## Annex I. Exploring a longer option expiration

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### I. Introduction — speed differential and negative drift
- LIC economies face a speed differential: frequent negative shocks versus insufficient average growth to build durable gains.
- Negative shocks include "rapidly changing climatic conditions", "security threats", and "COVID-19".
- LICs are subject to more frequent negative shocks than emerging or advanced economies (IMF, 2011).
- Institutional, policy, buffer, and infrastructure weaknesses increase vulnerability and cause reforms to lag the speed of shocks.
- Example: Niger (1980–2018)
  - Timeline: 19 negative shocks between 1980 and present.
  - Average frequency between two negative events: 1.5 years.
  - Ratio of positive to negative standard deviations of growth: 0.5 between 1980-2018.
  - Real per capita GDP has not caught up to its 1980 level (1980 = 100).
- Implication: recoveries after negative shocks are insufficient to make up lost ground, producing a negative drift — economies must achieve a minimum transformation rate over shorter spans to escape drift.

### Escape speed (ES) and the positive shock strategy
- Definition: "escape speed" (ES) — minimum economic growth rate necessary to escape the speed differential between two shocks.
- Policy objective: construct a "positive shock" that catalyzes a virtuous cycle and progressive steady gains over the routine negative shocks cycle.
- Conceptual model:
  - Agents choose between D1 (routine, incremental adjustments) and D2 (significant, risky transformation decisions modeled as exercising a real option).
  - Synchronized D2 decisions can create critical mass and an aggregate positive shock.
- Aggregate estimate of the premium on these riskier options represents the potential cost of the positive shock.
- Real options theory informs engaging the private sector bottom-up to generate transformation.

### II. Motivation for an option pricing approach
- Rationale: an option gives its holder the right, but not the obligation, to buy or sell an underlying asset at a fixed price before expiry; the underlying asset here is GDP.
- Option trades mirror economic decisions on direction (expansion/contraction) and speed (growth rate).
- Holding "real" options increases likelihood of successful positive shock via feedback and synchronization.
- Reasons option pricing is appropriate:
  1. Fractional price and synergy
     - Options allow investing a fraction of the underlying asset price to protect downside or partake in upside; real options let agents test directions with limited risk, producing transferable lessons even if unsuccessful.
  2. Speculative premium: micro re-orientation feeding macro transformation
     - Options capture re-orientation risk where D2 decisions are rarer and riskier.
     - Out-of-the-money options have inherent value zero; their value comes from a speculative premium reflecting agents' confidence in transformation.
     - Low speculative premium = low risk appetite; high speculative premium = greater likelihood of D2 activity and aggregate transformation.

### 3. Volatility — overview and literature
- Options thrive when underlying assets are expected to make large movements in shorter horizons with significant volatility; this fits the "positive shock" concept (large impact over short horizon).
- Higher post-pandemic uncertainty and new technologies create a more volatile environment where higher risk taking may be necessary.
- Negative drift is similar to hysteresis (Cerra et al, (2020)); repeated negative shocks and eroded savings make the plateau between shocks insufficient without a counter-veiling positive shock.
- Path dependence constrains ES via:
  - (D1) path dependence reinforcing existing patterns.
  - (D2) risk aversion toward transformation; influential agents may be content with status-quo, causing horizon mismatch and free-rider problems.
- The paper revives "Big Push" logic within an options framework: planning selects complementary strategies and implements them close in time; randomness (optionality) preserves flexibility and lesson accumulation.

### Volatility Ratio (VR) — statistical measure to measure drift
- Volatility of economic growth measured by standard deviation:
  - V(g) = √[ ∑_{i=1}^{n} (g_t − g_m)^2 / (n−1) ]
- Proposed ratio isolating positive to negative deviations:
  - V(g+) / V(g−) = VR
  - VR = √[ ∑_{i=1}^{n} (g_(t+) − g_m)^2 / (n−1) ] / √[ ∑_{i=1}^{n} (g_(t−) − g_m)^2 / (n−1) ]
- Hypothesis: over long stretches VR < 1 for LICs, meaning negative deviations overtake positive ones, producing negative drift.
- Considerations:
  - Balance: a few large negative deviations can dominate VR; ensure sufficient balance in number of observations between numerator and denominator.
  - Sustainability: VR alone does not indicate sustainability; combine VR with population growth and per capita GDP growth. Use per capita GDP growth when measuring VR for groups.

