Self-Financing Portfolios¶

In the discrete-time binomial model (Chapter 1), a replicating portfolio is rebalanced at each time step without injecting or withdrawing funds. The continuous-time analogue of this idea is the self-financing condition, which requires that all changes in portfolio value arise solely from changes in asset prices---not from external cash flows. This condition is the mathematical foundation of the hedging argument that leads to the Black-Scholes PDE.

This section defines trading strategies and the self-financing condition in continuous time, derives the wealth dynamics of self-financing portfolios, and establishes the connection to risk-neutral pricing through the discounted wealth process.

Learning Objectives

After completing this section, you should be able to:

Define a continuous-time trading strategy and state the required measurability conditions
State the self-financing condition in both differential and integral form
Derive the wealth dynamics of a self-financing portfolio holding a risky asset and a risk-free bond
Show that the discounted wealth process of a self-financing strategy is a local martingale under the risk-neutral measure
Explain the connection between self-financing portfolios and derivative replication

The Market Model¶

1. Assets¶

We work in the Black-Scholes market consisting of two traded assets on a filtered probability space $(\Omega, \mathcal{F}, \{\mathcal{F}_t\}_{0 \leq t \leq T}, \mathbb{P})$, where the filtration is generated by a standard Brownian motion $W_t$.

Risk-free bond (money market account): The bond price $B_t$ evolves deterministically according to

\[ dB_t = r\,B_t\,dt, \qquad B_0 = 1 \]

with solution $B_t = e^{rt}$, where $r \geq 0$ is the constant risk-free rate.

Risky asset (stock): The stock price $S_t$ follows geometric Brownian motion as developed in the preceding section on GBM:

\[ dS_t = \mu S_t \, dt + \sigma S_t \, dW_t, \qquad S_0 > 0 \]

where $\mu \in \mathbb{R}$ is the drift and $\sigma > 0$ is the volatility.

2. Trading Strategies¶

A trading strategy specifies the number of shares of each asset held at every point in time.

Definition: Trading Strategy

A trading strategy is a pair of stochastic processes $(\phi_t, \psi_t)_{0 \leq t \leq T}$ where:

$\phi_t$ = number of shares of the stock held at time $t$
$\psi_t$ = number of units of the bond held at time $t$

Both processes are required to be predictable (i.e., $\mathcal{F}_t$-adapted and left-continuous) and to satisfy the integrability conditions

\[ \int_0^T \phi_t^2 S_t^2 \, dt < \infty \quad \text{a.s.} \]

\[ \int_0^T |\psi_t| B_t \, dt < \infty \quad \text{a.s.} \]

The predictability condition means that the portfolio holdings at time $t$ are determined by information available just before time $t$. This prevents the trader from "looking ahead" and using future price information. In the discrete-time binomial model of Chapter 1, this corresponds to choosing the portfolio at node $(n, j)$ based on the filtration $\mathcal{F}_n$, before observing the $(n+1)$-st price move.

The integrability conditions ensure that the stochastic integrals appearing in the wealth process are well defined.

Portfolio Value and the Self-Financing Condition¶

1. Portfolio Value¶

Given a trading strategy $(\phi_t, \psi_t)$, the portfolio value (or wealth process) at time $t$ is

\[ X_t = \phi_t S_t + \psi_t B_t \]

This is simply the mark-to-market value of the portfolio: the number of shares times the stock price, plus the number of bond units times the bond price.

2. The Self-Financing Condition¶

The wealth changes over time for two distinct reasons: (i) asset prices change, and (ii) the trader rebalances the portfolio. The self-financing condition requires that all changes in wealth come from price changes alone---no external funding or withdrawal occurs.

Definition: Self-Financing Portfolio

A trading strategy $(\phi_t, \psi_t)$ is self-financing if

\[ dX_t = \phi_t \, dS_t + \psi_t \, dB_t \]

for all $t \in [0, T]$. In integral form:

\[ X_t = X_0 + \int_0^t \phi_u \, dS_u + \int_0^t \psi_u \, dB_u \]

where $X_0 = \phi_0 S_0 + \psi_0 B_0$ is the initial wealth.

