What exactly is/How exactly do we interpret the binomial model's Radon-Nikodym derivative?

Question

Related: Lewis' triviality result?

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As I recall the one-step binomial model goes like this:

1. The time periods are now $t=0$ and later $t=1$.

2. We have

• 2.1. a stock that pays off $u$ for going up or $d$ for going down for an initial investment of $S_0$, the price of 1 unit of the stock, where $u > d > 0$ and $S_0 > 0$.

• 2.2. a claim $X$ on the stock (where $X$ is, say, a European call option or something). The payoffs are $X_u$ for up and $X_d$ for down for an initial investment of 1 unit of the claim.

• 2.3. and a bond that pays $1+R$ for an investment of $1$ ($R$ is rate of return right?), for $R>0$

3. We assume both going up and going down have positive probability. We have real world probability $\mathbb P$ and risk-neutral probability $\mathbb Q$. The probabilities are $P(up)=p_u$, $P(down)=p_d$, $Q(up)=q_u$, $Q(down)=q_d$.

4. For no arbitrage we must have $d < 1+R < u$ (or $d \le 1+R \le u$ or something).

5. From no arbitrage we can compute $q_u$ and $q_d$ in terms of $u,d,R$ and then we don't need $p_u$ and $p_d$ except for the assumption that both real world probabilities are positive or something. (and then $q_u$ and $q_d$ are positive too or something.)

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1. Question 1: What exactly is/how exactly do we interpret the Radon-Nikodym derivative $\frac{d \mathbb Q}{d \mathbb P}$ ?

• 1.1. Its formula/equation/whatever appears to be $\frac{d \mathbb Q}{d \mathbb P} = \frac{q_u}{p_u}1_u + \frac{q_d}{p_d}1_d$, so it's some asset with payoffs $\frac{q_u}{p_u}$ and $\frac{q_d}{p_d}$, expected value of 1 and replicating portfolio $(x,y)$ of

$$x=\frac{1}{1+R}\frac{u(\frac{q_d}{p_d})-d(\frac{q_u}{p_u})}{u-d}$$ $$y=\frac{1}{S_0}\frac{\frac{q_u}{p_u}-\frac{q_d}{p_d}}{u-d}$$

This seems to be some replicating portfolio that is expected to payoff 1 at $t=1$.

• 1.2. Well, $(x,y)=(\frac{1}{1+R},0)$ seems to give the same payoff but with -100% lower risk

• 1.3. Guess: It's a hypothetical stock that pays off $\frac{q_u}{p_u}$ for up and $\frac{q_d}{p_d}$ for down. In particular, what's so hypothetical about this is that we don't necessarily know $p_u$ and $p_d$.

• 1.4. Guess: In re (2) below, I think $\frac{d \mathbb Q}{d \mathbb P}$ is like a specific case of $X\frac{d \mathbb Q}{d \mathbb P}$ with $X_u=1=X_d$ so like...we choose $X$ as like...a bond with rate of return $0$? idk

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2. For $X\frac{d \mathbb Q}{d \mathbb P}$ in the one-step binomial model...

we could say that the price of $X$ uses not

$$E[X] = X_up_u+X_dp_d$$

but rather

$$E^{\mathbb Q}[X] = E[X\frac{d \mathbb Q}{d \mathbb P}] = X_u\frac{q_u}{p_u}p_u + X_d\frac{q_d}{p_d}p_d$$

Question 2: So what exactly is/how exactly do we interpret '$X\frac{d \mathbb Q}{d \mathbb P}$', i.e. $X$ multiplied by the Radon-Nikodym derivative $\frac{d \mathbb Q}{d \mathbb P}$ ?

• 2.1. Its formula/equation/whatever appears to be $\frac{d \mathbb Q}{d \mathbb P} = X_u\frac{q_u}{p_u}1_u + X_d\frac{q_d}{p_d}1_d$, so it's some asset with payoffs $X_u\frac{q_u}{p_u}$ and $X_d\frac{q_d}{p_d}$

• 2.2. Not sure what's its replicating portfolio. Not sure we need one since we're using real world probabilities.

• 2.3. Its real world expected payoff is equal to $X$'s risk neutral expected payoff.

• 2.4. Guess: Once again, it's a hypothetical stock but this time the payoffs are $X_u\frac{q_u}{p_u}$ for up and $X_d\frac{q_d}{p_d}$ for down. Again, in particular, what's so hypothetical about this is that we don't necessarily know $p_u$ and $p_d$. So if the option/claim price is $\frac{1}{1+R}E^{\mathbb Q}[X]$, then I could tell you to, instead of buying the claim or investing in its replicating portfolio, buy a stock that will payoff $X_u\frac{q_u}{p_u}$ for up and $X_d\frac{q_d}{p_d}$ for down and has an initial price of $\frac{1}{1+R}E^{\mathbb Q}[X]$ or something. The thing is we don't necessarily know $p_u$ and $p_d$, so we can't quite identify a similar stock.

Answer 1

In economics, the Radon-Nikodym density $\frac{d \mathbb Q}{d \mathbb P}$ of the risk-neutral measure $\mathbb Q$ with respect to the physical measure $\mathbb P$ is the price of Arrow-Debreu securities. It is a price, not a claim.

In the binomial setting, there are two AD securities, $1_u$ and $1_d$. The former entitles the holder to 1 unit of numeraire if, and only if, state $u$ realizes. The no-arbitrage price of $1_u$ is $\frac{q_u}{p_u}1_u$ (say $R = 1$). Similarly for $1_d$.

In general, the price of the AD portfolio $1_{\Omega'}$---which pays off 1 unit of numeraire if, and only if, state $\omega \in \Omega'$ realizes---is $$ E^{\mathbb P}[ 1_{\Omega'} \frac{d \mathbb Q}{d \mathbb P}] = \mathbb Q (\Omega'). $$ So $$ E^{\mathbb P}[\frac{d \mathbb Q}{d \mathbb P}] = 1 $$ is then the price of a bond/risk-free security (discount accordingly if $r \neq 1$).

Extending from $1_{\Omega'}$ to a general $X$ gives the usual risk-neutral pricing formula $E^{\mathbb Q}[X]$, as you have written out.