y),y \in (0,1). Let Y be a uniformly distributed random variable defined on some probability space (\Omega,\mathcal{F},\mathbb{P}) and set X = N^{-1}(Y) then

\begin{align*} \mu_X[a,b] &= \mathbb{P}\{\omega \in \Omega ; a \leq X(\omega) \leq b\} \\ &= \mathbb{P}\{\omega \in \Omega ; a \leq N^{-1}(Y(\omega)) \leq b\}\\ &= \mathbb{P}\{\omega \in \Omega ; N(a) \leq Y(\omega) \leq N(b) \} \\ &= N(b) - N(a) \\ &= \int_a^b \phi(x) \, dx \end{align*}

Any random variable that has this distribution regardless of the probability space is called standard normal distribution.
Thing to note the use of uniformly distributed random variable for generating a standard random variable is called probability integral transform and this is commonly used in Monte Carlo simulation

Stochastic Processes

Weiner process

A fundamental stochastic process, which is also commonly used in the construction of stochastic differential equations (SDEs) to describe asset price movements, is the Wiener process, also called Brownian motion. Mathematically, a Wiener process, W(t), is characterized by the following properties:

a)\quad W(t\_{0}) = 0 (technically:\mathbb{P}[W(t_0)=0] =1)

b)\quad W(t) is\quad almost \quad surely \quad continous

c)\quad W(t) \quad has \quad independent \quad incerements \quad with \quad distribution\quad W(t)- W(s) \sim \mathcal{N}(0,t-s)

Black-Scholes model

Part I: Mathematical Foundations and Definitions

This section establishes all the mathematical machinery needed for the Black-Scholes proofs. We assume knowledge of measure theory and basic stochastic calculus but will define all financial and probabilistic concepts explicitly.

Financial Market Framework

Definition 1.1 (Financial Market)

A financial market consists of:

1. A probability space (\Omega, \mathcal{F}, \mathbb{P}) where \Omega represents all possible market scenarios
2. A filtration \{\mathcal{F}_t\}_{t \geq 0} representing information available at time t
3. A finite collection of traded assets with price processes \{S^i_t\}_{i=0,1,\ldots,n}

Definition 1.2 (Brownian Motion)

A stochastic process \{W_t\}_{t \geq 0} on (\Omega, \mathcal{F}, \mathbb{P}) is a standard Brownian motion if:

1. W_0 = 0 almost surely
2. W has independent increments: for 0 \leq s < t, W_t - W_s is independent of \mathcal{F}_s
3. W_t - W_s \sim N(0, t-s) for all 0 \leq s < t
4. W has continuous sample paths almost surely

Definition 1.3 (Natural Filtration of Brownian Motion)

The natural filtration of Brownian motion is \mathcal{F}_t^W = \sigma(W_s : 0 \leq s \leq t) the \sigma-algebra generated by the Brownian motion up to time t.

Definition 1.4 (Adapted Process)

A stochastic process \{X_t\}_{t \geq 0} is adapted to the filtration \{\mathcal{F}_t\} if X_t is \mathcal{F}_t-measurable for all t \geq 0.

Definition 1.5 (Geometric Brownian Motion)

A process \{S_t\}_{t \geq 0} follows geometric Brownian motion with parameters \mu \in \mathbb{R} and \sigma > 0 if it satisfies the stochastic differential equation: dS_t = \mu S_t \, dt + \sigma S_t \, dW_t with initial condition S_0 > 0.

Remark 1.1

The explicit solution to geometric Brownian motion is: S_t = S_0 \exp\left(\left(\mu - \frac{\sigma^2}{2}\right)t + \sigma W_t\right) This can be verified using Itô’s lemma on f(t,x) = \ln x.

Portfolio and Trading Strategy Concepts

Definition 1.6 (Trading Strategy)

A trading strategy is a pair of adapted processes (\phi^0_t, \phi^1_t) where:

• \phi^0_t represents the number of units of the bond held at time t
• \phi^1_t represents the number of shares of stock held at time t

Both processes must be adapted to the filtration \{\mathcal{F}_t\}.

Definition 1.7 (Portfolio Value)

The value of a portfolio with trading strategy (\phi^0_t, \phi^1_t) at time t is: V_t = \phi^0_t B_t + \phi^1_t S_t where B_t is the bond price and S_t is the stock price.

Definition 1.8 (Self-Financing Strategy)

A trading strategy (\phi^0_t, \phi^1_t) is self-financing if: dV_t = \phi^0_t dB_t + \phi^1_t dS_t This means no money is added or withdrawn from the portfolio; changes in value come only from price movements of held assets.

Definition 1.9 (Arbitrage Opportunity)

An arbitrage opportunity is a self-financing trading strategy with:

1. Initial value V_0 = 0
2. \mathbb{P}(V_T \geq 0) = 1 for some time T > 0
3. \mathbb{P}(V_T > 0) > 0

Definition 1.10 (Arbitrage-Free Market)

A market is arbitrage-free if no arbitrage opportunities exist.

