# Black–Scholes Option Pricing Revisited?††thanks: As we sometimes can hardly
believe our results, we wonder whether our analysis overlooks something. To
the best of our knowledge, however, our proofs are correct, and we therefore
share them with a broader audience so that they can be further scrutinized.
While any mistakes are ours, we are grateful to everyone who helped us by
commenting on our analysis or with more general advice. We thank especially
Karma Dajani, as well as Alex Boer, Ron Doney, Richard Gill, Jakob de Haan,
Jeroen van der Hoek, Roger Laeven, Iman van Lelyveld, Roger Lord, Rodney
Ramcharan, Edo Schets, Rogier Swierstra and Folkert van Cleef. Mark Mink
<EMAIL_ADDRESS>is at De Nederlandsche Bank (DNB), P.O. Box 98, 1000AB,
Amsterdam, The Netherlands. Frans de Weert<EMAIL_ADDRESS>is at
Rabobank, P.O. Box 17100, 3500HG Utrecht, The Netherlands. All views expressed
are ours and not those of our employers.

Mark Mink  Frans J. de Weert

###### Abstract

The hedging argument of Black and Scholes (1973) hinges on the assumption that
a continuously rebalanced asset portfolio satisfies the continuous-time self-
financing condition. This condition, which is a special case of the
continuous-time budget equation of Merton (1971), is believed to
mathematically formalize the economic concept of an asset portfolio that is
rebalanced continuously without requiring an inflow or outflow of external
funds. Although we sometimes find it hard to believe our results, we believe
that we show with three alternative mathematical proofs that the continuous-
time self-financing condition does not hold for rebalanced portfolios. In
addition, we pinpoint the mistake in the derivation that Merton (1971) uses to
motivate the continuous-time budget equation. Specifically, by inadvertently
equating a deterministic variable to a stochastic one, Merton (1971)
implicitly assumes that the portfolio rebalancing does not depend on changes
in asset prices. If correct, our results invalidate the continuous-time budget
equation of Merton (1971) and the hedging argument and option pricing formula
of Black and Scholes (1973).
Keywords: Self-financing condition, Black–Scholes option pricing, continuous-
time budget equation, binomial option pricing, probability theory.

##  1 Introduction

A key assumption that underlies the option pricing formula of Black and
Scholes (1973) is that the continuously rebalanced asset portfolio that
replicates the option satisfies the continuous-time self-financing
condition.111Harrison and Kreps (1979), who coined the term self-financing,
point out that “this restriction is implicit in the original treatment of
Black and Scholes (1973) and is explicitly displayed in Merton (1977).” Our
analysis too shows that the option pricing formula of Black and Scholes (1973)
implies the continuous-time self-financing condition, and the condition is
often used explicitly to derive the formula. For some textbook examples, see
Duffie (2001), Lamberton and Lapeyre (2011), Merton (1990) and Shreve (2004).
This condition is believed to mathematically formalize the economic concept of
an asset portfolio that is rebalanced continuously without requiring an inflow
or outflow of external funds. The continuous-time self-financing condition has
not been formally derived in continuous time, however, but is defined based on
a discrete-time motivation in Merton (1971). When he derives the continuous-
time budget equation of an investor, of which the continuous-time self-
financing condition is a special case, Merton (1971) points out that “it is
not obvious" how to derive the condition in continuous time directly. Instead,
“it is necessary to examine the discrete-time formulation of the model and
then to take limits carefully to obtain the continuous-time form." Merton
(1973b, 1977) and subsequent studies refer back to this argument when they use
the continuous-time self-financing condition, although Harrison and Pliska
(1981) add that “we have no doubt that these are the right definitions, but a
careful study of this issue is certainly needed." More recently, Björk (2009)
emphasizes that the reasoning behind the condition “has only been of a
motivating nature."

Although we sometimes find our results hard to believe, we think that our
analysis mathematically proves that the continuous-time self-financing
condition does not hold for rebalanced portfolios. When this condition was
introduced it made techniques from continuous-time stochastic calculus
available for the analysis of intertemporal consumption and portfolio
optimization problems (e.g., Merton 1971, 1973a). To be able to use these
techniques, however, it does not suffice mathematically to define a
continuous-time self-financing condition based on a discrete-time motivation.
In fact, with three alternative mathematical proofs, our analysis shows that
the continuous-time self-financing condition does not hold for rebalanced
portfolios. We also pinpoint the mistake in the derivation that Merton (1971)
uses to motivate the continuous-time budget equation, of which the continuous-
time self-financing condition is a special case. Specifically, we show that
Merton (1971) inadvertently equates a deterministic variable to a stochastic
one, and thereby implicitly assumes that the rebalancing does not depend on
changes in asset prices. If correct, by invalidating the continuous-time self-
financing condition, our analysis invalidates the continuous-time budget
equation of Merton (1971) as well as the hedging argument and option pricing
formula of Black and Scholes (1973).

To provide more context to the result that the option pricing formula of Black
and Scholes (1973) is incorrect, we discuss the argument of Cox, Ross and
Rubinstein (1979) that the continuous-time formula of Black and Scholes (1973)
is equal to the limit of their discrete-time binomial option pricing formula.
More specifically, we show that the binomial model of Cox, Ross and Rubinstein
(1979) implicitly assumes that options and their underlying stocks are only
exposed to systematic risk (i.e., to the market portfolio) and not to
idiosyncratic risk. We then show that this assumption directly implies the
partial differential equation of Black and Scholes and the difference equation
of Cox, Ross and Rubinstein (1979), without using the (continuous-time or
discrete-time) self-financing condition or a hedging argument. From a
mathematical perspective, the difference equation that underlies the binomial
option pricing formula therefore reflects an implicit assumption rather than a
hedging argument. From an economic perspective this assumption seems difficult
to rationalize, and our result that the option pricing formula of Black and
Scholes is incorrect implies that it does not hold in continuous time.

If our analysis is correct, we can only speculate about the sign and magnitude
of the mispricing that results when using the formula of Black and Scholes
(1973) to determine the value of an option.222The option pricing formula of
Black and Scholes (1973) has served as a theoretical benchmark for almost
fifty years. Over time, the literature documented several empirical
observations that seem at odds with this benchmark. While these findings were
typically regarded as evidence against the idealized capital market conditions
under which the benchmark was derived, our analysis implies that even under
these idealized conditions, the formula of Black and Scholes (1973) does not
provide the correct value of an option. Moreover, there would no longer be a
basis for the argument that hedging an option with the underlying stock yields
a risk-free portfolio. While option traders do not use the formula of Black
and Scholes (1973) directly, they typically use the underlying hedging
argument. This argument contributed to the development of option markets by
allowing traders to price options solely based on their view of the volatility
of the underlying stock, without the need to have a view on the expected
return of the stock and the expected return of the option. By disproving this
hedging argument, our analysis implies that trading options cannot be reduced
to trading the volatility of the underlying stock. Still, the value of an
option can be expressed as its expected payoff at maturity discounted by its
expected return. This expression resembles the option pricing formula of Black
and Scholes (1973), but contains the percentage drift on the stock and the
(time-varying) percentage drift on the option instead of the risk-free rate of
return.

##  2 Analysis

We first summarize a standard derivation of the replicating portfolio that
underlies the option pricing formula of Black and Scholes (1973). Staying
close to their original notation, consider a call option with value $w_{t}$ at
time $t$ that is written on a stock with value $x_{t}$. As we adopt a
continuous-time setting, we focus on an infinitesimal $dt\to 0$ throughout the
analysis. The value of the stock is assumed to follow a geometric Brownian
motion:

$\displaystyle\frac{dx_{t}}{x_{t}}=\mu dt+\sigma dW_{t}\hskip
7.11317pt\text{with $x_{0}>0$ given},$ (1)

where $\mu$ is the percentage drift, $\sigma>0$ is the percentage standard
deviation, and $\\{W_{t}\\}_{t\geq 0}$ is a Wiener process on a filtered
probability space $(\Omega,\mathcal{F},\\{\mathcal{F}_{t}\\}_{t\geq
0},\mathbb{P})$ satisfying the usual conditions. Black and Scholes derive a
pricing formula for this option by constructing a replicating portfolio. To
date, their replicating portfolio is typically replaced by the one defined by
Merton (1973b, 1977), who replicates the option with a portfolio of $w_{1,t}$
stocks with value $x_{t}$ and $\beta_{t}$ bonds with value
$b_{t}=\text{exp}\left(rt\right)b_{0}$ that earn a risk-free return
$db_{t}=b_{t}rdt$.333Bergman (1981) argues that the replicating portfolio
defined by Black and Scholes contains mathematical inaccuracies which can be
overcome by using the replicating portfolio of Merton (1973, 1977). MacDonald
(1997) therefore argues that these inaccuracies are an issue of terminology
rather than methodology. Specifically, Merton (1973, 1977) defines the option
value as:

$w_{t}=w_{1,t}x_{t}+\beta_{t}b_{t},$ (2)

where $w_{1,t}=\partial w_{t}/\partial x_{t}$ is the partial derivative of the
value of the option with respect to the value of the stock (i.e., delta). This
partial derivative changes over time as it depends on the value of the stock
and on the remaining maturity of the option. Black and Scholes thus assume
that the number of stocks changes over time as well, and refer to this trading
strategy as continuous rebalancing. Applying the product rule of stochastic
integration to equation (2) shows that the return on the option over an
infinitesimal $dt\to 0$ equals:

$dw_{t}=w_{1,t}dx_{t}+dw_{1,t}x_{t}+dw_{1,t}dx_{t}+\beta_{t}db_{t}+d\beta_{t}b_{t},$
(3)

which uses the fact that $db_{t}$ is deterministic so that
$d\beta_{t}db_{t}=O\left(dt^{2}\right)$ and can therefore be neglected. Merton
(1973a, 1977) assumes that for $dt\to 0$ the next equality also holds:

$dw_{1,t}x_{t}+dw_{1,t}dx_{t}+d\beta_{t}b_{t}=0,$ (4)

which he substitutes into equation (3) to obtain the return on the option:

$dw_{t}=w_{1,t}dx_{t}+\beta_{t}db_{t}.$ (5)

In summary, the replicating portfolio that underlies the option pricing
formula of Black and Scholes is described by equations (2) and (5), and hinges
on the validity of equation (4). This equation is known as the continuous-time
self-financing condition (a term coined by Harrison and Kreps, 1979). While it
is trivially obvious that the condition holds for an asset portfolio that is
not rebalanced (i.e., for $dw_{1,t}=d\beta_{t}=0)$, the condition is also
believed to hold for asset portfolios that are continuously rebalanced without
requiring an inflow or outflow of external funds, based on a motivation that
goes back to Merton (1971).444Specifically, the continuous-time self-financing
condition is equal to equation (9’) in Merton (1971) when consumption
$C_{t}dt=0$, in which case the rebalanced portfolio does not exhibit inflows
or outflows of funds. The next remark shows, however, that the analysis of
Merton (1971) inadvertently equates a deterministic variable to a stochastic
one, and thereby implicitly assumes that the portfolio rebalancing does not
depend on changes in asset prices.

