Self-financing and Black-Scholes-Merton formula

Question

Self-financing is an important concept in financial product replicating, normally used in pricing.

I read about several ways to derive Black-Scholes-Merton (BSM) formula. Seems some approaches actually rely on replicating, implying the self-financing prerequisite. But it turns out that such approaches missed to check the condition!

Is my understanding correct? Details as below.

Reference:

[German] Helyette Geman, Nicole El Karoui, Jean-Charles Rochet "Changes of Numeraire, Changes of Probability Measure and Option Pricing" Journal of Applied Probability, Vol. 32, No. 2 (Jun., 1995) , pp. 443-458.

[Shreve] Steven E. Shreve "Stochastic calculus for finance II", 2004.

[Hull] John Hull, "Options, Futures, and Other Derivatives", 2009.

Self-financing

According to [German], a portfolio $$\begin{equation*} V(t)=\sum_{k=1}^n w_k(t) S_k(t) \tag{G-1} \end{equation*}$$

is defined as self-financing if $$\begin{equation*} dV(t)=\sum_{k=1}^n w_k(t) \,d S_k(t) \tag{G-2} \end{equation*}$$

holds.

A simple example of non-self-financing product would be an American option. Hence, if a deriving of the BSM formula/equation does not exclude

American options, it probably neglected the self-financing prerequisite.

BSM Equation and Formula

As [Shreve] Chapter 4.5.3 put, BSM equation is: $$\begin{equation*} c_t(t,x)+rxc_x(t,x)+\frac{1}{2}\sigma^2x^2c_{xx}(t,x)=rc(t,x) \tag{S-4.5.14} \end{equation*}$$

for all $t\in[0,T)$ and $x\ge 0$, where $c(t,x)$ is the call option price at time $t$ for underlying asset $x$, and $c_t$, $c_x$ are partial differentials.

The solution of BSM Equation (Shreve 4.5.14) is BSM Formula, as [Shreve] provided in Ch 4.5.4 : $$\begin{equation*} c(t,x)=xN(d_+(T-t,x))-Ke^{-r(T-t)}N(d_-(T-t,x)) \tag{S-4.5.19} \end{equation*}$$

BSM Formula Derivation #1: from BSM Equation

[Shreve] provides (S-4.5.19) without touching the detailed steps solving the backward parabolic equation; it only use Exercise 4.9 to show that it fulfills (S-4.5.14). This is alright. The problem is how [Shreve] establish (S-4.5.14), as excerpted below.

Stock price modeled by the geometric Brownian motion is

$$\begin{equation*} \,\text{d}S(t) = \alpha S(t) \,\text{d}t + \sigma S(t) \,\text{d}W(t) \tag{S-4.5.1} \end{equation*}$$

Set up the replicating portfolio as $$\begin{equation*} X(t)=\Delta(t)S(t)+\Gamma(t)M(t) \tag{S-4.10.16} \end{equation*}$$

This means at each time t, the investor holds $\Delta(t)$ shares of stock. The position $\Delta(t)$ can be random but must be adapted to the filtration associated with the Brownian motion $W(t), t > 0$. The remainder of the portfolio value, $X(t) — \Delta(t)S(t)$, is invested in the money market account.

Assuming constant rate of interest as $r$, one has $$\begin{align} \,\text{d}X(t) &= \Delta\,\text{d}S + r(X - \Delta\,S )\,\text{d}t\\ &= \Delta (\alpha S \,\text{d}t + \sigma S \,\text{d}W) + r(X - \Delta S )\,\text{d}t\\ &= rX\,\text{d}t + \Delta\,(\alpha - r) S \,\text{d}t + \Delta\,\sigma S\,\text{d}W \tag{S-4.5.2} \\ \,\text{d}(e^{-rt}S(t)) &= -re^{-rt}S\,\text{d}t+e^{-rt}\,\text{d}S \\ &= (\alpha-r)e^{-rt}S\,\text{d}t + \sigma e^{-rt} S\,\text{d}W \tag{S-4.5.4} \\ \,\text{d}(e^{-rt}X(t)) &= -re^{-rt}X \,\text{d}t+e^{-rt}\,\text{d}X \\ &= \Delta\,(\alpha-r)e^{-rt}S\,\text{d}t + \Delta\,\sigma e^{-rt} S\,\text{d}W \\ &= \Delta\,\text{d}(e^{-rt}S(t)) \tag{S-4.5.5} \\ \end{align}$$

Let $c(t,x)$ denote the value of the call option at time $t$ if the stock price at that time is $S(t) = x$. [Shreve] says since " Black, Scholes, and Merton argued that the value of this call at any time should depend on the time (more precisely, on the time to expiration) and on the value of the stock price at that time", there is nothing random about the function $c(t,x)$ -- I understand this implies the option price is a Markov process.

