In this post we take the PDE approach to pricing derivatives in the Black-Scholes universe. In a subsequent approach we will cover the risk-neutral valuation approach; the two are essentially equivalent by the Feynman-Kac formula.
I. ASSUMPTIONS
We will assume that our market consists soley of an equity (stock ) $S$, a risk-free money market account (bond) $B$ with return $r\geq0$, and any number of derivatives with $S$ as the underlying. We will make several non-technical assumptions about $S$ and our market, collectively termed the "Black Scholes market."
1. Infinite liquidity. Market participants can buy or sell $S$ at any time.
2. Infinite depth. The buying and selling of $S$ does not affect the price of $S$, no matter the transaction size.
3. No friction. It costs nothing to buy or sell $S$ (i.e. trading $S$ incurs no transaction costs). In particular, the price paid by the buyer and seller is the same (i.e. bid-offer spread it $0$).
4. Constant risk-free rate. $r\equiv\text{const}.$
5. No arbitrage. There do not exist portfolios of assets where the portfolio is riskless and earns more than the risk-free money account $B$.
6. Infinite divisibility. Market participants can buy $S$ in any amount $\Delta\in\mathbb{R}$.
7. Short selling is possible. Market participants can short (borrow) $S$ at no cost.
8. No storage costs. Market participants can hold $S$ at no cost.
It is possible to relax several of these assumptions in various ways. Short selling is generally permitted in US markets (cf. SEC up-tick rule) and there are of course no storage costs for equities except for the payment of dividends if one has shorted a dividend paying stock.
One can factor variable risk-free interest rates directly into the model by allowing $r$ to be a deterministic function of time, or even to follow a stochastic process. There is a tremendous amount of on-going research in dealing with transaction costs, but let us just say that most institutional investors in derivative contracts (e.g. market makers/banks) take huge positions and therefore the effect of transaction costs is minimized (as we will see below, it is the frequency of trades in $S$ in a process termed dynamic hedging that is the source of transaction costs, not the quantity/volume of a single trade). For similar reasons, the divisibility assumption is also relatively minor when large numbers are involved, since the fractional component of the quantity in a transaction is small relative to the whole number quantity.
The assumption of infinite liquidity is generally not an issue for plain vanilla contracts that actively trade on exchanges, as there are always counter-parties available to take opposing positions. The assumption becomes more dubious when one moves to the over-the-counter market where exotic contracts are traded; however, the existence of market makers willing to take positions in essentially any contract that can be hedged essentially validates the assumption. The non-zero bid-offer spread of course then violates the no friction assumption.
The assumption which is most arguable, however, is that of infinite depth, which violates the law of supply and demand. At the end of the day however, our goal is to develop a model which is both simple and provides a good approximation to reality, and these assumptions will lead us to such a model which is generally good enough, and can be perfected in various ways as needed.
II. ASSET PRICE MODEL - QUANTITATIVE ANALYSIS APPROACH
In order to price derivatives dependent on a stock $S$ (or any underlying for that matter), it is necessary to first come up with a mathematical model for the price movements of $S$. This is where the approaches of fundamental and quantitative analysis diverge markedly. Whereas fundamental analysis attempts to predict stock (asset) price movements through a careful analysis of a company's financial statements and other qualitative sources of information like press releases, general market sentiment, new product launches, etc. (this process is often termed equity research and is the method used by traditional long-term value investors), quantitative analysis models stock (asset) price movements with a stochastic model which we now discuss, and attempts to make predictions on future stock price movements through statistical significance analysis/econometrics, which we will not discuss in any detail here. Incidentally, a third approach, known as technical analysis, attempts to use past data and trends in order to predict the price in the future. As we will see, the validity of such an approach represents a direct contradiction to the quantitative model we are about to develop, and we therefore assume it is invalid. There is in fact, little evidence to suggest that technical analysis leads to reliably predictable results of any degree.
We assume the existence of a probability space $(\Omega,\mathbb{P},\mathcal{F})$ and an associated filtration $\{\mathcal{F}\}_{t\geq0}$ that supports a Brownian motion $\{W(\omega,t)\}_{\omega\in\Omega,t\geq0}.$ We will assume that the reader is familiar with the construction and basic properties of $W(\omega,t)$.