######  Remark 1.

Equation (8) in Merton (1971) states that an investor with a portfolio of
$\alpha_{t}$ stocks with value $x_{t}$ can consume an amount $C_{t}\Delta
t=-\left(\alpha_{t}-\alpha_{t-\Delta t}\right)x_{t}$ over a discrete time
interval $\Delta t$. Merton then “increments" this equation by $\Delta t$ to
obtain $C_{t+\Delta t}\Delta t=-\left(\alpha_{t+\Delta
t}-\alpha_{t}\right)x_{t+\Delta t}$. Using this “incremented” new stochastic
variable, Merton argues that taking the limit $\Delta t\to 0$ of $C_{t+\Delta
t}\Delta t$ yields $C_{t}dt=-d\alpha_{t}\left(x_{t}+dx_{t}\right)$. The
implications of this argument become clear when taking the limit $\Delta t\to
0$ of the original deterministic variable $C_{t}\Delta t$ in (his) equation
(8), which yields $C_{t}dt=-\lim_{\Delta t\to
0}\left(\alpha_{t}-\alpha_{t-\Delta t}\right)x_{t}$. Merton therefore
implicitly argues that $\lim_{\Delta t\to 0}\left(\alpha_{t}-\alpha_{t-\Delta
t}\right)x_{t}=d\alpha_{t}\left(x_{t}+dx_{t}\right)$, which equates a
deterministic variable to a stochastic one unless $d\alpha_{t}$ is
deterministic for all $t$ (in which case $\partial\alpha_{t}/\partial
x_{t}=0$). His analysis therefore implicitly assumes that the portfolio
rebalancing is deterministic and does not depend on changes in asset prices.

The implicit assumption that the portfolio rebalancing does not depend on
changes in asset prices seems to enter the analysis of Merton (1971) when he
takes the limit of his discrete-time model to go to continuous time. As the
remark showed, before taking this limit, Merton (1971) “increments" all time
subscripts by $\Delta t$ to obtain the stochastic version of his model. In the
context of the self-financing condition (where the portfolio also includes
$\beta_{t}$ bonds and consumption is equal to zero), this procedure assumes
that because the discrete-time self-financing condition holds in realization
(i.e., ex-post) it must also hold stochastically (i.e., ex-ante). While our
proofs focus on the self-financing condition in continuous time, the next
remark shows that this assumption raises the question of how the investor
knows which Wiener process drives the stock returns, as he can only observe
the ex-post realizations of this process.

######  Remark 2.

For a portfolio of $\alpha_{t}$ stocks and $\beta_{t}$ bonds, the ex-post
(i.e., in realization) discrete-time self-financing condition is:

$[\Delta\alpha_{t}x_{t}+\Delta\alpha_{t}\Delta
x_{t}+\Delta\beta_{t}b_{t}+\Delta\beta_{t}\Delta b_{t}|\mathcal{F}_{t+\Delta
t}]=0,$ (6)

which Merton (1971) assumes to also hold stochastically (i.e., ex-ante), so
that:

$[\Delta\alpha_{t}x_{t}+\Delta\alpha_{t}\Delta
x_{t}+\Delta\beta_{t}b_{t}+\Delta\beta_{t}\Delta b_{t}|\mathcal{F}_{t}]=0.$
(7)

The assumption that equation (7) holds implies that an investor can ex-ante
choose the stochastic processes $[\Delta\alpha_{t}|\mathcal{F}_{t}]$ and
$[\Delta\beta_{t}|\mathcal{F}_{t}]$ so that they depend on the stochastic
process $[\Delta x_{t}|\mathcal{F}_{t}]$. The latter is, however,
fundamentally unknown to the investor. For example, in the context of equation
(1), the investor knows that the stock returns are driven by a Wiener process
$\\{W_{t}\\}_{t\geq 0}$, but he does not know which one out of the infinitely
many Wiener process this is, as he can only observe the ex-post realizations
of this process.

When we define $dw_{1,t}=d\alpha_{t}$, Merton (1971) obtains the continuous-
time self-financing condition in equation (4) by taking the limit of the
(stochastic) discrete-time self-financing condition in equation (7). He
thereafter observes in footnote 10 that “it is not obvious from the continuous
version directly" that equation (4) should contain the term $dw_{1,t}dx_{t}$,
but he argues that this term appears when one takes the limit of the discrete-
time self-financing condition to obtain its continuous-time form. The next
proposition shows, however, that the continuous-time self-financing condition
does not hold for the replicating portfolio of Black and Scholes (i.e., it
implies that $\partial w_{1,t}/\partial x_{t}=0$, which contradicts the fact
that the portfolio rebalancing depends on changes in asset prices).

######  Proposition 1.

The continuous-time self-financing condition does not hold for the replicating
portfolio of Black and Scholes.

###### Proof.

Dividing the continuous-time self-financing condition in equation (4) by
$b_{t}$ and subtracting $dw_{1,t}$ and $d\beta_{t}$ from both sides of the
equation yields:

$\frac{x_{t}-b_{t}}{b_{t}}dw_{1,t}+\frac{dw_{1,t}dx_{t}}{b_{t}}=-dw_{1,t}-d\beta_{t}.$
(8)

A Taylor expansion of $dw_{1,t}$ gives:

$dw_{1,t}=w_{11,t}dx_{t}+w_{12,t}dt+\frac{1}{2}w_{111,t}\sigma^{2}x_{t}^{2}dt,$
(9)

which can be substituted into the above equation to obtain:

$\frac{x_{t}-b_{t}}{b_{t}}w_{11,t}dx_{t}+\frac{\left(x_{t}-b_{t}\right)\left(w_{12,t}+\frac{1}{2}w_{111,t}\sigma^{2}x_{t}^{2}\right)+w_{11,t}\sigma^{2}x_{t}^{2}}{b_{t}}dt=-dw_{1,t}-d\beta_{t},$
(10)

where we used $dw_{1,t}dx_{t}=w_{11,t}\sigma^{2}x_{t}^{2}dt$. Using equation
(1) and defining:

$\displaystyle\kappa_{t}$
$\displaystyle=\frac{\left(x_{t}-b_{t}\right)\left(w_{12,t}+\frac{1}{2}w_{111,t}\sigma^{2}x_{t}^{2}+w_{11,t}x_{t}\mu\right)+w_{11,t}\sigma^{2}x_{t}^{2}}{b_{t}},$
(11) $\displaystyle\lambda_{t}$
$\displaystyle=\frac{x_{t}-b_{t}}{b_{t}}w_{11,t}x_{t}\sigma,$ (12)

allows us to write equation (10) as:

$\kappa_{t}dt+\lambda_{t}dW_{t}=-dw_{1,t}-d\beta_{t}.$ (13)

If we define:

$\theta_{t}=-w_{1,t}-\beta_{t},$ (14)

taking the integral over time of equation (13) while conditioning on
$\mathcal{F}_{t_{0}}$ yields:

$\theta_{T}=\theta_{t_{0}}+\int_{t_{0}}^{T}\kappa_{t}dt+\int_{t_{0}}^{T}\lambda_{t}dW_{t},$
(15)

where $T>t_{0}$ is the maturity date of the option. The option contract
implies that $\theta_{T}=-w_{1,T}-\beta_{T}=0$ (since $w_{1,T}=0$ and
$\beta_{T}=0$ if the option matures out of the money, and $w_{1,T}=1$ and
$\beta_{T}=-1$ if the option matures in the money), so that the left hand side
of equation (15) is deterministic. The right hand side of the equation must
therefore be deterministic as well, which due to the Markov property of the
Wiener process implies that $\lambda_{t}=0$ (as we formalize in the appendix)
and therefore $w_{11,t}=0$ for all $t_{0}<t<T$. However, this implication
contradicts the fact that $\partial w_{1,t}/\partial x_{t}=w_{11,t}>0$. ∎

Proposition 1 shows that the continuous-time self-financing condition does not
hold for the replicating portfolio of Black and Scholes. The next proposition
shows, in addition, that the option pricing formula of Black and Scholes only
holds if the continuous-time self-financing condition holds (see also Harrison
and Kreps, 1979). Taken together, these propositions therefore imply that the
option pricing formula is incorrect.

######  Proposition 2.

The option pricing formula of Black and Scholes only holds if the continuous-
time self-financing condition holds.

###### Proof.

The option pricing formula of Black and Scholes states that the value of a
call option with maturity date $T>t$ and strike price $k>0$ is equal to:

$w_{t}=N\left(d_{1,t}\right)x_{t}-\text{exp}\left(-r\left(T-t\right)\right)N\left(d_{2,t}\right)k,$
(16)

where
$d_{1,t}=\frac{1}{\sigma\sqrt{T-t}}\left(\ln\left(\frac{x_{t}}{k}\right)+\left(r+\frac{1}{2}\sigma^{2}\right)\left(T-t\right)\right)$
and $d_{2,t}=d_{1,t}-\sigma\sqrt{T-t}$, and where $N\left(\cdot\right)$ is the
standard normal cumulative distribution function. Since $w_{1,t}=\partial
w_{t}/\partial x_{t}=N\left(d_{1,t}\right)$, equation (16) can also be written
as:

$w_{t}=w_{1,t}x_{t}+\beta_{t}b_{t},$ (17)

where $\beta_{t}=-N\left(d_{2,t}\right)$ and
$b_{t}=\text{exp}\left(-r\left(T-t\right)\right)k$. Moreover, following Black
and Scholes, using Itô’s lemma implies that the option return over an
infinitesimal $dt\to 0$ equals:

$dw_{t}=w_{1,t}dx_{t}+w_{2,t}dt+\frac{1}{2}w_{11,t}\sigma^{2}x_{t}^{2}dt,$
(18)

where $w_{2,t}=\partial w_{t}/\partial t$ is the partial derivative of the
value of the option with respect to the remaining maturity of the option
(i.e., theta), and where $w_{11,t}=\partial^{2}w_{t}/\partial x_{t}^{2}$ is
the partial derivative of $w_{1,t}$ with respect to the value of the stock
(i.e., gamma). As is well-known, equation (16) implies that these partial
derivatives are equal to
$w_{2,t}=-\frac{x_{t}N^{\prime}\left(d_{1,t}\right)\sigma}{2\sqrt{T-t}}+\beta_{t}b_{t}r$
and $w_{11,t}=\frac{N^{\prime}\left(d_{1,t}\right)}{x_{t}\sigma\sqrt{T-t}}$,
where $N^{\prime}\left(\cdot\right)$ is the standard normal probability
density function. Substituting these partial derivatives in equation (18) and
using $b_{t}rdt=db_{t}$ yields:

$dw_{t}=w_{1,t}dx_{t}+\beta_{t}db_{t}.$ (19)

Equations (17) and (19) are the same as equations (2) and (5), and therefore
only hold if the continuous-time self-financing condition in equation (4)
holds. ∎

While Proposition 1 focused on the continuous-time self-financing condition in
the context of the replicating portfolio of Black and Scholes, two alternative
proofs in the appendix show that the continuous-time self-financing condition
does not hold for any rebalanced portfolio (the second proof applies
regardless of whether the rebalancing depends on changes in asset prices). As
the discussion around Remarks 1 and 2 illustrated, Merton (1971) obtains this
condition by taking the limit of the discrete-time self-financing condition,
and both proofs use this limiting procedure as well. In fact, to be more
precise, Merton (1971) does not derive the continuous-time self-financing
condition directly, but this condition is a special case of his continuous-
time budget equation. More specifically, for an investor with wealth $w_{t}$
held in a portfolio of $\alpha_{t}$ stocks and $\beta_{t}$ bonds, Merton
(1971) models the value of the portfolio as
$w_{t}=\alpha_{t}x_{t}+\beta_{t}b_{t}$ and models the return as
$dw_{t}=\alpha_{t}dx_{t}+\beta_{t}db_{t}$. When defining $\alpha_{t}=w_{1,t}$,
these equations are the same as equations (2) and (5), while equation (4)
describes the special case where consumption is equal to zero (see footnote
4). Our analysis therefore implies that the continuous-time budget equation of
Merton (1971) is incorrect as well.555When reading the paper of Merton (1971)
with this result in mind, we realized that his notation does not reflect his
economic narrative on the sequencing between changes in asset prices and
changes in portfolio weights. More specifically, Merton (1971) states that
“prices are known at the beginning of the period” (italics are his) and that
thereafter the new portfolio weights are chosen so that “trades are made at
(known) current prices.” His notation does not reflect this sequencing, since
discrete-time equation (8) in his paper gives the new portfolio weights the
same time subscript as the asset prices. In hindsight, if our analysis is
correct, this simplification does not seem innocuous.