Martingale Theory for Finance

Definition 1.11 (Martingale)

An adapted process \{M_t\}_{t \geq 0} is a martingale with respect to filtration \{\mathcal{F}_t\} and probability measure \mathbb{P} if:

1. \mathbb{E}[|M_t|] < \infty for all t \geq 0
2. \mathbb{E}[M_t | \mathcal{F}_s] = M_s for all 0 \leq s \leq t

Definition 1.12 (Equivalent Probability Measures)

Two probability measures \mathbb{P} and \mathbb{Q} on (\Omega, \mathcal{F}) are equivalent (written \mathbb{P} \sim \mathbb{Q}) if they have the same null sets: \mathbb{P}(A) = 0 \iff \mathbb{Q}(A) = 0 \text{ for all } A \in \mathcal{F}

Definition 1.13 (Radon-Nikodym Derivative)

If \mathbb{Q} \ll \mathbb{P} (Q is absolutely continuous with respect to P), then there exists a non-negative \mathcal{F}-measurable random variable Z such that: \mathbb{Q}(A) = \int_A Z \, d\mathbb{P} \text{ for all } A \in \mathcal{F} We write Z = \frac{d\mathbb{Q}}{d\mathbb{P}} and call Z the Radon-Nikodym derivative.

Theorem 1.1 (Girsanov’s Theorem - Statement)

Let \theta be an adapted process with \int_0^T \theta_s^2 \, ds < \infty almost surely. Define: Z_t = \exp\left(-\int_0^t \theta_s \, dW_s - \frac{1}{2}\int_0^t \theta_s^2 \, ds\right)

If \mathbb{E}[Z_T] = 1, then:

1. The process Z_t is a martingale
2. The measure \mathbb{Q} defined by \frac{d\mathbb{Q}}{d\mathbb{P}} = Z_T is a probability measure equivalent to \mathbb{P}
3. Under \mathbb{Q}, the process \tilde{W}_t = W_t + \int_0^t \theta_s \, ds is a Brownian motion

Definition 1.14 (Risk-Neutral Measure)

In a financial market with bond B_t = e^{rt} and stock following dS_t = \mu S_t \, dt + \sigma S_t \, dW_t a probability measure \mathbb{Q} equivalent to \mathbb{P} is called risk-neutral if the discounted stock price e^{-rt}S_t is a \mathbb{Q}-martingale.

Options and Derivatives

Definition 1.15 (European Option)

A European option is a financial contract that gives the holder the right (but not obligation) to:

• Call option: Buy an asset at strike price K at maturity time T
• Put option: Sell an asset at strike price K at maturity time T

Definition 1.16 (Option Payoff)

The payoff of a European option at maturity T is:

• Call: h(S_T) = (S_T - K)^+ = \max(S_T - K, 0)
• Put: h(S_T) = (K - S_T)^+ = \max(K - S_T, 0)

where S_T is the stock price at maturity.

Definition 1.17 (Option Price)

The price of an option at time t is denoted V(t, S_t) and depends on the current time t and current stock price S_t.

Definition 1.18 (Replicating Portfolio)

A replicating portfolio for an option is a self-financing trading strategy (\phi^0_t, \phi^1_t) such that the portfolio value at maturity equals the option payoff: V_T = \phi^0_T B_T + \phi^1_T S_T = h(S_T)

Fundamental Theorems of Asset Pricing

Theorem 1.2 (First Fundamental Theorem of Asset Pricing)

A market is arbitrage-free if and only if there exists a probability measure \mathbb{Q} equivalent to \mathbb{P} such that all discounted asset prices are \mathbb{Q}-martingales.

Theorem 1.3 (Second Fundamental Theorem of Asset Pricing)

An arbitrage-free market is complete if and only if the risk-neutral measure is unique.

Definition 1.19 (Complete Market)

A market is complete if every contingent claim (option payoff) can be replicated by a self-financing trading strategy.

Theorem 1.4 (Risk-Neutral Valuation Principle)

In an arbitrage-free complete market, the price of any attainable contingent claim h(S_T) at time t is: V(t, S_t) = e^{-r(T-t)} \mathbb{E}_\mathbb{Q}[h(S_T) | \mathcal{F}_t] where \mathbb{Q} is the unique risk-neutral measure.

Part II: The Black-Scholes Model Setup

Model Specification

We consider a financial market on a probability space (\Omega, \mathcal{F}, \mathbb{P}) with filtration \{\mathcal{F}_t\}_{t \geq 0} and two traded assets:

Definition 2.1 (Black-Scholes Market Model)

The Black-Scholes market consists of:

1. Risk-free bond: dB_t = rB_t \, dt with B_0 = 1, giving B_t = e^{rt}
2. Risky stock: dS_t = \mu S_t \, dt + \sigma S_t \, dW_t with S_0 > 0

where:

• r > 0 is the constant risk-free interest rate
• \mu \in \mathbb{R} is the stock’s expected return (drift)
• \sigma > 0 is the stock’s volatility
• \{W_t\}_{t \geq 0} is a standard Brownian motion adapted to \{\mathcal{F}_t\}

Definition 2.2 (European Call Option in Black-Scholes Model)

We consider a European call option with:

• Strike price K > 0
• Maturity time T > 0
  
• Payoff at maturity: h(S_T) = (S_T - K)^+

Our goal is to find the option price V(t, S_t) for 0 \leq t \leq T.