The economic interpretation is immediate: the change in portfolio value equals the gains from holding the stock ($\phi_t \, dS_t$) plus the interest earned on the bond position ($\psi_t \, dB_t$). There are no additional cash inflows or outflows.

3. Derivation and Interpretation¶

To understand why the self-financing condition takes this form, consider the general change in portfolio value:

\[ dX_t = d(\phi_t S_t + \psi_t B_t) \]

By the product rule for stochastic differentials:

\[ dX_t = \phi_t \, dS_t + S_t \, d\phi_t + d\phi_t \, dS_t + \psi_t \, dB_t + B_t \, d\psi_t + d\psi_t \, dB_t \]

Since $B_t$ is deterministic, $d\psi_t \, dB_t = 0$. The remaining cross term $d\phi_t \, dS_t$ may or may not vanish depending on the regularity of $\phi_t$. The self-financing condition $dX_t = \phi_t \, dS_t + \psi_t \, dB_t$ therefore imposes the constraint

\[ S_t \, d\phi_t + B_t \, d\psi_t + d\phi_t \, dS_t = 0 \]

In many standard treatments (where $\phi_t$ is of bounded variation or sufficiently smooth), the cross term $d\phi_t \, dS_t$ vanishes, and the self-financing condition simplifies to

\[ \boxed{S_t \, d\phi_t + B_t \, d\psi_t = 0} \]

This says that any purchase of stock must be financed entirely by selling bonds, and vice versa. The cost of rebalancing is zero in net terms---every dollar spent buying one asset is raised by selling the other.

Discrete-Time Analogue

In the binomial model, the self-financing condition at time $n$ states that the portfolio value just after rebalancing (using the new holdings $\phi_{n+1}, \psi_{n+1}$) equals the portfolio value just before rebalancing (using the old holdings $\phi_n, \psi_n$), both evaluated at time-$n$ prices:

\[ \phi_{n+1} S_n + \psi_{n+1} B_n = \phi_n S_n + \psi_n B_n \]

The continuous-time condition $S_t \, d\phi_t + B_t \, d\psi_t = 0$ is the infinitesimal version of this.

Wealth Dynamics in Terms of the Stock Price¶

1. Eliminating the Bond Holdings¶

For a self-financing portfolio, the bond position $\psi_t$ is determined by the stock position $\phi_t$ and the total wealth $X_t$:

\[ \psi_t = \frac{X_t - \phi_t S_t}{B_t} \]

Substituting $dB_t = r B_t \, dt$ into the self-financing condition:

\[ dX_t = \phi_t \, dS_t + \psi_t \, r B_t \, dt = \phi_t \, dS_t + r(X_t - \phi_t S_t)\,dt \]

Rearranging:

\[ \boxed{dX_t = r X_t \, dt + \phi_t (dS_t - r S_t \, dt)} \]

2. Interpretation¶

This expression has a clean economic interpretation:

The term $r X_t \, dt$ represents the risk-free growth of the total wealth at rate $r$
The term $\phi_t (dS_t - r S_t \, dt)$ represents the excess return from holding $\phi_t$ shares of stock, where $dS_t - r S_t \, dt$ is the stock's return in excess of the risk-free rate

Substituting the GBM dynamics $dS_t = \mu S_t \, dt + \sigma S_t \, dW_t$:

\[ dX_t = rX_t \, dt + \phi_t S_t\left[(\mu - r)\,dt + \sigma\,dW_t\right] \]

The quantity $\mu - r$ is the equity risk premium (the excess expected return per unit time), and $\phi_t S_t$ is the dollar amount invested in stock.

The Discounted Wealth Process¶

1. Discounting¶

The discounted wealth process is defined by

\[ \tilde{X}_t = \frac{X_t}{B_t} = e^{-rt} X_t \]

This expresses the portfolio value in units of the bond (the numeraire). Intuitively, $\tilde{X}_t$ measures the "real" wealth after removing the effect of risk-free interest accumulation.