Associated to any process $\Delta(\omega,t)\in L^{2}(\Omega\times\mathbb{R}_{t})$ is an integral called the Ito integral
$$I(\omega,t)=\int_{0}^{t}\Delta(\omega,t)\;dW(\omega,t),$$
which we also assume the reader is familiar with (it is defined for each fixed $\omega\in\Omega$ just like the usual Riemann-Stieltjes integral, except that the sample point in the approximating sum is always taken to be the left-end of each partition interval). In what follows, we will generally suppress reference to $\omega\in\Omega$ and sometimes use the more customary subscript notation $W_{t}, I_{t}, \Delta_{t},$ etc.
An Ito process is a stochastic process $\{X_{t}\}_{t\geq0}$ defined by the stochastic integral equation (SDE)
$$X(t)=X(0)+\int_{0}^{t}a(X(s),s)\;ds+\int_{0}^{t}b(X(s),s)\;dW(s).$$
Note that the terminology stochastic refers to the fact that the equation involves an Ito integral; however, the solution to an SDE is a random process, and so the method of solution of an SDE in principle does not involve any probability theory. It is the fact that $W(s)$ is almost surely non-differentiable that complicates any proposed solution method (this is expressed by Ito's lemma, which is a resultantly more complicated version of the chain rule).
For $\mu,\sigma\geq0$, if $a=\mu X$ and $b=\sigma X$, we get (with $S=X$)
$$S(t)=S(0)+\mu\int_{0}^{t}S(s)\;ds+\sigma\int_{0}^{t}S(s)\;dW(s).$$
This is our model for stock-price movements, and it is often called geometric Brownian motion. It is often expressed locally in "differential" form as
$$dS=\mu Sdt+\sigma SdW$$
or
$$\frac{dS}{S}=\mu dt+\sigma dW.$$
There is no harm in doing this as long as one understands vividly that the differential form has no rigorous meaning attached to it, whereas the integral form is well-defined mathematically. In fact, the entire topics of SDE's should be replaced by SIE's, since $W$ is no-where differentiable and so differential equations involving it really make no sense. Nonetheless, the convention pervades the subject, and in particular mathematical finance, and so one must get accustomed to it.
There are two reasons why the differential representation of geometric Brownian motion is used. First, it provides better intuition for why the model is a good model for stock price movements, as we will explain shortly. Second, as we will see through the remainder of this article, the it facilitates the computations which come up in quantitative finance, in particular those involving Ito's lemma.
Why is geometric Brownian motion a good model for stock price movements? The two parameters $\mu$ and $\sigma$ represent drift (expected return) and standard deviation per unit time (volatility), respectively.
Geometric Brownian motion captures the intuitive idea that asset prices should drift according to the expected return $\mu$; the riskier the asset, the higher the expected return demanded by investors, and therefore the greater drift in the asset price, regardless of any random fluctuations. The expected return on an asset is also independent of the stock price, i.e. investors will demand $\mu$ whether the asset trades at $50$ or $5$. This is modeled by the $\mu\int_{0}^{t}S(s)ds$ term, or $\mu Sdt.$ If there was no stochastic component to the model, then we would have
$$S(t)=S(0)+\mu\int_{0}^{t}S(s)\;ds,$$
or by the fundamental theorem of calculus
$$S'(t)=\mu S(t),$$
which has the solution
$$S(t)=S(0)e^{\mu t}.$$
This is how we would expect the price of a riskless asset with return $\mu$ to grow.
