No-Arbitrage Principle and Risk-Neutral Valuation › Understanding Risk-Neutral Probabilities

Risk-neutral probabilities are emphatically not real-world probabilities. This point deserves emphasis because it is a common source of confusion. They do not represent:

• The actual likelihood of the stock going up or down
• Any investor's subjective beliefs about future prices
• Historical frequencies of up and down moves

In our example, $q = 0.5751$ says nothing about whether the stock will rise or fall. If a stock analyst believes there's an 80% chance the stock goes up, that real-world probability $p = 0.80$ is irrelevant for pricing. The option price is $10.94 regardless of what anyone believes about the stock's direction. A bullish investor and a bearish investor, holding opposite views about the stock's prospects, must both agree on this price of $10.94. This is counterintuitive at first. How can the option price not depend on the probability the stock goes up? After all, a call option is worth more if the stock is more likely to rise. If you believe the stock has an 80% chance of reaching $120, shouldn't you pay more for the call than someone who thinks there's only a 40% chance?

The resolution lies in recognizing that the replication argument works regardless of probabilities because we're constructing a hedge that eliminates risk. When we hold $\Delta$ shares against the option, the portfolio is perfectly hedged: it produces the same value whether the stock goes up or down. The hedged position is completely insensitive to the stock's direction.

In equilibrium, the cost of this hedged portfolio must be the price of the derivative. The hedge completely eliminates exposure to the up/down uncertainty, so the underlying probabilities become irrelevant to the pricing. Since we can perfectly replicate the option using the stock and bond, the option's price is pinned down by the replication cost, not by probability assessments.

Another way to see this: risk-neutral probabilities implicitly adjust for risk preferences. In the real world, investors are typically risk-averse. They demand higher expected returns from risky assets to compensate for bearing risk. Stocks have expected returns above the risk-free rate precisely because investors require this risk premium. A stock that is expected to return 10% per year includes both the time value of money (perhaps 5%) and a risk premium (another 5%) that compensates for the possibility of loss.

Risk-neutral probabilities mathematically combine two effects:

1. Real-world probabilities of outcomes
2. Risk adjustments that discount risky payoffs more heavily

The net result is a probability measure under which all assets earn the risk-free rate. The risk premium has been baked into the probabilities rather than the discount rate. Instead of using high real-world probabilities and a high discount rate (reflecting risk aversion), we use lower risk-neutral probabilities and a low discount rate (the risk-free rate). Both approaches give the same answer because they capture the same economic reality from different angles.

We now have two mathematically equivalent ways to price derivatives:

Real-World Approach: Compute expected payoffs under actual probabilities and discount at a risk-adjusted rate

$$f_0 = e^{-\mu T} E^P[f_T]$$

where:

• $f_0$: derivative price today
• $\mu$: risk-adjusted discount rate
• $T$: time to expiration
• $E^P[\cdot]$: expectation under real-world probability measure $P$
• $f_T$: derivative payoff at time $T$

This approach requires knowing the real-world probabilities $P$ and determining the appropriate risk-adjusted discount rate $\mu$ for the specific payoff pattern. The discount rate must reflect the time value of money, the riskiness of the cash flows, and how that risk relates to the broader market.

Risk-Neutral Approach: Compute expected payoffs under risk-neutral probabilities and discount at the risk-free rate

$$f_0 = e^{-rT} E^Q[f_T]$$

where:

• $f_0$: derivative price today
• $r$: risk-free interest rate
• $T$: time to expiration
• $E^Q[\cdot]$: expectation under risk-neutral probability measure $Q$
• $f_T$: derivative payoff at time $T$

This approach uses the artificial risk-neutral probabilities $Q$ and always discounts at the risk-free rate $r$, regardless of the derivative's risk characteristics.

The risk-neutral approach is vastly more practical because:

• We don't need to estimate real-world probabilities (notoriously difficult)
• We don't need to determine the appropriate risk premium for each derivative
• The risk-free rate is directly observable

The entire complexity of risk preferences and probability estimation gets absorbed into finding the right risk-neutral measure, which is determined by no-arbitrage conditions. Rather than solving the hard problem of estimating probabilities and discount rates, we solve the tractable problem of finding the replicating portfolio or equivalently the risk-neutral measure.