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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?

MMFdW
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    I haven’t had time to read the paper yet (and perhaps I will never read it :d ), but I think the authors must be very confident in themselves. The level of math used in the paper makes me doubt a little about what these authors wrote. – NN2 May 27 '23 at 18:36
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    Does this new version address the points made by Kevin and Gordon? – Bob Jansen May 27 '23 at 18:42
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    I see two close votes for this question which is a shame. But I think there is no flaw in the Black Scholes replicating argument. The clearest explanation of the replicating argument is imo by Bjork in his book Arbitrage Theory in Continuous Time. If this question has not been closed by tomorrow I'll try to summarize Bjork's argument here. – Frido May 27 '23 at 23:51
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    Ever since no arbitrage theory was put on rigorous footing by using martingale theory papers popped up that proudly presented "flaws" in BSM's argumentation. Be it as it may. Here is one of the many papers that show that no damage was done in the end. BSM did what was possible with the rigor availlable at that time. – Kurt G. May 28 '23 at 08:03
  • Hi Bob, thank you for your question. The answer is yes, we take those points into account. Specifically, Kevin correctly points out that there are (well-known) mistakes in the original paper of Black and Scholes. Merton repaired these mistakes with the self-financing condition (although he did not give it this name, the term “self-financing” was coined by Harrison and Kreps). The preprint shows, however, that the self-financing condition is mathematically incorrect, which is the first mistake we refer to above. @bob – MMFdW May 28 '23 at 08:09
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    @MMFdW Well-known mistakes that have been fixed by other well-known papers. If there still is anything incorrect at the end of the first quarter of the 21st century I'd be happy to look into specific details If you spelled them out here formally. – Kurt G. May 28 '23 at 08:14
  • @Kurt In the preprint we carefully write down our arguments, the opening post is just a brief summary of them. The three mistakes we find are different from the well-known ones, but if you know any papers that discuss one of these three as well, please do let us know. – MMFdW May 29 '23 at 11:00
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    Just read the edit you made 3h ago. The equation $$ \int_0^tdw_s=\int_0^t\alpha_s,dx_s+\int_0^t\beta_s,db_s. $$ is the widely accepted definition of the trading strategy $(\alpha,\beta)$ to be self-financing. In the unlikely event that your preprint is correct this would not only invalidate BSM but the entire mathematical finance literature that has been written about self-financing trading strategies in continuous time since the seminal papers of Harrison, Kreps and Pliska. – Kurt G. May 30 '23 at 13:39
  • "This narrative suggests that the investor “comes into" time t with $α_{t−∆_t}$ stocks and “leaves" time $t$ with $\alpha_t$ stocks, by changing his stock position “instantaneously" at time $t$. The latter is mathematically impossible, however, because in discrete time the shortest period over which such a change can be implemented is defined as $∆t$." – fes Jun 19 '23 at 09:09
  • Pretty much all discrete time models in economics make this assumption. It is an approximation but not a mistake. You could also assume the trading decision is made at $t+\Delta \delta$, where $\delta \rightarrow 0+$. Think of it as trading at the end of each day/month. – fes Jun 19 '23 at 09:15
  • The replication argument in BS is of course problematic though since you cannot trade continuously, but this is well known. – fes Jun 19 '23 at 09:19
  • Actual price processes are also discrete so the stock cannot really follow a GBM – fes Jun 19 '23 at 09:56
  • @fes, thank you for sharing this. We have also wondered for a while whether this 'approximation' is actually just a minor issue. However, as we show in the next paragraphs of the preprint, the 'approximation' has serious consequences for the stochastic properties of Merton's model (even if it does not cause problems in a non-stochastic model). – MMFdW Jun 19 '23 at 19:14
  • Moreover, from a strictly mathematical perspective, in a discrete-time model where $\Delta t$ is one day, there exists no such thing as an infinitesimal period $\delta$. So your example of trading at the end of the day seems to imply a continuous-time environment with discrete rebalancing (e.g., the real world). This is not the model of Merton, however, as he explicitly starts in discrete time and claims to go to continuous time only when he takes the limit of his model. This limiting argument is an additional reason to carefully distinguishing between discrete and continuous time. – MMFdW Jun 19 '23 at 19:14

3 Answers3

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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.

