Risk neutrality: why real-world probabilities do not matter
Because the option can be hedged perfectly with the share, its price follows from no arbitrage alone. Investors' risk aversion and their views on the chance of an up move are already in the share price, so the option is valued with risk-neutral probabilities and the risk-free rate.
Before this: Pricing calls and puts with risk-neutral probabilities
On the exam
- Describe the concept of risk neutrality in derivatives pricing
Describe the concept of risk neutrality. Expect concept items: what π depends on (and what it does not), why discounting is at the risk-free rate, how an arbitrage appears when a price is off, and the effect of wider up and down moves.
How deeply to learn each part
- Know the idea
- Understand it in general. It frames other topics, but you will not calculate with it.
- Use it
- Understand it well enough to solve a question with it. Recognize when it applies.
- Memorize
- Learn it by heart. The exam will not give it to you.
You know this one. The full lesson is folded away. The takeaways, formulas and checkpoint below are your quick refresher, and the checkpoint is a fair test of whether you still do.
You tested strong here. The full lesson is folded away. The takeaways, formulas and checkpoint below are your quick refresher, and the checkpoint is a fair test of whether you still know it.
Why can we price options as if investors did not care about risk?
Because the option can be hedged perfectly with the share, so its price follows from no arbitrage alone. That is arbitrage-free pricing . It does not need to know investors’ risk aversion (how much they dislike risk) or the real-world probability of an up move. Those things are already reflected in today’s share price S₀, and the option is priced from S₀ and the two possible end prices.
What does risk neutrality actually say?
It says the option’s price equals its expected payoff, discounted at the risk-free rate, when the expectation uses the right probabilities: the ones under which the share itself is expected to earn exactly r. This is risk neutrality used as a pricing device. It does not say investors are risk-neutral.
Those probabilities are π and 1 − π from the last lesson. You can check them: under π, the share’s expected end value is its start value grown at the risk-free rate.
Because the share earns r under π, every derivative on the share can be valued by discounting its expected payoff at r. π depends on R_u, R_d and r only. That is all the model needs.
- Inputs that matter Memorize they sit inside the hedge
- Up and down gross returns Memorize wider spread = more volatility = higher call and put
- Risk-free rate r Memorize sets π and the discounting
- Exercise price and share price set the payoffs
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- Inputs that do not matter Memorize already in the share price
- Real-world chance of an up move, q Memorize no effect
- Investors' risk aversion Memorize no effect
- Expected return on the share no effect
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What goes wrong with the real-world probability?
You get a price the hedge can beat. Suppose analysts think the up move has a 65% chance and discount the call’s expected payoff at r.
The gap of 2.21 (11.04 − 8.83) is the profit today, which is 2.34 at expiry. A price that lets a trader lock in a riskless profit is not a no-arbitrage price, so the real probability is the wrong one to use.
Go deeper Could the real probability work with a different discount rate?
Yes, in principle. Discounting the real-world expected payoff of 0.65 × 18 = 11.70 gives 8.83 only at a rate of about 32.5%, because the call is a leveraged bet on the share. Finding the right risk-adjusted rate for each option is hard and changes with the share price and time. Risk-neutral pricing avoids that: the risk adjustment sits in the weights, and the rate is simply r.
How does volatility enter the tree?
Through the spread between R_u and R_d. A wider spread means a more volatile share, so options are more likely to finish well in the money and are worth more. The spread is a direct input to the price, unlike the chance of an up move. This is volatility in a binomial model .
Both options gain. This matches the factor table from the option pricing module: higher volatility raises calls and puts alike.
Exam tip Two exam rules cover most questions here. If an input is the real probability of an up move, an investor view or risk aversion, it does not change the price. If it is the spread of the moves, it raises both options.
Go deeper The same idea elsewhere
A forward price is also not a forecast. It comes from the cost of carrying the asset, not from where anyone thinks the price will go. Risk-neutral pricing is the same principle for options. It is also the basis of Monte Carlo option pricing in the simulation module, where simulated payoffs are averaged under risk-neutral growth and discounted at r.
Memorize this
MemorizeRisk neutrality, what to recall
| Question | Answer |
|---|---|
| What does π depend on? | , and only |
| Does it depend on or risk aversion? | No |
| Which rate discounts the expected payoff? | The risk-free rate |
| Do investors have to be risk-neutral? | No. It is a pricing device |
| Wider and spread? | Higher call and put values |
| Higher ? | Higher call, lower put |
Memory hook The option is priced from the share, not from a forecast. The hedge cancels every view and every taste for risk.
