Dynamic Trading with Realization Utility: Dai, Qin & Wang (2026)

What this is. The paper’s core theoretical results, model structure, and predictions: enough to know what it found without reading all 50 pages. To replicate or extend it, read the full source at the canonical DOI (paywalled).

TL;DR

Dai, Qin, and Wang build a continuous-time jump-diffusion model in which an investor receives utility bursts from realizing stock gains and losses at the individual stock level, while also managing a dynamic mental trading budget shared across all investment episodes. The key departure from prior realization-utility models (Barberis-Xiong 2012, Ingersoll-Jin 2013, He-Yang 2019) is that the investor is not forced to invest his entire budget in a single stock: he can save a fraction in the risk-free asset or use leverage. This intensive margin, combined with downward jumps in stock prices, generates two new predictions: (i) an investor with sufficient savings voluntarily sells a stock at an arbitrarily deep loss to reset his reference level, and (ii) an investor with low savings will not sell a deep loser but will sell it after its price rebounds just enough. Leverage amplifies the disposition effect; leverage constraints dampen it.

Core results

Magnitudes and thresholds are as reported. Locators point into the source PDF.

#	Result	Locator	Magnitude
R1	With sufficient savings ($w^* > 0$), the investor optimally saves 63.5% of his budget and allocates only 36.1% to the stock each trade	§III.A, Figure 2, p. 205-206	Baseline: $w^* = 1.76$; stock share $= 1/(1 + \theta_p + w^*) = 36.1\%$; value of saving option = 21% of mental trading budget ($\Delta(w^*) = 21\%$, Figure 2 Panel B)
R2	Savings cause the investor to realize losses sooner than in IJ (2013), reducing the disposition effect	Figure 3, p. 207	Loss-realization boundary $x^* = 0.69$ in baseline model vs 0.55 in IJ (2013); downside loss in dollars is one-third of the IJ (2013) model because savings absorb the hit at the trading-account level
R3	Leverage ($w^* < 0$) strengthens the disposition effect: loss-realization threshold falls and gain-realization threshold rises	Figure 4, p. 208	At $\sigma = 20\%$: $w^* = -0.36$; $x^*$ falls from 0.6 (IJ 2013) to 0.47; $\bar{x}$ rises from 1.03 to 1.04; the option to use leverage is worth 31% of the investor’s trading budget
R4	Binding leverage constraints mitigate the disposition effect by forcing earlier loss realization	Figure 5, p. 209	Tightening $\kappa$ from 0.79 to 0.59 raises loss-realization threshold $x^*$ from 0.47 to 0.52; gain-realization threshold $\bar{x}$ unchanged at 1.04
R5	Investors prefer stocks with high or low volatility, not intermediate volatility; leverage users prefer low-volatility stocks, savers prefer high-volatility stocks	Figure 6, p. 210	Scaled value $\hat{v}$ is U-shaped in $\sigma$ with minimum at $\sigma = 25\%$; investors use leverage when $\sigma < 25\%$ and save when $\sigma > 25\%$; $\hat{v}$ at $\sigma = 25\%$ equals 7.95 (same as IJ 2013, where no saving/leverage is available)
R6	With downward jumps and sufficient savings (Case A, $\sigma = 30\%$), the investor voluntarily realizes deep losses for all $x \in (0, 0.38)$; saving $w^* = 0.24$ means 19.2% of budget is in the risk-free asset, making deep-loss realization optimal	Figure 7, §IV.A, p. 213-214	Three-region solution: gain-realization at $x \geq 1.03$; holding for $x \in (0.38, 1.03)$; voluntary loss realization for all $x \in (0, 0.38)$ including $x$ near 0; payoff function $f(w^*, 0) = 2.3 > 0$
R7	With low savings (Case B, $\sigma = 24\%$), the investor holds a deep-loss stock (deep-loss holding region $x \in (0, 0.04)$) but sells after the price rebounds just enough to exit the deep-loss region	Figure 8, §IV.B, p. 214-215	Four-region solution: deep-loss holding for $x \in (0, 0.04)$; loss-realization for $x \in (0.04, 0.34)$; normal holding for $x \in (0.34, 1.03)$; gain-realization for $x \geq 1.03$; $w^* = 0.02$ (only 1.9% in savings)

Overall (paper’s conclusion). The two-layered mental account generates qualitatively new loss-realization predictions that diffusion-only models cannot produce. The sell-at-deep-loss (Case A) and sell-after-rebound (Case B) predictions arise from the interaction between the dynamic trading budget and downward jumps. Both predictions are consistent with observed retail investor behavior. Quantitatively, the option to save in the risk-free asset is worth over 20% of the investor’s total trading budget in calibrated diffusion models.

