Risk-Neutral Pricing, Replication, and the Change of Probability Measure

# Conceptual problem with risk neutrality-What is a 'risk-neutral world', exactly?


# Conceptual problem with risk neutrality-What is a 'risk-neutral world', exactly?












I have persistent, deep problems with the concept of 'risk-neutrality'. To make it more precise, let's look at the following explanation taken from a book:

> "In a world where investors are risk neutral, the expected return on all securities is the risk-free rate of interest, r. The reason is that risk-neutral investors do not require a premium to induce them to take risks. It is also true that the present value of any cash flow in a risk-neutral world can be obtained by discounting its expected value [in a risk neutral world] at the risk-free rate.” (Hull-Options, Futures, and Other Derivatives, page 245)

After tens of hours of studying and looking into all books/resources/notes that I could get my hands on, reading countless of explanations (including all relevant ones on this very forum), I must admit that I still seem to not understand the concept of ‘risk-neutrality’. At this point it's become very frustrating. I have masters degree in mathematics so the math isn't the problem; but, trying to get my head around financial math, I keep having problems with the concept of 'risk-neutrality'.

To make it very concrete, regarding the above-mentioned snippet from Hull's book I have the following questions.

- It's said: "In a world where investors are risk neutral, the expected return on all securities is the risk-free rate of interest, r." Why is this the case? I can't even tell whether this is true or false, since I lack the understanding of the semantics.



To me, these are just statements that someone makes; I am unable to verify whether or not this is true, since I am not able to grasp the semantics.

I.e., the core question is: What is a 'risk-neutral world'? Can someone provide me with a definition (or at least an explanation that unveils the semantics), and subsequently prove Hull's claims?

I'd be very grateful if someone had an explanation/perspective on this that could help me. Many thanks in advance!

## Answer by Ilmari Karonen (score 11)

https://quant.stackexchange.com/a/74328

> I have masters degree in mathematics so the math isn't the problem; but, trying to get my head around financial math, I keep having problems with the concept of 'risk-neutrality'.

I suspect you might simply be missing the definition of risk-neutrality, so let me try to summarize it:

A risk-neutral investor is one with risk-neutral preferences — that is, one who considers an uncertain investment's value to be equal to its mathematical expected value. For example, a perfectly risk-neutral person would consider a lottery ticket that has a 1% chance of winning \$100 (and a 99% chance of being worthless) to be worth exactly \$1, since 1% of \$100 is \$1.

(A risk-averse person would rather just have the guaranteed \$1, while a risk-seeking person would prefer to have the lottery ticket and might pay more than \$1 for it.)

And a "risk-neutral world" is simply a hypothetical, imaginary world where all investors are assumed to be risk-neutral. Obviously we do not actually live in such a world, but — as nbbo2's answer describes in more detail — sometimes making such an assumption anyway can be justifiable (e.g. because risk can be hedged away) and doing so can simplify the math.

## Answer by nbbo2 (score 10)

https://quant.stackexchange.com/a/74321

A little bit of history. This goes back to the early days when the Black Scholes formula for options was proposed but was still new and somewhat mysterious.

It was (and is) widely accepted in Finance that long term rates of return for stocks $r$ are higher than the return on T-Bills $r_F$ (also known as the risk free rate). This makes sense because stocks are risky and the extra return can be seen as compensation for risk. There is no free lunch: you can invest with no risk and low return or you can get higher return by accepting more risk. This is Financial Economics not Math. Everyone understood this at the time. You have to accept it (or go along with it) to understand what happened next.

In view of this well known fact a major mystery was why the Black Scholes formula involves the risk free rate $r_F$. After all options are risky, perhaps riskier than stocks in some sense! So shouldn't options be discounted at a high rate ?! In fact Black Scholes was not the first formula for options, there had been other attempts including by Samuelson and Merton and some of these formulas included risky rates of returns. (Even today quant.stackexchange users frequently ask why the BSM formula uses the risk free rate).