Of course, asset prices are anything but deterministic, and it is natural to assume the presence of an unbiased (i.e. centered at $0$) white noise weighted according to the perceived volatility of the asset (volatility is not a risk-meaure; it is, among other things, directly tied to the liquidity of the asset and how trading affects its price). The asset price swings should also be directly proportional the the price of the asset itself. For instance, a stock that trades around $1$ will have price swings markedly lower than an asset that trades around $100.$ This combined with the volatility of the asset is modeled by the $\sigma\int_{0}^{t}S(s)\;dW(s)$ term, or $\sigma SdW.$
If $\mu=0$, then we would have
$$S(t)=S(0)+\sigma\int_{0}^{t}S(s)\;dW(s).$$
There is no corresponding fundamental theorem of calculus for the Ito integral; however, we will see how to solve this below by using Ito's lemma. In any event, if we combine these two terms, we recover the geometric Brownian motion model
$$S(t)=S(0)+\mu\int_{0}^{t}S(s)\;ds+\sigma\int_{0}^{t}S(s)\;dW(s).$$
III. EXTENDING THE MODEL TO DERIVATIVES
The extension of the model to derivatives, that is functions of the asset price $S$ and time $t$, involves a technical theorem known as Ito's lemma. An expository article on its motivation and rigorous proof can be found on one of my previous posts entitled a rigorous proof of Ito's lemma.
If $V=V(S(t),t)$ is the value of a contingent claim on $S$, then we have from Ito's lemma (appropriately extended to handle geometric Brownian motion) that $V$ follows the process
$$\begin{align*}
V(t)&=V(0)+\int_{0}^{t}V_{t}(S(s),t)\;ds+\int_{0}^{t}V_{S}(S(s),s)\;dW(s)+\frac{1}{2}\int_{0}^{t}V_{SS}(S(s),s)\;ds\\
&=V(0)+\mu\int_{0}^{t}S(s)V_{S}(S(s),s)\;ds+\sigma\int_{0}^{t}S(s)V_{S}(S(s),s)\;dW(s)+\int_{0}^{t}V_{t}(S(s),s)\;ds+\frac{1}{2}\sigma^{2}\int_{0}^{t}S^{2}(s)V_{SS}(S(s),s)\;ds\\
&=V(0)+\int_{0}^{t}\left(\mu S(s)V_{S}+\frac{1}{2}\sigma^{2}S^{2}(s)V_{SS}(S(s),s)+V_{t}(S(s),s)\right)\;ds+\sigma\int_{0}^{t}S(s)V_{S}(S(s),s)\;dW(s).\end{align*}$$
In the more usual differential form, we have then
$$dV=\left(\mu SV_{S}+\frac{1}{2}\sigma^{2}S^{2}V_{SS}+V_{t}\right)dt+\sigma SV_{S}dW.$$
We again point out that the differential form is just a short-hand for the integral form above, which is the only mathematically meaningful expression.
VI. DERIVING A NO-ARBITRAGE CONDITION ON $V(S(t),t): THE BLACK-SCHOLES PDE
In this section $\Pi$ denotes the value of a portfolio consisting of $\Delta$ units of an asset $S$, whose value $\{S_{t}\}_{t\geq0}$ follows the geometric Brownian motion process discussed previously, and a short position in an derivative whose underlying is $S$.
As we will see, the exact nature of the derivative and its payoff (the value of $V$ at expiry, i.e. $V(S(T),T)$) function is unimportant, just that its value at time $0\leq t\leq T$ depends only $(S(t),t)$. In other words, $V$ is path independent. The PDE $V$ must satisfy is the same whether $V$ is the value function for an option, a forward agreement, future, whatever. In the subsequent posts we will discuss briefly on how one might extend the PDE to cover more exotic types of derivatives with path dependent payoffs or other exotic features written into their contracts like barriers and knock-outs. We will also assume that $S$ provides no income over the life of the derivative. In subsequent posts we will discuss how to incorporate income (i.e. dividends if $S$ is a stock), and in any event the modification is trivial (just replace $r$ with $r-q$ below, $q$ being the
The idea of the derivation is to determine the initial capital (cost) that must be put up to construct the portfolio $\Pi$ and hedge the position (risk) so as to ensure a riskless payoff, which must be equal to the payoff of the risk-free money market account $P_{0}e^{rt}$; otherwise there would exist simple arbitrage strategies. This puts a condition on $\Delta$, the quantity of the underlying we must hold at time $t$ in order to hedge against a short position in $V$. The strategy is known as continuous time delta hedging. The hedging eliminates the risk of the portfolio in real time, and as mentioned, means the portfolio must earn the risk free rate. It is really quite phenomenal that this is possible, and the basic reason that it is has to do with the fact that both $S$ and $V$ are affected by the same underlying uncertainty, namely the Brownian motion process $\{W_{t}\}_{t\geq0}$. Using Ito's lemma, we can eliminate the terms involving $W$ and therefore eliminate uncertainty. Indeed, it is impossible to do this if we assume more complicated asset price models different from geometric Brownian motion (this is an active area of research since there is much evidence to suggest that the true process followed by an asset $S$ is a Levy flight, or a fat tailed Brownian motion).
