How does the "risk-neutral pricing framework" work?

Question

I've struggled for a long time to understand this - What is this? And how does it affect you?

Yes I mean risk neutral pricing - Wilmott Forums was not clear about that.

Answer 1

I assume you mean risk neutral pricing? Think of it this way (beware, oversimplification ahead ;-)

You want to price a derivative on gold, a gold certificate. The product just pays the current price of an ounce in $.

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. Therefore the risk preferences did not matter (=risk neutrality) because this product is derived (= derivative) from an underlying product (=underlying).

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). 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...

So, you see, the basic concept of risk neutrality is quite natural and easy to grasp. Of course, the devil is in the details... but that is another story.

See also my answer to a similar question here: Why Drifts are not in the Black Scholes Formula

Answer 2

We bet on a fair coin toss -- heads you get $\$100$, tails you get $\$0$. So the expected value is $\$50$. But it is unlikely that you'll pay $\$50$ to play this game because most people are risk averse. If you were risk neutral, then you WOULD pay $\$50$ for an expected value of $\$50$ for an expected net payoff of $\$0$. A risk neutral player will accept risk and play games with expected net payoffs of zero. Or equivalently, a risk neutral player doesn't need a positive expected net payoff to accept risk.

Let's say that you would pay $\$25$ to play this game. 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 $\$0$.

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.

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.

Answer 3

Suppose that you and other bettors participate in a lottery with $N$ possible outcomes; event will occur with probability $\pi_n$. There are $N$ basic contracts available for purchase. Contract $n$ costs $p_n$ and entitles you to one dollar if outcome $n$ occurs, zero otherwise.

Now, imagine that you have a contingent claim that pays a complex payoff based on the outcome, say $f(n)$. The expected value of the payoff is

$$E(f(n))=\sum_n \pi_n f(n) =E(f)$$

Now, consider a portfolio of $f(1)$ units of basic contract $1$, $f(2)$ units of basic contract $2$, etc. This portfolio has exactly the same random payoff as the contingent claim. Because of the law of one price, it must have the same price as the contingent claim. Hence, the contingent claim has price equal to

$$\text{price}(f)=\sum_n p_n f(n)$$

Define $r= 1/(\sum_{i=1}^N p_i)$ and set $\tilde p_n := r p_n$, which is a probability measure, and you can rewrite

$$\text{price}(f)=r^{-1} \sum_n \tilde p_n f(n)=r^{-1} E^*(f)$$

So the risk-neutral probabilities are essentially the normalized prices of "state-contingent claims", i.e., outcome-specific bets. And the price of any claim is the discounted expectation according to this probability distribution. $r$ 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. Hence $r$ is the risk-free interest rate.

Where do these prices come from? There are three ways to think about price determination:

1. They are determined by a non-arbitrage condition, where no bettor can make something for nothing almost surely;
2. They are determined by an equilibrium condition, where all bettors optimize their utility;
3. They are determined by a single-agent utility optimization problem.

All conditions imply that the prices are strictly positive. For more information, Duffie's Dynamic Asset Pricing is still the standard reference.

This basic intuition behind this framework goes back 35 years to Cox-Rubinstein. Harrison-Kreps extended the result, and since then it has further extended. The most general forms to useless level of technicality are by Delbaen and Schachermayer.

Answer 4

The risk-netural measure has a massively important property which is worth making very clear:

The price of any trade is equal to the expectation of the trade’s winnings and losses under the risk-neutral measure.

This property gives us a scheme for pricing derivatives:

1. take a collection of prices of trades that exist in the market (eg swap rates, bond prices, swaption prices, cap/floor prices),
2. back out the set of risk-neutral probabilities that these prices imply,
3. calculate the expectation of the derivative trade’s payoff under these risk-neutral proabilities,
4. that is the price of the derivative.

The risk-neutral measure is in some sense the flip-side of the concept of risk premium.

Without even getting mixed up with stock and bond prices and suchlike, we can get a good sense of the risk-premium concept at work in a simple betting game.