Calculating $\,\text{d}c(t,S(t))$, one has $$\begin{align} \,\text{d}c(t,S(t)) &= \left[c_t+\alpha Sc_x+\frac{1}{2}\sigma^2S^2c_{xx}\right]dt + \sigma Sc_x dW(t) \tag{S-4.5.6} \\ \,\text{d}(e^{-rt}c(t,S(t)) &= -re^{-rt}c(t,S(t))\,\text{d}t + e^{-rt}\,\text{d}c(t,S(t)) \\ &= e^{-rt}[-rc+c_t+\alpha Sc_x+\frac{1}{2}\sigma^2 S^2 c_{xx}]\,\text{d}t \\ & + e^{-rt} \sigma Sc_x\,\text{d}W \tag{S-4.5.7} \\ \end{align}$$

Here the trick comes -- [Shreve] says, "a (short option) hedging portfolio starts with some initial capital $X(0)$ and invests in the stock and money market account so that the portfolio value $X(t)$ at each time $t\in[0, T]$ agrees with $c(t, S(t))$", this happens if and only if $X(0)=c(0,S(0))$ and
$$\begin{equation*} d(e^{-rt}X(t))=d(e^{-rt}c(t,S(t)),\forall t\in [0,T) \tag{S-4.5.8} \end{equation*}$$

Comparing (S-4.5.5) and (S-4.5.7), (S-4.5.8) leads to, $\forall t \in [0, T)$, $$\begin{align} \Delta(t) &= c_x(t,S(t)) \tag{S-4.5.11} \\ (\alpha-r)Sc_x &= -rc+c_t+\alpha Sc_x + \frac{1}{2}\sigma^2S^2c_{xx} \tag{S-4.5.12} \\ \end{align} $$

Finally (S-4.5.12) means (S-4.5.14), the BSM equation.

Now, there is a loophole - the self-financing of the replicating portfolio $X(t)$. Actually the first step of (S-4.5.2), or explicitly, $$\begin{equation*} \,\text{d}X(t) = \Delta\,\text{d}S + r(X - \Delta\,S )\,\text{d}t \tag{S-4.10.9} \end{equation*}$$ implies the self-financing condition (G-2).

[Shreve] Exercise 4.9.10 touches the self-financing topic but does not prove that $X(t)$ is self-financing. Instead, Exercise 4.9.10 only says that the self-financing condition, or (S-4.10.9), is equivalent to $$\begin{equation*} S(t)\,\text{d}\Delta(t)+\,\text{d}S(t)\,\text{d}\Delta(t)+M(t)\,\text{d}\Gamma(t)+\,\text{d}M(t)\,\text{d}\Gamma(t)=0 \tag{S-4.10.15} \end{equation*}$$

and with such condition the Black-Scholes formula (S-4.5.14) stands.

But, finding two alternative loopholes, doesn't solve either on of them!

To prove a call option fulfills either (S-4.5.9) or (S-4.5.15) need analyze the property of the call option. This was not done in [Shreve]; it only states call option price $c(t,S(t))$ is a Markov process.

In other words, [Shreve] presents method #1 in but doesn't exclude BSM equation (S-4.10.14) and consequently BSM Formula (S-4.5.19) to be applied to American options.

BSM Formula Derivation #2: as an expectation under the risk neutral measure

In [Shreve] Chapter 5.2.4 & 5.3.2 it explains how to derive it as an expectation under the risk neutral measure.

