Three mathematical mistakes in Black-Scholes-Merton option pricing?

Question

In this preprint on arXiv (a revised version of the one discussed in a post here) we show that there are three mathematical mistakes in the option pricing framework of Black, Scholes and Merton. As a result, the option pricing formula seems incorrect even under the idealized capital market assumptions of Black and Scholes. As the preprint shows in more detail, the three mathematical mistakes are:

i) The self-financing condition is misspecified (i.e., it does not express the concept of portfolio rebalancing without inflows or outflows of external funds);

ii) Even if one assumes that the self-financing condition is correctly specified (i.e., if one sidesteps mistake (i)), there is a circularity in the proof that Black and Scholes provide for their claim that a rebalanced portfolio of stocks and risk-free bonds can replicate an option;

iii) Even if one also assumes that the rebalanced portfolio replicates an option (i.e., if one sidesteps mistakes (i) and (ii)), the PDE of Black and Scholes implies that there are paths where the rebalanced portfolio is not self-financing or does not replicate an option.

To facilitate the discussion a little bit, let's focus on mistake (i) and set aside (ii) and (iii). Staying close to the notation of Black and Scholes, the preprint summarizes that derivations of the option pricing formula consider a replicating portfolio of $\alpha_{t}$ stocks with value $x_{t}$ and $\beta_{t}$ risk-free bonds with value $b_{t}$. These derivations define the value of this portfolio as: \begin{equation} w_{t}=\alpha_{t}x_{t}+\beta_{t}b_{t}, \end{equation} and define the return as: \begin{equation}\label{return} \int_{0}^{t}dw_{s}=\int_{0}^{t}\alpha_{s}dx_{s}+\int_{0}^{t}\beta_{s}db_{s}. \end{equation} Since applying the product rule of stochastic integration to the portfolio value yields: \begin{equation}\label{prsi} \int_{0}^{t}dw_s=\int_{0}^{t}\alpha_{s}dx_s+\int_{0}^{t}d\alpha_{s}x_{s}+\int_{0}^{t}d\alpha_{s}dx_{s}+\int_{0}^{t}\beta_{s}db_{s}+ \int_{0}^{t}d\beta_{s}b_{s}+ \int_{0}^{t}d\beta_{s}db_{s}, \end{equation} the above definition of the portfolio return implies that: \begin{equation}\label{ctsfc} \int_{0}^{t}d\alpha_{s}x_{s}+ \int_{0}^{t}d\alpha_{s}dx_{s}+\int_{0}^{t}d\beta_{s} b_{s}+ \int_{0}^{t}d\beta_{s}db_{s}=0, \end{equation} which is known as the continuous-time self-financing condition. This condition is believed to reflect that the portfolio is rebalanced without inflows or outflows of external funds, based on a motivation that goes back to Merton (1971). The preprint shows, however, that there is a timing mistake in the analysis of Merton, and that this mistake causes his self-financing condition to be misspecified. That is, the last equation does not reflect the concept of portfolio rebalancing without inflows or outflows of external funds (and the return on a portfolio that is rebalanced without inflows or outflows of external funds is therefore not equal to the second equation). Is our analysis of mistake (i) in the preprint correct, or do we make a mistake somewhere ourselves?

Answer 1

Don't take this as an answer per se, but as mentioned in my comment more a summary of imo Bjork's clear explanation that hopefully can convince you there is nothing wrong with the BS PDE and self-financing portfolio, even though the original Black-Scholes derivation may leave room for some doubt.

So let's assume that the market under $\mathbb P$ is $$ dS(t) = \mu S(t) dt + \sigma S(t) dW(t) \\ dB(t) = rB(t) dt $$ with $W(t)$ a standard Brownian motion, and following BMS' original assumptions $r, \sigma$ are constants.

The crux is I believe the following lemma:

Lemma Assume there exists a scalar process $F(t)$ such that $$ \frac{dF(t)}{F(t)} = w_B(t) \frac{dB(t)}{B(t)} + w_S(t) \frac{dS(t)}{S(t)} $$ where $w_B, w_S$ are adapted, and for all $t$ $$ w_B(t) + w_S(t) = 1 $$ Then the process defined by $$ V(t) = h_B(t) B(t) + h_S(t) S(t) \\ h_B(t) = w_B(t) \frac{ F(t)}{B(t)},\; h_S(t) = w_S(t) \frac{ F(t)}{S(t)} $$ is self-financing and $V(t) = F(t)$ for all $t$.

Proof That $V(t) = F(t)$ for all $t$ is clear from the definition, but that doesn't mean it's self-financing. But \begin{align} dV(t) &= d\left[ w_B(t) \frac{ F(t)}{B(t)} B(t) + w_S(t) \frac{ F(t)}{S(t)} S(t) \right] \\ &= d [w_B(t) F(t) + w_S(t)F(t) ] \\ &= dF(t) \end{align} because $w_B(t) + w_S(t) = 1$ for all $t$.