As an alternative to the option pricing formula of Black and Scholes, the
value of an option can be expressed as its expected payoff at maturity
discounted by its expected return. The next remark illustrates that when the
value of the stock evolves according to equation (1), this expression
resembles the option pricing formula of Black and Scholes, but contains the
percentage drift on the stock and the (time-varying) percentage drift on the
option instead of the risk-free rate of return.

######  Remark 3.

At time $t$, the expected payoff on an option at maturity date $T$ is:

$\displaystyle\mathbb{E}\left[w_{T}|\mathcal{F}_{t}\right]=\int_{k}^{\infty}(x-k)dF_{T}(x)=\text{exp}\left(\mu\left(T-t\right)\right)N\left(d^{*}_{1,t}\right)x_{t}-N\left(d^{*}_{2,t}\right)k,$
(20)

where $F_{T}(x)=\Pr\left(x_{T}\leq x\right|x_{t})$ is the cumulative
distribution function of $x_{T}$ conditional on $\mathcal{F}_{t}$. Note that
$d^{*}_{1,t}=\frac{1}{\sigma\sqrt{T-t}}\left(\ln\left(\frac{x_{t}}{k}\right)+\left(\mu+\frac{1}{2}\sigma^{2}\right)\left(T-t\right)\right)$
and $d^{*}_{2,t}=d^{*}_{1,t}-\sigma\sqrt{T-t}$ differ from $d_{1,t}$ and
$d_{2,t}$ in the formula of Black and Scholes unless $\mu=r$. The option value
$w_{t}$ is equal to the expected payoff at maturity times the discount factor
$\text{exp}\left(-\mu_{w,t}\left(T-t\right)\right)$, where $\mu_{w,t}$ is the
(time-varying) percentage drift of the option. If $\mu_{w,t}=\mu=r$ the
resulting expression is the same as the option pricing formula of Black and
Scholes.

###  2.1 Binomial option pricing

The previous analysis implies that the continuous-time option pricing formula
of Black and Scholes is incorrect. Cox, Ross and Rubinstein (1979) argue,
however, that the formula of Black and Scholes is a special limiting case of
their discrete-time option pricing formula. Together, these two observations
raise the question of how the mathematical commonality between both formulas
can be explained, which we examine in this section. Specifically, we show that
the binomial model of Cox, Ross and Rubinstein (1979) implicitly assumes that
the stock and the option are only exposed to systematic risk and not to
idiosyncratic risk. We then show that this assumption directly implies the
partial differential equation of Black and Scholes and the difference equation
of Cox, Ross and Rubinstein (1979), without using the (continuous-time or
discrete-time) self-financing condition or a hedging argument.

To derive their discrete-time option pricing formula, Cox, Ross and Rubinstein
(1979) assume that the stock price and the option price each follow a binomial
process, and point out that “three-state or trinomial stock price movements
will not lead to an option pricing formula based solely on arbitrage
considerations." An implicit restriction of the binomial process is that the
asset is only exposed to systematic risk and not to idiosyncratic risk.
Specifically, assuming binomial processes for the stock and the option implies
that each of them is exposed to just a single risk factor. Otherwise, if the
stock would be exposed to two risk factors, the number of potential outcomes
for its price would at least be equal to four instead of two, and likewise for
the option. Since Cox, Ross and Rubinstein (1979) assume that the expected
returns on the stock and the option exceed the risk-free return, their single
risk factor is priced and is therefore systematic. Propositions 3 and 4 show
that this assumption directly implies the partial differential equation of
Black and Scholes and the difference equation of Cox, Ross and Rubinstein
(1979).

######  Proposition 3.

The continuous-time partial differential equation of Black and Scholes is
directly implied (without using the self-financing condition or a hedging
argument) when assuming that the stock and the option are only exposed to
systematic risk.

###### Proof.

If a stock with value $x_{t}$ and an option with value $w_{t}$ are exposed to
systematic risk only, their excess rates of return are equal to:

$\displaystyle dx_{t}/x_{t}-rdt$
$\displaystyle=\beta_{x}\left(dm_{t}/m_{t}-rdt\right),$ (21) $\displaystyle
dw_{t}/w_{t}-rdt$ $\displaystyle=\beta_{w}\left(dm_{t}/m_{t}-rdt\right),$ (22)

where $dm_{t}/m_{t}-rdt$ is the rate of return on the market portfolio in
excess of the risk-free rate of return, and where $\beta_{x}$ and $\beta_{w}$
indicate the exposure of the stock and of the option to the market
portfolio.666In the capital asset pricing model (CAPM) these parameters are
referred to as ‘beta’, and are defined as the covariance of the return on the
asset with the return on the market portfolio, divided by the variance of the
return on the market portfolio. Defining
$\alpha_{t}=w_{t}\beta_{w}/x_{t}\beta_{x}$ and combining both equations
yields:

$dw_{t}=w_{t}rdt+\alpha_{t}dx_{t}-\alpha_{t}x_{t}rdt,$ (23)

Furthermore, restating equation (18) for the return on the option gives:

$dw_{t}=w_{1,t}dx_{t}+w_{2,t}dt+\frac{1}{2}w_{11,t}\sigma^{2}x_{t}^{2}dt,$
(24)

Subtracting equation (24) from equation (23) yields:

$\left(w_{1,t}-\alpha_{t}\right)dx_{t}=\left(w_{t}r-\alpha_{t}x_{t}r-w_{2,t}-\frac{1}{2}w_{11,t}\sigma^{2}x_{t}^{2}\right)dt.$
(25)

Since the right-hand side of this equation is deterministic the left-hand side
must be deterministic as well, which implies that:

$\alpha_{t}=w_{1,t}.$ (26)

Substituting this result in equation (25) and dropping $dt$ yields:

$w_{2,t}=rw_{t}-rw_{1,t}x_{t}-\frac{1}{2}w_{11,t}\sigma^{2}x_{t}^{2},$ (27)

which is the partial differential equation in equation (7) of Black and
Scholes. ∎

######  Proposition 4.

The discrete-time difference equation of Cox, Ross and Rubinstein (1979) is
directly implied (without using the self-financing condition or a hedging
argument) by their assumption that the stock and the option are only exposed
to systematic risk.

###### Proof.

Cox, Ross and Rubinstein (1979) start with a one-period binomial tree in
discrete time. Using their notation, the stock price at the start of the
period is $S$ and the option price at this time is $C$. At the end of the
period, the stock price is either equal to $uS$ or to $dS$, where
$u>r^{*}>d>0$ and $r^{*}=1+r$ (we added an ∗ to distinguish their notation
from our symbol $r$). Likewise the option price at the end of the period is
either $C^{u}$ or $C^{d}$. As this binomial tree implies that the stock and
the option are only exposed to systematic risk, their excess rates of return
are equal to:

$\displaystyle\left(uS-S\right)/S-(r^{*}-1)$
$\displaystyle=\beta_{S}\left(\left(m^{u}-m\right)/m-(r^{*}-1)\right),$ (28)
$\displaystyle\left(dS-S\right)/S-(r^{*}-1)$
$\displaystyle=\beta_{S}\left(\left(m^{d}-m\right)/m-(r^{*}-1)\right),$ (29)
$\displaystyle\left(C^{u}-C\right)/C-(r^{*}-1)$
$\displaystyle=\beta_{C}\left(\left(m^{u}-m\right)/m-(r^{*}-1)\right),$ (30)
$\displaystyle\left(C^{d}-C\right)/C-(r^{*}-1)$
$\displaystyle=\beta_{C}\left(\left(m^{d}-m\right)/m-(r^{*}-1)\right),$ (31)

where $\left(m^{u}-m\right)/m$ is the rate of return on the market portfolio
when the market goes up, $\left(m^{d}-m\right)/m$ is the rate of return when
it goes down, and $\beta_{S}$ and $\beta_{C}$ indicate the exposure of the
stock and of the option to the market portfolio. Combining these equations
yields:

$\displaystyle C^{u}-Cr^{*}$
$\displaystyle=\frac{C\beta_{C}}{S\beta_{S}}\left(uS-Sr^{*}\right),$ (32)
$\displaystyle C^{d}-Cr^{*}$
$\displaystyle=\frac{C\beta_{C}}{S\beta_{S}}\left(dS-Sr^{*}\right).$ (33)

Using one of these equations to substitute for $\beta_{C}/\beta_{S}$ in the
other gives:

$C=\left[\left(\frac{r^{*}-d}{u-d}\right)C^{u}+\left(\frac{u-r^{*}}{u-d}\right)C^{d}\right]/r^{*},$
(34)

which is the difference equation in equation (2) of Cox, Ross and Rubinstein
(1979). ∎

The previous two propositions show that the partial differential equation of
Black and Scholes and the difference equation of Cox, Ross and Rubinstein
(1979) are directly implied when one assumes that the stock and the option are
only exposed to systematic risk. While Black and Scholes do not make this
assumption, Cox, Ross and Rubinstein (1979) do so implicitly by modelling the
stock price and the option price as binomial processes. From a mathematical
perspective, their difference equation results from this assumption rather
than from a hedging argument. In continuous time, this same assumption implies
the partial differential equation of Black and Scholes, which explains why
both option pricing formulas are mathematically related. Hence, this
relationship between both formulas does not indicate that the continuous-time
hedging argument of Black and Scholes can be interpreted as the limit of a
discrete-time hedging argument by Cox, Ross and Rubinstein (1979). Moreover,
because Propositions 1 and 2 in the previous section show that the option
pricing formula of Black and Scholes is incorrect, the assumption that the
stock and the option are only exposed to systematic risk does not hold in
continuous time. Hence, in continuous time, changes in the value of an option
are partially idiosyncratic even if the underlying stock is only exposed to
systematic risk.

##  3 Conclusion

If our analysis is correct, the continuous-time self-financing condition does
not hold for rebalanced asset portfolios. This result implies that the
continuous-time budget equation of Merton (1971) and the hedging argument and
option pricing formula of Black and Scholes (1973) are incorrect. Moreover,
there would no longer be a basis for the argument that hedging an option with
the underlying stock yields a risk-free portfolio. Still, the value of an
option can be expressed as its expected payoff at maturity discounted by its
expected return. This expression resembles the option pricing formula of Black
and Scholes (1973), but contains the percentage drift on the stock and the
(time-varying) percentage drift on the option instead of the risk-free rate of
return.

## References

  * [Bergman, 1981] Bergman, Y. Z. (1981).  A characterization of self-financing portfolio strategies.  Technical report, University of California at Berkeley.
  * [Björk, 2009] Björk, T. (2009).  Arbitrage theory in continuous time.  Oxford university press.
  * [Black and Scholes, 1973] Black, F. and Scholes, M. (1973).  The pricing of options and corporate liabilities.  Journal of Political Economy, 81:637–554.
  * [Cox et al., 1979] Cox, J. C., Ross, S. A., and Rubinstein, M. (1979).  Option pricing: A simplified approach.  Journal of Financial Economics, 7(3):229–263.
  * [Duffie, 2001] Duffie, D. (2001).  Dynamic asset pricing theory.  Princeton University Press.
  * [Harrison and Kreps, 1979] Harrison, J. M. and Kreps, D. M. (1979).  Martingales and arbitrage in multiperiod securities markets.  Journal of Economic Theory, 20(3):381–408.
  * [Harrison and Pliska, 1981] Harrison, J. M. and Pliska, S. R. (1981).  Martingales and stochastic integrals in the theory of continuous trading.  Stochastic Processes and their Applications, 11(3):215–260.
  * [Lamberton and Lapeyre, 2011] Lamberton, D. and Lapeyre, B. (2011).  Introduction to stochastic calculus applied to finance.  Chapman and Hall/CRC.
  * [MacDonald, 1997] MacDonald, A. (1997).  The hypotheses underlying the pricing of options: a note on a paper by Bartels.  Proceedings of the 7th AFIR International Colloquium, Cairns, 2:617–630.
  * [Merton, 1971] Merton, R. (1971).  Optimum consumption and portfolio rules in a continuous-time model.  Journal of Economic Theory, 3:373–413.
  * [Merton, 1973a] Merton, R. (1973a).  An intertemporal capital asset pricing model.  Econometrica, 41:876–887.
  * [Merton, 1973b] Merton, R. (1973b).  Theory of rational option pricing.  The Bell Journal of Economics and Management Science, 4:141–183.
  * [Merton, 1977] Merton, R. (1977).  On the pricing of contingent claims and the Modigliani-Miller theorem.  Journal of Financial Economics, 5(2):241–249.
  * [Merton, 1990] Merton, R. (1990).  Continuous-time finance.  Blackwell Cambridge.
  * [Shreve, 2004] Shreve, S. E. (2004).  Stochastic calculus for finance II: Continuous-time models, volume 11.  Springer Science & Business Media.

