Speculative Bubbles

Speculative Bubbles (Harrison-Kreps 1978)

Harrison & Kreps (1978) extend Miller’s static model to a dynamic setting in which the identity of the optimists changes over time. Today’s optimists know they will not be tomorrow’s optimists — but they also know that whoever is tomorrow’s optimist will be willing to buy the asset at a price that exceeds today’s optimist’s own buy-and-hold valuation. So today’s optimist buys at a price above her own valuation in order to resell to a future optimist who will pay even more. The price exceeds the present value of dividends as computed by every current investor. The gap is the resale option value, and the equilibrium is called a speculative bubble.

The model. A risky asset in fixed supply of $1$ unit pays dividends $a_t$ following a multiplicative random walk: $a_{t+1}=s_{t+1}{a}_t$, with $s_{t+1}$ drawn i.i.d. from a distribution $F^{\text{true}}$ with mean ${\mu }^{\text{true}}=1$. At each date $t$ a new generation of risk-neutral traders enters with discount factor $1/(1+r)$. Within each generation, half are optimists ($j=1$, ${\mu }^1=1+\Delta$) and half are pessimists ($j=0$, ${\mu }^0=1-\Delta$) — but they are only optimistic about the next shock $s_{t+1}$; for shocks $s_{t+k}$ with $k\ge 2$, all traders agree on $F^{\text{true}}$. Short-selling is forbidden. The “this time is different” optimism rotates across generations.

Buy-and-hold valuations. Each generation-$t$ trader, if forced to hold forever, would value the asset at
$v^{1,\text{hold}}=\frac{a_t(1+\Delta )}{r},v^{0,\text{hold}}=\frac{a_t(1-\Delta )}{r}.$
The current optimist’s buy-and-hold valuation is the highest number any current trader would attach to the asset.

Equilibrium price. Conjecture a recursive price function $p(a)=\mathcal{P}\cdot a$. The optimist of generation $t$ owns the asset and faces capital gains because the next generation’s optimists are more optimistic about the next shock than she will be at $t+1$ (she will then be valuing $s_{t+2}$ rationally, while they are optimistic about $s_{t+2}$). The Bellman equation for the optimist is
$p(a)=\frac{1}{1+r}(a(1+\Delta )+{\mathbb{E}}^1[p(as)]).$
Substituting the conjecture and using ${\mathbb{E}}^1[s]=1+\Delta$ gives
$\mathcal{P}=\frac{1+\Delta }{r-\Delta },p(a)=\frac{a(1+\Delta )}{r-\Delta }.$

The resale option. The equilibrium price exceeds even the most optimistic current trader’s buy-and-hold valuation:
$\frac{a(1+\Delta )}{r-\Delta }>\frac{a(1+\Delta )}{r}=v^{1,\text{hold}}.$
The gap, which equals $\frac{a(1+\Delta )\Delta }{r(r-\Delta )}$, is the resale option value — the present value of being able to sell to a future, more optimistic generation. The gap is large when $r$ is low (resale options are far in the future and barely discounted) and when $\Delta$ is high (the future optimists pay a big premium).

Empirical hook. The Harrison-Kreps mechanism predicts a tight link between bubbles and trading volume: in each period the asset is sold from the current optimist to the next, generating $100\%$ turnover by construction. Hong & Stein (2007) document that during the dot-com episode, glamour stocks had monthly turnover several percentage points higher than value stocks, with the gap widening exactly when valuations peaked. The volume signature is the empirical fingerprint of the resale-option mechanism.

Intuition

Miller’s model produces a one-shot overvaluation but no trade — the optimist buys the asset once and holds it. Real bubbles look very different: they are loud, high-volume episodes where assets change hands rapidly. The Harrison-Kreps insight is that volume and overvaluation share a single source as soon as the model is dynamic. If the identity of the optimist rotates over time, then today’s optimist is not just buying for the dividend stream — she is buying because she expects to sell to someone else tomorrow at an even higher price. That someone else exists because tomorrow brings a new generation of “this time is different” traders who are optimistic about the next shock. Each generation’s optimists step into the role briefly, hold the asset for one period, and pass it on.

The key wedge from Miller is that the capital gain from selling the asset is part of its value, on top of the dividend stream. In Miller, the price equals the optimist’s buy-and-hold valuation. In Harrison-Kreps, the price exceeds it by the present value of the option to sell to a more optimistic future buyer. This “resale option” can be very large — in fact, it can be arbitrarily large if $\Delta \to r$, in which case the price diverges. The intuition is the same as a bubble that “feeds on itself”: each round of optimism adds another layer of capital-gain expectation, and the rounds compound.

Properties

(Resale Option Value) The price exceeds every current trader’s buy-and-hold valuation.

The closed-form solution $p(a)=a(1+\Delta )/(r-\Delta )$ exceeds the most optimistic buy-and-hold valuation $a(1+\Delta )/r$ by a factor of $r/(r-\Delta )>1$. The excess is the present value of the option to sell to a future buyer who values the asset more than today’s holders. This is qualitatively different from Miller’s static overvaluation: in Miller the price equals an existing investor’s valuation; in Harrison-Kreps the price exceeds every existing investor’s valuation. Speculative bubbles are therefore more extreme than belief-disagreement bubbles.

(Generation Rotation) The identity of the optimist must change over time.

The mechanism only works because the trader who is optimistic at $t$ knows she will not be optimistic at $t+1$. If the same trader were always the optimist, she would just hold the asset forever and the price would equal her buy-and-hold valuation (which is Miller’s static result). The rotation creates a sequence of distinct buyers, each willing to pay more than the last for one period. The “overlapping generations” framing is a clean way to formalize this; alternatives include any model where beliefs of the same individual update over time so that the marginal buyer at each date is different.