2. Dynamics Under the Physical Measure¶

Applying the product rule to $\tilde{X}_t = e^{-rt} X_t$:

\[ d\tilde{X}_t = e^{-rt}\,dX_t - r e^{-rt} X_t \, dt = e^{-rt}(dX_t - rX_t \, dt) \]

From the wealth dynamics derived above:

\[ d\tilde{X}_t = e^{-rt}\,\phi_t(dS_t - rS_t \, dt) = \phi_t \, d\tilde{S}_t \]

where $\tilde{S}_t = e^{-rt} S_t$ is the discounted stock price. Therefore:

\[ \boxed{d\tilde{X}_t = \phi_t \, d\tilde{S}_t} \]

The discounted wealth of a self-financing portfolio changes only through gains from the discounted stock---a remarkably simple expression.

3. Dynamics Under the Risk-Neutral Measure¶

By Girsanov's theorem, there exists an equivalent probability measure $\mathbb{Q}$ under which the discounted stock price $\tilde{S}_t$ is a martingale. Specifically, defining the Brownian motion under $\mathbb{Q}$ by

\[ dW_t^{\mathbb{Q}} = dW_t + \frac{\mu - r}{\sigma}\,dt \]

the discounted stock price satisfies

\[ d\tilde{S}_t = \sigma \tilde{S}_t \, dW_t^{\mathbb{Q}} \]

Since $d\tilde{X}_t = \phi_t \, d\tilde{S}_t$, the discounted wealth process becomes

\[ d\tilde{X}_t = \phi_t \sigma \tilde{S}_t \, dW_t^{\mathbb{Q}} \]

This is a stochastic integral with respect to a $\mathbb{Q}$-Brownian motion with zero drift, so $\tilde{X}_t$ is a local martingale under $\mathbb{Q}$.

Theorem: Discounted Wealth is a Local Martingale

If $(\phi_t, \psi_t)$ is a self-financing trading strategy, then the discounted wealth process $\tilde{X}_t = e^{-rt} X_t$ is a local martingale under the risk-neutral measure $\mathbb{Q}$.

If additionally the integrability condition $\mathbb{E}^{\mathbb{Q}}\!\left[\int_0^T \phi_t^2 \sigma^2 \tilde{S}_t^2 \, dt\right] < \infty$ holds, then $\tilde{X}_t$ is a true martingale, and

\[ \tilde{X}_0 = \mathbb{E}^{\mathbb{Q}}[\tilde{X}_T] \]

Equivalently, $X_0 = e^{-rT}\,\mathbb{E}^{\mathbb{Q}}[X_T]$.

Connection to Derivative Replication¶

1. The Replication Problem¶

A European contingent claim with maturity $T$ is a random variable $H = h(S_T)$ representing the payoff at time $T$. The central problem is to find a self-financing strategy $(\phi_t, \psi_t)$ with initial wealth $X_0$ such that

\[ X_T = H \quad \text{a.s.} \]

If such a strategy exists, $H$ is said to be replicable (or attainable), and its no-arbitrage price at time $0$ is the initial capital $X_0$ required to fund the replicating portfolio.

2. Pricing via the Martingale Property¶

If $H$ is replicable by a self-financing strategy satisfying the integrability condition, then the discounted wealth $\tilde{X}_t$ is a $\mathbb{Q}$-martingale, and

\[ X_0 = \tilde{X}_0 = \mathbb{E}^{\mathbb{Q}}[\tilde{X}_T] = \mathbb{E}^{\mathbb{Q}}[e^{-rT} H] \]

This is the risk-neutral pricing formula:

\[ \boxed{V_0 = e^{-rT}\,\mathbb{E}^{\mathbb{Q}}[H]} \]

The price of the claim equals the discounted expected payoff under $\mathbb{Q}$---precisely because the self-financing and no-arbitrage conditions force this equality.