From the way we have constructed our portfolio and the previous sections, we have at time $0\leq t\leq T$ we have
$$\begin{align*}
\Pi(t)&=\Delta(t)S(t)-V(S(t),t)\\
&=\Delta(t)\left\{S(0)+\int_{0}^{t}\mu S(s)\;ds+\int_{0}^{t}\sigma S(s)\;dW(s)\right\}\\
&\;\;\;\;-\left\{V(0)+\int_{0}^{t}\left(\mu S(s)V_{S}(S(s),s)+\frac{1}{2}\sigma^{2}S^{2}(s)V_{SS}(S(s),s)+V_{t}(S(s),s)\right)\;ds+\int_{0}^{t}\sigma S(s)V_{S}(S(s),s)\;dW(s)\right\}\\
&=\Delta(t)S(0)-V(0)\\
&\;\;\;\;+\int_{0}^{t}\left[\Delta(t)\mu S(s)-\left(\mu S(s)V_{S}(S(s),s)+\frac{1}{2}\sigma^{2}S^{2}(s)V_{SS}(S(s),s)+V_{t}(S(s),s)\right)\right]ds\\\
&\;\;\;\;+\int_{0}^{t}\left[\Delta(t)\sigma S(s)-\sigma S(s)V_{S}(S(s),s)\right]dW(s).
\end{align*}$$
We seek to eliminate the uncertainty in $\Pi$, so we match for each time
$$\Delta(t)=V_{S}(S(t),t).$$
(Observe very carefully below how this substitution works!)
Thus, $\Pi$ being riskless, we must have in order to preclude arbitrage opportunities with this portfolio,
$$\begin{align*}
\Pi(t)&=V_{S}(S(0),0)S(0)-V(0)\\
&\;\;\;\;+\int_{0}^{t}\left[\left(\mu V_{S}(S(s),s)S(s)-\mu S(s)V_{S}(S(s),s)\right)-\left(\frac{1}{2}\sigma^{2}S^{2}(s)V_{SS}(S(s),s)+V_{t}(S(s),s)\right)\right]ds\\\
&\;\;\;\;+\int_{0}^{t}\left[\sigma V_{S}(S(s),s)S(s)-\sigma S(s)V_{S}(S(s),s)\right]dW(s).\\
&=V_{S}(S(0),0)S(0)-V(0)-\int_{0}^{t}\left[\frac{1}{2}\sigma^{2}S^{2}(s)V_{SS}(S(s),s)+V_{t}(S(s),s)\right]ds\\
&=\Pi(0)e^{rt}\\
&=\text{value of risk free money market account at time}\;t\;\text{with initial investment}\;\Pi(0).
\end{align*}$$
Note that with the non-differentiable Brownian motion now gone, we can legitimately pass to the differential form of this expression by using the fundamental theorem of calculus. Differentiating with respect to $t$ we have the localized version of $\Pi$ given by
$$\Pi_{t}=-\left(V_{t}+\frac{1}{2}\sigma^{2}S^{2}V_{SS}\right)=r\Pi(0)e^{rt}=r\Pi=r(SV_{S}-V).$$
Consequently,
$$V_{t}+\frac{1}{2}\sigma^{2}S^{2}V_{SS}+rSV=rV.$$
Note that we also have
$$V_{t}+\frac{1}{2}\sigma^{2}S^{2}V_{SS}=-r\Pi_{0}e^{rt}.$$
Of coruse we don't know $\Pi_{0}$ a priori, so we appeal to the previous PDE, which is known as the Black-Scholes PDE. Observe that despite $S$ being present, there is nothing random about the PDE and $S$ can be regarded as as dummy variable. It is the no-arbitrage requirement that leads to a deterministic outcome, and at each time/asset price $(S(t),t)$ we have the price given by the solution to the PDE, with remaining time to expiry beign $T-t$.