Frido
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  • Hi Frido, thank you for sharing your thoughts. We refer to the textbook of Björk in the preprint, because to his credit he is one of the few authors who point out that there is no formal proof for the continuous-time self-financing condition. But as he still uses the condition, mistake (i) above applies to his analysis as well. – MMFdW May 29 '23 at 10:58
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    @MMFdW Hi Mark/Frans, I'll read your paper again but I still do not see why the self-financing condition is mis-specified. Assuming the existence of $F$ (which exists as a solution to the PDE) then by the lemma $dV = h_S dS + h_B dB$, which automatically implies by Ito's lemma that $ dh_S(S+dS) + dh_B (B+dB) = 0$ which is the condition that there is no external financing. – Frido May 29 '23 at 11:16
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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

Jon Ingersoll
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  • Hi Jon, the preprint shows that Merton (1977) uses the self-financing condition too. In addition, the preprint has a separate section on the binomial formula, which shows that the underlying difference equation is mathematically correct but does not reflect a hedging argument. Instead, the equation follows mechanically from the implicit assumption in the model that stocks exposed to systematic risk are not exposed to idiosyncratic risk. Hence, the hedging argument of BS cannot be interpreted as the limit of a binomial hedging argument (as there is no binomial hedging argument to begin with). – MMFdW Jun 15 '23 at 17:54
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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} $$\tag*{$\Box$} \quad $$

Kurt G.
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  • Hi Kurt, thank you for this, and we agree with what you show: the Black-Scholes formula and PDE are consistent with the self-financing condition (this is no surprise indeed, since the condition is used in deriving the formula). However, the preprint shows that the self-financing condition does not reflect portfolio rebalancing without inflows or outflows of external funds. Unless there is a mistake in the preprint, the self-financing condition as well as the option pricing formula are therefore both incorrect (although both are indeed consistent with each other). – MMFdW May 30 '23 at 16:22
  • @MMFdW It seems to me that you are trying to make the following point: In continuous time the strategy $\alpha_t$ does not have finite variation. Therefore, writing $B=e^{rt},,$ we have $$dC=\underbrace{S,d\alpha+B,d\beta}{(A)}+\underbrace{\alpha,dS+\beta dB}{(B)}+d\langle \alpha,S\rangle,.$$ The terms (A) and (B) have clear economic interpretations in discrete time. The covariation term spoils this. I have not given much though to this yet. If this is a case for research I'd recommend – Kurt G. May 30 '23 at 17:11
  • to focus on that rather than writing a BSM centric preprint in non standard notation such as $[\text{some process}|{\cal F}_t],.$ – Kurt G. May 30 '23 at 17:13
  • @MMFdW The matter with infinite variation is discussed in this old MSE post. As far as I see, modern continuous time finance defines the cumulative cost of a strategy to be (in the notation from above) $$C(S_t,t)-\int_0^t\alpha_u,dS_u-\int_0^t\beta_u,dB_u,.$$ This leads to the self-financing condition $$dC=\alpha,dS+\beta,dB$$ that I was promoting. Cheap trick? Maybe. To me the bottom line is that in practice nobody will be able to trade with infinite variation of $\alpha_t,.$ No option can be hedged perfectly. What counts is – Kurt G. May 30 '23 at 18:14
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    the insight given by BSM that the risk-neutral measure should be used for pricing. – Kurt G. May 30 '23 at 18:14
  • Hi Kurt, indeed, the question of how to interpret the covariance term was one of the things that started our thinking as well. Interestingly, Merton (1971) describes in footnote 10 that he also struggled a bit with this. While this question kept us busy for some time, in the end it turned out that the appearance of the covariance term is a symptom of the timing mistake that Merton makes earlier in his analysis. We show that this mistake causes more problems with the self-financing condition (and therefore also with the definition of the cumulative cost of a strategy, as one implies the other). – MMFdW May 31 '23 at 09:56