3. The Hedging Strategy¶

For a European call with payoff $H = (S_T - K)^+$, the option price at time $t$ is the function $V(S_t, t)$ solving the Black-Scholes PDE. The replicating strategy is

\[ \phi_t = \frac{\partial V}{\partial S}(S_t, t) \qquad \text{(the delta)} \]

\[ \psi_t = \frac{V(S_t, t) - \phi_t S_t}{B_t} \qquad \text{(the bond position)} \]

This strategy is self-financing by construction: the Black-Scholes PDE ensures that the portfolio value $X_t = V(S_t, t)$ satisfies $dX_t = \phi_t \, dS_t + \psi_t \, dB_t$ at all times.

Example: Self-Financing Portfolio of Stock and Bond¶

To make the self-financing condition concrete, consider a simple strategy that maintains a constant number of shares $\phi_t = \phi$ (a buy-and-hold stock position). The remaining wealth is invested in the bond.

Setup: Initial wealth $X_0$, holding $\phi$ shares of stock.

Bond position: $\psi_t = (X_t - \phi S_t)/B_t$.

Wealth dynamics: From the general formula,

\[ dX_t = rX_t \, dt + \phi(dS_t - rS_t \, dt) \]

Substituting $dS_t = \mu S_t \, dt + \sigma S_t \, dW_t$:

\[ dX_t = rX_t \, dt + \phi S_t[(\mu - r)\,dt + \sigma\,dW_t] \]

Verification of self-financing: Since $\phi$ is constant, $d\phi_t = 0$. The bond position adjusts as the stock price changes, but no external cash is added:

\[ S_t \, d\phi_t + B_t \, d\psi_t = 0 + B_t \, d\psi_t \]

Since all wealth changes come from asset price movements, the condition $B_t \, d\psi_t = -S_t \, d\phi_t = 0$ is automatically satisfied when $d\phi_t = 0$, confirming that $d\psi_t = 0$ as well. Wait---this is not quite right when $\phi$ is constant: $\psi_t$ does change because the stock position $\phi S_t$ changes in value while $X_t$ also changes. Let us verify directly.

We have $\psi_t B_t = X_t - \phi S_t$, so

\[ d(\psi_t B_t) = dX_t - \phi \, dS_t = (rX_t \, dt + \phi\,dS_t - r\phi S_t\,dt) - \phi\,dS_t = r(X_t - \phi S_t)\,dt = r\psi_t B_t\,dt \]

This gives $d(\psi_t B_t) = r\psi_t B_t \, dt$, which is consistent with $\psi_t \, dB_t + B_t \, d\psi_t = r\psi_t B_t \, dt$ and therefore $B_t \, d\psi_t = 0$. The bond position $\psi_t$ remains constant, and the bond value grows at rate $r$. The strategy is indeed self-financing. $\square$

Admissibility¶

1. The Need for Admissibility¶

Not all self-financing strategies are economically meaningful. Without further restrictions, a trader could employ doubling strategies (the continuous-time analogue of the martingale betting system) that generate arbitrage from nothing by allowing unbounded losses at intermediate times.

2. Admissibility Condition¶

Definition: Admissible Strategy

A self-financing strategy $(\phi_t, \psi_t)$ is admissible if there exists a constant $a > 0$ such that the wealth process satisfies

\[ X_t \geq -a \quad \text{a.s. for all } t \in [0, T] \]

That is, the strategy has bounded credit: the trader's losses are bounded below.

Under admissibility, the local martingale $\tilde{X}_t$ is bounded below by $-a e^{-rT}$ and is therefore a supermartingale under $\mathbb{Q}$. This rules out doubling strategies and ensures that the no-arbitrage pricing framework is consistent.

In many treatments, the stronger condition $X_t \geq 0$ (no borrowing beyond the initial capital) is imposed, which makes $\tilde{X}_t$ a non-negative local martingale and hence a supermartingale.