Theory / model

Two-layered mental accounts. The investor has a trading account with budget $\Pi_t > 0$ at time $t$, used solely for realization-utility optimization. At each trade he allocates a fraction to a risky stock and saves the rest in the risk-free asset (the intensive margin $w$). Prior models (BX 2012, IJ 2013, HY 2019) force $w = 0$ at all times; here $w$ is a choice variable.

State variables and dynamics. Three state variables: risk-free wealth $W_t$, risky wealth $X_t$, and reference level $B_t$ (eq. 1, p. 198):

$$\frac{dP_{n,t}}{P_{n,t}} = \mu \, dt + \sigma \, dZ_{n,t}, \qquad t > 0 \tag{1}$$

Between two consecutive trading times $(\tau_i, \tau_{i+1})$, risky wealth follows the same GBM (eq. 2, p. 198):

$$dX_t = \mu X_t \, dt + \sigma X_t \, dZ_{n,t}, \qquad t \in (\tau_i, \tau_{i+1}) \tag{2}$$ $$dW_t = r W_t \, dt, \qquad t \in (\tau_i, \tau_{i+1}) \tag{3}$$

The mental budget at $t$ is (eq. 4, p. 198):

$$\Pi_t = W_t + (1 - \theta_s) X_t \tag{4}$$

where $\theta_s$ is the proportional sale cost. Post-purchase wealth satisfies (eq. 5, p. 199):

$$W_{\tau_i^+} = \Pi_{\tau_i} - (1 + \theta_p) X_{\tau_i^+} \tag{5}$$

The leverage constraint (eq. 6, p. 199):

$$X_t \geq -W_t / \kappa, \qquad \text{where } 0 < \kappa < 1 - \theta_s \tag{6}$$

Reference level and realization utility. The reference level grows at the risk-free rate (eq. 7-8, p. 199):

$$dB_t = r B_t \, dt \qquad \text{for } t \in (\tau_i, \tau_{i+1}) \tag{7}$$ $$B_{\tau_i^+} = X_{\tau_i^+} \tag{8}$$

Realized gain (loss) at $\tau_i$ is (eq. 9-10, p. 199-200):

$$G_{\tau_i} = (1 - \theta_s) X_{\tau_i} - B_{\tau_i} \tag{9}$$ $$g_{\tau_i} = G_{\tau_i} / B_{\tau_i} \tag{10}$$

Utility burst (eq. 11, p. 200):

$$U(G, B) = B^{\beta} u(G/B) = B^{\beta} u(g) \tag{11}$$

The scaled utility function is S-shaped CPT (eq. 12, p. 200):

$$u(g) = \begin{cases} g^{\alpha_+} & \text{if } g \geq 0 \\ -\lambda (-g)^{\alpha_-} & \text{if } g < 0 \end{cases} \tag{12}$$

with $\lambda \geq 1$ (loss aversion), $\alpha_+, \alpha_- \in (0, 1]$ (diminishing sensitivity), and $\beta \leq \min\{\alpha_+, \alpha_-\}$ (eq. 13, p. 200) to ensure $|U(G,B)|$ decreases in $B$ for fixed $G$.

Optimization problem. The investor chooses trading times $\{\tau_i \geq t\}$ and stock allocations $X_{\tau_i^+}$ to maximize (eq. 14, p. 201):

$$\max \; \mathbb{E}_t \!\left[ \sum_{i=1}^{\infty} e^{-\delta(\tau_i - t)} U(G_{\tau_i}, B_{\tau_i}) \mathbf{1}_{\tau_i < \tau_L} + e^{-\delta(\tau_L - t)} U(G_{\tau_L}, B_{\tau_L}) \right] \tag{14}$$

subject to the leverage constraint (6) and dynamics (2), (3), (5), (7), (8), where $\delta > 0$ is the subjective discount rate and $\tau_L$ is the liquidity-shock arrival time (exogenous Poisson with rate $\xi$).