### Empirical Ratios (Table 1: Volatility Ratios for Selected Groups (1980-2018), PPP Real GDP Growth)
- Table 1 results (1980-2018):
  - SSA: Ratio 0.72 ; Balance* 0.95
  - SSA LIC: Ratio 0.93 ; Balance* 0.86
  - SSA Fragile Countries: Ratio 0.77 ; Balance* 0.70
  - SSA Middle Income: Ratio 0.60 ; Balance* 0.70
  - WAEMU: Ratio 1.01 ; Balance* 0.95
  - Emerging LICs: Ratio 2.75 ; Balance* 1.17
- Notes:
  - Source: WEO and author's calculations.
  - 1/ Country groups are obtained from IMF SSA REO.
  - *Balance represents the ratio of number of positive to number of negative deviations regardless of the size of the deviation.
- Observation: SSA middle-income countries show a ratio significantly below 1, consistent with a middle income trap.

### The Model — structure and objectives
A. Growth as a Function of Decisions
- Economy comprised of private sector agents p_i and government G.
- Production results from accumulation of decisions and interactions:
  - Y_t = f(dp_i, dG, dp_i dp_ii, dG dp_i)   (2)
- Two decision types:
  - D1: routine decisions.
  - D2: structural, non-linear, riskier decisions.
- Growth components:
  - Linear component driven by D1.
  - Transformative component (speculative premium) driven by D2 (purchase and exercise of options).
- Probabilistic assumptions:
  - D1 ~ N(0,1).
  - D2 has perceived probability of success P(D2S) and probability decision is taken P(D2I), with Bayes relation:
    - P(D2I / D2S) = P(D2I and D2S) / P(D2S)   (3)
  - P(D2S)_tk = f_t(PS, Σ_{k=1}^{n} D2k)   (4)
  - PD2I governed by binomial distribution.
- Key inference:
  - Absent PS, as number of agents increases for a given t, the joint probability of agents purchasing options for significant impact tends to zero:
    - ∏_{k=1}^{n} P_t(D2I) → 0 as n → a   (5)
    - where a represents a sufficiently large number such that it has significant impact on P(D2S).
  - Conclusion: absent exogenous intervention, probability of meaningful impact from these decisions tends to zero — formalization of path dependence.
- Expected production for agent at time ti:
  - EP(ti) = Pdensity * qt(dil) + P * dsnl * Psuccess * qt(dsnl)
- Aggregate expected GDP:
  - E(GDP) = Σ_{n=1}^{nag} (EP(tn))   (6)
- Aggregate option value:
  - Σ_{n=1}^{k} (C_n) = C(gdp)   (7)
    - C_n is a call option available to agent n; C(gdp) is the sum representing overall call option value on the economy.

B. Model Objectives
- Primary objective 1: Determine minimum expected overall economic growth rate, ES, at which agents find it profitable to purchase real options — enabling more risk taking toward transformation.
- Primary objective 2: Propose framework to value aggregation of individual options at the expected growth rate, yielding a hypothetical overall option on the economy representing the speculative premium.
  - Estimation purposes:
    - Estimate speculative premium for country-level relative ranking.
    - Provide minimum threshold for overall size of the positive shock: aggregate option value indicates size of subsidy/support needed to induce option purchase.
  - Policy interpretation:
    - Cost of positive shock can be represented as a generic subsidy for option purchase; gap between aggregate premium and profitability threshold indicates additional cost of positive shock.
    - Example instrument: a pool of funds saved in good times to bridge the profitability threshold.

C. General Assumptions (to apply Black-Scholes)
- Baseline: use Black-Scholes to find closed-form solution for ES. Assumptions:
  1. Agents face D1 (routine, N(0,1)) and D2 (purchase of real options).
  2. Strike price K is current per capita GDP, g, factored by propensity c (consumption) the agent is willing to pay.
  3. Current value of project underlying the option, S_t, is K expected to grow at rate g, discounted by real interest rate.
  4. Agent splits net resources between linear activity and purchase of option.
  5. Given assumption 3, unless implied return of option relative to current income outweighs return on prevailing growth g, agent will allocate all post-consumption resources to routine behavior.