Define Discount Process as
$$\begin{equation*} D(t):=exp\left\{-\int_0^tR(s)\,\text{d}s\right\} \tag{S-5.2.17} \end{equation*}$$ , where $R(t)$ is the adapted interest rate process. Then one has
$$\begin{equation*} \,\text{d}D(t) = -R(t)D(t)\,\text{d}t \tag{S-5.2.18} \end{equation*}$$

Let $V(T)$ is the pay-off of a derivatives at time $T$. The target is to set up the replicating process $$\begin{equation*} X(t)=\Delta(t)S(t)+\Gamma(t)M(t) \tag{S-4.10.16} \end{equation*}$$ such that $$\begin{equation*} X(T) = V(T), a. s. \tag{S-5.2.28} \end{equation*}$$

Assuming (S-5.2.28) is possible, similar to (S-4.5.2), one has $$\begin{align} \,\text{d}X(t) &= \Delta(t)\,\text{d}S(t) + R(t)(X(t)-\Delta(t)S(t))\,\text{d}t \\ &= RX \,\text{d}t + \Delta (\alpha(t)-R(t))S \,\text{d}t+\Delta \sigma S \,\text{d}W \\ &= RX \,\text{d}t + \Delta \sigma S [\Theta(t)\,\text{d}t + \,\text{d}W(t)]\tag{S-5.2.25} \\ \end{align}$$ , where $$\begin{equation*} \Theta(t)=\frac{\alpha(t)-R(t)}{\sigma(t)} \tag{S-5.2.21} \end{equation*}$$ is the market price of risk.

Also, similarly, comparable to (S-4.5.5) $$\begin{align} \,\text{d} (D(t)X(t)) &= \Delta(t) \sigma(t) D(t) S(t) [\Theta(t)S(t)\,\text{d}t + \,\text{d}W(t)] \\ &= \Delta(t) \,\text{d} (D(t)S(t)) \tag{S-5.2.26} \\ \end{align}$$

By Girsanov Theorem, one define measure $\tilde {\mathbb{P}}$ by the Radon-Nikodym derivative process $$\begin{equation*} Z(t) = exp \left\{ -\int_0^t \Theta(u) \,\text{d}W(u) - \frac{1}{2}\int_0^t \Theta^2(u) du \right\} \tag{S-5.2.11} \end{equation*}$$

, under which $\,\text{d}\tilde{W}$ defined as

$$\begin{align} \tilde{W} (t) &= W(t) + \int_0^t \Theta(u) du \tag{S-5.2.12}\\ d\tilde{W}(t) &= dW(t) + \Theta(t) dt \\ \end{align}$$

is a Brownian Motion.

Then (S-5.2.26) becomes $$\begin{equation*} d(D(t)X(t)) = \Delta(t)\sigma(t)D(t)S(t)d\tilde{W}(t) \tag{S-5.2.27} \end{equation*}$$ , which means under measure $\tilde{\mathbb{P}}$, $D(t)X(t)$ is a martingale.

So due to Martingale Representation Theorem, (S-5.2.28) is achievable; at the same time,

$$\begin{equation*} D(t)X(t) = \tilde{\mathbb{E}}[D(T)X(T)\mid \mathscr{F}(t)] = \tilde{\mathbb{E}}[D(T)V(T)\mid \mathscr{F}(t)] \tag{S-5.2.29} \end{equation*}$$

, so one can from (S-5.2.29) to define $V(t)$ as $$\begin{equation*} D(t)V(t) = \tilde{\mathbb{E}}[D(T)V(T) \mid \mathscr{F}(t)], 0\le t\le T \tag{S-5.2.30} \end{equation*}$$ or $$\begin{equation*} V(t) = \tilde{\mathbb{E}}\left[ exp\left\{-\int_t^TR(u)du\right\}V(T) \mid \mathscr{F}(t)\right], 0\le t\le T \tag{S-5.2.31} \end{equation*}$$

Applying (S-5.2.31) to call option, notice in this product $V(T) = (S(T)-K)^+$, one has $$\begin{equation*} c(t, S(t)) = \tilde{\mathbb{E}}[e^{-r(T-t)} (S(T)-K)^+ \mid \mathscr{F}(t)] \tag{S-5.2.32} \end{equation*}$$

Furthermore, stock price geometric Brownian motion assumption $$\begin{equation*} \,\text{d}S(t) = \alpha S(t) \,\text{d}t + \sigma S(t) \,\text{d}W(t) \tag{S-4.5.1} \end{equation*}$$ leads to $$S(t)=S(0)e^{\sigma \tilde{W}(t) + (r-\frac{1}{2}\sigma^2)}$$ Then after applying such result of $S(t)$ and make use of the normal distribution properties, BSM formula is derived.

The problem of Deriving BSM Formula method #2 is similar to the problem in Deriving BSM Formula method #1 -- that from $X(t) = \Delta(t)S(t)+\Gamma(t)M(t)$ one cannot directly get the first step of (S-5.2.25): $$dX(t) = \Delta(t)dS(t) + R(t)(X(t)-\Delta(t)S(t))dt $$

Here actually implies the self-financing condition, which is not checked.