Baseline calibration (Table II, p. 205): $\alpha_+ = \alpha_- = 0.5$, $\lambda = 1.5$, $\beta = 0.3$, $r = 3\%$, $\delta = 5\%$, $\mu = 9\%$, $\sigma = 30\%$, $\theta_s = \theta_p = 1\%$, $\kappa = 0.79$, $\xi = 0$.

Jump-diffusion extension (Section IV, p. 211-216): stock prices follow (eq. 29, p. 212):

$$\frac{dP_{n,t}}{P_{n,t^-}} = \mu \, dt + \sigma \, dZ_{n,t} - (1 - Y) \, dJ_{n,t}, \qquad P_0 > 0 \tag{29}$$

where $J_n$ is a Poisson process with arrival rate $\rho = 0.73/\text{year}$ and jump size $Y \in [0,1]$ drawn from cdf $\Omega(Y) = Y^{\psi}$ with $\psi = 6.3$, implying expected price drop $\mathbb{E}[1-Y] = 1/(\psi+1) = 14\%$ per jump. In scaled variables (eq. 30, p. 212):

$$\frac{dx_t}{x_{t^-}} = (\mu - r) \, dt + \sigma \, dZ_{n,t} - (1 - Y) \, dJ_{n,t} \tag{30}$$

Method

Homogeneity reduction. Using the homogeneity of the value function $V(W,X,B) = B^{\beta} v(w,x)$ and payoff function $F(W,X,B) = B^{\beta} f(w,x)$ (p. 202), the three-state problem reduces to a two-state scaled problem with scaled variables (eq. 19, p. 203):

$$w_t = W_t / B_t \qquad \text{and} \qquad x_t = X_t / B_t \tag{19}$$

Since $w_t$ is constant between trades ($dw_t = 0$, eq. 21, p. 203), the investor optimally picks a constant target ratio $w^*$ at each trade. The scaled value with budget one is (eq. 24-25, p. 203):

$$\hat{v} = \max_{w \geq -\kappa} m(w) \tag{24}$$ $$m(w) = \left(\frac{1}{w + 1 + \theta_p}\right)^{\!\beta} v(w, 1) \tag{25}$$

The simplified scaled optimization problem is (eq. 22, p. 203):

$$v(w_t, x_t) = \max_{\tau} \; \mathbb{E}_t \!\left[ e^{-\delta_e(\tau - t)} f(w_{\tau}, x_{\tau}) \mathbf{1}_{\tau < \tau_L} + e^{-\delta_e(\tau_L - t)} u\!\left((1-\theta_s)x_{\tau_L} - 1\right) \right] \tag{22}$$

where $\delta_e = \delta - \beta r$ is the effective discount rate and $f(w, x) = u((1-\theta_s)x - 1) + [(1-\theta_s)x + w]^{\beta} \hat{v}$ (eq. 23, p. 203).

HJB / variational inequality. In the holding domain the scaled value satisfies (eq. 26, p. 204 for the diffusion model; eq. 31, p. 212 for jump-diffusion):

$$\delta_e v(w,x) = \tfrac{1}{2} \sigma^2 x^2 v_{xx} + (\mu - r) x v_x + \xi \!\left[u\!\left((1-\theta_s)x - 1\right) - v(w,x)\right] \tag{26}$$ $$\delta_e v(w,x) = \frac{\sigma^2 x^2}{2} v_{xx} + (\mu - r) x v_x + \rho \!\left(\mathbb{E}[v(w, Yx)] - v(w,x)\right) \tag{31}$$

The full variational inequality (Appendix A, pp. 218-219). The unscaled form is (A.1); after applying the homogeneity reduction the scaled variational inequality is (A.4):

$$\max \!\left\{ \mathcal{L} v(w,x),\; f(w,x) - v(w,x) \right\} = 0 \tag{A.4}$$ $$\mathcal{L} v = \tfrac{1}{2} \sigma^2 x^2 v_{xx} + (\mu - r) x v_x - \delta_e v + \xi \!\left[u\!\left((1-\theta_s)x - 1\right) - v\right] \tag{A.5}$$

When the leverage constraint (6) binds ($w = -\kappa x$): $v(w,x) = f(w,x)$ (eq. A.3, p. 219).

