![]() This property gives us a scheme for pricing derivatives: The price of any trade is equal to the expectation of the trade’s winnings and losses under the risk-neutral measure. The risk-netural measure has a massively important property which is worth making very clear: The most general forms to useless level of technicality are by Delbaen and Schachermayer. Harrison-Kreps extended the result, and since then it has further extended. This basic intuition behind this framework goes back 35 years to Cox-Rubinstein. For more information, Duffie's Dynamic Asset Pricing is still the standard reference. Where do these prices come from? There are three ways to think about price determination:Īll conditions imply that the prices are strictly positive. ![]() Hence $r$ is the risk-free interest rate. $r$ is easy to identify: if the contingent claim is 1 dollar for any outcome, then it's price is the discounted value of a dollar using the risk-free interest rate. And the price of any claim is the discounted expectation according to this probability distribution. ![]() So the risk-neutral probabilities are essentially the normalized prices of "state-contingent claims", i.e., outcome-specific bets. Hence, the contingent claim has price equal toĭefine $r= 1/(\sum_ E^*(f)$$ Because of the law of one price, it must have the same price as the contingent claim. This portfolio has exactly the same random payoff as the contingent claim. Now, consider a portfolio of $f(1)$ units of basic contract $1$, $f(2)$ units of basic contract $2$, etc. Now, imagine that you have a contingent claim that pays a complex payoff based on the outcome, say $f(n)$. Contract $n$ costs $p_n$ and entitles you to one dollar if outcome $n$ occurs, zero otherwise. There are $N$ basic contracts available for purchase. Suppose that you and other bettors participate in a lottery with $N$ possible outcomes event will occur with probability $\pi_n$. So if you can find the risk neutral measure for an asset based on a set of outcomes, then you can use this measure to easily price other assets as an expected value. So if we can convert from the risk probability measure $(1/2, 1/2)$ to a risk neutral probability measure $(1/4, 3/4)$, then we can price this asset with a simple expectation. That means if you were risk-neutral, that you'd be assigning probabilities of 1/4 to heads and 3/4 to tails for an expected value of $\$25$ and an expected net payoff of $\$0$. Let's say that you would pay $\$25$ to play this game. Or equivalently, a risk neutral player doesn't need a positive expected net payoff to accept risk. A risk neutral player will accept risk and play games with expected net payoffs of zero. If you were risk neutral, then you WOULD pay $\$50$ for an expected value of $\$50$ for an expected net payoff of $\$0$. But it is unlikely that you'll pay $\$50$ to play this game because most people are risk averse. We bet on a fair coin toss - heads you get $\$100$, tails you get $\$0$. See also my answer to a similar question here: Why Drifts are not in the Black Scholes Formula So, you see, the basic concept of risk neutrality is quite natural and easy to grasp. As soon as the price of the gold certificate diverges from the original price a shrewd trader would just buy/sell the underlying and sell/buy the certificate to pocket a risk free profit - and the price will soon come back again. This is because all of the different risk preferences of the market participants is already included in the price of the underlying and the derivative can be hedged with the underlying continuously (at least this is what is often taken for granted). Therefore the risk preferences did not matter (=risk neutrality) because this product is derived (= derivative) from an underlying product (=underlying). Now, how would you price it? Would you think about your risk preferences? No, you won't, you would just take the current gold price and perhaps add some spread. The product just pays the current price of an ounce in $. You want to price a derivative on gold, a gold certificate. I assume you mean risk neutral pricing? Think of it this way (beware, oversimplification ahead -)
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