### D. Solution to the model: Black-Scholes Formula Applied
- Black-Scholes for agent at time t:
  - C(St, t) = N(d1) St – N(d2) PV(K) (8)
- In framework:
  - St = [(i c l) (1+g*)] e^{-r (T-t)} (9)
    - i = GDP per capita
    - c = consumption factor less than 1
    - l = routine behavior factor less than 1
    - g = expected economic growth rate
    - r = risk-free real interest rate
    - T-t = time to maturity
  - K = i c l (10)
- Re-formulated option value:
  - C(St, t) = N(d1) [(i c l) (1+g*)] e^{-r (T-t)} - N(d2) (i c l) e^{-r (T-t)} (11)
- Option attractive condition (must be ≥ present value of prevailing-return investment):
  - N(d1) [(i c l) (1+g*)] e^{-r (T-t)} - N(d2) (i c l) e^{-r (T-t)} ≥ [(i c l) (1+g)] e^{-r (T-t)} (12)
- Standard Black-Scholes d1 and d2:
  - d1 = [1 / (σ (√T-t))] [ln (St / K) + (r + σ^2 /2) (T -t)] (13)
  - d2 = d1 – σ (√T-t) (14)
  - σ represents volatility.
- Solving for minimum g*:
  - g* = [ ((1+g) + N(d2)) / N(d1)] -1 (15)
  - N(d1) expanded via series:
    - N(d1) = [( 1 / (√2π ) (d1 – (1/6) d1^3 + (1/40)d1^5 – (1/336) d1^7 ....) ) ] (16)
- One-year option life assumption (T-t = 1) for interpretability and alignment with LIC/fragile contexts.
  - Under one-year horizon: d2 = d1 - σ (17)
  - d1 rewritten as: [1 / σ] [ln (St / K) + (r + σ^2 /2)] and substituting St and K leads to expanded N(d1) and N(d2).

### Key model variables (quadratic outcome variables)
- 1) icl — resources available to the agent net of consumption and routine decisions.
- 2) r — real “risk-free” interest rate on a government paper (one-year security minus average inflation).
- 3) g — “prevailing” economic growth rate (latest realized growth rate).
- 4) σ — standard deviation of growth in the past 20 years to represent a realistic expectation for the agent.
- 5) g* — the variable solved for: growth rate needed so option premium is valuable enough to beat prevailing growth rate; ES.
- 6) The difference in valuation of the option using g* and g from existing nominal GDP base indicates the additional “premium” per agent (potential overall cost of the positive shock).

### Numerical simulation approach and robustness
- Simulations used Mathematica and Newton’s method to approximate g* across combinations of the above variables for selected countries, regions, and groupings.
- Simulations varied icl; results were orthogonal to levels of icl (outcomes did not materially depend on nominal level of icl) — noted as potential model weakness/interpretation nuance.
- Baseline σ uses 20 years of growth data. Footnote: using shorter-horizon volatility increases the gap between ES, g* and prevailing growth rate by 13 percent for fragile countries in SSA when positive volatility decreased in the last 10 years.

### Simulation results and key statistics (grouping-level outcomes)
- For LICs in general:
  - ES is nearly 8.3 percent higher than the prevailing growth rate or more than 0.4 percentage points.
- For LICs in SSA:
  - Differential rises to 68 percent or 2.8 percentage points.
- For fragile economies in SSA:
  - ES is nearly 88 percent higher than the prevailing growth rate or 2.1 percentage points.
- Longer-term option (9-year maturity) results:
  - For SSA LICs, growth differential rises to 16.8 percentage points over the longer horizon.
  - For fragile economies in SSA, differential is 8.5 percentage points vs. prevailing medium-term growth assumption.
- Interpretation: longer maturities increase required g* because more uncertainties accumulate over time.