## Appendix

######  Proposition A.1.

If
$\theta_{T}=\theta_{t_{0}}+\int_{t_{0}}^{T}\kappa_{t}dt+\int_{t_{0}}^{T}\lambda_{t}dW_{t}$
and $\theta_{T}=0$ then $\lambda_{t}=0$ for all $t_{0}<t<T$.

###### Proof.

As
$\theta_{T}=\theta_{t_{0}}+\int_{t_{0}}^{t_{1}}\kappa_{t}dt+\int_{t_{1}}^{T}\kappa_{t}dt+\int_{t_{0}}^{t_{1}}\lambda_{t}dW_{t}+\int_{t_{1}}^{T}\lambda_{t}dW_{t}$,
the following equality holds:

$\displaystyle\mathop{\mathbb{C}ov}\left(\int_{t_{0}}^{t_{1}}\kappa_{t}dt,\int_{t_{0}}^{t_{1}}\kappa_{t}dt+\int_{t_{0}}^{t_{1}}\lambda_{t}dW_{t}\right)$
$\displaystyle=-\mathop{\mathbb{C}ov}\left(\int_{t_{0}}^{t_{1}}\kappa_{t}dt,\theta_{t_{0}}+\int_{t_{1}}^{T}\kappa_{t}dt+\int_{t_{1}}^{T}\lambda_{t}dW_{t}\right),$
$\displaystyle=-\mathop{\mathbb{C}ov}\left(\int_{t_{0}}^{t_{1}}\kappa_{t}dt,\int_{t_{1}}^{T}\kappa_{t}dt\right),$
(35)

where the first line uses $\theta_{T}=0$, and the second line uses the facts
that $\theta_{t_{0}}$ is deterministic and that the Markov property of the
Wiener process implies
$\mathop{\mathbb{C}ov}\left(\int_{t_{0}}^{t_{1}}\kappa_{t}dt,\int_{t_{1}}^{T}\lambda
dW_{t}\right)=0$ (i.e., the Wiener process does not depend on past events).
Moreover,
$\theta_{T}=\theta_{t_{1}}+\int_{t_{1}}^{T}\kappa_{t}dt+\int_{t_{1}}^{T}\lambda_{t}dW_{t}$
for $\theta_{T}=0$ implies that
$\mathop{\mathbb{C}or}\left(\int_{t_{1}}^{T}\kappa_{t}dt,\int_{t_{1}}^{T}\lambda_{t}dW_{t}\right)=-1$.
Because this perfect correlation is not stochastic, it is equal to $-1$
irrespective of whether one conditions on $\mathcal{F}_{t_{1}}$ or on
$\mathcal{F}_{t_{0}}$. Combining this perfect correlation with the above
result that
$\mathop{\mathbb{C}ov}\left(\int_{t_{0}}^{t_{1}}\kappa_{t}dt,\int_{t_{1}}^{T}\lambda
dW_{t}\right)=0$ implies that
$\mathop{\mathbb{C}ov}\left(\int_{t_{0}}^{t_{1}}\kappa_{t}dt,\int_{t_{1}}^{T}\kappa_{t}dt\right)=0$,
so that equation (Proof.) becomes:

$\mathop{\mathbb{C}ov}\left(\int_{t_{0}}^{t_{1}}\kappa_{t}dt,\int_{t_{0}}^{t_{1}}\kappa_{t}dt+\int_{t_{0}}^{t_{1}}\lambda_{t}dW_{t}\right)=0.$
(36)

Furthermore, we can also use
$\mathop{\mathbb{C}ov}\left(\int_{t_{0}}^{t_{1}}\kappa_{t}dt,\int_{t_{1}}^{T}\kappa_{t}dt\right)=0$
to write:

$\mathop{\mathbb{V}ar}\left(\int_{t_{0}}^{T}\kappa_{t}dt\right)=\mathop{\mathbb{V}ar}\left(\int_{t_{0}}^{t_{1}}\kappa_{t}dt\right)+\mathop{\mathbb{V}ar}\left(\int_{t_{1}}^{T}\kappa_{t}dt\right),$
(37)

and since the Markov property implies that
$\mathop{\mathbb{C}ov}\left(\int_{t_{0}}^{t_{1}}\lambda
dW_{t},\int_{t_{1}}^{T}\lambda_{t}dW_{t}\right)=0$, we also have:

$\mathop{\mathbb{V}ar}\left(\int_{t_{0}}^{T}\lambda_{t}dW_{t}\right)=\mathop{\mathbb{V}ar}\left(\int_{t_{0}}^{t_{1}}\lambda_{t}dW_{t}\right)+\mathop{\mathbb{V}ar}\left(\int_{t_{1}}^{T}\lambda_{t}dW_{t}\right).$
(38)

For $\theta_{T}=0$, the equality
$\theta_{T}=\theta_{t_{1}}+\int_{t_{1}}^{T}\kappa_{t}dt+\int_{t_{1}}^{T}\lambda_{t}dW_{t}$
implies that
$\mathop{\mathbb{V}ar}\left(\int_{t_{1}}^{T}\kappa_{t}dt\right)=\mathop{\mathbb{V}ar}\left(\int_{t_{1}}^{T}\lambda_{t}dW_{t}\right)$,
and likewise the equality
$\theta_{T}=\theta_{t_{0}}+\int_{t_{0}}^{T}\kappa_{t}dt+\int_{t_{0}}^{T}\lambda_{t}dW_{t}$
implies that
$\mathop{\mathbb{V}ar}\left(\int_{t_{0}}^{T}\kappa_{t}dt\right)=\mathop{\mathbb{V}ar}\left(\int_{t_{0}}^{T}\lambda_{t}dW_{t}\right)$.
Combining equations (37) and (38) therefore yields:

$\mathop{\mathbb{V}ar}\left(\int_{t_{0}}^{t_{1}}\kappa_{t}dt\right)=\mathop{\mathbb{V}ar}\left(\int_{t_{0}}^{t_{1}}\lambda_{t}dW_{t}\right).$
(39)

Using this result the following equality holds:

$\displaystyle\mathop{\mathbb{V}ar}\left(\int_{t_{0}}^{t_{1}}\kappa_{t}dt+\int_{t_{0}}^{t_{1}}\lambda_{t}dW_{t}\right)$
$\displaystyle=2\mathop{\mathbb{V}ar}\left(\int_{t_{0}}^{t_{1}}\kappa_{t}dt\right)+2\mathop{\mathbb{C}ov}\left(\int_{t_{0}}^{t_{1}}\kappa_{t}dt,\int_{t_{0}}^{t_{1}}\lambda_{t}dW_{t}\right),$
$\displaystyle=2\mathop{\mathbb{C}ov}\left(\int_{t_{0}}^{t_{1}}\kappa_{t}dt,\int_{t_{0}}^{t_{1}}\kappa_{t}dt+\int_{t_{0}}^{t_{1}}\lambda_{t}dW_{t}\right),$
$\displaystyle=0,$ (40)

where we used equation (36) in the last step. This equality states that the
integral $\int_{t_{0}}^{t_{1}}\kappa_{t}dt+\lambda_{t}dW_{t}$ has zero
variance and is therefore deterministic for all $t_{0}<t_{1}<T$. The integrand
must therefore be deterministic as well, which implies that $\lambda_{t}=0$. ∎

######  Proposition A.2.

The continuous-time self-financing condition does not hold if the portfolio
rebalancing depends on changes in asset prices.

###### Proof.

According to Merton (1971), the continuous-time self-financing condition can
be derived by starting with the ex-post discrete-time self-financing
condition, which for a portfolio of $\alpha_{t}$ stocks and $\beta_{t}$ bonds
is equal to:

$\left[\Delta\alpha_{t}x_{t}+\Delta\alpha_{t}\Delta
x_{t}+\Delta\beta_{t}b_{t}+\Delta\beta_{t}\Delta
b_{t}|\mathcal{F}_{T}\right]=0.$ (41)

While this condition holds ex-post when trades have taken place (i.e.,
conditional on $\mathcal{F}_{T}$, where $T\geq t+\Delta t$ is the maturity
date of the option), Merton argues that equation (41) also holds ex-ante
(i.e., conditional on $\mathcal{F}_{t}$). He then takes the limit $\Delta t\to
0$ to obtain:

$\lim_{\Delta t\to 0}\left[\Delta\alpha_{t}x_{t}+\Delta\alpha_{t}\Delta
x_{t}+\Delta\beta_{t}b_{t}|\mathcal{F}_{t}\right]=0,$ (42)

where we used the fact that $\Delta b_{t}$ is deterministic because $b_{t}$ is
a risk-free bond, so that $\lim_{\Delta t\to 0}\left[\Delta\beta_{t}\Delta
b_{t}|\mathcal{F}_{t}\right]$ is of higher order than the other terms and can
therefore be neglected. Likewise, taking the limit $\Delta t\to 0$ of equation
(41) yields:

$\lim_{\Delta t\to 0}\left[\Delta\alpha_{t}x_{t}+\Delta\alpha_{t}\Delta
x_{t}+\Delta\beta_{t}b_{t}+\Delta\beta_{t}\Delta
b_{t}|\mathcal{F}_{T}\right]=0,$ (43)

which is conditioned on $\mathcal{F}_{T}$ so that all terms are deterministic
(i.e., they are numbers) and not stochastic. Because of this conditioning,
$\lim_{\Delta t\to 0}\left[\Delta\alpha_{t}\Delta
x_{t}|\mathcal{F}_{T}\right]$ and $\lim_{\Delta t\to
0}\left[\Delta\beta_{t}\Delta b_{t}|\mathcal{F}_{T}\right]$ go to zero faster
than $\lim_{\Delta t\to 0}\left[\Delta\alpha_{t}x_{t}|\mathcal{F}_{T}\right]$
and $\lim_{\Delta t\to
0}\left[\Delta\beta_{t}b_{t}|\mathcal{F}_{T}\right]$.777To avoid confusion, we
point out that $\lim_{\Delta t\to 0}\left[\Delta
x_{t}|\mathcal{F}_{T}\right]\neq dx_{t}$ since $\lim_{\Delta t\to
0}\left[\Delta x_{t}|\mathcal{F}_{T}\right]$ is conditioned on
$\mathcal{F}_{T}$ and is therefore deterministic, while $dx_{t}$ as defined in
equation (1) is conditioned on $\mathcal{F}_{t}$ and is therefore stochastic.
Neglecting these higher order terms yields:

$\lim_{\Delta t\to
0}\left[\Delta\alpha_{t}x_{t}+\Delta\beta_{t}b_{t}|\mathcal{F}_{T}\right]=0.$
(44)