Robert C. Merton in particular had to decide if the new Black Scholes formula was right, which would mean that the Merton-Samuleson idea of using a risky-rate to value options was NOT right. To his great credit Merton quickly decided that Black Scholes was right and did a lot to convince people of this. (Later on someone would joke that Merton understood the BS formula before B and S did). And nowadays it is called the BSM formula.

Merton said, we can imagine a world where all investors are risk-neutral. It is not the world we live in, where investors are risk averse. In such a world in equilibrium there is no compensation for equity risk, and stocks (and risky assets in general) earn the risk free rate. Because option risk can be hedged it does not matter if we value an option in the Risk Free world or the Real World. The Dynamic Hedging mechanism eliminates risk so it does not matter if the investors who use it are risk averse or not. For convenience, we can assume a Risk Neutral World when valuing options. This leads to a simple derivation of the BS formula which uses $r_F$. Then we can say that the formula also holds in the Real World. This idea of using the Risk Neutral World assumption in the derivation was picked up by Hull and many others. It convinced people that the use of a risk free rate in option valuation was justified.

The "RNW Assumption" provided both a simple theory of option value (option value is the expected value of the payoff in a Risk Neutral World) and justified the $R_F$ term in the BSM formula.

Operationally the "RNW Assumption" just says "If you come across the rate of return on risky assets in the calculation of Expected Value, just replace it with the risk free rate $R_F$ and proceed onward, you will arrive at the same destination as Black and Scholes".

## Answer by Freelunch (score 2)

https://quant.stackexchange.com/a/74330

Forget about risk-neutrality for a second and think about what the definition of a $\mathbb{Q}$-measure. It is some probability measure (not necessary unique) which reproduces market prices such that any asset $Z$ when discounted in some strictly positive numéraire $X$ (usually a bank account) is a $\mathbb{Q}$-martingale, i.e.,

\begin{equation*} \frac{Z_t}{X_t} = E^\mathbb{Q} \Bigg[ \frac{Z_T}{X_T} \Big| \mathcal{F}_t \Bigg] \end{equation*}

The point here is that we can easily derive a fair arbitrage-free price for any contingent claim without even thinking about how it can be replicated.

Regardless of what numéraire we use, the drift under the $\mathbb{Q}$-measure will be equal to the risk-free rate. To see this consider the price of a forward contract on $Z_T$. This is the price agreed upon today to buy or sell $Z$ at some future date $T$, which is $E^\mathbb{Q} \Big[ Z_T \big| \mathcal{F}_t\Big]$. The present money value of this is $e^{-r(T-t)} E^\mathbb{Q} \Big[ Z_T \big| \mathcal{F}_t\Big]$, but of course spending money now to buy delivery in the future is equivalent to just buying at spot price $Z_t$ hence

\begin{equation*} Z_t = e^{-r(T-t)} E^\mathbb{Q} \Big[ Z_T \big| \mathcal{F}_t\Big] \end{equation*}

or \begin{equation*} E^\mathbb{Q} \Big[ Z_T \big| \mathcal{F}_t\Big] = e^{r(T-t)}Z_t \end{equation*}

This is exactly what you get when replicating the forward contract. You would borrow $Z_t$, buy the asset and at some future date owe exactly $e^{r(T-t)}Z_t$.

## Answer by EYC.CHANG (score 2)

https://quant.stackexchange.com/a/77471

One definition of Risk Neutral is where the marginal utility is constant, i.e., u'(x) = c, for all x. A direct consequence of this is that the stochastic discount factor $m$ $(:=\beta\frac{u'(c_{t+1})}{u'(c_t)})$ in the basic consumption based asset pricing formula $p = E[mx]$ is a constant equal to $\beta$ the discounting factor. Now, apply this fact to the Hansen-Jagannathan bounds $|E[R_i] - R^f| \leq \frac{\sigma(m)}{E[m]}\sigma(R_i)$, we see that $\sigma(m)$ will just be 0, as m is a constant. Thus the RHS is just 0, we get $E[R_i] = R^f$ as a result. You can find these formulas in Cochrane's asset pricing textbook.

