The proof of the Black Scholes and Merton partial differential equation for options is one of the most likely to be asked in any interview in Quantitative Finance. Let's go through it together.
Modern option pricing theory was developed in the late 70s by different economists, including Black, Scholes and Merton. You can read a bit more about their story on Wikipedia. The closed form for the European option in particular is credited as one of the most significant developments in Quantitative Finance.Scholes and Merton, who survived Black, derived immense prestige in finance for their work. They received in 1997 the Nobel Prize 🏆 (Black did not, because it cannot be awarded posthumously).Their reputation was tarnished however with their involvement in LTCM, a hedge fund that led to one of the most spectacular collapse on Wall Street, and its infamous US$3.6 billion bailout in 1998. It's been used as a case-in-point that financial theorists do not necessarily have an edge over practitioners (but that's another debate!).
There are actually two main results from the work of Black, Scholes and Merton.
We'll run through the full proof of the PDE equation, which is quite standard for students in quantitative finance 🎓. For the the European call option result, we'll discuss the resolution assumptions but not the proof details.
The Black Scholes Merton model makes two groups of assumptions.
We note \(r\) the risk-free rate, \(T\) the time at maturity, \(S_t\) the price of the underlying stock at time \(t\) and \(K\) the strike price.
The Black Scholes equation describes the price \(V\)of an option.
\[\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2S^2\frac{\partial^2 V}{\partial S^2}+rS\frac{\partial V}{\partial S}-rV=0\]
By convention, we express the solution for the European call option. The reason why only the call is provided is because of put call parity, which allows you to infer the put.
\[C(S_t, t; r , T, K) = \mathcal{N}\left(d_+\right)\times S_t - \mathcal{N}\left(d_-\right)\times K \times e ^ {-r(T - t)}\]
One of the most powerful features of the Black Scholes Merton model is that every parameter is observable, except for volatility.
Its assumptions are rather strong, but they are fairly realistic and have become standard assumptions over time.
You can read more about the Goemetric Brownian Motion on my page on the proof of its closed form here.
It might seem intimidating, but once you got the gist of it, the proof for the PDE is not that difficult. You need prior knowledge of Ito's Lemma, and nothing else!
Disclaimer. There are several variations of the proof and notations. I don't pretend the below one is the absolute best one, but it is the one that has worked best for me!
We recall the general equation of the price of an asset that follows a geometric Brownian motion, using the drift \(\mu\) and the volatility \(\sigma\). \(W\) is the Wiener process (i.e. the Brownian motion itself). \[dS = \mu Sdt+\sigma SdW\]
We will use Ito's lemma to derive the Black Scholes PDE. We call \(V(S,t)\) the price of an option and express it as a function of the underlying stock \(S(t)\) and maturity \(T\).
\[dV = \left(\frac{\partial V}{\partial t} + \frac{1}{2}S^2\sigma^2\frac{\partial^2 V}{\partial S^2}\right)dt + \frac{\partial V}{\partial S}dS\]
We then replace with the definition of the geometric Brownian motion and refactor the terms. Note the right hand side term is now \(dW\).
\[dV = \left(\mu S\frac{\partial V}{\partial S} + \frac{\partial V}{\partial t} + \frac{1}{2}S^2\sigma^2\frac{\partial^2 V}{\partial S^2}\right)dt + \sigma S\frac{\partial V}{\partial S}dW\]
The trick is now to represent a portfolio of value \(\Pi\) with a short option of value \(V\) that is delta-hedged in stock \(S\). We can do the other way around but it simplifies the computations later to short the option and long the stock.Delta-hedging means that we create a riskless portfolio where \(\Delta = \frac{\partial \Pi}{\partial S} = 0\). That delta is one of the "greeks". It follows that for such portfolio, with \(b \geq 0\).
\[\begin{align}\Pi &= -V + bS\\ \frac{\partial \Pi}{\partial S}&= -\frac{\partial V}{\delta S} + b\\ 0 &= -\frac{\partial V}{\partial S} + b\\ b &= \frac{\partial V}{\delta S}\end{align}\] and therefore we construct the portfolio such that the amount of stock \(S\) is \(\frac{\partial V}{\partial S}\), which is the delta of the option of price \(V\).