Because equation (44) holds for all realized paths of $\Delta\alpha_{t}$ and
$\Delta\beta_{t}$, it also holds stochastically (i.e., conditional on
$\mathcal{F}_{t}$) so that:

$\lim_{\Delta t\to
0}\left[\Delta\alpha_{t}x_{t}+\Delta\beta_{t}b_{t}|\mathcal{F}_{t}\right]=0,$
(45)

which can be subtracted from equation (42) to obtain:

$\lim_{\Delta t\to 0}\left[\Delta\alpha_{t}\Delta
x_{t}|\mathcal{F}_{t}\right]=0.$ (46)

Merton (1971) argues that equation (42) yields the continuous-time self-
financing condition in equation (4) when defining $\alpha_{t}=w_{1,t}$. This
argument among others implies that:

$\lim_{\Delta t\to 0}\left[\Delta\alpha_{t}\Delta
x_{t}|\mathcal{F}_{t}\right]=d\alpha_{t}dx_{t},$ (47)

which requires that $\lim_{\Delta t\to 0}\left[\Delta
x_{t}|\mathcal{F}_{t}\right]$ converges to the geometric Brownian motion
$dx_{t}$ as defined in equation (1). Moreover, using the Taylor expansion
$d\alpha_{t}=(\partial\alpha_{t}/\partial
x_{t})dx_{t}+(\partial\alpha_{t}/\partial
dt)dt+0.5(\partial^{2}\alpha_{t}/\partial^{2}x_{t})dx_{t}^{2}$ and neglecting
higher order terms gives:

$d\alpha_{t}dx_{t}=\frac{\partial\alpha_{t}}{\partial
x_{t}}\sigma^{2}x_{t}^{2}dt,$ (48)

where we used the fact that $dx_{t}^{2}=\sigma^{2}x_{t}^{2}dt>0$ is the
quadratic variation of the geometric Brownian motion. Combining equations (46)
to (48) yields:

$\frac{\partial\alpha_{t}}{\partial x_{t}}\sigma^{2}x_{t}^{2}dt=0.$ (49)

When the portfolio rebalancing depends on changes in the stock price, however,
we have $\partial\alpha_{t}/\partial x_{t}\neq 0$, in which case equation (49)
contradicts the fact that the geometric Brownian motion has positive quadratic
variation. ∎

######  Proposition A.3.

The continuous-time self-financing condition does not hold if the portfolio is
continuously rebalanced.

###### Proof.

Merton (1971) argues that the continuous-time self-financing condition in
equation (4) is obtained when conditioning on $\mathcal{F}_{t}$ and taking the
limit $\Delta t\to 0$ of his discrete-time self-financing condition:

$\lim_{\Delta t\to 0}\left[\Delta\alpha_{t}x_{t}+\Delta\alpha_{t}\Delta
x_{t}+\Delta\beta_{t}b_{t}+\Delta\beta_{t}\Delta
b_{t}|\mathcal{F}_{t}\right]=0,$ (50)

where $\alpha_{t}$ and $\beta_{t}$ are the numbers of stocks and bonds in the
portfolio. The discrete-time self-financing condition implies that the change
in the value of the portfolio at $t$ equals:

$\Delta w_{t}=\alpha_{t}\Delta x_{t}+\beta_{t}\Delta b_{t}.$ (51)

Likewise, since the discrete-time self-financing condition also holds ex-post,
we have:

$\left[\Delta\alpha_{t-\Delta t}x_{t-\Delta t}+\Delta\alpha_{t-\Delta t}\Delta
x_{t-\Delta t}+\Delta\beta_{t-\Delta t}b_{t-\Delta t}+\Delta\beta_{t-\Delta
t}\Delta b_{t-\Delta t}|\mathcal{F}_{t}\right]=0,$ (52)

so that conditional on $\mathcal{F}_{t}$, the change in the value of the
portfolio over the period between $t-\Delta t$ and $t$ equals:

$\Delta w_{t-\Delta t}=\alpha_{t-\Delta t}\Delta x_{t-\Delta
t}+\beta_{t-\Delta t}\Delta b_{t-\Delta t}.$ (53)

Furthermore, the value of the portfolio at $t+\Delta t$ equals:

$\displaystyle w_{t+\Delta t}$ $\displaystyle=w_{t}+\Delta w_{t},$
$\displaystyle=w_{t-\Delta t}+\Delta w_{t-\Delta t}+\Delta w_{t},$ (54)

and the value at $t-\Delta t$ equals:

$w_{t-\Delta t}=\alpha_{t-\Delta t}x_{t-\Delta t}+\beta_{t-\Delta
t}b_{t-\Delta t}.$ (55)

Substituting equations (51), (53) and (55) into equation (Proof.) gives:

$\displaystyle w_{t+\Delta t}$ $\displaystyle=\alpha_{t-\Delta t}x_{t-\Delta
t}+\beta_{t-\Delta t}b_{t-\Delta t}+\alpha_{t-\Delta t}\Delta x_{t-\Delta
t}+\beta_{t-\Delta t}\Delta b_{t-\Delta t}+\alpha_{t}\Delta
x_{t}+\beta_{t}\Delta b_{t},$ $\displaystyle=\alpha_{t-\Delta t}x_{t+\Delta
t}+\beta_{t-\Delta t}b_{t+\Delta t}+\Delta\alpha_{t-\Delta t}\Delta
x_{t}+\Delta\beta_{t-\Delta t}\Delta b_{t},$ (56)

where we simplified the second line by using $x_{t+\Delta t}=x_{t-\Delta
t}+\Delta x_{t-\Delta t}+\Delta x_{t}$, and likewise for $b_{t+\Delta t}$.
Taking the limit $\Delta t\to 0$ of equation (Proof.) conditional on
$\mathcal{F}_{t}$ gives:

$\displaystyle\lim_{\Delta t\to 0}\left[w_{t+\Delta
t}|\mathcal{F}_{t}\right]=\lim_{\Delta t\to 0}\left[\alpha_{t-\Delta
t}x_{t+\Delta t}+\beta_{t-\Delta t}b_{t+\Delta t}|\mathcal{F}_{t}\right],$
(57)

which uses the fact that $\lim_{\Delta t\to 0}\left[\Delta\alpha_{t-\Delta
t}\Delta x_{t}+\Delta\beta_{t-\Delta t}\Delta b_{t}|\mathcal{F}_{t}\right]$ is
of higher order than the other terms and can therefore be neglected. Equation
(57) implies, however, that the portfolio weights of a point in time smaller
than $t$ can be used to replicate the portfolio value at a point in time
larger than $t$, which contradicts the fact that the portfolio is continuously
rebalanced (including at time $t$). ∎


# MARTINGALES

ROHAN SHAH  Milton Academy, Milton, MA, 02186<EMAIL_ADDRESS>
(Date: August 28, 2024)

Abstract. This paper covers martingales with introduction to measure theory
concepts, and other concepts, including the Lebesgue Integration and
Conditional Expectation. It follows up with proofs on Kolmogorov’s Theorem on
conditional expectations, the Martingale Property, and the Pythagorean Theorem
on Martingales. Finally, it ends with martingales’ application in finance.

###### CONTENTS

  1. 1 INTRODUCTION
  2. 2 PROBABILITY AND MEASURE THEORY
  3. 3 LEBESGUE INTEGRATION
  4. 4 CONDITIONAL EXPECTATION
  5. 5 MARTINGALES
  6. 6 APPLICATIONS IN FINANCE
  7. 7 ACKNOWLEDGEMENTS

##  1\. INTRODUCTION

Let’s consider a bettor participating in a game involving the flipping of a
fair coin. In this game, the bettor earns $1 for a heads outcome and loses $1
for a tails outcome. This means that if the better flips the coin once and it
lands on tails, they would lose a dollar. Conversely, if it lands on heads,
they would gain a dollar.

Over many run-throughs of flipping the coin, the bettor’s total earnings,
represented as T, could be either positive or negative, depending on the
outcomes of the flips. Each individual coin flip carries an equal probability
of 50%, meaning there is an equal chance that the next flip will result in
either heads or tails. Consequently, with each flip, the bettor’s total
earnings will either increase by $1 (if heads) or decrease by $1 (if tails).

After many flips, the bettor’s total earnings will result from random
fluctuations due to the equal probability of heads or tails. Over many flips,
the total earnings T will show a random pattern of gains and losses, yet the
expectation of total earnings remains zero. This means that at one point,
despite the variances in time, basic probability states that the final value
will be 0. This is an example of a fair game, where the expected earnings
after the next turn would equal the current earnings. A martingale shares a
similar idea. Introduced by Paul Lévy in 1934 and later by Ville in 1939, a
martingale is a probability model based on a fair game. This property can be
translated to concepts outside regular games, including the stock market. In
simple terms, a martingale is a process where the conditional expectation of
the next value is equal to the present value. Since there are no predictable
trends in a fair game, a martingale can effectively visualize the randomness
and fairness of such games. This paper covers the mathematical definition of
martingale with the requirement of knowledge of elementary probability,
specifically measure theory ($\sigma$-Algebra, measurable spaces, Lebesgue
measure, Borel sets).

Before we begin, I will introduce the notation for a martingale and explain
each part throughout the paper.

######  Definition 1.1.

$X$ is a martingale if

$X_{n}=E[X_{n+1}|\mathcal{F}_{n}]$

##  2\. PROBABILITY AND MEASURE THEORY

We begin with definitions regarding the basics of measure theory, including
sets and events. We are given a sample space denoted by $\Omega$ as the set of
all possible outcomes of a random experiment. An event is a subset of the
sample space $\Omega$. For instance, if $\Omega$ represents the possible
outcomes of a dice roll, then (2,4,6) could be an event representing the roll
of an even number. We begin by looking at the first approximation, the power
set of omega, $P(\Omega)$ which equals the set of all subsets of $\Omega$. To
restrict this power set, we introduce our first definition.

######  Definition 2.1.

A system $A\subseteq P(X)$, where A is a collection of elements of subsets of
X. This collection is called $\sigma$-algebra. A $\sigma$-algebra is a
collection of subsets of the sample space that includes the empty set and the
sample space itself. However, to be named a $\sigma$-algebra (an important
measure theory definition), it must follow these three rules:

  * •

$\varnothing,X\subseteq P(X)$

  * •

If $B\in A$ then $B^{C}=X/B\in A$. This means that for any set in the
$\sigma$-algebra, its complement is also in the $\sigma$-algebra.

  * •

$B,D\in A\to B\cup D\in A$. Additionally, if the set B and D are in set, A,
the intersection of set B and D will also be in A.

  * •

Then $(X,A)$ is a measurable space.

Measurable spaces are essential to provide a framework in which measures are
defined. In the definition above, $(X,A)$ is a measurable space with set X and
the collection of subsets, the sigma-algebra, A. The sigma-algebra includes
the subsets of X, that are considered measurable.

######  Definition 2.2.

  * •

For $M\subseteq P(X)$, there is a smallest sigma algebra that contains M:

$\sigma(M)=:\bigcap_{\begin{subarray}{c}A\supseteq M\end{subarray}}|$ $A$ is a
$\sigma$-algebra generated by set M.

  * •

E.g. $X=\\{a,b,c,d\\},M=\\{\\{a\\},\\{b\\}\\}$

  * •

Therefore,
$\sigma(\mathcal{M})=\left\\{\begin{array}[]{r}\phi,X,\\{a\\},\\{b\\},\\{a,b\\},\\{b,c,d\\},\\{a,c,d\\},\\{c,d\\}\\}\end{array}\right.$

By defining $\sigma(M)$ as the intersection of all sigma-algebras containing
$M$, we guarantee that $(\sigma(M))$ includes all necessary sets to be a
sigma-algebra while being minimal in the sense that it does not include any
extraneous sets not required by the sigma-algebra properties and the inclusion
of $M$.