Closed-form solution (diffusion, no liquidity shocks). In the holding region, the value function has the form (Appendix B, eq. B.1, p. 222):

$$v(w,x) = C_1(w)\, x^{\eta_1} + C_2(w)\, x^{\eta_2} \tag{B.1}$$

where $\eta_1 > 0$ and $\eta_2 < 0$ are the two roots of the fundamental quadratic (eq. B.2, p. 222):

$$h(\eta) = \frac{\sigma^2}{2} \eta(\eta - 1) + (\mu - r)\eta - \delta_e = 0 \tag{B.2}$$

The optimal $w^*$ is found by (eq. B.4, p. 222):

$$w^* = \operatorname*{argmax}_{w \geq -\kappa} \frac{C_1(w) + C_2(w)}{[w + (1 + \theta_p)]^{\beta}} \tag{B.4}$$

Value-matching and smooth-pasting conditions at the two boundaries $\bar{x}(w)$ (gain) and $x^*(w)$ (loss) give a system of four equations (B.5)-(B.8) jointly with the FOC (B.9), p. 223.

For the jump-diffusion model, the variational inequality (B.10, p. 225):

$$\max \!\left\{ \mathcal{L}^J v(w,x),\; f(w,x) - v(w,x) \right\} = 0 \qquad \text{for } x \geq 0,\; w \geq 0 \tag{B.10}$$

is solved numerically via a penalty method (Appendix B.3).

Empirical specifications

This is a purely theoretical paper. There are no regression equations, no portfolio sorts, and no empirical datasets used. All quantitative results derive from numerical solution of the variational inequality or its closed-form analogue under the baseline and jump-diffusion calibrations.

Calibration targets (Table II, p. 205 and §IV parameter choices, p. 213):

• $\alpha_+ = \alpha_- = 0.5$, $\lambda = 1.5$, $\beta = 0.3$ match the CPT parameter values in IJ (2013), enabling direct comparison.
• $\mu = 9\%$, $r = 3\%$ target a 6% risk premium consistent with U.S. equity estimates (Hansen and Singleton 1982; Mehra and Prescott 1985).
• $\kappa = 0.79$ targets an 80% maximum loan-to-value ratio.
• Jump parameters $\rho = 0.73/\text{year}$, $\psi = 6.3$ (implying 14% expected price drop per jump) follow Barro and Jin (2011) and the rare-disaster literature.

Comparative statics are conducted by varying one parameter at a time (Figures 2-9, pp. 205-217): $\sigma$ from 10% to 50% (Fig. 6); $\kappa$ from 0.79 to 0.59 (Fig. 5); $\sigma$ from 30% (Case A) to 24% (Case B) in the jump-diffusion extension (Figs. 7-8).

Model predictions are discussed qualitatively against empirical findings in Barber et al. (2019), Heimer and Imas (2022), An et al. (2024), and Hartzmark (2015) but the paper does not run statistical tests against data.

Datasets used

This paper is purely theoretical. No empirical dataset is used; all results are derived analytically or via numerical solution of the model. No data tags apply.

Dataset	Role in paper	Wiki page
None	Theory and calibration only; parameter values ($\lambda$, $\mu$, $\sigma$, $r$, etc.) are taken from prior literature (Andersen et al. 2022, standard equity-premium estimates)	N/A

When to read the full paper

Read the original if you are: extending the realization-utility framework to allow saving or leverage; studying the disposition effect under jump risk; looking for the closed-form solution procedure (Appendix B) or the variational-inequality proofs (Appendix A); or evaluating the model’s quantitative calibration against empirical disposition-effect magnitudes. The locators above point to the exact figures. For “what did this paper find,” the table above is sufficient.

Attribution and rights

Source: peer-reviewed, The Journal of Finance 81(1), February 2026, pp. 189–238. DOI: 10.1111/jofi.13472. © 2026 the American Finance Association. This distillation was extracted by an LLM on 2026-05-31 and is not human-verified or independently reproduced. The article is paywalled; no open-access or CC licence was found in Crossref metadata. This page contains only extracted findings (extract-only).

Dai, Min, Cong Qin, and Neng Wang. “Dynamic Trading with Realization Utility.” The Journal of Finance 81, no. 1 (February 2026): 189–238. DOI: 10.1111/jofi.13472.