1 Black-Scholes-Merton Model
References: Columbia Note
The Black-Scholes-Merton (BS) model is a constant volatility model for a financial market. The model is a fundamental in mathematical finance, particularly in derivatives pricing where it is used to describe the dynamics of the underlying assets of derivatives such as options.
The Black-Scholes model consists of two assets, a risk free asset \(B\) and a risky asset (stock) \(S\) with price dynamics given by \[ \begin{align}dB_{t}&=rB_{t}dt \\dS_{t}& =\alpha S_{t}dt+\sigma S_{t}dW_{t},\end{align} \] where \(r\), \(\alpha\) and \(\sigma\) are deterministic constants. Here \(r\) is the risk-free interest rate, \(\alpha\) is the drift coefficient and \(\sigma\) is the diffusion coefficient.
2 Black-Scholes Model Option Pricing
In our note on Derivatives Pricing we discuss methods for pricing derivatives in which we always assume our underlying assets follow some kind of volatility model. Assuming an asset has dynamics given by the BS Model we are able to compute the fair market price of derivatives of these assets using one of two approaches. The first approach comes from Partial Differential Equations where one builds a self-financing replicating portfolio which under the no-arbitrage assumption allows us to construct the Black-Scholes PDE (BSPDE). The other approach is probabilistic in that it involves constructing a risk-neutral measure and computing the discounted expected payoff under this measure.
To introduce some notation, assume that an option on asset \(S\) has payoff at maturity time \(T\) given by \(\Phi(T,S)\)
2.1 Black-Scholes Partial Differential Equation (BSPDE)
We construct a self-financing (i.e. no money is added or removed) replicating portfolio using a stock \(S\) and a bond \(B\) which has exactly the same payoff as the option.
The idea is that one can build a self-financing replicating portfolio using stock and bond which has exactly the same payoff as the option. Assuming no arbitrage, the portfolio and the claim must have the same price.
To be specific, assume the self-financing replicating portfolio \(h_{t}\) consists of \(a_{t}\) units of stocks and \(b_t\) units of bond at time \(t\) (assuming the bond has initial price \(1\) at time \(0\)) so the portfolio value process is \[ V_{t}^h=a_{t}S_{t}+b_{t}e^{rt}, \] where \(r\) is the short interest rate. From Itô Formula we have that the change in the value of the portfolio is given by \[ dV_{t}=a_{t}dS_{t}+S_{t}da_{t}+d\langle S,a\rangle_{t}+rb_{t}e^{rt}dt+e^{rt}d_{t} \] Inspecting this we see \(S_{t}da_{t}+e^{rt}db_{t}\) denotes the change in the value of the portfolio that is due to the change of the position in stock and bond and \(d\langle S,a\rangle_{t}\) is the cross variation term. By definition, a self-financing portfolio satisfies \[ S_{t}da_{t}+d\langle S,a\rangle_{t}+e^{rt}db_{t}=0. \] Hence, the change in value of the process reduces to \[ \begin{align} dV_{t} & =a_{t}dS_{t}+rb_{t}e^{rt}dt \\ & = a_{t}(\alpha S_{t}dt+\sigma S_{t}dW_{t})+rb_{t}e^{rt}dt\tag{$\star$} \end{align} \] with the assumption that \(V_{t}=u(t,S_{t})\) has the Markovian form with deterministic time-independent feedback function \(u\). Applying Itô Formula we have that \[ \begin{align} dV_{t} & =\partial_{t}udt+\partial_{x}udS_{t}+ \frac{1}{2}\partial_{x x} ud\langle S,S\rangle_{t} \\ & = \left( \partial _{t}u+\mu S_{t}\partial_{x}u+ \frac{1}{2}\sigma^2S_{t}^2 \partial_{x x}u \right)dt+\sigma S_{t}\partial_{x}udW_{t} \tag{$\star\star$} \end{align} \] Matching the coefficients of \((\star)\) and \((\star\star)\) we obtain the following system \[ \begin{cases} \sigma a_{t}S_{t}=\sigma S_{t}\partial_{x}u \\ \mu A_{t}S_{t}+rb_{t}e^{rt}=\partial_{t}u+\mu S_{t}\partial_{x}u+\frac{1}{2}\sigma^2S_{t}^2\partial_{x x}u, \end{cases} \] hence the replicating portfolio \(h_t\) is satisfies \[ \begin{cases} a_{t}=\partial_{x}u(t,S_{t}) \\ b_{t}=\frac{\partial_{t}u(t,S_{t})+\frac{1}{2}\sigma^2S_{t}^2\partial_{x x}u(t,S_{t})}{\mathrm{re}^{rt}}. \end{cases} \] Our only remaining problem is to find the PDS characterization for \(u=u(t,x)\) to solve for \(u\). We recall that the self-financing property requires \[ a_{t}S_{t}+b_{t}e^{rt}=V_{t}=u(t,S_{t}) \] into which we plug in \(a_{t}\) and \(b_{t}\) computed above to get the Black-Scholes PDE (BSPDE) as a characterization for \(u\) \[ \partial_{t}u+rx\partial_{x}u_+\frac{1}{2}\sigma^2x^2\partial_{x x}u-ru=0 \] paired with the natural terminal condition that \[ u(T,x)=h(x), \] to match the payoff of the option. As a result, the option price at time \(0\) is given by \(V_{0}=u(0,S_{0})\).
2.1.1 Risk Neutral Pricing
The arbitrage-free price of a financial derivative is not equal to its discounted expected payoff since one has to compensate the buyers for the risk. However, if we assume the investor is risk-neutral, he has no positive or negative preference on taking any risk thus the price of the derivative is the expected payoff.
To construct such a risk neutral space we have to make sure that the discounted stock price \(\{e^{-rt}S_{t}\}\) is a martingale. As a result, the drift term in the dynamics of \(\{S_{t}\}\) shall be \(rS_{t}\) and all other extra drift terms shall be absorbed into the diffusion term which leads to a change of measure (Girsanov's Theorem).
We call the real world probability measure \(\mathbb{P}\) and the risk-neutral measure \(\mathbb{Q}\). We can manipulating our expression to read \[ \begin{align} dS_{t} & =rS_{t}dt+((\mu-r)S_{t}dt+\sigma S_{t}dW_{t} \\ & =rS_{t}dt+\sigma S_{t}\left(\frac{\mu-r}{\sigma}dt+dW_{t} \right) \\ & =rS_{t}dt+\sigma S_{t}dW_{t}^\mathbb{Q}, \end{align} \] where \(dW_{t}^\mathbb{Q}=\frac{\mu-r}{\sigma}dt+dW_{t}\) is a BM under \(\mathbb{Q}\).
Girsanov's Theorem enables us to find the connection between \(\mathbb{P}\) and \(\mathbb{Q}\). We write \(L_{t}=-\frac{\mu-r}{\sigma}W_{t}\) such that \(\langle W,L\rangle_{t}=-\frac{\mu-r}{\sigma}t,\) then \[ W_{t}^\mathbb{Q}=W_{t}-\langle W,L\rangle_{t} \] and so by Girsanov the Radon-Nikodym Derivative is given by \[ \frac{d\mathbb{Q}}{d\mathbb{P}}=\exp\left\{ L_{T}-\frac{1}{2}\langle L,L\rangle_{T} \right\}=\exp\left\{ - \frac{{\mu-r}}{\sigma}W_{T}- \frac{1}{2}\left( \frac{{\mu-r}}{\sigma} \right)^2T \right\}. \] It is simple to check that the RN derivative has expectation 1 so its not only a local martingale but also a martingale, which satisfies the condition of Girsanov’s theorem. In this risk neutral world, the price of the option at time 0 is just the discounted expected payoff \[ \exp\{-rT\}\mathbb{E}_{\mathbb{Q}}[h(S_{T})]. \] ### Key Results - Black-Scholes PDE (BSPDE) - European Call Option Price - European Put Option Price