Risk-Neutral Pricing
1 No-Arbitrage
Let \((B_t)\) be a risk-free bond and \(S=(S^1,\dots,S^d)\) the price processes of \(d\) risky assets on a probability space \((\Omega,\mathcal{F},\mathbb{P})\). A portfolio holds \(a_t^i\) units of asset \(i\) and \(b_t\) units of the bond.
A portfolio with value \(V_t = \sum_i a_t^iS_t^i + b_tB_t\) is self-financing if its value changes only through changes in the asset prices, not through adding or removing money,
\[ dV_t = \sum_i a_t^i\,dS_t^i + b_t\,dB_t. \]
An arbitrage is a self-financing portfolio with \(V_0 = 0\), \(V_T \ge 0\) almost surely and \(\mathbb{P}(V_T > 0) > 0\). That is, a profit with no initial investment and no risk of loss. A market is arbitrage-free if no arbitrage exists.
The principle of no arbitrage is the basis of derivatives pricing: two portfolios with the same payoff must have the same price. It is tied to the existence of a risk-neutral measure (defined below) by the following result.
A market is arbitrage-free if and only if there exists an equivalent martingale measure \(\mathbb{Q}\). The market is complete (every payoff can be replicated) if and only if \(\mathbb{Q}\) is unique.
(This is the version for general markets with an appropriate no-arbitrage condition, such as no free lunch with vanishing risk.)
2 Risk-Neutral Measure
The risk neutral measure (equilibrium measure, equivalent martingale measure) is a probability measure such that each share price is exactly equal to the discounted expectation of the share price under this measure. The risk neutral measure is used frequently in option pricing due to the fundamental theorem of asset pricing above, which implies that in a complete market, a derivative’s price is the discounted expected value of the future payoff under the unique risk-neutral measure, which exists if and only if the market is arbitrage-free.
Intuitively, the risk-neutral measure is the probability measure of a transformed random variable, typically the utility function of the payoff. The risk-neutral measure is the measure corresponding to an expectation of the payoff with a linear utility.
Let \(S\) be a \(d\)-dimensional market representing the price processes of the risky assets. \(B\) the risk-free bond and \((\Omega, \mathcal{F}, \mathbb{P})\) the underlying probability space. The measure \(\mathbb{Q}\) is called an equivalent (local) martingale measure if: 1. \(\mathbb{Q}\approx \mathbb{P}\) (measure equivalence); and 2. the processes \(\left( \frac{{S_{t}^i}}{B_{t}} \right)_{t}\) are (local) martingales with respect to \(\mathbb{Q}\) for all \(i=1, \dots, d\).
Suppose at a future time \(T\) a derivative has value equal to the payoff \[ V(T)=h(S_{[0,T]}) \] which is a random variable on the probability space describing the market. Further, suppose that the discount factor from \(T\) to \(0\) is \(D_{T}\). Then today’s fair value of the derivative is \[ V(0)=D_{T}\mathbb{E}_{\mathbb{Q}}[h(S_{[0,T]})]=D_{T}\mathbb{E}_{\mathbb{P}}\left[ \frac{d\mathbb{Q}}{d\mathbb{P}}\,h(S_{[0,T]}) \right], \] where \(\frac{d\mathbb{Q}}{d\mathbb{P}}\) is the Radon-Nikodym Derivative of \(\mathbb{Q}\) with respect to \(\mathbb{P}\), and therefore is still a martingale.
2.1 Example - Geometric Brownian Motion Stock Price Model
Let \(S_{t}\) be the price of a stock following Geometric Brownian Motion \[ dS_{t}=\mu S_{t}dt + \sigma S_{t}dW_{t}^\mathbb{P}, \] where: \(\mu\) is the drift term (expected return under real measure \(\mathbb{P}\)), \(\sigma\) is the diffusion term and \(W_{t}^\mathbb{P}\) is a Wiener process under the real measure. The solution is known to be \[ S_{t}=S_{0}\exp\left( \left( \mu-\frac{1}{2}\sigma^2 \right)t+\sigma W_{t}^\mathbb{P} \right). \] We wish to define new measure \(\mathbb{Q}\) such that the discounted stock price \(e^{-rt}S_{t}\) is a martingale under \(\mathbb{Q}\). Girsanov Theorem allows us to define a new Brownian motion under \(\mathbb{Q}\) such that \[ dW_{t}^\mathbb{Q}=dW_{t}^\mathbb{P}+\theta dt. \] We define the Radon-Nikodym Derivative \(Z_{t}\) of \(\mathbb{Q}\) with respect to \(\mathbb{P}\) as \[ Z_{t}= \frac{d\mathbb{Q}}{d\mathbb{P}}\Big|_{\mathcal{F}_{t}}=\exp\left( -\theta W_{t}^\mathbb{P}- \frac{1}{2}\theta^2 t \right), \] where for \(Z_t\) to be a true martingale (so \(\mathbb{Q}\) is well defined) we typically require [[novikovs-condition]]. Rewriting our expression in terms of our new BM \(W_{t}^\mathbb{Q}\) we have \[ dS_{t}=\mu S_{t}dt+\sigma S_{t}(dW_{t}^\mathbb{Q}-\theta dt)=(\mu-\sigma \theta)S_{t}dt+\sigma S_{t}dW_{t}^\mathbb{Q}, \] and selecting \(\theta\) such that our drift term is equal to the risk-free interest rate gives \[ \mu-\sigma \theta=r \implies \theta=\frac{{\mu-r}}{\sigma}. \] Thus the stock price process is given by \[ dS_{t}=\mu S_{t}dt+\sigma S_{t}\left( dW_{t}^\mathbb{Q}-\frac{{\mu-r}}{\sigma} dt \right)=rS_{t}dt+\sigma S_{t}dW_{t}^\mathbb{Q}, \] and discounted stock price is a \(\mathbb{Q}\)-martingale \[ d(e^{-rt}S_{t})=e^{-rt}\sigma S_{t}dW_{t}^\mathbb{Q}. \]