It is backwards parabolic, and so requires a terminal condition $f$ and time $t=T$, the payoff for the specific derivative under consideration. Additional boundary conditions will lead to prices for other more exotic options discussed in subsequent posts.
If $V=V(S(t),t)$ is the value of a contingent claim on $S$, then we have from Ito's lemma (appropriately extended to handle geometric Brownian motion) that $V$ follows the process
$$\begin{align*}
V(t)&=V(0)+\int_{0}^{t}V_{t}(S(s),t)\;ds+\int_{0}^{t}V_{S}(S(s),s)\;dW(s)+\frac{1}{2}\int_{0}^{t}V_{SS}(S(s),s)\;ds\\
&=V(0)+\mu\int_{0}^{t}S(s)V_{S}(S(s),s)\;ds+\sigma\int_{0}^{t}S(s)V_{S}(S(s),s)\;dW(s)+\int_{0}^{t}V_{t}(S(s),s)\;ds+\frac{1}{2}\sigma^{2}\int_{0}^{t}S^{2}(s)V_{SS}(S(s),s)\;ds\\
&=V(0)+\int_{0}^{t}\left(\mu S(s)V_{S}+\frac{1}{2}\sigma^{2}S^{2}(s)V_{SS}(S(s),s)+V_{t}(S(s),s)\right)\;ds+\sigma\int_{0}^{t}S(s)V_{S}(S(s),s)\;dW(s).\end{align*}$$
In the more usual differential form, we have then
$$dV=\left(\mu SV_{S}+\frac{1}{2}\sigma^{2}S^{2}V_{SS}+V_{t}\right)dt+\sigma SV_{S}dW.$$
We again point out that the differential form is just a short-hand for the integral form above, which is the only mathematically meaningful expression.
VI. DERIVING A NO-ARBITRAGE CONDITION ON $V(S(t),t): THE BLACK-SCHOLES PDE
In this section $\Pi$ denotes the value of a portfolio consisting of $\Delta$ units of an asset $S$, whose value $\{S_{t}\}_{t\geq0}$ follows the geometric Brownian motion process discussed previously, and a short position in an derivative whose underlying is $S$.
As we will see, the exact nature of the derivative and its payoff (the value of $V$ at expiry, i.e. $V(S(T),T)$) function is unimportant, just that its value at time $0\leq t\leq T$ depends only $(S(t),t)$. In other words, $V$ is path independent. The PDE $V$ must satisfy is the same whether $V$ is the value function for an option, a forward agreement, future, whatever. In the subsequent posts we will discuss briefly on how one might extend the PDE to cover more exotic types of derivatives with path dependent payoffs or other exotic features written into their contracts like barriers and knock-outs. We will also assume that $S$ provides no income over the life of the derivative. In subsequent posts we will discuss how to incorporate income (i.e. dividends if $S$ is a stock), and in any event the modification is trivial (just replace $r$ with $r-q$ below, $q$ being the
The idea of the derivation is to determine the initial capital (cost) that must be put up to construct the portfolio $\Pi$ and hedge the position (risk) so as to ensure a riskless payoff, which must be equal to the payoff of the risk-free money market account $P_{0}e^{rt}$; otherwise there would exist simple arbitrage strategies. This puts a condition on $\Delta$, the quantity of the underlying we must hold at time $t$ in order to hedge against a short position in $V$. The strategy is known as continuous time delta hedging. The hedging eliminates the risk of the portfolio in real time, and as mentioned, means the portfolio must earn the risk free rate. It is really quite phenomenal that this is possible, and the basic reason that it is has to do with the fact that both $S$ and $V$ are affected by the same underlying uncertainty, namely the Brownian motion process $\{W_{t}\}_{t\geq0}$. Using Ito's lemma, we can eliminate the terms involving $W$ and therefore eliminate uncertainty. Indeed, it is impossible to do this if we assume more complicated asset price models different from geometric Brownian motion (this is an active area of research since there is much evidence to suggest that the true process followed by an asset $S$ is a Levy flight, or a fat tailed Brownian motion).