### VI. Policy Implications — convergence vs divergence and incentives
- Two possible macro outcomes when more options are purchased:
  - Convergence to a critical mass: options end up in the money, increasing overall option premium and raising next-period option values.
  - Failure to reach critical mass: options end up out of the money; continuation of linear path or convergence back to status-quo.
- Without an initial coordinated push, probability of sufficient transformative decisions (D2) converging to critical mass is practically zero.
- To incentivize agents to purchase riskier options and empower D2, policy must provide:
  - i) conditions conducive to more risk-taking for transformative micro-level behaviors;
  - ii) an initial macro-level push via strategic risks with prioritization and asymmetric concentration (avoid equally spreading resources — the “Christmas tree approach”);
  - iii) project credibility, specificity, and speed of reform to match short-term private-agent horizons (agents bet on both direction and speed of reforms);
  - iv) simultaneity — complementary projects and measures implemented relatively close in time to maximize synergy.

### VII. General Design Principles for a Positive Shock (real option theory applied)
- Three policy axes for designing the Positive Shock:
  1. Policy tools — range from policy incentives to governance reforms to lift private-sector constraints.
  2. Administrative actions — ad-hoc measures and direct support mechanisms (e.g., purchasing options for agents, training, grant competitions).
  3. Sectoral support — strategic selection of emerging sectors for targeted support.
- Project/measure selection criteria to maximize optionality:
  - a. Abandon/modification quality — prefer measures easy to modify or abandon so lessons accrue and option value rises through exercisability.
  - b. Lesson value and synergy — prefer measures that provide lessons and create synergies even if abandoned.
  - c. Asymmetric concentration — do not split resources equally; choose low-hanging fruits plus selected strategic bets; invest in transferable technical skills (e.g., coding) and technology-enhancing skills.
  - d. Simultaneity and speed — reforms must have a speed element and matrix of discrete actions within limited timeframe; pursue components simultaneously to preserve synergy and credibility to private agents. Speed increases the time-value component of the option and induces private purchases.

### VII. Conclusion — implications and summary findings
- Long-term development approach alone may not suffice to escape cycles of negative shocks. Options framework identifies substantial gaps between ES and prevailing growth:
  - Gap more than 88 percent for fragile SSA economies.
- Options and optionality tend to widen divergence: economies with more flexibility and space for risk-taking will better exploit profitable real options.
- Economies subject to routine negative shocks risk stagnation unless they can leverage positive shocks.
- Synergy between private agents and governments can accelerate non-linear decisions needed to achieve ES. Waiting for full institutional development may not be necessary; dynamic private-sector development could drive institution building.
- Broader implication: economies need a short-term growth strategy (positive shock) before transitioning to a longer-term virtuous cycle.

### Annex — Exploring a longer option expiration (9-year maturity)
- Objective: isolate g* from g* = [ ((1+g) + N(d2)) / N(d1) ] -1 when g* enters definitions of d1 and d2 via N(·), using series expansion and a hypothetical option life of 9 years.
- Series expansions used:
  - N(d1) = [( 1 / (√2π) (d1 – (1/6) d1^3 + (1/40)d1^5 – (1/336) d1^7 ....) ) ]
  - N(d2) = [(1 / (√2π) ((d1 - σ) – (1/6) (d1 - σ)^3 + (1/40) (d1 - σ)^5 – (1/336) (d1 - σ)^7 ....))]
- Substitutions for T - t = 9 introduce factors:
  - appearances of 9, e^{-9r}, [1 / 3σ], and [1 / 9σ] due to √9 and scaling in d1 and d2.
- Final algebraic expression for g* with 9-year maturity remains implicit and nested because g* appears inside logarithmic and normal CDF expansions; derivation accounts for g* inside N(d1) and N(d2) using polynomial terms up to odd powers (d1^3, d1^5, d1^7, ...).

*Source: wpiea2023152-print-pdf - Annex I. Exploring a longer option expiration*

### Annex I. Exploring a longer option expiration ..........................................................................

### Annex I. Exploring a longer option expiration

### I. Introduction — speed differential and negative drift
- LIC economies face a speed differential: frequent negative shocks versus insufficient average growth to build durable gains.
- Negative shocks include "rapidly changing climatic conditions", "security threats", and "COVID-19".
- LICs are subject to more frequent negative shocks than emerging or advanced economies (IMF, 2011).
- Institutional, policy, buffer, and infrastructure weaknesses increase vulnerability and cause reforms to lag the speed of shocks.
- Example: Niger (1980–2018)
  - Timeline: 19 negative shocks between 1980 and present, most unique to Niger.
  - Average frequency between two negative events: 1.5 years.
  - Ratio of positive to negative standard deviations of growth: 0.5 between 1980-2018.
  - Real per capita GDP has not caught up to its 1980 level (1980 = 100).
- Implication: recoveries after negative shocks are insufficient to make up lost ground, producing a negative drift — economies must achieve a minimum transformation rate over shorter spans to escape drift.

