Assume the price of a stock $S_t$ paying continuous dividend $a$ satisfies $$ dS_t = S_t\left((\mu - a)dt + \sigma dW_t\right). $$ The risk-neutral pricing formula states that if $\mathbb{Q}$ is any probability measure such that $e^{-rt}S_t$ is a $\mathbb{Q}$-martingale (MG), then the value of any self-financing replicating strategy $V_t$ that replicates a payoff $X = f(S_T)$ is $$ V_t = \mathbb{E}_{\mathbb{Q}}\left[e^{-r(T-t)}X|\mathcal{F}_t\right]. $$

So, discount $S_t$ and compute the differential $$ dZ_t := d(e^{-rt}S_t) = Z_t\left((\mu - a - r)dt + \sigma dW_t\right). $$ Define $\frac{d \mathbb{P}}{d \mathbb{Q}} = \exp\left(-\theta W_t -\frac{1}{2} \theta^2 t\right)$ and $\tilde{W}_t = W_t + \theta t$ for some $\theta$, which we determine by replacing $W_t$ with $\tilde{W}_t$ in $dZ_t$ and making it driftless: \begin{align*} dZ_t & = Z_t\left((\mu - a - r)dt + \sigma (d\tilde{W}_t - \theta dt)\right) \\ & = Z_t\left((\mu - a - r - \theta \sigma)dt + \sigma d\tilde{W}_t \right) \\ & = \sigma Z_td\tilde{W}_t, \end{align*} where we set $\theta = \frac{\mu - a - r}{\sigma}$. We have found a $\mathbb{Q}$ such that $Z_t$ is a $\mathbb{Q}$-MG, and so can price as usual.

However, Shreve (Stochastic Calculus for Finance II) on p. 235 instead derives the discounted portfolio process (with reinvested dividends), and finds the market price of risk $\theta$ for that process, showing that when this is done, the discounted stock process is indeed not a martingale. Why not do it the way I did?


1 Answer 1


What do you mean by "so we can price as usual"? What you showed is that for every $c \in \mathbb R$ we can find a probability measure such that the drift of $S$ is $c$. But that does not really say anything about pricing.

You can easily see that $V_t = E^Q_t[e^{-r(T-t)} \Phi_T]$ does not give arbitrage free prices with your choice of $Q$. Indeed if $\Phi_T = S_T$, then $E^Q_t[e^{-r(T-t)} S_T] = S_0$. But if I invest $S_0$ in the stock at time $0$, then at time $T$, I will own $S_T$ plus all the dividends I earned.

What matters in arbitrage free pricing is the notion of self-financing portfolio (= strategy). Two self-financing portfolio giving the same cashflows must have equal price. The problem is that, in the presence of dividends, a portfolio $V_t = S_t$ is not self-financing.

Edit: You may be confused by the fact that a risk neutral measure is often defined as a measure under which the discounted underlyings price processes are martingales. If that was the case your measure would be a risk neutral measure. But the right definition of a risk neutral measure is a measure under which the self-financing portfolios discounted price processes are martingales (this is mentionned by Shreve p. 234). This is important because being self-financing is independent of the measure you consider so you can reason under the historical measure to decide whether a portfolio is self-financing or not and then use risk neutral pricing to actually price it.

This is where the pricing formula comes from: if

  1. the payoff $\Phi_T$ can be replicated by a self-financing portfolio $V_T = \Phi_T$,
  2. there exists a measure under which self-financing portfolios discounted prices are martingales then $$ V_t= E_t^Q[e^{-r(T-t)}V_T]= E_t^Q[e^{-r(T-t)}\Phi_T]. $$ which gives us the price of the cashflow.

Consider a market with underlying assets $(S^1,\ldots,S^d)$ and money account $S^0_t = e^{rt}$.

Claim: In the abscence of dividend, if all the discounted $S^i$ are martingales under a probability measure $\mathbb{Q}$ then so are all the discounted self-financing portfolios (so $\mathbb{Q}$ is a risk neutral measure).

Proof: Consider a portfolio $$ V_t = \delta^0_t S^0_t + \sum_{i=1}^d \delta_t^i S^i_t $$ Notice that $\delta^0_t S^0_t = V_t - \sum_{i=1}^d \delta_t^i S^i_t$. Plug this in the self-financing equation \begin{eqnarray*} dV_t &=& \delta^0_t dS^0_t + \sum_{i=1}^d \delta_t^i dS^i_t \\ dV_t &=& \delta^0_t rS^0_tdt + \sum_{i=1}^d \delta_t^i dS^i_t \\ dV_t &=& r(V_t - \sum_{i=1}^d \delta_t^i S^i_t)dt + \sum_{i=1}^d \delta_t^i dS^i_t \\ dV_t - rV_t &=& \sum_{i=1}^d \delta_t^i (dS^i_t - rS^i_t dt) \\ d(e^{-rt}V_t) &=& \sum_{i=1}^d \delta_t^i d(e^{-rt}S^i_t) \end{eqnarray*} Since all the $(e^{-rt}S^i_t)$ are $\mathbb{Q}$-martingales, so is $(e^{-rt}V_t)$.

Conclusion What makes risk neutral measure useful for pricing is that discounted self-financing portfolios are martingales. In the abscence of dividends, this is equivalent to the discounted underlyings being martingales. But a dividend paying asset by itself is not a self financed portfolio so it is not useful to find a probability measure under which it is a martingale.

  • $\begingroup$ thanks for the response, but would you mind elaborating? It seems like you gave an example of a non-self-financing replicating strategy, but I don't quite see where I went wrong in developing my question. $\endgroup$
    – bcf
    Commented May 25, 2015 at 17:24
  • $\begingroup$ Your math is correct but the conclusion "so we can price as usual" is false. I edited to elaborate on why portfolios are key. $\endgroup$
    – AFK
    Commented May 26, 2015 at 0:59
  • $\begingroup$ added a proof that risk neutral measure definition in terms of self-financing portfolio reduces to the usual definition in the absence of dividends. $\endgroup$
    – AFK
    Commented May 28, 2015 at 0:21

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