Naturally, the whole deriving does not exclude American options to arrive (S-5.2.32) and consequently BSM Formula (S-4.5.19) to be applied.

BSM Formula Derivation #3: by Feynman-Kac Theorem

[Shreve] introduced this option in Ch 6.4.

First, Discounted Feynman-Kac Theorem is introduced.

Theorem 6.4.3 Consider the stochastic differential equation $$\begin{equation*} dX(u) = \beta(u, X(u)) du + \gamma(u, X(u)) dW(u). \tag{S-6.2.1} \end{equation*}$$ Let $h(y)$ be a Borel-measurable function and let $r$ be constant. Fix $T > 0$, and let $t \in [0,T]$ be given. Define the function $$\begin{equation*} f(t, x) = \mathbb{E}^{t,x} [e^{-r(T-t)}h(X(T))]. \tag{S-6.4.3} \end{equation*}$$ (We assume that $\mathbb{E}^{t,x} \mid h(X(T)) \mid < \infty$ for all $t$ and $x$.) Then $f(t,x)$ satisfies the partial differential equation $$\begin{equation*} f_t(t, x) + \beta(t, x)f_x(t, x) + \frac{1}{2}\gamma^2(t, x)f_{xx}(t, x) = rf(t, x) \tag{S-6.4.4} \end{equation*}$$ and the terminal condition $$\begin{equation*} f(T, x) = h(x), \forall x \tag{S-6.4.5} \end{equation*}$$

With Discounted Feynman-Kac Theorem ready, it's easy to go ahead. For $V(T) $ defined, and $V(t)=\tilde{\mathbb{E}}[e^{-r(T-t)}h(S(T)) \mid \mathscr{F}(t)]$, naturally there's a $v(t,x)$ so that $V(t)=v(t,S(t))$ and $v(t,x)$ fulfills BSM equation $$\begin{equation*} v_t+rxv_x+\frac{1}{2}\sigma^2x^2v_{xx}=rv \tag{S-6.4.9} \end{equation*}$$

Now if one applies $V(T) = (S(T)-K)^+$ , the call option boundary constraint, it's easy to see the rest is like BSM Formula Derivation #1 -- to solve a backward parabolic equation.

The problem in BSM Formula Derivation #3 is, we can define such a $v(t,x)$, however, we still need self-financing condition to say that $c(t,x) = v(t,x)$.

Summary

So, it seems, all 3 approaches in [Shreve] deriving BSM formula actaully implies prerequisite of self-financing condition, and neglected it.

I've also checked John Hull's book [Hull], it's even worse.

So, Is my understanding correct that [Shreve]'s BSM formula missed the self-financing prerequisite? If so, is there any book on pricing covered this part?

---

Added after reading the replies below

Thank you all, especially Brian B. and emcor's long post.

Thanks but I've to say you guys missed my point.

Let me explain why self-financing is pre-required during Shreve's arguments.

In short, Shreve methods 1 assumes self-financing implicitly, and derives BSM Equation (BSM PDE), based on whicch BSM Formula is derived, or: Self-financing => BSM PDE => BSM Formula.

My original post was asking: how to prove call option is self-financing? without this first step we can't reach BSM PDE nor BSM Formula.

That's why I can't accept @emcor's post as basically actually says "self-financing is fulfilled under the assumed Black-Scholes PDE" , or BSM PDE => Self-financing. Now this makes up a circular argument: BSM PDE => Self-financing (emcore) and Self-financing => BSM PDE => BSM Formula (Shreve).

Let's examine a bit details in Shreve's step of portfolio replication.

Basically Shreve is saying the a portfolio is set up as

$$X(t) = \Delta(t) S(t) + \Gamma(t) M(t)$$

, and this replicates the call option, so

$$X(t) = c(t)$$

. Then from here after use ito's lemma and argue there's no $dt$ item BSM PDE is set up.

For $M(t)$, we always has $$dM(t) = r M(t) dt$$

Now let's check time $t_1$ and the time $t_2$ shortly after $t_1$.