(Look up any graduate microeconomics textbook, like MWG, and you will learn the formal definition of risk neutrality in decision theory chapters, not some sloppy "economics intuition" answers.)

PS @Ilmari Karonen provides the answer in terms of preference, but under some assumptions (see MWG), we can actually find utility representation of any preference. In most cases, working with the utility representation is easier to do the analysis and interpretation.

## Answer by dm63 (score 1)

https://quant.stackexchange.com/a/74320

What is a 'risk-neutral world'? It is not a real world- it is a purely mathematical construct which is defined by your statements (the present value of cashflows can be correctly found bydiscounting at risk free rates). We build this construct purely to make it easier to price certain financial contracts, chiefly derivatives. Is that what you are asking ?

## Answer by Arshdeep (score 1)

https://quant.stackexchange.com/a/79431

There is no proof, a world where everything grows at risk free rate is defined as risk neutral world. Its just a term; there is no utility so far. I can define a "risk loving" world where investors have 100X leverage on everything. I've attached a term to a world, it has no consequence.

The utility/consequence comes when you realise you can price derivatives with this framework i.e. assuming a world where things grow at risk free rate. There is no intuition for this other than this is the Feynman Kac theorem. If this were obvious then the Feynman Kac theorem would be obvious.

## Answer by spinthebs (score 0)

https://quant.stackexchange.com/a/74357

I'm going to be slightly contrary and not give an actual answer here. The term Risk Neutral is tacked onto various other words and used to mean either something precise like the Risk Neutral Measure, or something a bit more general like Risk Neutral Pricing. And it can sometimes be difficult to tell which case you are in. This, in my opinion, is an example of slightly misleading terminology that stuck around from a developing area of mathematics/economics that, again, in my opinion, still isn't fully mature. Risk Neutral does mean something, but often you are trying to use the concepts to try and price/value something and the idea of risk neutrality is just incidental or possibly philosophical. It might be helpful for some people and tie nicely in with the idea of hedging in complete markets, but I feel that this isn't really the key insight, at least not for me.

Confounding this is the fact that this stuff is so widely used and a lot of texts try to dumb down some quite difficult mathematics, some people seem okay with this, but if you're like me you'll be more confused by the omissions. It took reading through the Fundamental Theory of Asset Pricing, Girsanov's Theorem and the Martingale Representation Theorem for me to feel comfortable with the techniques and reasoning behind pricing in this way. Understanding the statements of these theorems should be enough. In a lot of texts these theorems are glossed over because this is where the maths starts to get really deep. You can try and understand the intuition behind the term Risk Neutral, but in my opinion, there's really a bit of mathematical slight of hand going on that is justified by some quite deeps maths.

## Answer by levzettelin (score 0)

https://quant.stackexchange.com/a/79411

A RNW (risk-neutral world) is a hypothetical world in which all actors are risk-neutral, which means that they value an asset that pays $S_T$ at time $T$ as having the present value

\begin{equation} S_0 = \mathbb{E} \left[ S_T \right] e^{-rT} \end{equation}

where $r > 0$ is the risk-free rate.

The RNW assumption can be used to derive $\mathbb{E} \left[ S_T \right]$ when $S_0$ is given (called "forward propagation" here). It can also be used to derive $S_0$ when $\mathbb{E} \left[ S_T \right]$ is given (called "backward propagation" here).

RNV (risk-neutral valuation) can then be used to price derivatives by using "forward propagation" on the price of the underlying followed by "backward propagation" on the price of the derivative.