######  Definition 2.3.

Let A be an algebra, $\mu:A\to[0,\infty]$ is a finitely additive measure on A,
if

  * •

For any two sets $B,D\in A$ with $B\cap D=\varnothing:\mu(B\cup
D)=\mu(B)+\mu(D)$.

A finitely additive measure is a function that assigns a non-negative value to
each set, representing its size, in a way that the measure of the union of two
non-overlapping sets equals the sum of their measures. This property ensures
consistent and additive measurements, allowing us to determine the size of
complex shapes by summing the measures of their non-overlapping parts.

###### Proof.

Using Definition 2.1 and 2.3:

  * •

Then $A_{1},A_{2},A_{3}\cdots A_{n}\in A:A_{1}\cup A_{2}\cup\cdots A_{n}\in A$

  * •

If $A_{i}\cup A_{j}=\varnothing 1<i\neq j<n$ , then
$\mu\bigcup_{i=1}^{n}A_{i}$ = $\sum_{i=1}^{n}\mu(A_{i})$.

∎

The proof shows that any finite union of sets in a sigma-algebra $A$ is also
in $\mathcal{A}$. It then uses the countable additivity property of measures
to show that the measure of a finite union of disjoint sets is the sum of the
measures of the individual sets.

######  Definition 2.4.

A measure $\mu$ on an algebra, A is a set function from $A\to[0,\infty]$ if it
follows the following properties:

  * •

If we take any sequence of sets: $A_{1},A_{2},A_{3}\cdots A_{n}\in A$ such
that $\cup_{i=1}^{n}A_{i}$ and $A_{i}\cap A_{j}=\varnothing$ if $i\neq j$ then
$\mu(\cup_{i}A_{i})=\sum_{i}\mu(A_{i})$.

Note that this property is denoted as $\sigma$-additivity. However if the set
function, A is in the interval $[0,1]$, then these properties won’t apply as
it is a probability measure rather than a measure. Now, we can define what a
probability measure is.

######  Definition 2.5.

A probability measure (P) on the sample space satisfies $P(\Omega)=1$.
Therefore, the triple $(\Omega,\mathcal{F},P)$ is called a probability space,
where $\mathcal{F}$ is the sigma algebra and $P$ is the probability measure.

With these terms defined, we can introduce what random variables are.

######  Definition 2.6.

Given a measurable space $(\Omega,\mathcal{F})$, a function
$X:\Omega\to\mathbb{R}$ is said to be a measurable function or a random
variable if for every Borel Set, B $\in\mathbb{B}(\mathbb{R})$, the pre-image
of that set, $X^{-1}(B)\in\mathcal{F}$. This implies that the function $X$
maps events in the sample space $\Omega$ to real numbers in a way that
preserves the structure of $\sigma$-algebra.

######  Definition 2.7.

A random variable is a measurable function from the sample space to the real
numbers. The distribution of a random variable $X$ is a function $X:B\to[0,1]$
such that $X(B)=P(X^{-1}(B))$ for all Borel sets $B$. Note that Borel sets
(subset) are important as a way of showing that if open sets are measurable,
then so are all other sets of interest (like closed sets, countable unions,
and intersections).

##  3\. LEBESGUE INTEGRATION

If X, a integrable random variable takes on an infinite number of values,
calculating the weighted average requires integration to handle this infinite
sum. However, the standard Riemann integral is often unsuitable in this
context. The Riemann integral approximates the integral by splitting the
domain into contiguous intervals to form ”rectangles,” then summing the areas
of these rectangles. This approach might not make sense for an abstract sample
space, and some measurable functions may not meet the continuity requirements
for Riemann integration. Therefore, we use a more generalized form of the
Riemann integral called the Lebesgue integral. However, before we define the
integral, we need understand simple functions.

######  Definition 3.1.

Let $A$ be a subset of $\Omega$. The indicator function of $A$ is the function
$\mathbf{1}_{A}:\Omega\rightarrow\mathbb{R}$ defined as:

$\mathbf{1}_{A}(\omega)=\left\\{\begin{array}[]{lll}1&\text{ if }&\omega\in
A\\\ 0&\text{ if }&\omega\notin A\end{array}.\right.$

######  Theorem 3.2.

The indicator function of $A$ is measurable in every $\sigma$-algebra
$\mathcal{F}$ containing $A$.

###### Proof.

Every possible Borel set $B$ falls into one of four categories:

  * •

$1\in B,0\notin B$. Then $\mathbf{1}_{A}^{-1}(B)=A$;

  * •

$0\in B,1\notin B$. Then $\mathbf{1}_{A}^{-1}(B)=A^{c}$;

  * •

$0,1\in B$. Then $\mathbf{1}_{A}^{-1}(B)=\Omega$;

  * •

$0,1\notin B$. Then $\mathbf{1}_{A}^{-1}(B)=\varnothing$.

By definition, if $A\in\mathcal{F}$, then
$\Omega,\varnothing,A^{c}\in\mathcal{F}$. Thus, by Definition
$3.1,\mathbf{1}_{A}$ is measurable. ∎

######  Definition 3.3.

Simple functions are measurable functions that take on a finite number of
values. They can be written as:

$g=\sum_{i=1}^{n}a_{i}\mathcal{X}A_{i}$

In this notation above, $a_{i}$ are real numbers, $A_{i}$ are measurable sets,
and ${}_{\mathcal{X}A_{i}}$.

###### Proof.

If we integrate the following simple function using the linearity property:
1\. Define the Simple Function:

$g=\sum_{i=1}^{n}a_{i}\mathcal{X}_{A_{i}}$

2\. Set Up the Integral: To integrate $g$ with respect to a measure $\mu$, we
use the definition:

$\int gd\mu=\int\sum_{i=1}^{n}a_{i}\mathcal{X}_{A_{i}}d\mu$

3\. Use Linearity of the Integral: By the linearity property of the integral,
we can move the summation outside the integral:

$\int\sum_{i=1}^{n}a_{i}\mathcal{X}_{A_{i}}d\mu=\sum_{i=1}^{n}a_{i}\int\mathcal{X}_{A_{i}}d\mu$

4\. Evaluate Each Integral: The integral of the characteristic function
$\mathcal{X}_{A_{i}}$ of a set $A_{i}$ with respect to $\mu$ is simply the
measure of $A_{i}$ :

$\int\mathcal{X}_{A_{i}}d\mu=\mu\left(A_{i}\right)$

5\. Combine the Results: Substitute the measure of each $A_{i}$ into the sum:

$\int gd\mu=\sum_{i=1}^{n}a_{i}\mu\left(A_{i}\right)$

So, the integral of the simple function $g$ with respect to the measure $\mu$
is:

$\int\mathrm{gd}\mu=\sum_{\mathrm{i}=1}^{\mathrm{n}}a_{i}\mu(Ai),\text{ where
}\mu\text{ is a measure on }\Omega$

.

∎

Properties of Lebesgue Integral:

  * •

Linearity:
$\int(\mathrm{af}+\mathrm{bg})\mathrm{d}\mu=\mathrm{a}\int\mathrm{fd}\mu+\mathrm{b}\int\mathrm{gd}\mu$

  * •

Monotonicity: If $\mathrm{f}\leq\mathrm{g}$, then
$\int\mathrm{fd}\mu\leq\int\mathrm{gd}\mu$

  * •

Countable Additivity:
$\int\sum_{i=1}^{\infty}f_{i}\mathrm{\leavevmode\nobreak\
d}\mu=\sum_{i=1}^{\infty}\int f_{i}\mathrm{\leavevmode\nobreak\ d}\mu$

The utility of the Lebesgue integral becomes clearer with this definition. The
Lebesgue integral of a simple function is simply the sum of its possible
values weighted by the measure of the subsets of the domain corresponding to
those values. In the context of a probability measure, this is equivalent to
averaging the possible values of a random variable weighted by the probability
of the events that produce those values—that is, the expectation. For more
complex, non-simple functions, the integral can be defined by approximating
the function with simple functions. This leads to the following theorem, which
facilitates this process.

######  Theorem 3.4.

Let $f:\Omega\rightarrow[0,\infty]$ be a measurable function on the space
$(\Omega,\mathcal{F},\mu)$. Then, there exists an increasing sequence of
simple functions $\left(x_{i}\right)_{i=1}^{\infty}$ that converges point-wise
to $f$.

This is known as the Simple Function Approximation Theorem. As we covered a
little about expectation, we can know understand both that and conditional
expectation.

##  4\. CONDITIONAL EXPECTATION

Conditional expectation, denoted $E[Y|X]$, represents the expected value of a
random variable $Y$ given the knowledge of another variable $X$. It is used to
update our expectation of $Y$ based on the additional information provided by
$X$. To understand this better, we begin by defining simple random variables,
then move to expectation.

######  Definition 4.1.

A random variable, X is simple if there exists
$n>0,X_{1},X_{2},\cdots,X_{n}\in\mathbb{R}$,
$A_{1},A_{2},\ldots,A_{n}\in\mathcal{F}:$

$X=\sum_{Xi=1}^{n}X_{\varepsilon}\mathbbm{1}_{A_{\varepsilon}}$

This means that X is a function from $\Omega\to\mathbb{R}$. $A_{\varepsilon}$
are sets in the $\sigma$-algebra, meaning that they are subsets in $\Omega$.

######  Theorem 4.2.

For every random variable X, there exist $X_{1},X_{2},\cdots$ that are known
as simple random variables such that:

$|X_{n}|\leq|X|$

For every $\omega\in$ sample space, $\Omega$.

$X_{n}(w)\vec{n}_{\infty}X(w)$

######  Theorem 4.3.

If $X(\omega)\geq 0$ for all $\omega\in\Omega$, then $X_{n}(\omega)$ can
chosen to be non-decreasing in $n$ for every $\omega$.

”Non-decreasing for n” means that as the sequence progresses, the values do
not decrease for any outcome in the sample space. In other words, the values
either stay the same or increase as the index increases, ensuring the sequence
never gets smaller for each outcome.

After these definitions of random variables, we can now define the expectation
of a random variable.

######  Definition 4.4.

If $X=\sum_{Xi=1}^{n}X_{\varepsilon}\mathbbm{1}_{A_{\varepsilon}}$ is simple,
then $\mathbb{E}X$ (expected value)
$=\sum_{Xi=1}^{n}X_{\varepsilon}\mathbb{P}(A_{\varepsilon})$

This is Step 1, only applying if the random variable, X is a simple, following
the properties defined in Definition 3.1. Now, we will discuss Step 2 when
$X\geq 0$ (not negative).

######  Definition 4.5.

If $X\geq 0$ and a random variable, then $X_{1},X_{2},\ldots$ is a simple
random variable such that $X_{i}$ converges to $X$.

$\mathbb{E}X=lim_{X\to i}\mathbb{E}X_{i}\in[0,\infty]$

######  Definition 4.6.

If $X$ is a random variable, then $X:=X^{+}-X^{-}$

This second part of the definition is stating that the positive part of the
variable, $X^{+}$ plus the negative part of the variable, $X^{-}$ equals to
the variable itself. For instant the positive part of 4 is 4, but the negative
part of 4 is 0. Adding these two numbers, we get the number 4 itself. This may
seem intuitive, but if we do this for every $\omega$, we can define a
function, where both the positive and negative part of the variable is not
negative. If we take the expectation of this equation we get the following:

$\mathbb{E}X:=\mathbb{E}X^{+}-\mathbb{E}X^{-}$

unless

$\mathbb{E}X^{+}=\infty=\mathbb{E}X^{-}$

Making $\mathbb{E}[X]$ not exist, which is known as the Cauchy random
variable.
Now we move to defining what conditional expectation is.