Summary¶

Self-financing portfolios formalize the idea that dynamic hedging strategies fund themselves through trading, without external cash injections or withdrawals.

1. Trading strategies: A pair $(\phi_t, \psi_t)$ of predictable processes specifying stock and bond holdings, with $X_t = \phi_t S_t + \psi_t B_t$.

2. Self-financing condition: $dX_t = \phi_t \, dS_t + \psi_t \, dB_t$, equivalently $S_t \, d\phi_t + B_t \, d\psi_t = 0$. Rebalancing is cost-neutral.

3. Wealth dynamics: $dX_t = rX_t \, dt + \phi_t(dS_t - rS_t \, dt)$. Portfolio growth decomposes into risk-free growth and excess return from the stock position.

4. Discounted wealth: $d\tilde{X}_t = \phi_t \, d\tilde{S}_t$, which is a local martingale under $\mathbb{Q}$.

5. Pricing connection: If a claim $H$ is replicable by a self-financing strategy, its no-arbitrage price is $V_0 = e^{-rT} \mathbb{E}^{\mathbb{Q}}[H]$.

6. Admissibility: Bounded-credit conditions prevent pathological doubling strategies and ensure the pricing framework is consistent.

These results establish the mathematical infrastructure for the Black-Scholes PDE derivation in the next section, where the specific self-financing strategy that replicates a European option is constructed explicitly.

Exercises¶

Exercise 1. Let $\phi_t = 1$ (hold one share of stock) and $X_0 = S_0 + B$ for some constant $B > 0$. Determine the bond position $\psi_t$ and verify directly that the strategy is self-financing by checking that $S_t \, d\phi_t + B_t \, d\psi_t = 0$.

Solution to Exercise 1

With $\phi_t = 1$ (one share) and $X_0 = S_0 + B$ for some constant $B > 0$:

The bond position is $\psi_t = (X_t - \phi_t S_t)/B_t = (X_t - S_t)/e^{rt}$.

At $t = 0$: $\psi_0 = (S_0 + B - S_0)/1 = B$.

Since $\phi_t = 1$ is constant, $d\phi_t = 0$, and we need to verify $S_t\,d\phi_t + B_t\,d\psi_t = 0$.

The first term is $S_t \cdot 0 = 0$. For the second term, we compute $\psi_t$ explicitly.

The wealth dynamics with $\phi_t = 1$ are:

\[ dX_t = rX_t\,dt + 1 \cdot (dS_t - rS_t\,dt) = rX_t\,dt + dS_t - rS_t\,dt \]

The bond wealth is $\psi_t B_t = X_t - S_t$. Compute its differential:

\[ d(\psi_t B_t) = dX_t - dS_t = rX_t\,dt + dS_t - rS_t\,dt - dS_t = r(X_t - S_t)\,dt = r\psi_t B_t\,dt \]

Since $d(\psi_t B_t) = \psi_t\,dB_t + B_t\,d\psi_t = r\psi_t B_t\,dt + B_t\,d\psi_t$, we get:

\[ B_t\,d\psi_t = r\psi_t B_t\,dt - r\psi_t B_t\,dt = 0 \]

Therefore $d\psi_t = 0$, meaning $\psi_t = \psi_0 = B$ for all $t$. Both the stock holding and bond holding are constant, confirming:

\[ S_t\,d\phi_t + B_t\,d\psi_t = 0 + 0 = 0 \quad \checkmark \]

The strategy is self-financing. The portfolio simply holds one share and $B$ units of the bond, with both positions unchanged over time. $\square$

Exercise 2. Consider a self-financing portfolio with wealth dynamics $dX_t = rX_t \, dt + \phi_t(dS_t - rS_t \, dt)$. Suppose the trader invests a constant fraction $\pi$ of wealth in the stock, so $\phi_t S_t = \pi X_t$. Show that the wealth process satisfies

\[ dX_t = X_t\left[(r + \pi(\mu - r))\,dt + \pi\sigma\,dW_t\right] \]

and solve this SDE explicitly for $X_t$.