Key Properties of the Model

Lemma 2.1 (Stock Price Solution)

The solution to the stock price SDE dS_t = \mu S_t \, dt + \sigma S_t \, dW_t is: S_t = S_0 \exp\left(\left(\mu - \frac{\sigma^2}{2}\right)t + \sigma W_t\right)

Proof of Lemma 2.1

Let Y_t = \ln S_t. By Itô’s lemma with f(x) = \ln x:

\begin{align} dY_t &= f'(S_t) dS_t + \frac{1}{2}f''(S_t)(dS_t)^2 \\ &= \frac{1}{S_t} dS_t + \frac{1}{2}\left(-\frac{1}{S_t^2}\right)(dS_t)^2 \\ &= \frac{1}{S_t}(\mu S_t \, dt + \sigma S_t \, dW_t) - \frac{1}{2S_t^2}(\sigma S_t)^2 dt \\ &= \mu \, dt + \sigma \, dW_t - \frac{\sigma^2}{2} dt \\ &= \left(\mu - \frac{\sigma^2}{2}\right) dt + \sigma \, dW_t \end{align}

Integrating from 0 to t: Y_t = Y_0 + \left(\mu - \frac{\sigma^2}{2}\right)t + \sigma W_t

Since Y_t = \ln S_t and Y_0 = \ln S_0: \ln S_t = \ln S_0 + \left(\mu - \frac{\sigma^2}{2}\right)t + \sigma W_t

Therefore: S_t = S_0 \exp\left(\left(\mu - \frac{\sigma^2}{2}\right)t + \sigma W_t\right)

Corollary 2.1 (Log-Normal Distribution of Stock Price)

Under \mathbb{P}, we have: \ln(S_t/S_0) \sim N\left(\left(\mu - \frac{\sigma^2}{2}\right)t, \sigma^2 t\right)

Part III: Proof via Martingale Approach

Step 1: Construction of Risk-Neutral Measure

Theorem 3.1 (Existence of Risk-Neutral Measure in Black-Scholes)

There exists a unique probability measure \mathbb{Q} equivalent to \mathbb{P} such that the discounted stock price e^{-rt}S_t is a \mathbb{Q}-martingale.

Proof of Theorem 3.1

Step 1: Define the market price of risk: \theta = \frac{\mu - r}{\sigma}

This is the constant that will appear in Girsanov’s theorem.

Step 2: Define the Radon-Nikodym density process: \begin{align} Z_t &= \exp\left(-\theta W_t - \frac{1}{2}\theta^2 t\right) \\ &= \exp\left(-\frac{\mu - r}{\sigma} W_t - \frac{1}{2}\left(\frac{\mu - r}{\sigma}\right)^2 t\right) \end{align}

Step 3: Verify that Z_t is a martingale and \mathbb{E}[Z_T] = 1.

Since \theta is constant, we can compute: \mathbb{E}[Z_t] = \mathbb{E}\left[\exp\left(-\theta W_t - \frac{1}{2}\theta^2 t\right)\right]

Since W_t \sim N(0,t) under \mathbb{P}: \begin{align} \mathbb{E}[Z_t] &= \int_{-\infty}^{\infty} \exp\left(-\theta w - \frac{1}{2}\theta^2 t\right) \frac{1}{\sqrt{2\pi t}} \exp\left(-\frac{w^2}{2t}\right) dw \\ &= \exp\left(-\frac{1}{2}\theta^2 t\right) \int_{-\infty}^{\infty} \frac{1}{\sqrt{2\pi t}} \exp\left(-\frac{w^2 + 2\theta tw}{2t}\right) dw \\ &= \exp\left(-\frac{1}{2}\theta^2 t\right) \int_{-\infty}^{\infty} \frac{1}{\sqrt{2\pi t}} \exp\left(-\frac{(w + \theta t)^2 - \theta^2 t^2}{2t}\right) dw \\ &= \exp\left(-\frac{1}{2}\theta^2 t\right) \exp\left(\frac{1}{2}\theta^2 t\right) \int_{-\infty}^{\infty} \frac{1}{\sqrt{2\pi t}} \exp\left(-\frac{(w + \theta t)^2}{2t}\right) dw \\ &= 1 \end{align}

The last integral equals 1 since it’s the integral of a normal density.

Step 4: Define \mathbb{Q} by \frac{d\mathbb{Q}}{d\mathbb{P}} = Z_T.

Step 5: Apply Girsanov’s theorem. Under \mathbb{Q}, the process: \tilde{W}_t = W_t + \int_0^t \theta \, ds = W_t + \theta t is a \mathbb{Q}-Brownian motion.