######  Theorem 4.7.

Kolmogorov’s Theorem on conditional expectations.
Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, $X$ is an
integrable random variable; $\mathbb{E}|X|<\infty$. Let
$\mathcal{G}\subseteq\mathcal{F}$ be a sub sigma-algebra. Take random variable
$Y$ such that it follows these properties:

  * •

$Y$ is $\mathcal{G}$-measurable.

  * •

$\mathbb{E}|Y|<\infty$

  * •

$\forall A\in\mathcal{G}$

  * •

$\int_{A}XdP=\int_{A}YdP=\int_{A}\mathbb{E}(X|\mathcal{G})dP$

The random variable $Y$ is denoted by $\mathbb{E}(X|\mathcal{G})$ and is
called the conditional expectation of $X$ given $\mathcal{G}$.

The best way to explain this is through a proof, where we shall require the
use of Radon-Nikodym Theorem.

###### Proof.

Define a new measure, $Q$ on $(\Omega,\mathcal{G})$ by:

$Q(A)=\int_{A}XdP$

for all $A\in\mathcal{G}$.
Since $X$ is integrable, $Q$ is a finite measure on $(\Omega,\mathcal{G})$:

$Q(\Omega)=\int_{\Omega}XdP=\mathbb{E}[X]<\infty$

The Radon-Nikodym theorem states that since $Q$ is a sigma-finite measure on
$(\Omega,\mathcal{G})$ and $\left.P\right|_{\mathcal{G}}$ is the restriction
of the probability measure $P$ to $\mathcal{G}$, there exists a
$\mathcal{G}$-measurable function $Y$ such that:

$Q(A)=\int_{A}YdP\quad\text{ for all }A\in\mathcal{G}$

From here, since $A\in\mathcal{G}$:

$\int_{A}XdP=\int_{A}YdP$

Define $Y=\mathbb{E}[X\mid\mathcal{G}]$. By construction, $Y$ is
$\mathcal{G}$-measurable and satisfies:

$\int_{A}XdP=\int_{A}\mathbb{E}[X\mid\mathcal{G}]dP\quad\text{for
all}A\in\mathcal{G}$

To show uniqueness, suppose $Y^{\prime}$ is another $\mathcal{G}$-measurable
function that satisfies the same integral property. Then for all
$A\in\mathcal{G}$ :

$\int_{A}\mathbb{E}[X\mid\mathcal{G}]dP=\int_{A}Y^{\prime}dP$

Taking $A=\left\\{\mathbb{E}[X\mid\mathcal{G}]>Y^{\prime}\right\\}$, we get:

$\int_{\left\\{\mathbb{E}[X\mid\mathcal{G}]>Y^{\prime}\right\\}}\left(\mathbb{E}[X\mid\mathcal{G}]-Y^{\prime}\right)dP=0$

Since $\mathbb{E}[X\mid\mathcal{G}]-Y^{\prime}\geq 0$ on
$\left\\{\mathbb{E}[X\mid\mathcal{G}]>Y^{\prime}\right\\}$, this implies
$P\left(\left\\{\mathbb{E}[X\mid\mathcal{G}]>Y^{\prime}\right\\}\right)=$ 0\.
A similar argument holds for
$\left\\{\mathbb{E}[X\mid\mathcal{G}]<Y^{\prime}\right\\}$. Thus,
$\mathbb{E}[X\mid\mathcal{G}]=Y^{\prime}$ almost surely.
The Radon-Nikodym theorem provides a $\mathcal{G}$-measurable function $Y$
that satisfies the integral property, ensuring the existence of the
conditional expectation $\mathbb{E}[X\mid\mathcal{G}]$. The uniqueness follows
from the fact that any two $\mathcal{G}$-measurable functions satisfying this
property must be equal almost surely. Thus, Kolmogorov’s theorem on
conditional expectations is proved. ∎

Let’s move on to an important property of Conditional Expectation, the tower
rule.

######  Definition 4.8.

$(\Omega,\mathcal{F},\mathbb{P})$;
$\mathcal{G}\subseteq\mathcal{H}\subseteq\mathcal{F}$ (These are all
$\sigma$-algebras). Both $\mathcal{G}$ and $\mathcal{H}$ are sub-sigma-
algebras. Let $X$ be a random variable, $\mathbb{E}[X]<\infty$.
Then the Tower rule states:

$\mathbb{E}(\mathbb{E}(X|\mathcal{G})|\mathcal{H})=\mathbb{E}[X|\mathcal{G}]$

The Tower Rule essentially tells us that if we first condition on a larger
sigma-algebra $\mathcal{G}$ and then on a smaller sigma-algebra $\mathcal{H}$,
the result is the same as if we had conditioned on the smaller sigma-algebra
$\mathcal{H}$ directly.

##  5\. MARTINGALES

We are given a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and a
filtration as defined below.

######  Definition 5.1.

A filtration $({\mathcal{F}_{t}})_{t\geq 0}$ is an increasing family of
$\sigma$-algebras:

$\mathcal{F}_{s}\subseteq\mathcal{F}_{t}\subseteq\mathcal{F}\text{ for }s\leq
t$

From this we can define adaptation.

######  Definition 5.2.

A sequence of random variables $\left\\{X_{n};n\geq 0\right\\}$ is adapted if:

$X_{n}\in\mathcal{F}_{n}$

In Adaptation, the data $X_{n}$ only depends on information until instant $n$.

######  Definition 5.3.

We consider a sequence of random variables $X=\left\\{X_{n};n\geq 0\right\\}$
such that

  * •

$\left\\{X_{n};n\geq 0\right\\}$ is adapted.

  * •

$X_{n}\in L^{1}(\Omega)$ for all $n\geq 0$.

Then

  * •

$X$ is a martingale if
$X_{n}=\mathrm{E}\left[X_{n+1}\mid\mathcal{F}_{n}\right]$.

  * •

$X$ is a supermartingale if
$X_{n}\geq\mathrm{E}\left[X_{n+1}\mid\mathcal{F}_{n}\right]$.

  * •

$X$ is a submartingale if $X_{n}\leq
E\left[X_{n+1}\mid\mathcal{F}_{n}\right]$.

We see that the definition of a martingale was shared in definition $1.1$,
this is the martingale property. This equation emphasizes the current value,
$X_{n}$ is the expected value of the next step, given the information up to
$n$. However, this definition is in discrete time, meaning that it follows the
adaptation and integrability condition. To better understand this idea, we can
use an example of the fair game shared in the introduction. Using the same
rules, we can now prove this definition.

######  Example.

Let $X_{n}$ represent the player’s total winnings after $n$ games. We will
show that $({X_{n}})_{n\geq 0}$ is a martingale.
Filtration:
Define the filtration $\left\\{\mathcal{F}_{n}\right\\}_{n\geq 0}$ where
$\mathcal{F}_{n}$ represents the information up to the $n$th game.
Specifically, $\mathcal{F}_{n}$ contains the outcomes of the first $n$ coin
tosses.
Martingale Property:
To prove that $\left\\{X_{n}\right\\}_{n\geq 0}$ is a martingale with respect
to $\left\\{\mathcal{F}_{n}\right\\}_{n\geq 0}$, we need to verify the three
conditions:

  * •

Adaptation: $X_{n}$ is $\mathcal{F}_{n}$-measurable because the total winnings
after $n$ games depend only on the outcomes of the first $n$ games.

  * •

Integrability: The expectation $\mathbb{E}\left[\left|X_{n}\right|\right]$ is
finite. Since $X_{n}$ is the sum of $n$ independent bets of $\pm 1$, its
absolute value grows linearly with $n$, ensuring finite expectation.

  * •

Martingale Property:

$\mathbb{E}\left[X_{n+1}\mid\mathcal{F}_{n}\right]=X_{n}$

###### Proof.

Given that $X_{n}$ is the player’s total winnings after $n$ games, the total
winnings after $n+1$ games, $X_{n+1}$, can be written as:

$X_{n+1}=X_{n}+Y_{n+1}$

where $Y_{n+1}$ is the result of the $(n+1)$ th game

  * •

$Y_{n+1}=1$ if the player wins the $(n+1)$ th game (heads).

  * •

$Y_{n+1}=-1$ if the player loses the $(n+1)$ th game (tails).

The expected value of $Y_{n+1}$ given $\mathcal{F}_{n}$ is:

$\mathbb{E}\left[Y_{n+1}\mid\mathcal{F}_{n}\right]=0$

because the coin toss is fair, and the expected outcome of a fair coin toss is
zero (since

$\left.P\left(Y_{n+1}=1\right)=P\left(Y_{n+1}=-1\right)=1/2\right)$

Therefore,

$\mathbb{E}\left[X_{n+1}\mid\mathcal{F}_{n}\right]=\mathbb{E}\left[X_{n}+Y_{n+1}\mid\mathcal{F}_{n}\right]=X_{n}+\mathbb{E}\left[Y_{n+1}\mid\mathcal{F}_{n}\right]=X_{n}+0=X_{n}$

This confirms that the process $\left\\{X_{n}\right\\}_{n\geq 0}$ is a
martingale with respect to the filtration
$\left\\{\mathcal{F}_{n}\right\\}_{n\geq 0}$ ∎

From this example, we can move to another stochastic process like martingales,
stopping time.

######  Definition 5.4.

Given a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and a filtration
$\left\\{\mathcal{F}_{t}\right\\}_{t\geq 0}$, a random variable
$\tau:\Omega\rightarrow[0,\infty]$ is a stopping time if for every $t\geq 0$,

$\\{\tau\leq t\\}\in\mathcal{F}_{t}.$

######  Lemma 5.5.

Let T be a stopping time If there exists a not random variable, $N$ and
$\varepsilon>0$ such that $\mathbb{P}\left(T\leq
n+N\mid\mathcal{F}_{n}\right)>\varepsilon\quad\forall n\geq 0$, then
$\mathbb{E}[T]<\infty$.

###### Proof.

$\mathbb{P}(T>kN)=\mathbb{P}(T>kN\cup T>(k-1)N)$

$=\mathbb{P}(T>kN|T>(k-1)N)*\mathbb{P}(T>(k-1)N))$
$\leq(1-\varepsilon)*\mathbb{P}(T>(k-1)N)\quad\leq\quad(1-\varepsilon)*\mathbb{P}(T>(k-2)N)\leq\cdots\leq(1-\varepsilon)^{k}$
$\mathbb{E}[T]=\sum_{l=0}^{\infty}\mathbb{P}(T>l)\leq
N\sum_{k=0}^{\infty}\mathbb{P}(T>kN)\leq
N\sum_{k=0}^{\infty}(1-\varepsilon)^{k}=\frac{N}{1-(1-\varepsilon)}=\frac{N}{\varepsilon}<\infty$

∎

This shows that the expectation of a $T$, a stopping time is finite. Now, we
can move into the martingale transform.

######  Definition 5.6.

We assume we have a probability space with a filtration. $C_{n}$ is
predictable if $C_{n}$ is $\mathcal{F}_{n-1}$ measurable $\forall n$. Imagine
that $C_{n}$ is the strategy that you are making at time $n$, but to make
prediction at time $n$, we used the information at time $n-1$. Once we have
this base information, we can move to the martingale transform of $X$.

$Y_{n}=\sum_{1\leq k\leq n}C_{k}*(X_{k}-X_{k-1})$

######  Theorem 5.7.

If $C_{n}$ is predictable, and $0\leq C_{n}\leq k$. $\forall(n,\omega)$ and X
is a supermartingale, then so is the martingale transform, $C\cdot X$. If
$C_{n}$ is predictable, and $|C_{n}|\leq K$ $\forall(n,\omega)$ and $X$ is a
martingale, then so is $C\cdot X$

######  Remark 5.8.

If $C_{n}$ has $\mathbb{E}[C_{n}^{2}]<\infty$, and
$\mathbb{E}[X_{n}^{2}]<\infty$ $\forall n$, then Theorem $5.7$ still works
without the bound, $|C_{n}|\leq K$.

