Binomial Option Pricing and Risk-Neutral Probabilities
Summary
The document asks why a binomial option-pricing derivation uses a portfolio combining a long option with a short position in the underlying stock. In its one-step example, the stock starts at a stated price and can move up or down by one unit; the call payoff is one in the up state and zero in the down state. The question contrasts an expected payoff using the stated real-world probabilities with the option value obtained through a no-arbitrage argument.
The included answer says those subjective probabilities do not determine the option price. Instead, it chooses risk-neutral probabilities so the expected future stock price equals its current price when interest rates are zero; this gives equal probabilities for the two stock outcomes and the stated option value. The reply offers the pricing principle and calculation, but does not fully explain how the half-share hedge position arises or generalize the method beyond the simple one-step setting.
Key ideas
- Subjective probabilities for stock moves do not directly set the no-arbitrage option price.
- Risk-neutral probabilities make the expected future stock price equal its present value after accounting for interest.
- In the stated zero-rate example, the risk-neutral probabilities assign equal weight to the up and down moves.
- The answer gives the option value but leaves the hedge ratio intuition largely unresolved.
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# Intuition behind short 1/2 stock in option value - Paul Wilmott Quant Finance Chapter 3.3 # Intuition behind short 1/2 stock in option value - Paul Wilmott Quant Finance Chapter 3.3 I don't get the intuition behind the construction of long option + short 1/2 stock portfolio for finding the value of an option using binomial model. In Paul Wilmott `Introduces Quantitative Finance`, he uses the binomial asset model to find the option value. There are three assumption of his model - We will have a stock, and a call option on that stock expiring tomorrow. - The stock can either rise or fall by a known amount between today and tomorrow. - Interest rates are zero. And the underlying is currently worth \$100 and can rise to \$101 or fall to \$99 between today and tomorrow, where the probability of a given underlying to rise or fall are 0.6 and 0.4, respectively. Based on this simple model, we have to find the value of such an option. Usually as a beginner in quant finance we will think the value of an option should be 0.6 since the net return is 1 if the stock rises and 0 if the stock falls. Then the expectation values of the value of the option is 0.6. However, Paul Wilmott says this is not correct and the correct option price should be 0.5. To show the correct option value, he constructs a portfolio to long an option and short 1/2 stock. After some simple maths, he shows that by the risk-neutrality and no-arbitrage principle, the correct option value is 0.5. However, where does the intuition of constructing such a portfolio come from? Why is 1/2 instead of other numbers? Why do we have to short the stock instead of long? ## Answer by achirikhin (score 2, accepted) https://quant.stackexchange.com/a/79423 This is a classic interview question, btw. Probabilities you mention are "subjective" hence are irrelevant for option pricing. You need to use risk neutral stock migration probabilities, such that expectation of discounted values of the stock equal the present value. Since rates are zero, expected value of stock should equal the present value. Denoting $p$ the probability of going to 101, $101p + 99(1-p) = 100$ gives $p=0.5$. You can now use it to price the option.
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