Step 6: Transform the stock price dynamics. Under \mathbb{Q}: \begin{align} dS_t &= \mu S_t \, dt + \sigma S_t \, dW_t \\ &= \mu S_t \, dt + \sigma S_t (d\tilde{W}_t - \theta \, dt) \\ &= \mu S_t \, dt + \sigma S_t \, d\tilde{W}_t - \sigma S_t \theta \, dt \\ &= (\mu - \sigma\theta) S_t \, dt + \sigma S_t \, d\tilde{W}_t \\ &= \left(\mu - \sigma \cdot \frac{\mu - r}{\sigma}\right) S_t \, dt + \sigma S_t \, d\tilde{W}_t \\ &= r S_t \, dt + \sigma S_t \, d\tilde{W}_t \end{align}

Step 7: Verify the martingale property. The discounted stock price is: e^{-rt}S_t

Its differential is: \begin{align} d(e^{-rt}S_t) &= -re^{-rt}S_t \, dt + e^{-rt} dS_t \\ &= -re^{-rt}S_t \, dt + e^{-rt}(rS_t \, dt + \sigma S_t \, d\tilde{W}_t) \\ &= e^{-rt}\sigma S_t \, d\tilde{W}_t \end{align}

Since this has no dt term, e^{-rt}S_t is a \mathbb{Q}-martingale.

Step 2: Stock Price Under Risk-Neutral Measure

Lemma 3.1 (Stock Price Solution Under \mathbb{Q})

Under the risk-neutral measure \mathbb{Q}, the stock price has the explicit form: S_t = S_0 \exp\left(\left(r - \frac{\sigma^2}{2}\right)t + \sigma \tilde{W}_t\right) where \tilde{W}_t is a \mathbb{Q}-Brownian motion.

Proof of Lemma 3.1

Under \mathbb{Q}, the stock follows: dS_t = rS_t \, dt + \sigma S_t \, d\tilde{W}_t

Using the same technique as in the original measure, let Y_t = \ln S_t: \begin{align} dY_t &= \frac{1}{S_t} dS_t - \frac{1}{2S_t^2}(dS_t)^2 \\ &= \frac{1}{S_t}(rS_t \, dt + \sigma S_t \, d\tilde{W}_t) - \frac{1}{2S_t^2}(\sigma S_t)^2 dt \\ &= r \, dt + \sigma \, d\tilde{W}_t - \frac{\sigma^2}{2} dt \\ &= \left(r - \frac{\sigma^2}{2}\right) dt + \sigma \, d\tilde{W}_t \end{align}

Integrating: Y_t = Y_0 + \left(r - \frac{\sigma^2}{2}\right)t + \sigma \tilde{W}_t

Therefore: S_t = S_0 \exp\left(\left(r - \frac{\sigma^2}{2}\right)t + \sigma \tilde{W}_t\right)

Step 3: Risk-Neutral Valuation

Theorem 3.2 (Risk-Neutral Option Pricing)

The price of the European call option at time t is: V(t, S_t) = e^{-r(T-t)} \mathbb{E}_\mathbb{Q}[(S_T - K)^+ | \mathcal{F}_t]

Proof of Theorem 3.2

This follows directly from the risk-neutral valuation principle. In an arbitrage-free complete market, the option price equals the discounted expected payoff under the risk-neutral measure.

Step 4: Computing the Expectation

Theorem 3.3 (Black-Scholes Formula via Martingale Method)

The European call option price is: V(t, S_t) = S_t \Phi(d_1) - K e^{-r(T-t)} \Phi(d_2) where: \begin{align} d_1 &= \frac{\ln(S_t/K) + (r + \sigma^2/2)(T-t)}{\sigma\sqrt{T-t}} \\ d_2 &= d_1 - \sigma\sqrt{T-t} = \frac{\ln(S_t/K) + (r - \sigma^2/2)(T-t)}{\sigma\sqrt{T-t}} \end{align} and \Phi is the cumulative distribution function of the standard normal distribution.

Proof of Theorem 3.3

Step 1: Express the stock price at maturity. Under \mathbb{Q}, using the strong Markov property: S_T = S_t \exp\left(\left(r - \frac{\sigma^2}{2}\right)(T-t) + \sigma (\tilde{W}_T - \tilde{W}_t)\right)

Since \tilde{W}_T - \tilde{W}_t \sim N(0, T-t) under \mathbb{Q}, let Z \sim N(0,1). Then: S_T = S_t \exp\left(\left(r - \frac{\sigma^2}{2}\right)(T-t) + \sigma\sqrt{T-t} \cdot Z\right)

Step 2: Set up the expectation. \begin{align} V(t, S_t) &= e^{-r(T-t)} \mathbb{E}_\mathbb{Q}[(S_T - K)^+ | \mathcal{F}_t] \\ &= e^{-r(T-t)} \mathbb{E}_\mathbb{Q}\left[\left(S_t e^{(r-\sigma^2/2)(T-t) + \sigma\sqrt{T-t} Z} - K\right)^+\right] \end{align}