In order to replicate $c(t)$, it's required

$$ \begin{cases} X(t_1) = \Delta(t_1) S(t_1) + \Gamma (t_1) M(t_1)\\ X(t_2) = \Delta(t_2) S(t_2) + \Gamma (t_2) M(t_2) \end{cases} $$

In otherwords,

$$dX(t_1) = X(t_2) - X(t_1) = d(\Delta(t_1)S(t_1)) + d(\Gamma(t_1)M(t_1)) \\ =S(t_1)d(\Delta(t_1)) + \Delta(t_1)d(S(t_1) +d(S(t_1))\cdot d(\Delta(t_1)) \\ + \Gamma(t_1)d(M(t_1)) + M(t_1)d(\Gamma(t_1)) + d(M(t_1)) \cdot d(\Gamma(t_1)) $$

On the other hand, after the portfolio is adjusted to $X(t_1) = \Delta(t_1) S(t_1) + \Gamma (t_1) M(t_1)$, during the short period from $t_1$ to $t_2$, the portfolio becomes

$$X'(t_2) = \Delta(t_1) S(t_2) + \Gamma(t_1) M(t_2)$$

, or $$dX'(t_1) = X'(t_2) - X(t_1) = \Delta(t_1) dS(t_1) + \Gamma(t_1) d M(t_1) $$

To make such replicating portfolio $X(t)$ makes sense, it must be able to "rebalance" at time $t_2$ from $X'(t_2)$ to $X(t_2)$, in otherwords, $X(t_2)$ shall be equals to $X'(t_2)$, or $$dX'(t_1) = dX(t_1)$$

This leads to

$$S(t_1)d(\Delta(t_1)) + \Delta(t_1)d(S(t_1) +d(S(t_1))\cdot d(\Delta(t_1)) \\ + \Gamma(t_1)d(M(t_1)) + M(t_1)d(\Gamma(t_1)) + d(M(t_1)) \cdot d(\Gamma(t_1)) = \Delta(t_1) dS(t_1) + \Gamma(t_1) d M(t_1) $$

so,

$$S(t_1)d(\Delta(t_1)) +d(S(t_1))\cdot d(\Delta(t_1)) + M(t_1)d(\Gamma(t_1)) + d(M(t_1)) \cdot d(\Gamma(t_1)) = 0 $$

Replacing $t_1$ to $t$ one gets

$$S(t)d(\Delta(t)) +d(S(t)) d(\Delta(t)) + M(t)d(\Gamma(t)) + d(M(t)) d(\Gamma(t)) = 0 $$ This is exactly (S-4.10.15).

If (S-4.10.15) holds, we can say that $X(t) = X'(t), \forall t$, or all along the way from time $0$ to $T$, the replicating portfolio can rebalance.

Only under such condition, we can say $$dX(t) = dX'(t) = \Delta(t) dS(t) + \Gamma(t) dM(t) = \Delta(t) dS(t) + \Gamma(t) r M(t) dt \\ = \Delta(t) dS(t) + r (X(t) - \Delta(t) S(t)) dt$$

The last one is exactly (S-4.10.9).

So, what is self-financing? It means the car does not have leaking tyres; it means the portfolio is well enclosed, the pay-off are up to the market movements, but no one is going to "steal" money from the portfolio or have to add in more money to the portfolio -- it just evolves. The rebalance is only continuously change the composition of the portfolio, not its expected value.

To summarize, (S-4.10.9) and (S-4.10.15) are equivalent, they means self-financing. And self-financing is the prerequisite for deriving BSM Equation.

Answer 1

It is true that the self-financing property of the replicating portfolio seems not explicitly presumed nor shown in Shreve's derivation of the Black-Scholes formula. One may note that a replicating portfolio is by definition a self-financing portfolio which replicates the payoff.

The problem as I see is that Shreve is just suggesting some portfolio and solves the PDE, without actually showing that it is replicating and self-financing. He omits the expression for the weight of the riskfree asset.

Self-financing property is fulfilled, and can be shown as follows:

A portfolio $V_t=\alpha_tS_t+\beta_t B_t$ (for stock $S_t$ and riskfree bond $B_t$) is self-financing iff $$dV_t=\alpha_tdS_t+\beta_tdB_t$$

Shreve's definition of self-financing in (G-2) "$V(t)=\sum_{k=1}^n w_k(t) S_k(t)$" is rather unclear, the sum here means $n=2$ with $S_1(t)=B_t$, the first asset being a bond, and $S_2(t)=S_t$, the stock.