For example, let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a derivative of $S_T$, i.e., let it have payoff $f(S_T)$. Under the RNW assumption, the derivative then has present value

\begin{equation} f_0 = \mathbb{E} \left[ f ( S_T ) \right] e^{-rT}. \end{equation}

#### Interpretation

If the payoff of a derivative can be exactly replicated in a risk-free way by investing in $S_T$ and some risk-free bond that pays $r$ interest, then the value of the derivative in the RNW should be the same as in the real world. In other words, we can use RNV to price derivatives in the real world.

#### Example 1

Assume that $S_0 = 10$ and that

\begin{equation} S_T = \begin{cases} 8 & \text{ with probability } p \\ 12 & \text{ with probability } 1 - p \end{cases} \end{equation}

for some probability $p \in [0, 1]$. Assume that there also exists a call option $C$ on $S$ with strike $10$; so at $T$ this call option pays

\begin{equation} C_T = \begin{cases} 0 & \text{ if } S_T = 8 \\ 2 & \text{ if } S_T = 12 \end{cases} \end{equation}

Then a portfolio with guaranteed payoff at $T$ can be created by buying $\Delta = \tfrac{1}{2}$ of the stock and shorting a call option for a total payoff of

\begin{eqnarray} \Delta S_T - C_T & = & \begin{cases} \Delta 8 - 0 & \text{ with probability } p \\ \Delta 12 - 2 & \text{ with probability } 1 - p \end{cases} \\ & = & 4 \end{eqnarray}

So since $\mathbb{E} [ \Delta S_T - C_T ] = 4$ and using $r = 0.1$ and $T = 1$, backward propagation leads to $\Delta S_0 - C_0 = e^{-rT} 4$ and thus

\begin{equation} C_0 = \Delta S_0 - 4 e^{-0.1} \approx 1.38. \end{equation}

The same result can be obtained using RNV proper, though. For this, first note that

\begin{eqnarray} \mathbb{E} [ S_T ] & = & 8 \cdot p + 12 \cdot ( 1 - p ) \\ & = & 12 - 4 \cdot p \\ \end{eqnarray}

Then forward propagation gives the equation

\begin{equation} 10 = S_0 = \mathbb{E} [ S_T ] e^{-0.1} = (12 - 4 \cdot p) e^{-0.1} \end{equation}

which solves to $p \approx 0.237$. Thus,

\begin{equation} \mathbb{E} [ C_T ] = p \cdot 0 + ( 1 - p ) \cdot 2 = 1.526. \end{equation}

which via backward propagation also gives

\begin{equation} C_0 = \mathbb{E} [ C_T ] e^{-0.1} \approx 1.38. \end{equation}

#### Example 2

Consider an out-of-the-money call option with strike 110

\begin{equation} f(S_T) = \begin{cases} 0 & \text{ if } S_T \lt 110 \\ S_T - 110 & \text{ if } S_T \ge 110 \end{cases} \end{equation}

on an underlying that is currently valued at $S_0 = 100$. Assume that the time to expiry is $T = 10$ and the risk-free rate is $r = 0.1$. Make the additional assumption that $S_T \sim \mathcal{N} \left( \mu, 10 \right)$ is normally-distributed with known variance of $10$.

Forward propagation then shows that

\begin{equation} \mu = \mathbb{E} \left[ S_T \right] = S_0 e^{rT} = 100 \cdot e \approx 271.8 \end{equation}

Using the assumption about the distribution of $S_T$ we find

\begin{eqnarray} \mathbb{E} \left[ f(S_T) \right] & = & \int_{110}^\infty \left( S_T - 110 \right) \frac{1}{\sqrt{10 \cdot 2 \pi}} \exp\left( -\frac{(S_T - 100 \cdot e)^2}{10} \right) \\ & \approx & 114.43 \end{eqnarray}

and thus

\begin{eqnarray} f_0 & = & \mathbb{E} \left[ f(S_T) \right] \frac{1}{e} \approx 42. \end{eqnarray}