Step 3: Find the exercise region. The option is exercised when S_T > K, i.e., when: S_t e^{(r-\sigma^2/2)(T-t) + \sigma\sqrt{T-t} Z} > K

Taking logarithms: \begin{align} &(r-\sigma^2/2)(T-t) + \sigma\sqrt{T-t} Z > \ln(K/S_t) \\ &Z > \frac{\ln(K/S_t) - (r-\sigma^2/2)(T-t)}{\sigma\sqrt{T-t}} = -d_2 \end{align}

Step 4: Evaluate the integral. \begin{align} V(t, S_t) &= e^{-r(T-t)} \int_{-d_2}^{\infty} \left(S_t e^{(r-\sigma^2/2)(T-t) + \sigma\sqrt{T-t} z} - K\right) \\ &\quad \times \frac{1}{\sqrt{2\pi}} e^{-z^2/2} dz \\ &= e^{-r(T-t)} S_t e^{(r-\sigma^2/2)(T-t)} \int_{-d_2}^{\infty} e^{\sigma\sqrt{T-t} z} \frac{1}{\sqrt{2\pi}} e^{-z^2/2} dz \\ &\quad - e^{-r(T-t)} K \int_{-d_2}^{\infty} \frac{1}{\sqrt{2\pi}} e^{-z^2/2} dz \end{align}

Step 5: Evaluate the first integral. For the first integral, complete the square in the exponent: \sigma\sqrt{T-t} z - \frac{z^2}{2} = -\frac{1}{2}(z - \sigma\sqrt{T-t})^2 + \frac{\sigma^2(T-t)}{2}

Therefore: \begin{align} &\int_{-d_2}^{\infty} e^{\sigma\sqrt{T-t} z} \frac{1}{\sqrt{2\pi}} e^{-z^2/2} dz \\ &= e^{\sigma^2(T-t)/2} \int_{-d_2}^{\infty} \frac{1}{\sqrt{2\pi}} e^{-(z-\sigma\sqrt{T-t})^2/2} dz \\ &= e^{\sigma^2(T-t)/2} \int_{-d_2-\sigma\sqrt{T-t}}^{\infty} \frac{1}{\sqrt{2\pi}} e^{-u^2/2} du \\ &= e^{\sigma^2(T-t)/2} \Phi(d_2 + \sigma\sqrt{T-t}) \\ &= e^{\sigma^2(T-t)/2} \Phi(d_1) \end{align}

where we used the substitution u = z - \sigma\sqrt{T-t} and the fact that: d_2 + \sigma\sqrt{T-t} = d_1

Step 6: Evaluate the second integral. \int_{-d_2}^{\infty} \frac{1}{\sqrt{2\pi}} e^{-z^2/2} dz = \Phi(d_2)

Step 7: Combine the results. \begin{align} V(t, S_t) &= e^{-r(T-t)} S_t e^{(r-\sigma^2/2)(T-t)} e^{\sigma^2(T-t)/2} \Phi(d_1) - e^{-r(T-t)} K \Phi(d_2) \\ &= S_t e^{-r(T-t)} e^{(r-\sigma^2/2)(T-t)} e^{\sigma^2(T-t)/2} \Phi(d_1) - K e^{-r(T-t)} \Phi(d_2) \\ &= S_t e^{-r(T-t) + r(T-t) - \sigma^2(T-t)/2 + \sigma^2(T-t)/2} \Phi(d_1) - K e^{-r(T-t)} \Phi(d_2) \\ &= S_t \Phi(d_1) - K e^{-r(T-t)} \Phi(d_2) \end{align}

Part IV: Proof via PDE Approach

Step 1: Delta Hedging Strategy

Definition 4.1 (Delta Hedging Portfolio)

Consider a portfolio \Pi_t consisting of:

• Long position: 1 unit of the option with value V(t, S_t)
• Short position: \Delta_t shares of the stock
• Cash position: The remaining value invested in bonds

The portfolio value is: \Pi_t = V(t, S_t) - \Delta_t S_t

Definition 4.2 (Delta)

The delta \Delta_t represents the hedge ratio - the number of shares of stock to short for each option held. We will determine \Delta_t to make the portfolio risk-free.

Step 2: Application of Itô’s Lemma to Option Value

Lemma 4.1 (Stochastic Differential of Option Value)

Assume the option price has the form V(t, S_t) where V \in C^{1,2}([0,T] \times (0,\infty)). Then by Itô’s lemma: \begin{align} dV &= \frac{\partial V}{\partial t} dt + \frac{\partial V}{\partial S} dS_t + \frac{1}{2}\frac{\partial^2 V}{\partial S^2} (dS_t)^2 \\ &= \left[\frac{\partial V}{\partial t} + \mu S_t \frac{\partial V}{\partial S} + \frac{1}{2}\sigma^2 S_t^2 \frac{\partial^2 V}{\partial S^2}\right] dt + \sigma S_t \frac{\partial V}{\partial S} dW_t \end{align}

Proof of Lemma 4.1

Since dS_t = \mu S_t \, dt + \sigma S_t \, dW_t, we have: (dS_t)^2 = (\mu S_t \, dt + \sigma S_t \, dW_t)^2 = \sigma^2 S_t^2 (dW_t)^2 = \sigma^2 S_t^2 dt

where we used the fact that (dW_t)^2 = dt, dt \cdot dW_t = 0, and (dt)^2 = 0.