To replicate a claim $C(S_t,t)$ by a self-financing portfolio of stock and bond: $dV_t=dC_t$

The dynamics of $dC$ can be specified using Ito Lemma on $C(S_t,t)$ (naturally assuming $C\in C^2$):

$dC=\partial_tCdt+\partial_sCdS+\frac{1}{2}\sigma^2S_t^2\partial_{SS}Cdt=\partial_SCdS_t+(\partial_tC+\frac{1}{2}\sigma^2S_t^2\partial_{SS}C)dt$

Now we equate this expression for the option dynamics to explicitly show the self-financing weights: $dC=\alpha_tS_t+\beta_t B_t$

We show that the Black-Scholes PDE is equivalent to following existing self-financing portfolio, by:

$\partial_tC+\frac{1}{2}\sigma^2S_t^2\partial_{SS}C=rC-rS_t\partial_S C$ by BS-PDE.

So when $C$ satisfies the BS-PDE, this means existence of self-financing portfolio, inserting the PDE into $dC$:

$dC=\partial_SCdS_t+(C-S_t\partial_SC)rdt$

In reference to Shreve's notation, this is equivalent to $dX(t) = \Delta dS_t +r(X_t-\Delta S_t )dt $ with $\Delta(t)=\partial_SC$ and $X(t)=C_t$.

We further have the bond-dynamics $dB_t=B_trdt$, so:

$dC=\partial_SCdS_t+(\frac{C_t}{B_t}-\frac{S_t}{B_t}\partial_SC)dB_t$

Finally, the factors for $dS_t$ and $dB_t$ are exactly the replicating portfolio weights:

$\left(\alpha_t=\partial_SC_t,\,\beta_t=\dfrac{C_t}{B_t}-\dfrac{S_t}{B_t}\partial_SC_t\right)$

Hence, the Black-Scholes PDE implies the above existing self-financing portfolio (which by the boundary condition replicates the final payoff aswell), and the Black-Scholes formula remains valid replicating price (for all 3 approaches).

For American options, the PDE would stay the same, but its boundary conditions would change (as @BrianB posted below), including an optimal exercise free boundary function, at which the option is exercised prior maturity when reached by the stockprice. In particular for an American Put, there is no closed form solution under the required boundary conditions, so the replicating portfolio cannot be determined. For an American Call (without dividends), the solutions are the same as for European Call, because one would not exercise an American Call prior maturity, but rather just sell it to capture the positive time value.

As per request, the proof can also be written the other way round:

Let $\left(\alpha_t=\partial_SC_t,\,\beta_t=\dfrac{C_t}{B_t}-\dfrac{S_t}{B_t}\partial_SC_t\right)$ some self-financing portfolio $V$ to replicate an option $C(S_t,t)$.

Then, using the bond-dynamics, $dB_t=B_trdt$:

$dV=\partial_SCdS_t+(\frac{C_t}{B_t}-\frac{S_t}{B_t}\partial_SC)dB_t=\partial_SCdS_t+(C-S_t\partial_SC)rdt$

From Ito Lemma, we find the dynamics of $C(S_t,t)$ aswell:

$dC=\partial_tCdt+\partial_sCdS+\frac{1}{2}\sigma^2S_t^2\partial_{SS}Cdt=\partial_SCdS_t+(\partial_tC+\frac{1}{2}\sigma^2S_t^2\partial_{SS}C)dt$

Equating $dC=dV$:

$\partial_SCdS_t+(\partial_tC+\frac{1}{2}\sigma^2S_t^2\partial_{SS}C)dt=\partial_SCdS_t+(C-S_t\partial_SC)rdt$

This implies $\partial_tC+\frac{1}{2}\sigma^2S_t^2\partial_{SS}C=rC-rS_t\partial_S C$, which is the BS-PDE.

So the above self-financing portfolio for $C(S_t,t)$ implies BS-PDE.

The specific form of $C(S_t,t)$ depends on its particular boundary condition (e.g. $C(S_T,T)=(S_T-K)^+)$ for a call) to make the portfolio replicating, then $C$ can be determined as solution of BS-PDE. For American Put, the PDE has boundary conditions that cannot be solved, such that self-financing weights do not exist in closed form.

