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The answer is probably obvious, but interestingly enough, I was not able to find it in explicit form in the mathematical finance textbooks.

So when Shreve says in paragraph 5.2.2 of SCF-II:

Consider a stock price process whose differential is $$ dS(t) = \alpha(t)S(t)dt+ \sigma(t)S(t)dW(t), \quad 0 \leq t \leq T $$

does he refer to a nominal price or a real price of the stock?

The meaning of real and nominal is described there

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  • $\begingroup$ Not sure what you mean. $S(t)$ represents the spot price of the stock at time $t$. In other words, if you were to turn to the market to purchase the stock at that time $t$, it would cost you $S(t)$. Most models assume no friction so no bid/ask spread + continuous trading so you can trade at any $t$. $\endgroup$
    – Quantuple
    Commented Feb 26, 2018 at 9:48
  • $\begingroup$ @Quantuple The question is inspired by comments there quant.stackexchange.com/q/38459/15154 $\endgroup$
    – zer0hedge
    Commented Feb 26, 2018 at 9:54
  • $\begingroup$ If you have in mind a reduced form model, then you are free to assume that $S(t)$ is either nominal or real. It is true that, at least in principle, people should care about real quantities and therefore once you set up an equilibrium model you should care about a real $S(t)$. The point is that the volatility of inflation has been pretty low - at least since the 90s - compared to the volatility of stock and, at least, long term bonds: for this reason this source of risk can be neglected. Notice that the level of inflation should not matter for your portfolio choice. $\endgroup$
    – fni
    Commented Feb 26, 2018 at 19:26
  • $\begingroup$ The process stated by Shreve works for nominal or real S(t). It is a general form. If S(t) should be interpreted as nominal or real should be stated by the one that is proposing the model. I included a answer with this point $\endgroup$
    – Andre
    Commented Jun 2, 2023 at 20:37

3 Answers 3

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$S(t)$ is the stock nominal price.

Nothing precludes you from modeling a stochastic differential equation for the stock real price, but that would not be practical for pricing derivatives, as options fixed strike prices would have to be divided by the CPI to be converted to real prices, thus requiring joint modeling of the stock real price and the CPI.

Also discounting would have to be done at real interest rates, which would require bootstrapping the inflation ZC curve in addition to the standard OIS curve bootstrap.

So in the end even simple vanilla options prices would be expressed as a function of a lot of parameters, a complicated departure from the standard implied volatility surface representation, and not very practical for setting up hedges.

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Quant finance almost exclusively deals with the risk neutral assumption (as opposed to the real world measures you will usually find in acturial science). Wikipedia has a great entry on the History: Q versus P. Since risk-neutral measures discount forward (nominal) prices to the present, the risk free rate contains information both on the rate of return and rate of inflation. Therefore, we model nominal prices, but (almost???) always express present value using today's dollars. This practice is consistent with standard interpretation of risk free rates to correspond to yields on interest bearing liabilities. I.e., a risk-free security compensates the holder for time value and inflation.

Given Itô's lemma, and under fixed variance, we have:

(1) $S_t = S_0 \,\text{exp} \left[ ({\int_{0}^{t}{a_t}\,dt}) - \frac{\sigma^2}{2}t +Z \,\sigma \sqrt{t} \right]$

If ${\int_{0}^{t}{a_t}\,dt} \ne r\,t$, then we could go short one and long the other for an arbitrage trade. Since this is a no-no in quant finance, we revert to the risk neutral measure where upon (1) $\to $ (2):

(2) $S_t = S_0 \,\text{exp} \left[ (r - \frac{\sigma^2}{2})t +Z \,\sigma \sqrt{t} \right]$

Thus, for a probability density function, $f(x;S_t)$, we have:

(3) $\mathbb{E}^Q \left[ S_t \right] = \int_{-\infty}^{\infty} f(u)S(u)\, \mathcal{d} u = S_0 \exp \left[ r\,t \right] $

(Note, the same logic applied to a contingent payoff, $V$, results in Black-Scholes since from Itô, we have:

$dV = \frac{\partial V}{\partial t}dt + \frac{1}{2}\sigma^2S^2\frac{\partial ^2V}{\partial S^2}dt + \frac{\partial V}{\partial S}dS$)

None of this is to say that one could not model real prices, but to do so, one would also have to use a risk-free rate which is deconflates the effect of inflation. In practice, an individual might consider using rates on TIPs versus standard treasuries.

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The process stated by Shreve is general for nominal or real $S(t)$. If $S(t)$ will be assumed to be nominal or real depends on the objectives of the researcher.

For example, let us say that the process is for an asset that has expected real returns equal to zero, but returns follow a Wiener process. An option to model $S(t)$ is

$dS(t)=\sigma(t)S(t)dW(t)$,

where $S(t)$ denotes real prices, that is, prices adjusted for inflation.

On the other hand, if $S(t)$ are nominal prices, but real expected returns are still zero, then the specification with $\alpha(t)=0$ does not work well as nominal prices should be adjusted for inflation to imply zero expected real returns. We can then write a more general specification

$dS(t)=\alpha(t)S(t)dt + \sigma(t)S(t)dW(t)$.

In this case, if $S(t)$ denote nominal prices, then $\alpha(t)$ will change to accomodate changes in nominal asset prices caused by changes in inflation. It is more general than this, $\alpha(t)$ can accomodate a component for real returns plus inflation adjustments.

In the end, if $S(t)$ denotes nominal or real prices is a modeling choice.

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