Applying Itô’s lemma: \begin{align} dV &= \frac{\partial V}{\partial t} dt + \frac{\partial V}{\partial S} dS_t + \frac{1}{2}\frac{\partial^2 V}{\partial S^2} (dS_t)^2 \\ &= \frac{\partial V}{\partial t} dt + \frac{\partial V}{\partial S} (\mu S_t \, dt + \sigma S_t \, dW_t) + \frac{1}{2}\frac{\partial^2 V}{\partial S^2} \sigma^2 S_t^2 dt \\ &= \left[\frac{\partial V}{\partial t} + \mu S_t \frac{\partial V}{\partial S} + \frac{1}{2}\sigma^2 S_t^2 \frac{\partial^2 V}{\partial S^2}\right] dt + \sigma S_t \frac{\partial V}{\partial S} dW_t \end{align}

Step 3: Portfolio Dynamics

Lemma 4.2 (Portfolio Value Dynamics)

The change in portfolio value is: \begin{align} d\Pi_t &= dV - \Delta_t dS_t \\ &= \left[\frac{\partial V}{\partial t} + \mu S_t \left(\frac{\partial V}{\partial S} - \Delta_t\right) + \frac{1}{2}\sigma^2 S_t^2 \frac{\partial^2 V}{\partial S^2}\right] dt \\ &\quad + \sigma S_t \left(\frac{\partial V}{\partial S} - \Delta_t\right) dW_t \end{align}

Proof of Lemma 4.2

We have: \begin{align} d\Pi_t &= dV - \Delta_t dS_t \\ &= \left[\frac{\partial V}{\partial t} + \mu S_t \frac{\partial V}{\partial S} + \frac{1}{2}\sigma^2 S_t^2 \frac{\partial^2 V}{\partial S^2}\right] dt + \sigma S_t \frac{\partial V}{\partial S} dW_t \\ &\quad - \Delta_t (\mu S_t dt + \sigma S_t dW_t) \\ &= \left[\frac{\partial V}{\partial t} + \mu S_t \frac{\partial V}{\partial S} + \frac{1}{2}\sigma^2 S_t^2 \frac{\partial^2 V}{\partial S^2} - \Delta_t \mu S_t\right] dt \\ &\quad + \sigma S_t \left(\frac{\partial V}{\partial S} - \Delta_t\right) dW_t \\ &= \left[\frac{\partial V}{\partial t} + \mu S_t \left(\frac{\partial V}{\partial S} - \Delta_t\right) + \frac{1}{2}\sigma^2 S_t^2 \frac{\partial^2 V}{\partial S^2}\right] dt \\ &\quad + \sigma S_t \left(\frac{\partial V}{\partial S} - \Delta_t\right) dW_t \end{align}

Step 4: Eliminating Risk

Definition 4.3 (Delta-Neutral Portfolio)

To eliminate the stochastic component, we choose: \Delta_t = \frac{\partial V}{\partial S}

This choice makes the portfolio instantaneously risk-free.

Lemma 4.3 (Risk-Free Portfolio Dynamics)

With \Delta_t = \frac{\partial V}{\partial S}, the portfolio dynamics become: d\Pi_t = \left[\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S_t^2 \frac{\partial^2 V}{\partial S^2}\right] dt

Step 5: No-Arbitrage Condition

Theorem 4.1 (Risk-Free Rate Condition)

Since the portfolio is risk-free, it must earn the risk-free rate to prevent arbitrage: d\Pi_t = r\Pi_t dt

Substituting \Pi_t = V - \Delta_t S_t = V - \frac{\partial V}{\partial S} S_t: d\Pi_t = r\left(V - \frac{\partial V}{\partial S} S_t\right) dt

Proof of Theorem 4.1

If a risk-free portfolio earned more than the risk-free rate, we could borrow at rate r, invest in the portfolio, and earn a risk-free profit (arbitrage). If it earned less, we could short the portfolio, lend at rate r, and again earn risk-free profit.