The classic example, a game of coin tossing:

1. a player hands over some money, say £X, to play,
2. the host tosses an unbiased coin,
3. if it comes up heads then the player is given £2,
4. but if it comes up tails then nothing is given back.

A textbook on probability will tell you that the price of £1 per go is fair for this game because the concept of fair is defined in probability textbooks to mean that the price paid should equal the value of the expected winnings. Clearly it does for this example.

But let’s get savvy, step back from the theory, and ask how much would different players be prepared to pay for this game. Consider two different players:

1. person A that has £1.50 in their pocket but is under pressure from a traffic warden to pay £2 for a parking ticket (and nothing less than £2 will do),
2. person B that has £10 in their pocket and doesn’t really need anything more than that.

Don’t you think you could convince person A to pay up to their whole £1.50 for this game? Person B might be a harder sell, but perhaps they’d come around if we charged something like 50p a go and advertised the game as ‘potential 4 times returns on your investment’?

The important point is that the theoretical fair price may well be £1 for this game, but the actual price at which we sell the game may be something different since it will depend on the circumstances of the players we are selling it to.

The difference between the actual and theoretical price is called the risk premium. Throwing in a bit of market language, we can write this as:

the risk premium is the amount of premium (or discount) that needs to be added to the theoretical fair price in order to match the actual price of the trade in the market.

Remark: The risk-neutral measure is risk-neutral because in this alternative reality the price paid by player A for the game contains no risk premium — the price is exactly equal to the value of the expected winnings of the game.

I have written a little bit more on this in my blog if you want to go see.

Answer 5

A market is said to be complete if any contingent claim can be replicated by an admissible (i.e. with value process bounded from below) self-financing (i.e. all gains and losses exactly offset each other) trading strategy, a so-called replicating strategy. This strategy being constructed from primary securities - the market prices of which are unique - it must be that its price is identical to everyone, and the strategy is therefore independent of any assumptions on risk aversion. Any discrepancy between the replicating strategy's price and its underlying primary securities would be wiped out by arbitrage trades by market participants, regardless of their risk preferences.

Now, suppose you want to price a contingent claim, e.g. a European option on an equity security. Assuming the market is complete, the payoff of this security can be perfectly replicated using existing securities. Again, by the same arguments as above, the market price of the option and of the replicating strategy must be exactly the same under a no-arbitrage condition, regardless of risk preferences. Therefore, neither a positive nor a negative risk premium can be embedded into the equilibrium market price of the option, or equivalently of the replicating strategy (actually, a sort of "aggregate" risk premium is already included in the prices of the replicating strategy's primary securities, but no further risk premium is added when pricing the contingent claim).

We have shown that if the market has no arbitrage opportunities and is complete, then it must be that the option's market price is exactly equal to that of the replicating strategy, and that this price is in fact unique. This is essentially what the (Second) Fundamental Theorem of Asset Pricing (FTAP) says. Since the replicating strategy does not depend on any assumptions concerning risk preferences, it does not matter what assumptions are made on the risk preferences of the market participants. Therefore, the price in the real-world market (where risk-averse, risk-neutral and risk-seeking participants meet) must equal that in a risk-neutral market. Since it is much more convenient (and mathematically powerful, e.g. martingale theory) to work in a risk-neutral world, this is the standard pricing approach used in mathematical finance.

Answer 6

I learned something about decision sciences, stochastic processes and mathematical modeling in college before I learned something about quantitative finance, so I had struggled to grasp the concept which is so familiar and yet so alien. Here is my two bits. I might overlap with some of the previous answers at some parts, but the approach is different.

Before explaining risk neutral or martingale measure framework we need to clarify something. The objective of option pricing is to find a fair price. The definition of the fair price is the value which both side of the contract (long and short) should make the exact amount of money as if they were agreeing on a deposit with a fixed amount of interest rate (called risk-free rate with common abbreviation r). Plus some of the fine print; no friction (no taxes, no spread, borrow and lend at the same rate r).

