What is Risk Neutral Valuation?
Risk neutral valuation involves valuing options based on their expected payoffs, discounted from expiration to present, with the assumption of average growth at the risk-free rate.
Option value = Expected present value of payoff (under a risk-neutral random walk).
As a result, the average growth rate of the underlying asset has no impact on its value. Volatility, or the standard deviation of the underlying return, is important to consider. with actuality, estimating average growth is more difficult than estimating volatility. However, with derivatives, we only need to predict volatility, which is pretty constant. Hedging an option with the underlying eliminates exposure to the stock’s direction, regardless of whether it goes up or down. Eliminating risk removes the need to consider its value. As a result, it’s as if no one values risk and all assets increase at the risk-free rate.
If a derivative product can be hedged dynamically and perfectly (assuming no defaults and known volatility), the portfolio loses its randomness and acts like a bond.
A Short Example
A stock worth $44.75 has an average annual growth rate of 15%. The volatility is 22%. The interest rate is 4%. Calculate the value of a call option with a $45 strike and a two-month expiration. What can you do?
First and foremost, the 15% average growth is completely unimportant. The value of derivatives remains unaffected by the stock’s rise or direction. To predict where a stock will be in two months, simulate many future routes with an average annual growth rate of 4% and a volatility of 22%. Calculate the call payout for each of these pathways. Calculate the average across all paths by presenting each number back to today. That is your option value.
A Deep Dive into Risk Neutral Valuation
Risk-neutral pricing of derivatives takes use of the perfect correlation between changes in an option’s value and the underlying asset. This correlation should be perfect if the underlying random factor is the only one present. If an option increases in value with a rise in the stock, a long option and a sufficiently short stock position should not experience random fluctuations, indicating that the stock is hedging the option. The resulting portfolio is risk-free.
To sell a stock short, you must first have its proper number. The term ‘delta’ typically refers to a model. Delta hedging is not completely risk-free in practice due to the need for a mathematical model to calculate the delta and the limitations of quantitative finance models. There are various flaws with risk-neutral valuation. First, the hedge must be rebalanced on a constant basis.
To maintain a risk-free position, it’s necessary to regularly buy or sell shares when Delta fluctuates.
Obviously, this is impossible in practice. Second, model correctness is crucial. The underlying must be consistent with certain assumptions, such as Brownian motion without leaps and known volatility.
Risk neutral pricing allows us to evaluate derivatives by simulating their risk-neutral course and calculating payoffs. The payoffs are then discounted to the present and averaged. This average represents the contract’s fair value.
More Explanations on Risk Neutral Pricing
Explanation 1
In the Black-Scholes model, proper hedging eliminates all risk. If there is no risk, we should not expect remuneration for it. We can use a measure that assumes growth at the risk-free interest rate.
Explanation 2
If the asset model is 
Explanation 3
Two measures are considered equivalent if they share the same sets of zero probability. A portfolio is an arbitrage under one measure only if it is one under all equivalent measures, as zero probability sets remain constant. A price is non-arbitrable in the real world only if it is also non-arbitrable in the risk-neutral environment. The risk-neutral pricing is always non-arbitrable. Arbitrage is impossible if all assets have a martingale-like discounted price process. After discounting, all fundamental assets, like as stocks and bonds, become martingales. Defining the option price as the discounted expectation also results in a martingale. In a risk-neutral environment, everything follows the martingale distribution. Therefore, there is no arbitrage in the real world.
Explanation 4
For calls with a continuous distribution of strikes from zero to infinity, it is possible to synthesise any payout with the same expiration. These calls define the risk-neutral probability density function for each expiry, allowing us to understand the synthesised option as risk-neutral random walks. Static replication allows for model-independent pricing of complex derivatives using vanillas. However, the requirement for continuous distribution somewhat undermines this reasoning.
Risk neutral valuation assumes continuous hedging, zero transaction costs, and continuous asset routes. We may discover that simplifying our reality is ineffective once we leave it.
Related Readings
- Put-Call Parity: All You Need To Know
- Modelling Approaches in Quantitative Finance: All You Need To Know
- Central Limit Theorem In Finance: Power Your Portfolio Now
great!