Step 6: Deriving the Black-Scholes PDE

Theorem 4.2 (Black-Scholes Partial Differential Equation)

The option price V(t,S) satisfies the Black-Scholes PDE: \frac{\partial V}{\partial t} + rS \frac{\partial V}{\partial S} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} - rV = 0

with terminal condition: V(T,S) = (S-K)^+

Proof of Theorem 4.2

Equating the two expressions for d\Pi_t: \frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S_t^2 \frac{\partial^2 V}{\partial S^2} = r\left(V - \frac{\partial V}{\partial S} S_t\right)

Rearranging: \frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S_t^2 \frac{\partial^2 V}{\partial S^2} = rV - rS_t \frac{\partial V}{\partial S}

Therefore: \frac{\partial V}{\partial t} + rS_t \frac{\partial V}{\partial S} + \frac{1}{2}\sigma^2 S_t^2 \frac{\partial^2 V}{\partial S^2} - rV = 0

The terminal condition comes from the option payoff at maturity.

Step 7: Solving the PDE

To solve this PDE, we use a change of variables to transform it into the heat equation.

Definition 4.4 (Variable Transformation)

Let: \begin{align} \tau &= T - t \quad \text{(time to maturity)} \\ x &= \ln(S/K) \quad \text{(log-moneyness)} \\ u(\tau, x) &= e^{r\tau} \frac{V(T-\tau, Ke^x)}{K} \end{align}

Lemma 4.4 (Transformed PDE)

The function u(\tau, x) satisfies the diffusion equation: \frac{\partial u}{\partial \tau} = \frac{1}{2}\sigma^2 \frac{\partial^2 u}{\partial x^2} + \left(r - \frac{\sigma^2}{2}\right) \frac{\partial u}{\partial x}

with initial condition: u(0, x) = (e^x - 1)^+

Proof of Lemma 4.4

We need to compute the partial derivatives of V in terms of u. We have: V(t,S) = K e^{-r(T-t)} u(T-t, \ln(S/K))

Let \tau = T-t and x = \ln(S/K). Then:

\begin{align} \frac{\partial V}{\partial t} &= K e^{-r\tau} \left[r u - \frac{\partial u}{\partial \tau}\right] \\ \frac{\partial V}{\partial S} &= K e^{-r\tau} \frac{1}{S} \frac{\partial u}{\partial x} \\ \frac{\partial^2 V}{\partial S^2} &= K e^{-r\tau} \frac{1}{S^2} \left[\frac{\partial^2 u}{\partial x^2} - \frac{\partial u}{\partial x}\right] \end{align}

Substituting into the Black-Scholes PDE and simplifying (after dividing by K e^{-r\tau}): -\frac{\partial u}{\partial \tau} + \left(r - \frac{\sigma^2}{2}\right) \frac{\partial u}{\partial x} + \frac{1}{2}\sigma^2 \frac{\partial^2 u}{\partial x^2} = 0

Therefore: \frac{\partial u}{\partial \tau} = \frac{1}{2}\sigma^2 \frac{\partial^2 u}{\partial x^2} + \left(r - \frac{\sigma^2}{2}\right) \frac{\partial u}{\partial x}

The initial condition at \tau = 0 (i.e., t = T) is: u(0, x) = \frac{(Ke^x - K)^+}{K} = (e^x - 1)^+

Theorem 4.3 (Solution via Fundamental Solution)

The solution to the transformed PDE is: u(\tau, x) = \int_{-\infty}^{\infty} (e^y - 1)^+ G(\tau; x, y) dy

where G(\tau; x, y) is the fundamental solution: G(\tau; x, y) = \frac{1}{\sqrt{2\pi\sigma^2\tau}} \exp\left(-\frac{(y - x - (r-\sigma^2/2)\tau)^2}{2\sigma^2\tau}\right)

Step 8: Computing the Solution

Theorem 4.4 (Explicit Solution of the PDE)

After computing the integral (which follows the same steps as in the martingale approach), we obtain: u(\tau, x) = e^x \Phi(d_1) - \Phi(d_2)

where: \begin{align} d_1 &= \frac{x + (r + \sigma^2/2)\tau}{\sigma\sqrt{\tau}} \\ d_2 &= \frac{x + (r - \sigma^2/2)\tau}{\sigma\sqrt{\tau}} = d_1 - \sigma\sqrt{\tau} \end{align}

Theorem 4.5 (Black-Scholes Formula via PDE Method)

Transforming back to the original variables, the European call option price is: V(t,S) = S \Phi(d_1) - K e^{-r(T-t)} \Phi(d_2)

where: \begin{align} d_1 &= \frac{\ln(S/K) + (r + \sigma^2/2)(T-t)}{\sigma\sqrt{T-t}} \\ d_2 &= d_1 - \sigma\sqrt{T-t} \end{align}

Proof of Theorem 4.5

We have: V(t,S) = K e^{-r(T-t)} u(T-t, \ln(S/K))

The transformation from the u solution to the V solution follows the same integration techniques as in the martingale approach. The key insight is that both methods yield identical results, demonstrating the equivalence of risk-neutral valuation and PDE approaches.