###### Proof.

To prove $Y$ is a martingale, then we must prove the following equation:
$\mathbb{E}[(Y_{n}-Y_{n-1})\mathcal{F}_{n-1}]=0$. To prove this, we must find
what the difference of $(Y_{n}-Y_{n-1})$ is and it should equal
$C_{n}*(X_{n}-X_{n-1})$, where $C_{n}$ is $\mathcal{F}_{n-1}$ measurable.
Let’s begin by factoring out the original equation.

$=C_{n}\mathbb{E}(X_{n}-X_{n-1}|\mathcal{F}_{n-1})$
$=C_{n}(\mathbb{E}(X_{n}|\mathcal{F}_{n-1})-\mathbb{E}(X_{n-1}|\mathcal{F}_{n-1}))$

If $X$ is a martingale, $X_{n-1}=0$, which is proved here. ∎

######  Theorem 5.9.

Let $T$ be a stopping time, if $X$ is a supermartingale, then so is $X^{T}$.

###### Proof.

$C_{n}:=\mathbbm{1}(n\leq T)=\mathbbm{1}(n-1<T)$, which is $\mathcal{F}_{n}$
measurable.

$(C\cdot X)_{n}=\sum_{k=1}^{n}C_{k}(X_{k}-X_{k-1})$
$=\left\\{\begin{array}[]{l}T\geq
n:\sum_{k=1}^{n}\left(x_{k}-x_{k-1}\right)=x_{n}-x_{0}\\\
T<n:\sum_{k=1}^{T}\left(x_{k}-x_{k-1}\right)=x_{T}-x_{0}\end{array}\right.$

$=X_{T\wedge n}-X_{0}={X^{T}}_{n}-X_{0}$

∎

Now we can move to the Optional Stopping Theorem, the relation between
stopping time and martingales.

######  Remark 5.10.

If $M_{n}$ is a martingale, then
$\mathbb{E}[M_{n}]=\mathbb{E}(\mathbb{E}[M_{n}|\mathcal{F}_{n-1}])=\mathbb{E}[M_{n-1}]=\cdots=\mathbb{E}[M_{0}]$
(from the Tower Property).

######  Remark 5.11.

If $T$ is a stopping time, $M_{n}$ is a martingale, then
$\mathbb{E}[M_{T\wedge n}]=\mathbb{E}_{M_{0}}$.

######  Theorem 5.12.

Doob’s Optional Stopping: Let $T$ be a stopping time, $X$ is a
supermartingale. If it follows these conditions:

  * •

T is bounded OR X is bounded and T is infinite almost surely OR
$\mathbb{E}[T]<\infty$ and $|X_{n}-X_{n-1}|\leq k$

Then $\mathbb{E}[X_{T}]\leq\mathbb{E}[X_{0}]$

###### Proof.

If $T$ is not almost surely bounded, approximate it by a sequence of bounded
stopping times $T_{n}=\min(T,n)$. Then we can use the martingale property to
show that for each
$T_{n},\mathbb{E}\left[X_{T_{n}}\right]=\mathbb{E}\left[X_{0}\right]$ Finally,
apply the Dominated Convergence Theorem to take the limit as
$n\rightarrow\infty$ and conclude that $\mathbb{E}\left[X_{T}\right]=$
$\mathbb{E}\left[X_{0}\right]$ ∎

After talking a bit about convergence in supermartingales, we can move to
understanding Doob’s Forward Convergence Theorem.

######  Definition 5.13.

$X\in L_{p}$, if $\mathbb{E}[|X|^{p}]<\infty$;
$|X|^{p}:=(\mathbb{E}[|X|^{p}])^{1/p}$, where p is positive.

######  Definition 5.14.

$X_{n}$ is bounded in $L^{p}$, if we take $sup_{n}{||X_{n}||_{p}<\infty}$ This
means if we take the p-norms, it will still be bounded regardless if we take
the largest value of n. However, there is an uniform bound, $||X_{n}||_{p}\leq
k<\infty$ for all $n$.

######  Lemma 5.15.

Using Doob’s upcrossing Lemma, if $X_{n}$ is a supermartingale, then
$(b-a)\mathbb{E}[U_{n}]$ on the interval $[a,b]\leq\mathbb{E}[X_{n}-a]^{-}$,
we can do a collarary.

######  Corollary 5.16.

Let $X$ be a supermartingale, bounded in
$L^{1}:sup_{n}\mathbb{E}[|X_{n}|]<\infty$. If we fix $a<b$, then
$(b-a)\mathbb{E}[U_{\infty}]$ on the interval
$[a,b]\leq|a|+sup_{m}\mathbb{E}[|X_{m}|]<\infty$. In particular,
$\mathbb{P}(U_{\infty}[a,b]=\infty)=0$.

###### Proof.

$(b-a)\mathbb{E}[U_{n}][a,b]\leq\mathbb{E}[X_{n}-a]^{-}\leq\mathbb{E}|X_{n}-a|\leq\mathbb{E}|X_{n}|+|a|\leq
sup_{n}\mathbb{E}|X_{n}|+|a|$. $U_{n}[a,b]$ is defined as number of
upcrossings until time N, so if we know that it is non-decreasing in N, then
it has a limit. $lim_{n\to\infty}:U_{\infty}[a,b]$. We need to make sure that
this expression is finite, proving the corollary. To begin this, we take
$(b-a)\mathbb{E}|U_{\infty}|[a,b]$ then use the monotone convergence theorem
to find the limit. This means that it equals to
$(b-a)lim_{n\to\infty}\mathbb{E}U_{n}[a,b]$, which is $\leq k+|a|$ due to the
first statement of the proof. ∎

Now, we can move to what Doob’s forward convergence theorem states.

######  Theorem 5.17.

Doob’s forward convergence: Let $X$ be a supermartingales, bounded in $L^{1}$.
Then $X_{\infty}:=lim_{n}X_{n}$ exists almost surely and is almost surely
finite.

###### Proof.

We want to prove that $X_{n}$ does not convergence. To prove that
$X_{\infty}:=lim_{n}X_{n}$ exists almost surely, we begin that
$=\bigcup_{a<b}liminfX_{n}<a<b<limsupX_{n}\subseteq\bigcup_{a<b}U_{\infty}[a,b]=\infty$,
so the probability of this latter part $=0$.

$\mathbb{P}(X_{n}\text{ does not converge})=0$

For the second part of the theorem, of ”almost surely finite,” we state that
$\mathbb{E}|X_{\infty}|=\mathbb{E}[\text{lim inf}]|X_{n}|\leq$ (using Fatou’s
Lemma) $[\text{lim inf}]\mathbb{E}|X_{n}|\leq sup\mathbb{E}|X_{n}|<\infty$.
Therefore, knowing that X is a supermartingale is bounded in $L^{1}$, we get
$|X_{\infty}|<\infty$ almost surely. ∎

######  Remark 5.18.

If $X_{n}\geq 0$ is a supermartingale, then the $L^{1}$ bound is automatic:
$\mathbb{E}|X_{n}|=\mathbb{E}[X_{n}]\leq\mathbb{E}[X_{0}]<\infty$

From here we are able to understand convergence in $L^{p}$, but take a case of
a $L^{2}$ martingale. $L^{2}$ martingales: $M_{n}\in L^{2}\quad\forall n$.
Then $M_{v}-M_{u}\perp L^{2}(\mathcal{F}_{u})$ In this situation, what does
orthogonal exactly mean? Let’s assume we have a square integrable random
variable that is finite, we can define a similar scalar product.
$\mathbb{E}(X\cdot Y)="\langle x,y\rangle"$ In the sense that these random
variable are almost like vectors, we can fully understand what orthogonality
means. Going back to the first first expression, $\forall v>u>t>s$, then
following is true, $\mathbb{E}[(M_{t}-M_{s})*(M_{v}-M_{u})]=0$ This is saying
that the increment of my martingale from s to t and u to m are orthongal in
the sense that my scalar product is 0. The proof of this is the following.

###### Proof.

$\mathbb{E}[\mathbb{E}[(M_{t}-M_{s})*(M_{v}-M_{u})]|\mathcal{F}_{u}]$ Because
$v>u>t>s$, $M_{t}-M_{s}$ are $\mathcal{F}$-measurable. That means our new
expression is,
$\mathbb{E}[(M_{t}-M_{s})*\mathbb{E}[(M_{v}-M_{u})]|\mathcal{F}_{u}]$ And by
the martingale property, our latter expectation equals 0, this implies a
theorem. ∎

######  Theorem 5.19.

An $L^{2}$-martingale $M_{n}$ is bounded in $L^{2}$ if.
$\sum_{k=1}^{\infty}\mathbb{E}\left(M_{k}-M_{k-1}\right)^{2}<\infty$. In this
case, $M_{n}\rightarrow M_{0}$ almost surely in $L^{2}$.

###### Proof.

We can show proof of the Pythagorean Theorem.
$\mathbb{E}[M^{2}_{n}]=||M_{n}||^{2}_{2}$

$=\mathbb{E}[M_{0}+\sum_{k=1}^{\infty}(M_{k}-M_{k-1})]^{2}$
$=\mathbb{E}[M^{2}_{0}]+\sum_{k=1}^{n}\mathbb{E}[M_{k}-M_{k-1}]^{2}+0$

Therefore, $M$ is bounded in

$L^{2}:\mathbb{E}[M_{n}^{2}]=\mathbb{E}[M_{0}^{2}]+\sum_{k=1}^{n}\mathbb{E}[M_{k}-M_{k-1}]^{2}\leq\mathbb{E}[M_{0}^{2}]+\sum_{k=1}^{\infty}\mathbb{E}[M_{k}-M_{k-1}]^{2}<\infty$

Also, bounded in $L^{1}$ because of Doob’s Forward Convergence, meaning that

$M_{n}\to^{\text{almost surely}}M_{\infty}$

The second norm states the following.

$\mathbb{E}[M_{\infty}-M_{n}]^{2}\leq\text{lim
inf}\mathbb{E}[M_{n+r}-M_{n}]^{2}=\text{lim
inf}\sum_{k=n+1}^{n+r}\mathbb{E}[M_{k}-M_{k-1}]^{2}=\sum_{k=n+1}^{\infty}\mathbb{E}[M_{k}-M_{k-1}]^{2}$

and as $n$ approaches $\infty$, this expression equals 0. ∎

##  6\. APPLICATIONS IN FINANCE

This martingale theory can be applied to stock market strategies. For
instance, the Martingale Strategy suggests doubling your investment when you
incur a loss. In the context of the stock market, if you purchase a stock at
10 dollars and its price drops to 9 dollars, you would buy another share at
the new price. If the stock price falls further to 8 dollars, you would buy
two additional shares. Eventually, when the stock price rebounds to 10
dollars, you not only recover your initial investment but also make a profit
from the shares purchased at the lower prices. This approach reflects the
martingale theory’s essence, where the future expected value of the process
remains equal to the present value, despite short-term fluctuations.

##  7\. ACKNOWLEDGEMENTS

I want to thank my mentor, Paulina Paiz for the valuable feedback and tips
throughout this process. I would like to also thank Simon Rubenstein-Salzedo
for organizing the Euler Circle.

## References

  * [1] Ishaan Goel, _INTRODUCTION TO MARTINGALES WITH AN APPLICATION IN FINANCE_. UChicago, 2022\.
  * [2] Sheffield, _Martingales, risk neutral probability, and Black-Scholes option pricing_. MIT.
  * [3] Dr. Samy Tindel, _Martingales_. Purdue University.
  * [4] David Williams, _Probability with Martingales_. Cambridge University Press, 1991\.