To make things even easier with an example assume r is 0 and we live in a perfectly deterministic world. The price of asset A is 100 today and will be 120 in three months. Finally assume that I can only do a transaction (buy or sell the asset itself) after three months. Call it a rigged game if you like. If I were to write a European Call option contract with strike price 100 and maturity three months the fair price of the contract would be 20. Because in three months I will need to sell the asset for 100 to the counterparty (the person who bought the contract from me) and I have to buy it from the market for 120.

There we go. The buyer gave me 20 at t=0 and I bought the asset from the market for 120 and gave it to the buyer for 100. Then the buyer sold the asset to the market for 120. I have nothing in my hands and the buyer got his 20 back. In other words, we only killed time by doing a bunch of transactions and we are on square one. This is called equilibrium.

Of course real world is harsher. First of all real world is quite complex and stochastic (at least to us). You can sell and buy assets most of the time. And risk free rate can be different than 0.

Risk neutral pricing framework is only a way to estimate the fair price, albeit a popular one. The basic trick is to replace the drift with the risk-free rate. Then you discount your prediction on the asset by the risk free rate. The expected value of your outcome is the same as your current position. In other words, on average you don't get an extra dime than putting your money into a deposit or a solid bond.

You can also see that in the classical CAPM model. They essentially say 'No matter how or what you trade, on average you can't make more or less than the risk free rate.' This is the equilibrium state. If you are familiar with the concept it is similar to a steady-state markov chain. You can also relate with the common belief 'You can't beat the market (or index).'

Complete markets assumption is the core part of the option pricing (at least the distinguished BS pricing). It simply says (by the fundamental theorems of asset pricing - Shreve's book) the market is arbitrage-free (otherwise it would be trivial and there won't) and risk-neutral measure is unique. Oh, there can be more than one (enter the Levy processes or GARCH pricing), it is unsurprisingly called incomplete markets.

All those paragraphs and I haven't mentioned hedging yet. Recall the assumption you can't trade before three months. If you relax that assumption the price of the option drops to zero. Because since I know it will be 120 in three months and risk free rate is 0, I can immediately buy the asset for 100 and hedge myself completely. If I can sell the option for more than 0 to a sucker I make extra money (in other words arbitrage).

Now if we relax the deterministic part, in a complete market you can do the hedging by buying and selling the underlying asset continuously as the price of the underlying changes. It is also called delta hedging. The result will be the same though, no extra money to neither side on average.

"On average" or "expectation" are fundamental concepts that you need to set straight. I especially like Chevalier de Mere example or casino example ('House always wins') in such cases.

If you want me to speak more enigmatic so my words ring more true I quote Wikipedia:

"In economics, a complete market (or complete system of markets) is one in which the complete set of possible gambles on future states-of-the-world can be constructed with existing assets without friction."

Some footnotes.

Risk neutrality in decision sciences indicates the indifference between two games (or lotteries); one pays a fixed amount say 10, and the other indicates a win (more than 10 with probability p<1) and a loss (less than 10 wp 1-p) but the expected value of the game is the same amount of the risk free payoff 10. Risk seekers take the bet, risk averse people take the fixed amount.

Option pricing, portfolio optimization, risk management and similar areas all have the same objective, modeling and predicting the future value of an asset. But they usually differ in methodology. There are some bridges though, see Gerber and Shiu's paper on using an actuarial method called Esscher transform and come up with the BS model.

The popularity of the risk neutral pricing or complete markets comes from you don't need to think about the preferences (whether the agents are risk seeking or risk averse, so you have an 'objective' assessment).

Answer 7

I found that all of the answers under this post involves probability and randomness. In my humble opinion, risk neutral pricing does not have to involve randomness, and the notion comes so natural that we are applying it in our everyday life. So I would like to give an answer to simplify the concept of risk neutral pricing.

Think of your bank account. You have a 3-year term deposit that pays you \$5 at the end of every year and also pays back the principal amount of $100 at the end of the 3-year period. How would you price this asset?

