Overview
Risk neutral pricing is one of the primary pricing frameworks for derivative pricing. It can be used within the context of both discrete or continuous time models.
Intuition
The name risk neutrality is a reference to utility based choice theory in micro-economics. A risk neutral world is one in which all the participants have zero risk aversion. That is, the prices of assets do not reflect the riskiness of the asset in any sense. In such a world, agents are indifferent between two assets with the same expected pay-offs, regardless of risk.
The arbitrage arguments used to price derivatives do not make any reference to the expected returns of any of the assets involved. That is, when constructing the replicating portfolio, the trader does not need to know the expected returns of the assets involved, only the risks.
That means that derivatives have the same price in the risk neutral world as the real world.
Price and Numeraire
Price is generally thought to be the number of dollars required to buy a given asset. That is, the price is given in terms of dollars. But there is no reason why price cannot be denoted in terms of some other unit. For example, one could ask what the price of IBM stock is in terms of Microsoft stock. That is, how many units of Microsoft are required to be exchanged in order to receive IBM stock.
The asset in which a price is quoted is referred to as the numeraire. That is, one could ask what the price of IBM stock is in terms of the numeraire Microsoft.
When we assign a variable to a given asset, say {% X(t) %} (the price of the asset as a function of time), the default unit is assumed to be dollars. When we require a different numeraire, say and asset denoted by {% Y(t) %}, we will write {% X_Y(t) %}.
{% X_Y(t) = \frac{X(t)}{Y(t)} %}
Fundamental Theorem of Asset Pricing
Given an asset {% Y(t) %}, If there exists a measure {% \mathbb{Q} %} such that for any arbitrary asset {% X(t) %}, the price process {% X_Y(t) %} is a martingale, the there is no arbitrage in the market.
Martingale Pricing
Martingale pricing utilizes the fundamental theorem to calculate the prices of derivatives or other assets. Starting with an asset {% Y(t) %}, called the numeraire, and a real world measure {% \mathbb{P} %} the fundamental theorem asserts that there is an equivalent measure {% \mathbb{Q} %} such that
{% \frac{X(0)}{Y(0)} = \mathbb{E}_{\mathbb{Q}}(\frac{X(T)}{Y(T)}) %}
Sometimes a discount process, {% D(t) %} is defined such that
{% D(t) = \frac{1}{Y(t)} %}
Then the pricing equation becomes,
{% \frac{X(0)}{Y(0)} = \mathbb{E}_{\mathbb{Q}}(D(T)X(T)) %}
If the asset {% Y(t) %} is chosen such that {% Y(0) = 1 %}, then this equation becomes
{% X(0) = \mathbb{E}_{\mathbb{Q}}[D(T)X(T)] %}
Martingale pricing can be extended to describe the value of the discounted
price process at any point along its price path as follows
{% D(t)X(t) = \mathbb{E}_{\mathbb{Q}}[D(T)X(T)|\mathcal{F}(t)] %}
where {% \mathcal{F} %}
is the
filtration
of the sample space.
Frameworks
Given the general risk neutral framework, several different models can be created within that framework.
Examples
- Single Geometric Brownian Motions and Risk Free
- Multiple Geometric Brownian Motions and Risk Free
- Risk Neutral for Fixed Income
Extracting Risk Neutral Probabilities
The risk neutral framework provides a way to calculate derivative prices from a risk neutral probability distribution. However, in the real world, we are given a set of market prices, without a risk neutral distribution. It is often necessary to extract a risk neutral distribution from the set of market prices (a form of calibration) first, which is then used to price other derivatives.