(100% Turnover) The asset is sold every period.

Because each generation buys at $t$ and sells at $t+1$, the equilibrium has trading volume equal to the entire supply every period. This is a sharp prediction: speculative bubbles should coincide with extreme turnover. Hong & Stein (2007) verify this in the dot-com bubble — monthly turnover in glamour stocks reached $4$–$5\%$ in $2000$, several times the historical average and far above the turnover in value stocks. The volume signature distinguishes speculative bubbles from rational-bubble or static-Miller alternatives, both of which predict no trade.

(Sensitivity to Discount Rate) The bubble is largest when $r$ is small relative to $\Delta$.

The closed-form $\mathcal{P}=(1+\Delta )/(r-\Delta )$ blows up as $\Delta \to r$ from below. Low interest rates and high disagreement together produce extreme bubbles. This matches the empirical pattern that bubbles tend to occur in low-rate environments — the late-1990s tech bubble followed several years of declining real rates, the mid-2000s housing bubble coincided with the Greenspan low-rate era, and the 2020–2021 cryptocurrency and meme-stock episodes followed the pandemic-era zero rates. Low discount rates make resale options extremely valuable.

(Requires Short-Sale Constraints) Without short-selling restrictions, the bubble disappears.

If short-selling is allowed, current pessimists can short the asset until the price falls back to a consensus valuation, just as in Miller. The Harrison-Kreps mechanism inherits the same dependency on short-sale constraints — the dynamic story is built on top of the static story. Empirically this is why speculative bubbles concentrate in asset classes where shorting is hard: IPO-locked tech stocks, meme stocks during the GameStop episode (where short interest exceeded available float), and cryptocurrencies before futures markets opened.

(Belief Source Left Open) The model takes “this time is different” optimism as given.

Harrison-Kreps does not explain why a fresh generation of optimists keeps appearing. The model just assumes it does. Two leading microfoundations have been proposed: (i) extrapolative beliefs (Bordalo et al., 2018), where each new generation extrapolates from recent returns and concludes the asset is in a new regime, and (ii) overconfidence (Scheinkman-Xiong, 2003), where each trader overweights her own private signal and therefore disagrees with the previous generation about how to interpret new data. Either microfoundation rationalizes the “this time is different” assumption.

Examples

The dot-com bubble and the resale option

By late $1999$, internet stocks were trading at price-earnings ratios above $200$ — far above any reasonable estimate of even an optimistic investor’s buy-and-hold value. Yet investors continued to buy, in many cases explicitly stating they did not expect to hold the stocks long: they expected to sell at higher prices to a wave of new retail investors entering through online brokerages. Monthly turnover in tech stocks reached $5$–$10\%$ — roughly five times the historical average. The Harrison-Kreps interpretation: each cohort of buyers was paying not for the discounted earnings of Pets.com, but for the option to resell to the next cohort of more-optimistic buyers. When the resale chain broke in March $2000$ — when no fresh cohort of buyers appeared — prices collapsed by more than $80\%$ over the next two years.

Solving the recursive equilibrium

Setup. Dividends evolve as $a_{t+1}=a_t{s}_{t+1}$ with $s_{t+1}$ i.i.d. and ${\mathbb{E}}^{\text{true}}[s]=1$. The current generation’s optimist believes ${\mathbb{E}}^1[s_{t+1}]=1+\Delta$ but agrees with the truth about $s_{t+k}$ for $k\ge 2$. Risk-neutral traders, discount factor $1/(1+r)$, no short sales.

Bellman equation. The optimist owns the asset at date $t$, collects the dividend $a_{t+1}$ at $t+1$, and sells at price $p(a_{t+1})$ to the next generation’s optimist. Her valuation is
$p(a)=\frac{1}{1+r}{\mathbb{E}}^1[as+p(as)]=\frac{1}{1+r}(a{\mathbb{E}}^1[s]+{\mathbb{E}}^1[p(as)]).$

Guess and verify. Try $p(a)=\mathcal{P}a$ for some constant $\mathcal{P}$. Then ${\mathbb{E}}^1[p(as)]=\mathcal{P}a{\mathbb{E}}^1[s]=\mathcal{P}a(1+\Delta )$. Substituting:
$\mathcal{P}a=\frac{1}{1+r}(a(1+\Delta )+\mathcal{P}a(1+\Delta ))⟹\mathcal{P}(1+r)=(1+\Delta )(1+\mathcal{P}).$
Solving:
$\mathcal{P}=\frac{1+\Delta }{r-\Delta }.$

Resale option size. The optimist’s buy-and-hold value (if she could not resell) would be $v^{1,\text{hold}}=a(1+\Delta )/r$. The market price exceeds this by
$p(a)-v^{1,\text{hold}}=\frac{a(1+\Delta )}{r-\Delta }-\frac{a(1+\Delta )}{r}=\frac{a(1+\Delta )\Delta }{r(r-\Delta )}.$
This is the resale option value: zero when $\Delta =0$, growing without bound as $\Delta \to r$.

See Also

• 8.7. Bubbles — parent hub.
• 8.7.2. Heterogeneous Beliefs and Overvaluation — the static Miller model that this dynamic version generalizes.
• 8.7.4. Overconfidence and Trading Volume — Scheinkman-Xiong’s microfoundation for the rotating optimism.
• 8.7.1. Rational Bubbles — the rational benchmark this behavioral model replaces.
• 8.3. Challenges to Rational Pricing — the volume puzzle this model uniquely matches.