Easy, right? We would discount the cash flows with risk-free rate because money in the bank is essentially risk-free (it’s not actually risk-free, but we will talk about it later). Let’s suppose we have chosen the yield of U.S. Treasury Bills as the risk-free rate. Assume that the 1-year Treasury is currently at 1.5%, 2-year Treasury at 2.0%, and 3-year at 2.5%. The present value (PV) of your bank account is, therefore, the PV of the first year’s cash flow 5/(1+1.5%) + PV of the second year’s cash flow 5/(1+2.0%)^2 + PV of the third year’s cash flow (100+5)/(1+2.5%)^3.

Hold on for a second! But isn’t the sum of these three numbers just the value of the Treasury Bills that you need to purchase in order to replicate the cash flows of your bank account? Yes! You would have to purchase 5/(1+1.5%)=\$4.9261 of Treasury Bills for replicating the first year’s \$5, because it takes 1 year for \$4.9261 of 1-year Treasury Bills to grow to \$5 (this is just the definition of yield). Similarly, it takes 2 years for 5/(1+2.0%)^2 =\$4.8058 of 2-year Treasury Bills to grow to \$5 in order to replicate the second year’s cashflow. And it takes 3 years for (100+5)/(1+2.5%)^3 =\$97.5029 of 3-year Treasury Bills to grow to $105 in order to replicate the third year’s cashflow. Through no-arbitrage argument, the PV of your bank account has to be equal to the sum of PVs of the replicating Treasury assets. As a result, the PV of your bank account is \$4.9261 + \$4.8058 + \$97.5029, and we have just priced your bank account using the market pricing of Treasury Bills (yield is the market pricing of Treasury Bills).

This is called risk-neutral pricing! But you may ask what is the “risk” here? Well. If you think about it, your bank accounts and treasury bills are not actually risk-free. Of course, there is no credit risk in them, but there is still interest risk. In other words, when interest rates or yields rise, the price of your bank account or the Treasury Bills will fall. The further in the future the cash flow is, the higher this interest risk (you can try to add 1 percentage point to the discounting factor in each year’s cash flow, and you will see that the third year’s cash flow has the largest percentage drop).

What is more important, this interest rate risk has already been priced in the Treasury yields, and that is partially why you see that the longer treasuries have higher yields, because the investors require a larger compensation for higher interest rate risk (This is also why the yield curve is usually upward-sloping even the interest rate expectation is flat). Thus, when you price your bank account, you have already taken into consideration the interest rate risk inherent in your bank account. In other words, you have “risk neutralized” interest risk in your pricing!

See? The essence of risk neutral pricing is to price one asset through cash flow replication with other assets whose prices we already know. In doing so, we will be able to price in the risks using the market prices of these other assets, as the market has already priced in the risks with the prices that the market collectively believes as fair. Being able to replicate cash flows is massively important, as you can see that the argument in the bank account example will fall apart if you are not able to replicate your bank cash flows with Treasury Bills.

Note that our example does not involve any randomness, in contrast to what is being traditionally taught in the classroom. However, the same thought process holds when you are pricing an asset whose price is random (for example, a call option, as opposed to a bank account). You just need to find a portfolio of assets to replicate (or hedge) the cash flows of the call option, and the initial value of this portfolio (which consists of the underlying stock and cash) will be the fair price of call option through no arbitrage argument. The replication (or hedging) just needs to be dynamic in this case due to randomness. This is exactly what Black-Scholes is trying to do.

Answer 8

I like this point of view on risk-neutral pricing: risk-neutral probability $q$ is such a probability that the expected possible price of the option at $t=T$ calculated with this probability and then discounted gives you today price at $t=t_0$

It is derived from today price under the assumption that all the time holding portfolio of option (buy) and instrument (sell) you are delta hedged, so its value is known and the same in each case (rise, fall).

Other nice view is: future price of the option (expected with risk-free rate) is equal to its expected value, i.e if today price is $V$ and option price tomorrow might be $V^+$or $V^-$ and risk-free rate is $r$ then you can retrieve $q$ from this equation:

$(1+rdt)V=qV^++(1-q)V^-$