Answer 2

You seem particularly frustrated that these formula derivations are not excluding, e.g., American options. But keep in mind that, up through the derivation of the PDE, there is nothing in them that assumes a particular payoff condition. The PDEs such as

$$\begin{equation*} f_t(t, x) + \beta(t, x)f_x(t, x) + \frac{1}{2}\gamma^2(t, x)f_{xx}(t, x) = rf(t, x) \tag{S-6.4.4} \end{equation*}$$

apply to all payoffs (satisfying a few conditions), including American options and other exotics. In particular, it is just as possible to replicate American option value (or, at least, optimally-exercised American option value) with stocks and bonds as it is to replicate European option value within the BSM framework. I refer you to emcor's carefully constructed answer (which I have upvoted, and recommend you accept) on some of those aspects.

Now, here's something that will really drive you nuts: what happens when we extend the BSM SDE

$$\begin{equation*} \,\text{d}S(t) = \alpha S(t) \,\text{d}t + \sigma S(t) \,\text{d}W(t) \end{equation*}$$

to include random jumps

$$\begin{equation*} \,\text{d}S(t) = \alpha S(t) \,\text{d}t + \sigma S(t) \,\text{d}W(t) - S(t) \, j(Z(t)) \, d \, \Pi(t) \end{equation*}$$

for a poisson process $\Pi$ and a function of some extra unobserved random process $Z$?

Here, replication becomes impossible due to the unknown jump sizes. Yet to price options we still use effectively the same PDE derivation. The economic argument for validity becomes one not of exact replication, but rather of replication converging in distribution to options portfolio value for a dealer holding an $N$-instrument portfolio as $N$ grows large. (One can also apply economic equilibrium argument with essentially the same mathematical consequences).

Answer 3

Let me begin to say that this was one of the most interesting and well written questions I've read in a long time. Even though you have already answered your own question I would like to clear out some terminology and also present my theory as for why many people make the same mistake as Shreve.

First I would like to point out that it makes no sense to say that a product (e.g. an American option) is self-financing. The concepts of self-financing and replication applies to (portfolio) strategies only. For example in (G-1) $\ w(t) = \left(w_1(t), w_2(t), ..., w_n(t)\right)_{t \geq 0} \ $ is the portfolio strategy, and $V_w(t)$ is the corresponding value process. And it is $w$ which is self-financing, not the value process $V_w$.

Now to the important point. Shreve's derivation in §4 follow the same argument as Black-Scholes original proof. The idea is to assume many things (e.g. self-financing) and later show that they are true. With this direct approach you save a lot of work, since the only thing that matters is that your candidate strategy $(\Delta, \Gamma)$ is self-financing, and hence replicating. I guess many people read through B-S paper but didn't wait for the finishing touches.

This is the formal way if ending Shreve's §4-derivation. To prove that the portfolio strategy $(\Delta, \Gamma)$ is self-financing is actually a direct consequence of Itô's Lemma. This might be another reason why Shreve left it out?

$$\int_0^t \Delta(u) dS(u) + \int_0^t \Gamma(u) dB(u) \\ = \int_0^t\frac{\partial C}{dS}(u,S(u)) dS(u) + \int_0^t \left(\frac{\partial C}{dt}(u,S(u))+\frac{\sigma^2}{2}S^2(u)\frac{\partial^2 C}{dS^2}(u,S(u)) \right)du \\ = C(t,S(t)) - C(0,S(0)).$$

This is exactly the integral form of (G-2) and hence proves that the strategy $(\Delta, \Gamma)$ is self-financing.

Answer 4

very deep, elaborate question. It seems to me, though, that you're somewhat missing the point: The fundamental theorem of asset pricing is that in a no-arbitrage environment, asset prices divided by some numeraire are martingales (with respect to a measure dependent on the numeraire, with the measure being unique if the market is complete with respect to the asset).

Now, an asset (in this context) is something you can buy and sell at its price, denominated in domestic currency, with no intermediate cash flows. The "self-financing" condition means just that: no intermediate cash flows.

As such, contrary to what some people here said, it does make sense to speak of an asset or product as "self-financing": it's just a portfolio composed of (a quantity of one) of one asset, and with $V = 1 \cdot P(t)$ we have $dV = 1 \cdot dP(t)$, that is: the value of the portfolio is the price, and the change in value is the change in price. That sounds trivial, but it excludes dividends, interest, etc: Then the change in value would be the change in price plus the cash flow.