Know it cold Price = risk-neutral expected payoff, discounted at r. Real-world q never enters.
- Inputs that matter Memorize they sit inside the hedge
- Up and down gross returns Memorize wider spread = more volatility = higher call and put
- Risk-free rate r Memorize sets π and the discounting
- Exercise price and share price set the payoffs
-
- Inputs that do not matter Memorize already in the share price
- Real-world chance of an up move, q Memorize no effect
- Investors' risk aversion Memorize no effect
- Expected return on the share no effect
-
Quick example
Share 50 goes to 60 or 45, r = 3%, π = 0.4333. 0.4333 × 60 + 0.5667 × 45 = 26.00 + 25.50 = 51.50 = 50 × 1.03. Risk-free growth.
- 0.433333×60+(0.566667×45)= 51.5
- 50×1.03= 51.5
- 0.433333ENTER60×0.566667ENTER45×+ 51.5
- 50ENTER1.03× 51.5
ENTER enter (on the HP, puts the number on the stack)
How it is asked
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What does π depend on?
The risk-neutral probability in a one-period model depends on…
Up and down gross returns and the risk-free rate. Trap: investors' risk aversion or the real probability.
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Change the real probability
The real probability of an up move rises. What happens to the call?
Nothing. Trap: it rises.
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Widen the tree
R_u rises or R_d falls, all else equal. Effect on call and put?
Both rise (more volatility). Trap: call up, put down.
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Overpriced call
Call trades above its binomial value. How do you profit?
*Sell the call, buy h shares, borrow** at r. Trap: buy the call.
Exam traps
- Saying risk-neutral pricing assumes investors are risk-neutral. It does not: it is a way to price off a hedge.
- Saying a higher real probability of an up move raises the call's price in the model. It has no effect.
- Discounting at the share's expected return or a risk-adjusted rate instead of the risk-free rate.
- Thinking more volatility helps calls only. Wider moves raise puts as well.
Checkpoint
5 exam-style questions. About 90 seconds each.
1In the one-period binomial model, the risk-neutral probability depends on:
2Risk-neutral pricing most likely assumes that:
3All else equal, widening the spread between the up and down moves in a binomial tree will most likely:
4A call option trades at a higher price than the one-period binomial model implies. An investor can earn a riskless profit by:
5In a one-period binomial model, an increase in the risk-neutral probability of an up move, all else equal, will most likely cause the value of a put to:
Key takeaways
- Risk-neutral probabilities make the share earn r. Derivatives are then discounted at r.
- Real probabilities and risk preferences are already in the share price.
- Using a real probability gives a price the hedge can beat, so arbitrage is available.
- More volatility (a wider tree) raises both call and put values.
Quick example
Share 50 goes to 60 or 45, r = 3%, π = 0.4333. 0.4333 × 60 + 0.5667 × 45 = 26.00 + 25.50 = 51.50 = 50 × 1.03. Risk-free growth.
- 0.433333×60+(0.566667×45)= 51.5
- 50×1.03= 51.5
- 0.433333ENTER60×0.566667ENTER45×+ 51.5
- 50ENTER1.03× 51.5
ENTER enter (on the HP, puts the number on the stack)
All Derivatives formulas, every lesson
Practice exam-style questions on The One-Period Binomial Model
Questions about this lesson?
Where this comes back
- Fixed-Income Cash Flows and Types Fixed Income
- Introduction to Risk Management Portfolio Construction
- Simulating Prices and Returns Quantitative Methods
Questions candidates ask
What is risk-neutral pricing?
Pricing a derivative as its expected payoff, using probabilities that make the underlying earn exactly the risk-free rate, discounted at the risk-free rate. The probabilities are a pricing device built from the tree and r, not a forecast.
Does risk-neutral pricing assume investors are risk-neutral?
No. The option can be hedged perfectly with the share, so its price follows from no arbitrage whatever investors feel about risk. The method just gives a convenient way to calculate that price.
Why don't real probabilities matter in option pricing?
The hedge cancels the share's movement in both states, so the chance of each state drops out. Beliefs about the up move and attitudes to risk are already in today's share price, which is the input the option is priced from.
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