If the portfolio is risk-free, we know the returns are a function of the risk free rate \(r\) and time \(t\).
\[d\Pi = r\Pi dt\]
We first compute the left-hand side. \[\begin{align}\Pi &= -V + \frac{\partial V}{\partial S}S\\ d\Pi &= -dV + \frac{\partial V}{\partial S}dS\\\end{align}\]
We recall the equation of the geometric Brownian motion for \(dS\) and compute as follows. \[\begin{align}\frac{\partial V}{\partial S}dS &= \mu S\frac{\partial V}{\delta S}dt + \sigma S \frac{\partial V}{\partial S}dW\\ \frac{\partial V}{\partial S}dS -dV&= \mu S\frac{\partial V}{\delta S}dt + \sigma S \frac{\partial V}{\partial S}dW\\ &- \left(\mu S\frac{\partial V}{\partial S} + \frac{\partial V}{\partial t} + \frac{1}{2}S^2\sigma^2\frac{\partial^2 V}{\partial S^2}\right)dt - \sigma S\frac{\partial V}{\partial S}dW \end{align}\]
And you notice that the \(dW\) terms cancel each-other out: that's why we say delta-hedging is risk-free. We are just left with the below. \[\begin{align}d\Pi&= - \left(\frac{\partial V}{\partial t} + \frac{1}{2}S^2\sigma^2\frac{\partial^2 V}{\partial S^2}\right)dt\\ r\Pi dt&= - \left(\frac{\partial V}{\partial t} + \frac{1}{2}S^2\sigma^2\frac{\partial^2 V}{\partial S^2}\right)dt\\ r\left(-V + \frac{\partial V}{\partial S}S\right) dt&= - \left(\frac{\partial V}{\partial t} + \frac{1}{2}S^2\sigma^2\frac{\partial^2 V}{\partial S^2}\right)dt\\ \end{align}\]
When you simplify by \(-dt\) on both sides you obtain the Black Scholes PDE.
\[\boxed{\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2S^2\frac{\partial^2 V}{\partial S^2}+rS\frac{\partial V}{\partial S}-rV=0}\]
I find it easier to re-derive the PDE than to learn it by heart, the trick in that version of the proof is the delta-hedge portfolio. The PDE derivation is commonly asked in interviews.The idea of the resolution for the European call option is to look at the boundary conditions and then to apply a change of variables. It's not very straight forward I think, and you can find better proofs online (like Wikipedia's.)
I'd prefer to spend more time explaining the boundary conditions and the idea behind them. Once you understand them, the proof is just about knowing the change of variables and calculus.
The idea of finding the closed form solution of a PDE with the boundary conditions is the same as in many problems in Physics. In our case, we are bound by time \(t\in[0; T]\) and the stock price \(S\geq0\).
While not perfect, the Black Scholes Merton model is still applied by practitioners because of its relative simplicity. The Black Scholes volatility, implied from the model, is used by traders.The most notable "gap" between the theoretical model and the observed prices is the volatility smile. In a nutshell, the Black Scholes Merton model assumes a flat volatility (when plotting the implied volatility as a function of the option strike price), but in practice it's more like a parabola. And because the second derivative is positive, it's nicknamed a smile 😉.
Another insight is that you can theoretically fully hedge the risk of an option, that's actually the same trick as the one I used in the proof by introducing \(\Delta=\frac{\partial V}{\partial S}\). However, if you are fully hedged under no-arbitrage, you can only attain the risk-free rate: no free lunch!
We can identify the option greeks \(\Theta, \Gamma, \Delta\) in the PDE equation as follows. \[V = \frac{1}{r}\Theta + S\left(\frac{1}{2r}\sigma^2S\times\Gamma + \Delta\right)\]
If we go back to our delta-hedged portfolio, we have \(\Pi = -V + \Delta S\) which is actually the portfolio of the option issuer: one sold the option and bought the stock to hedge themselves. \[\begin{align}\Pi &= -\frac{1}{r}\Theta - S\left(\frac{1}{2r}\sigma^2S\times\Gamma + \Delta\right)+ \Delta S\\ &= -\frac{1}{r}\Theta - \frac{1}{2r}\sigma^2S^2\Gamma\\ \Theta &\propto -\Gamma\\ \end{align}\] The inverse relationship between \(\Theta\) and \(\Gamma\) means that the faster an option value decreases when it gets closer to maturity, the more the convexity will increase when the stock price is close to the strike price. In practice, traders will also gamma-hedge their portfolios to control hedging costs.
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