Now, a good number of things are not assets under these criteria, for example a dividend paying stock, or a unit of cash (pays interest), or of course a unit of a foreign currency. However, a stock with continuous dividends reinvested in the stock $S(t) e^{qt}$, or a unit of the money market account $\beta(t)$, or a unit of the foreign money market account multiplied with the FX rate $\beta_f(t)\,X(t)$ - those are "self financing", i.e. with no intermediate cash flow, thus the fundamental theorem of asset pricing holds for them, and they're martingales (properly discounted by the numeraire). This allows pricing of dividend paying stocks, composite options, quanto options, etc. (Note that strictly speaking, you could hold a unit of cash as cash and not receive interest, but that of course is an arbitrage opportunity, because you could put it in the money market account instead receiving interest; thus cash, for better or for worse, is not an asset in the derivatives pricing world... :-)

Now, an option is obviously self-financing, it has no intermediate cash flows. Thus, it's a martingale (discounted), thus we can price it with Black Scholes (in an arbitrage free market). We don't need to show that it's self-financing, it just is. Now, of course it's possible to conceive of some weird special options with intermediate cash flows, and those would not be self financing, and, lo and behold, they would have a different price. If you have an option that gives you some sort of dividend or interest occasionally, you'd rather have that, and it would command a higher price than the "vanilla" option.

One more point: I did not understand the distinction you are trying to make between European and American options. Both are self-financing, with no intermediate cash flows, and both can be priced with the fundamental theorem of asset pricing under the appropriate boundary conditions.

tl;dr: Self-financing just means "no intermediate cash flow", and it is indeed a prerequisite for the fundamental theorem of asset pricing to hold, but it applies to options (both European and American), and is not sneakily/implicitly assumed, or something that ought to be "proven" from some prior axioms, but a defining characteristic of the product we're trying to price. Without that characteristic, the price changes and the Black Scholes formula doesn't hold.

Answer 5

OK, I think now I got the point, after comparing to Shreve's "Stochastic calculus for finance I, The binomial asset pricing model", the simpler case.

The pricing theory in continuous time is:

1. Define the wealth process $X(t)$, by definition, it is self-financing: $$d X(t) = \Delta(t) dS(t) + r (X(t) - \Delta(t)S(t)) dt$$
2. Define risk-neutral measure $\widetilde{\mathbb{P}}$. Then $D(t)X(t)$ is a martingale under $\widetilde{\mathbb{P}}$.
3. Define price of the derivative security $V(t)$ via conditional expectation of discounted final paying off. $$D(t)V(t) := \widetilde{\mathbb{E}}[D(T)V(T) \mid \mathscr{F}(t)], 0\le t\le T \tag{S04b-5.2.30}$$ By denition $D(t)V (t)$ is a martingale.
4. By Martingale Representation Theorem (Theorem S04b-5.3.1), $X(t) = V (t)$ can be achieved via choosing $$X(0)=V(0)$$ and $$\Delta (t) = \frac{\widetilde{\Gamma}(t)}{\sigma(t)D(t)S(t)}, 0\leq t \leq T \tag{S04b-5.3.8}$$ So the replication prole is set up.

Shreve04b is slightly different from Shreve04a as in binomial tree part it's easy to define $V_i$ during backward replicating.

Shreve04b $\S 4$ deriving BSM formula omitted some points, that's true; but $\S 5$ covered all that up.

So Shreve's book is correct (how smart it could be to find that out? :p)

Answer 6

Shreve did not in my opinion approach the problem rigorously. The only economically meaningful portfolio is one that is self-financing (in Shreve's words, you cannot decided tomorrow what to invest in today). However he did not explicitly state in his derivation that the portfolio was self-financing, and that is where I feel he lacks rigor.

Shreve starts his derivation with a portfolio X with dynamics $$dX=\Delta dS_t +r(X_t-\Delta S_t)dt $$

A more rigorous approach is the following:

Consider a market with the following assets:

$$ S_t, \,\, M_t $$ which satisfy $$ dS=\mu Sdt+\sigma S dW_t, \,\, dM=r M dt $$ Let X be a portfolio containing S and M:

$$X_t = \Delta S_t + \Gamma M_t $$

Further, let $$d X_t = \Delta dS_t +\Gamma dM_t\,\,(\mathrm{self}\,\, \mathrm{financing}) $$ $$=\Delta dS_t + r\Gamma M_t dt $$ Rearranging, $$ \Gamma M_t= X_t-\Delta S_t $$ Substituting this into the previous equation yields $$dX_t = \Delta dS_t +r(X_t-\Delta S_t )dt $$

Shreve's derivation then follows.