Solution to Exercise 2

Given $\phi_t S_t = \pi X_t$ (constant fraction $\pi$ of wealth in stock), substitute into the wealth dynamics:

\[ dX_t = rX_t\,dt + \phi_t(dS_t - rS_t\,dt) \]

Since $\phi_t = \pi X_t / S_t$, the excess return term becomes:

\[ \phi_t(dS_t - rS_t\,dt) = \frac{\pi X_t}{S_t}(dS_t - rS_t\,dt) \]

Substituting $dS_t = \mu S_t\,dt + \sigma S_t\,dW_t$:

\[ = \frac{\pi X_t}{S_t}\left[(\mu - r)S_t\,dt + \sigma S_t\,dW_t\right] = \pi X_t\left[(\mu - r)\,dt + \sigma\,dW_t\right] \]

Therefore:

\[ dX_t = rX_t\,dt + \pi X_t(\mu - r)\,dt + \pi X_t\sigma\,dW_t = X_t\left[(r + \pi(\mu - r))\,dt + \pi\sigma\,dW_t\right] \]

This is a GBM with drift $r + \pi(\mu - r)$ and volatility $\pi\sigma$. The solution is:

\[ X_t = X_0\exp\!\left[\left(r + \pi(\mu - r) - \frac{1}{2}\pi^2\sigma^2\right)t + \pi\sigma W_t\right] \]

This is the Merton portfolio result. The log-return is normally distributed with mean $(r + \pi(\mu - r) - \frac{1}{2}\pi^2\sigma^2)t$ and variance $\pi^2\sigma^2 t$. The optimal $\pi$ that maximizes the expected log-wealth growth rate is $\pi^* = (\mu - r)/\sigma^2$ (the Kelly criterion).

Exercise 3. Prove that if two self-financing portfolios $X_t$ and $Y_t$ satisfy $X_T = Y_T$ almost surely, then $X_t = Y_t$ for all $t \in [0,T]$ almost surely. Use the martingale property of the discounted wealth under $\mathbb{Q}$. Explain why this result is essential for the uniqueness of derivative prices.

Solution to Exercise 3

Suppose $X_t$ and $Y_t$ are both self-financing portfolios with $X_T = Y_T$ a.s.

Define $Z_t = X_t - Y_t$. Then $Z_t$ is also a self-financing portfolio (the self-financing condition is linear), and $Z_T = 0$ a.s.

The discounted wealth $\tilde{Z}_t = e^{-rt}Z_t$ is a local martingale under $\mathbb{Q}$ (since both $\tilde{X}_t$ and $\tilde{Y}_t$ are local martingales under $\mathbb{Q}$, and their difference is also a local martingale).

Assuming both strategies are admissible (bounded credit), $\tilde{Z}_t$ is bounded below by some constant $-a$. A local martingale bounded below is a supermartingale. Similarly, $-\tilde{Z}_t$ is bounded below, so $-\tilde{Z}_t$ is also a supermartingale, which means $\tilde{Z}_t$ is a submartingale.

A process that is both a supermartingale and a submartingale is a martingale. Therefore $\tilde{Z}_t$ is a true $\mathbb{Q}$-martingale.

For a martingale with terminal value $\tilde{Z}_T = 0$:

\[ \tilde{Z}_t = \mathbb{E}^{\mathbb{Q}}[\tilde{Z}_T \mid \mathcal{F}_t] = \mathbb{E}^{\mathbb{Q}}[0 \mid \mathcal{F}_t] = 0 \quad \text{for all } t \in [0, T] \]

Therefore $Z_t = e^{rt}\tilde{Z}_t = 0$ for all $t$, so $X_t = Y_t$ for all $t \in [0, T]$ a.s. $\square$

Importance for uniqueness: This result guarantees that if a contingent claim $H$ is replicable, then the replication cost is unique. If two different self-financing strategies both replicate $H$ (i.e., $X_T = Y_T = H$), they must have the same initial cost $X_0 = Y_0$ and indeed the same value at all intermediate times. This ensures a unique arbitrage-free price for every replicable claim.