Now, it's pretty clear I think that the following Theorem holds (I'll use subscripts to denote partial derivatives):

Theorem Given the market under $\mathbb P$ as above and define $F$ as the solution to $$ F_t + rSF_S + \tfrac12 \sigma^2 S^2 F_{SS} = rF \quad (*)\\ F(T,S(T)) = \Phi(S(T) $$ then the process $V(t) = h_B(t) B(t) + h_S(t) S(t)$ with $$ h_B(t) = \frac{F(t) - S(t)F_S(t)}{B(t)}, \; h_S(t) = F_S(t) $$ is self-financing and for all $t$ we have $V(t) = F(t)$.

Proof Again it is clear that $V(t) = F(t)$, and we just need to demonstrate that it is self-financing. By an application of Ito's lemma we can write $$ \frac{dF}{F} = \frac{F_t + \tfrac12 \sigma^2 S^2 F_{SS}}{rF} \frac{dB}{B} + \frac{SF_S}{F} \frac{dS}{S} $$ Since $F$ satisfies the PDE (*) we can write this as $$ \frac{dF}{F} = \frac{rF - rSF_S}{rF} \frac{dB}{B} + \frac{SF_S}{F} \frac{dS}{S} $$ So it is clear that $$ \frac{rF - rSF_S}{rF} + \frac{rSF_S}{rF} = 1 $$ and therefore $V$ is self-financing as per the Lemma above.

Answer 2

The Black-Scholes formula is the proper limit of the binomial formula, and there seems little doubt that the derivation of the binomial model is correct as is uses no math beyond a bit of algebra. It seems to me, therefore, that the Black-Scholes formula must be correct if we assume that the binomial model converges to a diffusion. And if it doesn't, a lot more than Black-Scholes is in trouble.

You might also see Merton (1977) "On the pricing of contingent claims and the Modigliani-Miller theorem" which uses a completely different derivation

Answer 3

There is no doubt that the the Black & Scholes formula for the European call $$\tag{1} C(S_t,t)=S_t\Phi(d_1)-e^{-r(T-t)}K\Phi(d_2) $$ where $$\tag{2} d_{1,2}=\frac{\log(S_t/K)+r(T-t)\pm\sigma^2(T-t)/2}{\sigma\sqrt{T-t}} $$ satisfies the Black & Scholes PDE $$\tag{3} \partial_tC+\frac{1}{2}\sigma^2S^2\partial_{SS}C+rS\partial_SC-rC=0\,. $$ Proof. Take (1) and differentiate. Use $$\tag{4} S_t\Phi'(d_1)-e^{-r(T-t)}K\Phi'(d_2)=0\,. $$ $$\tag*{$\Box$} \quad $$ This proof also shows $$\tag{5} \partial_SC(S_t,t)=\Phi(d_1)\,. $$ The trading strategy to hold $$\tag{6} \alpha_t:=\Phi(d_1)=\partial_S C(S_t,t) $$ units of the stock $S_t$ and $$\tag{7} \beta_t:=\frac{C(S_t,t)-\alpha _tS_t}{e^{rt}}=e^{-rT}K\Phi(d_2) $$ units of the money market account $e^{rt}$ is self-financing.

Proof. By (1) the portfolio value $\alpha_t S_t+\beta_t e^{rt}$ is equal to the call price. By the widely accepted definition of the trading strategy to be self-financing we therefore have to verify that the call price satisfies $$\tag{8} dC=\alpha_t\,dS_t+\beta_t \,d(e^{rt})=\alpha_t\,dS_t+r\beta_t\,e^{rt}\,dt\,. $$ By Ito's formula and the Black-Scholes PDE (3) and using $$\tag{9} dS_t=\sigma S_t\,dW_t+rS_t\,dt $$ we have \begin{align} dC&\stackrel{\text{Ito}}{=}\partial_t C\,dt+\underbrace{\partial_SC}_{\alpha_t}\,dS_t+\frac{1}{2}\partial_{SS}C\,d\langle S\rangle_t\\ &\stackrel{(9)}{=}\partial_t C\,dt+\alpha_t\,dS_t+\frac{1}{2}\sigma^2S_t^2\partial_{SS}C\,dt\\ &\stackrel{(3)}{=}rC\,dt-r\alpha_tS_t\,dt+\alpha_t\,dS_t\\ &\stackrel{(7)}=r\beta_t\,e^{rt}\,dt+\alpha_t\,dS_t\,. \end{align} $$\tag*{$\Box$} \quad $$