Verification and Conclusion

Theorem 5.1 (Equivalence of Both Methods)

Both the martingale approach and the PDE approach yield the identical Black-Scholes formula: V(t,S) = S \Phi(d_1) - K e^{-r(T-t)} \Phi(d_2)

This demonstrates the deep connection between:

• Risk-neutral valuation (martingale theory)
• Dynamic hedging and partial differential equations
  
• Stochastic calculus and deterministic analysis

Economic Interpretation

Remark 5.1 (Economic Interpretation)

The Black-Scholes formula has a clear economic interpretation:

• S \Phi(d_1): Expected value of the stock if exercised, weighted by probability of exercise
• K e^{-r(T-t)} \Phi(d_2): Present value of strike price, weighted by probability of exercise
  
• \Phi(d_2): Risk-neutral probability that option finishes in-the-money
• \Delta = \frac{\partial V}{\partial S} = \Phi(d_1): Hedge ratio (number of shares to hold per option)

Key Insights

Remark 5.2 (Key Insights)

The derivation reveals several fundamental insights:

1. The option price depends only on the risk-free rate r, not the actual expected return \mu of the stock
2. Perfect hedging is possible in continuous time with continuous trading
3. The same formula emerges from completely different mathematical approaches
4. The risk-neutral measure transforms the pricing problem into an expectation calculation

Greeks and Risk Management

Definition 5.1 (The Greeks)

The Greeks measure sensitivities of the option price to various parameters:

Delta: \Delta = \frac{\partial V}{\partial S} = \Phi(d_1) - Measures sensitivity to stock price changes - Represents the hedge ratio

Gamma: \Gamma = \frac{\partial^2 V}{\partial S^2} = \frac{\phi(d_1)}{S\sigma\sqrt{T-t}} - Measures rate of change of delta - Important for portfolio convexity

Theta: \Theta = \frac{\partial V}{\partial t} = -\frac{S\phi(d_1)\sigma}{2\sqrt{T-t}} - rKe^{-r(T-t)}\Phi(d_2) - Measures time decay - Always negative for long call options

Vega: \nu = \frac{\partial V}{\partial \sigma} = S\phi(d_1)\sqrt{T-t} - Measures sensitivity to volatility changes - Always positive for long options

Rho: \rho = \frac{\partial V}{\partial r} = K(T-t)e^{-r(T-t)}\Phi(d_2) - Measures sensitivity to interest rate changes

where \phi(x) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2} is the standard normal density.

Extensions and Limitations

Remark 5.3 (Model Assumptions and Limitations)

The Black-Scholes model relies on several strong assumptions:

Assumptions: - Constant risk-free rate and volatility - No dividends - European exercise only - No transaction costs - Continuous trading possible - Log-normal stock price distribution

Limitations: - Volatility smile/skew not captured - Jump risk ignored - Early exercise features not handled - Transaction costs can be significant - Liquidity constraints in practice

Extensions: - American options (early exercise) - Stochastic volatility models (Heston, etc.) - Jump-diffusion models (Merton) - Dividend-paying stocks - Time-dependent parameters

Computational Implementation

Example 5.1 (Python Implementation)

import numpy as np
from scipy.stats import norm

def black_scholes_call(S, K, T, r, sigma):
    """
    Calculate Black-Scholes call option price
    
    Parameters:
    S: Current stock price
    K: Strike price  
    T: Time to expiration
    r: Risk-free rate
    sigma: Volatility
    """
    d1 = (np.log(S/K) + (r + 0.5*sigma**2)*T) / (sigma*np.sqrt(T))
    d2 = d1 - sigma*np.sqrt(T)
    
    call_price = S*norm.cdf(d1) - K*np.exp(-r*T)*norm.cdf(d2)
    
    return call_price

def calculate_greeks(S, K, T, r, sigma):
    """Calculate the Greeks"""
    d1 = (np.log(S/K) + (r + 0.5*sigma**2)*T) / (sigma*np.sqrt(T))
    d2 = d1 - sigma*np.sqrt(T)
    
    delta = norm.cdf(d1)
    gamma = norm.pdf(d1) / (S*sigma*np.sqrt(T))
    theta = (-S*norm.pdf(d1)*sigma/(2*np.sqrt(T)) 
             - r*K*np.exp(-r*T)*norm.cdf(d2))
    vega = S*norm.pdf(d1)*np.sqrt(T)
    rho = K*T*np.exp(-r*T)*norm.cdf(d2)
    
    return {'delta': delta, 'gamma': gamma, 'theta': theta, 'vega': vega, 'rho': rho}

# Example usage
S0 = 100    # Current stock price
K = 105     # Strike price
T = 0.25    # 3 months to expiration
r = 0.05    # 5% risk-free rate  
sigma = 0.2 # 20% volatility

price = black_scholes_call(S0, K, T, r, sigma)
greeks = calculate_greeks(S0, K, T, r, sigma)

print(f"Call option price: ${price:.2f}")
print(f"Delta: {greeks['delta']:.4f}")
print(f"Gamma: {greeks['gamma']:.4f}")

This complete treatment of the Black-Scholes model provides both theoretical rigor and practical applicability, showing how mathematical finance bridges pure mathematics and real-world financial applications.