Exercise 4. Define the doubling strategy informally and explain how it appears to generate arbitrage in continuous time. Then explain precisely how the admissibility condition $X_t \geq -a$ prevents this strategy from being used.

Solution to Exercise 4

The doubling strategy (informally): Start with initial wealth 0. At each step, bet on the outcome of a fair coin flip, doubling the stake after each loss. Specifically, bet $1; if you lose, bet $2; if you lose again, bet $4; and so on. The first win recovers all previous losses plus a $1 profit. Since you eventually win with probability 1, this appears to generate a sure profit from nothing.

In continuous time, this can be formalized: construct a self-financing strategy that starts with $X_0 = 0$ and achieves $X_T > 0$ with positive probability and $X_T \geq 0$ almost surely. This would constitute an arbitrage.

How it works mechanically: The strategy invests increasingly large amounts in the risky asset after losses, funded by borrowing. The wealth process $X_t$ can become arbitrarily negative at intermediate times (the trader accumulates potentially enormous debts before the "winning" trade occurs).

How admissibility prevents it: The admissibility condition requires $X_t \geq -a$ for some finite constant $a$ and all $t \in [0, T]$. This bounds the maximum allowable loss. The doubling strategy requires $X_t \to -\infty$ along certain paths (when the "winning" trade is delayed), which violates $X_t \geq -a$ for any finite $a$.

Mathematically, admissibility ensures that the discounted wealth $\tilde{X}_t$ is a supermartingale under $\mathbb{Q}$ (not merely a local martingale). For a supermartingale starting at $\tilde{X}_0 = 0$:

\[ \mathbb{E}^{\mathbb{Q}}[\tilde{X}_T] \leq \tilde{X}_0 = 0 \]

Combined with $X_T \geq 0$ (no-bankruptcy at maturity), this forces $X_T = 0$ a.s. --- no profit is possible. The admissibility condition thus closes the loophole that makes the doubling strategy appear to work.

Exercise 5. For a European call with Black-Scholes price $V(S_t, t) = S_t \mathcal{N}(d_1) - Ke^{-r(T-t)}\mathcal{N}(d_2)$, write down the replicating portfolio $(\phi_t, \psi_t)$ explicitly. Verify that at $t = 0$ the portfolio value equals $V(S_0, 0)$, and at $t = T$ the portfolio value equals $(S_T - K)^+$.

Solution to Exercise 5

The Black-Scholes call price is $V(S_t, t) = S_t\mathcal{N}(d_1) - Ke^{-r(T-t)}\mathcal{N}(d_2)$ where:

\[ d_1 = \frac{\ln(S_t/K) + (r + \frac{1}{2}\sigma^2)(T-t)}{\sigma\sqrt{T-t}}, \quad d_2 = d_1 - \sigma\sqrt{T-t} \]

Replicating portfolio:

\[ \phi_t = \frac{\partial V}{\partial S}(S_t, t) = \mathcal{N}(d_1) \]

\[ \psi_t = \frac{V(S_t, t) - \phi_t S_t}{B_t} = \frac{S_t\mathcal{N}(d_1) - Ke^{-r(T-t)}\mathcal{N}(d_2) - \mathcal{N}(d_1)S_t}{e^{rt}} = \frac{-Ke^{-r(T-t)}\mathcal{N}(d_2)}{e^{rt}} \]

\[ = -Ke^{-rT}\mathcal{N}(d_2) \]

Verification at $t = 0$: The portfolio value is:

\[ \phi_0 S_0 + \psi_0 B_0 = \mathcal{N}(d_1^0)S_0 + (-Ke^{-rT}\mathcal{N}(d_2^0)) \cdot 1 = S_0\mathcal{N}(d_1^0) - Ke^{-rT}\mathcal{N}(d_2^0) = V(S_0, 0) \quad \checkmark \]

Verification at $t = T$: As $t \to T^-$, there are two cases:

If $S_T > K$: Then $d_1, d_2 \to +\infty$, so $\mathcal{N}(d_1) \to 1$ and $\mathcal{N}(d_2) \to 1$. The portfolio value is $1 \cdot S_T + (-K \cdot 1) = S_T - K = (S_T - K)^+$. $\checkmark$
If $S_T < K$: Then $d_1, d_2 \to -\infty$, so $\mathcal{N}(d_1) \to 0$ and $\mathcal{N}(d_2) \to 0$. The portfolio value is $0 \cdot S_T + 0 = 0 = (S_T - K)^+$. $\checkmark$

In both cases, $X_T = (S_T - K)^+$, confirming exact replication. $\square$

Exercise 6. Show that the discounted stock price $\tilde{S}_t = e^{-rt}S_t$ satisfies

\[ d\tilde{S}_t = (\mu - r)\tilde{S}_t \, dt + \sigma \tilde{S}_t \, dW_t \]

under the physical measure $\mathbb{P}$. Explain why $\tilde{S}_t$ is not a martingale under $\mathbb{P}$ (assuming $\mu \neq r$) and identify the Girsanov change of measure that makes it a martingale.

Solution to Exercise 6

Apply the product rule to $\tilde{S}_t = e^{-rt}S_t$:

\[ d\tilde{S}_t = e^{-rt}\,dS_t + S_t\,d(e^{-rt}) = e^{-rt}\,dS_t - rS_te^{-rt}\,dt \]

\[ = e^{-rt}(\mu S_t\,dt + \sigma S_t\,dW_t) - re^{-rt}S_t\,dt \]

\[ = (\mu - r)e^{-rt}S_t\,dt + \sigma e^{-rt}S_t\,dW_t \]

\[ = (\mu - r)\tilde{S}_t\,dt + \sigma\tilde{S}_t\,dW_t \quad \square \]

Why $\tilde{S}_t$ is not a martingale under $\mathbb{P}$: A martingale must have zero drift. Under $\mathbb{P}$, $\tilde{S}_t$ has drift $(\mu - r)\tilde{S}_t$, which is nonzero when $\mu \neq r$. Consequently:

\[ \mathbb{E}^{\mathbb{P}}[\tilde{S}_t \mid \mathcal{F}_s] = \tilde{S}_s\,e^{(\mu - r)(t - s)} \neq \tilde{S}_s \]

so the martingale property fails.

Girsanov change of measure: Define the market price of risk $\theta = (\mu - r)/\sigma$ and the new Brownian motion:

\[ dW_t^{\mathbb{Q}} = dW_t + \theta\,dt = dW_t + \frac{\mu - r}{\sigma}\,dt \]

Under the risk-neutral measure $\mathbb{Q}$, defined by the Radon-Nikodym derivative:

\[ \frac{d\mathbb{Q}}{d\mathbb{P}}\bigg|_{\mathcal{F}_t} = \exp\!\left(-\theta W_t - \frac{1}{2}\theta^2 t\right) \]

the process $W_t^{\mathbb{Q}}$ is a standard Brownian motion (by Girsanov's theorem), and:

\[ d\tilde{S}_t = (\mu - r)\tilde{S}_t\,dt + \sigma\tilde{S}_t\,(dW_t^{\mathbb{Q}} - \theta\,dt) \]

\[ = (\mu - r)\tilde{S}_t\,dt + \sigma\tilde{S}_t\,dW_t^{\mathbb{Q}} - \sigma\theta\tilde{S}_t\,dt \]

\[ = (\mu - r)\tilde{S}_t\,dt - (\mu - r)\tilde{S}_t\,dt + \sigma\tilde{S}_t\,dW_t^{\mathbb{Q}} = \sigma\tilde{S}_t\,dW_t^{\mathbb{Q}} \]

The drift vanishes, so $\tilde{S}_t$ is a local martingale (and, by standard integrability arguments for GBM, a true martingale) under $\mathbb{Q}$. $\square$