From the way we have constructed our portfolio and the previous sections, we have at time $0\leq t\leq T$ we have
$$\begin{align*}
\Pi(t)&=\Delta(t)S(t)-V(S(t),t)\\
&=\Delta(t)\left\{S(0)+\int_{0}^{t}\mu S(s)\;ds+\int_{0}^{t}\sigma S(s)\;dW(s)\right\}\\
&\;\;\;\;-\left\{V(0)+\int_{0}^{t}\left(\mu S(s)V_{S}(S(s),s)+\frac{1}{2}\sigma^{2}S^{2}(s)V_{SS}(S(s),s)+V_{t}(S(s),s)\right)\;ds+\int_{0}^{t}\sigma S(s)V_{S}(S(s),s)\;dW(s)\right\}\\
&=\Delta(t)S(0)-V(0)\\
&\;\;\;\;+\int_{0}^{t}\left[\Delta(t)\mu S(s)-\left(\mu S(s)V_{S}(S(s),s)+\frac{1}{2}\sigma^{2}S^{2}(s)V_{SS}(S(s),s)+V_{t}(S(s),s)\right)\right]ds\\\
&\;\;\;\;+\int_{0}^{t}\left[\Delta(t)\sigma S(s)-\sigma S(s)V_{S}(S(s),s)\right]dW(s).
\end{align*}$$
We seek to eliminate the uncertainty in $\Pi$, so we match for each time
$$\Delta(t)=V_{S}(S(t),t).$$
(Observe very carefully below how this substitution works!)
Thus, $\Pi$ being riskless, we must have in order to preclude arbitrage opportunities with this portfolio,
$$\begin{align*}
\Pi(t)&=V_{S}(S(0),0)S(0)-V(0)\\
&\;\;\;\;+\int_{0}^{t}\left[\left(\mu V_{S}(S(s),s)S(s)-\mu S(s)V_{S}(S(s),s)\right)-\left(\frac{1}{2}\sigma^{2}S^{2}(s)V_{SS}(S(s),s)+V_{t}(S(s),s)\right)\right]ds\\\
&\;\;\;\;+\int_{0}^{t}\left[\sigma V_{S}(S(s),s)S(s)-\sigma S(s)V_{S}(S(s),s)\right]dW(s).\\
&=V_{S}(S(0),0)S(0)-V(0)-\int_{0}^{t}\left[\frac{1}{2}\sigma^{2}S^{2}(s)V_{SS}(S(s),s)+V_{t}(S(s),s)\right]ds\\
&=\Pi(0)e^{rt}\\
&=\text{value of risk free money market account at time}\;t\;\text{with initial investment}\;\Pi(0).
\end{align*}$$
Note that with the non-differentiable Brownian motion now gone, we can legitimately pass to the differential form of this expression by using the fundamental theorem of calculus. Differentiating with respect to $t$ we have the localized version of $\Pi$ given by
$$\Pi_{t}=-\left(V_{t}+\frac{1}{2}\sigma^{2}S^{2}V_{SS}\right)=r\Pi(0)e^{rt}=r\Pi=r(SV_{S}-V).$$
Consequently,
$$V_{t}+\frac{1}{2}\sigma^{2}S^{2}V_{SS}+rSV=rV.$$
Note that we also have
$$V_{t}+\frac{1}{2}\sigma^{2}S^{2}V_{SS}=-r\Pi_{0}e^{rt}.$$
Of coruse we don't know $\Pi_{0}$ a priori, so we appeal to the previous PDE, which is known as the Black-Scholes PDE. Observe that despite $S$ being present, there is nothing random about the PDE and $S$ can be regarded as as dummy variable. It is the no-arbitrage requirement that leads to a deterministic outcome, and at each time/asset price $(S(t),t)$ we have the price given by the solution to the PDE, with remaining time to expiry beign $T-t$.
It is backwards parabolic, and so requires a terminal condition $f$ and time $t=T$, the payoff for the specific derivative under consideration. Additional boundary conditions will lead to prices for other more exotic options discussed in subsequent posts.