### Escape speed (ES) and the positive shock strategy
- Definition: "escape speed" (ES) — minimum economic growth rate necessary to escape the speed differential between two shocks.
- Policy objective: construct a "positive shock" that catalyzes a virtuous cycle and progressive steady gains over the routine negative shocks cycle.
- Conceptual model:
  - Agents choose between:
    - D1: routine, incremental adjustments to existing activities.
    - D2: significant risk toward transformation (modeled as exercising a real option).
  - Sufficient synchronized D2 decisions in close time periods can create a critical mass and an aggregate positive shock.
- Aggregate estimate of the premium on these riskier options represents the potential cost of the positive shock.
- Real options theory informs how to engage the private sector bottom-up to generate transformation.

### II. Motivation for an option pricing approach
- Rationale: an option is a derivative giving its holder the right, but not the obligation, to buy or sell an underlying asset at a fixed price before expiry. Here the underlying asset is broadly the economy (specified later to be GDP).
- Two bets made by an option trader mirror economic decisions:
  - Direction of change (expansion or contraction).
  - Speed of change (growth rate).
- Holding "real" options by private agents increases the likelihood of a successful positive shock via feedback and synchronization.
- Example of a real option: acquiring a new skill — the time and resources invested are the option premium; potential future job gains justify the premium.

- Reasons why option pricing is appropriate for designing the positive shock:
  1. Fractional price and synergy
     - Options allow investing a fraction of the underlying asset price to protect against downside or partake in upside without fundamental portfolio re-orientation.
     - Real options let agents test directions with limited risk; even unsuccessful trials provide transferable lessons.
  2. Speculative premium: micro re-orientation feeding macro transformation
     - Options capture re-orientation risk where D2 decisions are rarer and riskier.
     - Out-of-the-money options have inherent value zero; their value comes from speculative premium reflecting agents' confidence in transformation.
     - Low speculative premium = low risk appetite; high speculative premium = greater likelihood of D2 activity and aggregate transformation.

### Organization and next steps (as described)
- Remaining sections in the paper (as outlined) will:
  - Motivate option pricing framework further.
  - Survey literature including "Big Push Theory".
  - Introduce a statistical measure: ratio of positive to negative standard deviations of growth to represent negative drift.
  - Present the model, assumptions, quantitative results, and interpretation.
  - Discuss policy implications and design of the positive shock using real options principles.
  - Conclude with final remarks.

*Source: wpiea2023152-print-pdf - Annex I. Exploring a longer option expiration*

### 3. Volatility

### 3. Volatility

### Overview
- Options thrive when underlying assets are expected to make large movements in shorter horizons with significant volatility; this framework is appropriate to describe the "positive shock" as defined in the chapter: a large impact over a short horizon.
- The world economy is facing higher uncertainty as it emerges from the pandemic; new challenges and the emergence of technologies contribute to an inherently more volatile environment where higher risk taking may be necessary to survive.

### Literature Review
- The concept of the negative drift is similar to hysteresis (Cerra et al, (2020)). This paper extends the pull beyond business cycles and negative shocks: a stable equilibrium combined with repeated negative shocks renders the plateau between shocks insufficient to resist the pull from the previous negative shock unless there is a counter-veiling positive shock, partly because of eroded savings during the negative shock.
- Path dependence constrains achievement of "escape speed" (ES):
  - Two reinforcing features: (D1) path dependence that reinforces existing economic patterns; (D2) risk aversion toward transformation.
  - Influential agents with potential to effectuate transformation may be content with the status-quo and are averse to marginal risk, leading to a horizon mismatch and free-rider problems that reduce aggregate risk appetite.
- The notion of escape speed (ES) requires a concerted push to overcome inertia inherent in path dependence.
- Historical and theoretical antecedents:
  - "Big push model" (Paul Rosenstein-Rodan, early 1940s). Murphy, Shleifer and Vishny (1989) used game theory to show equilibria where coordinated action (including initial government intervention) can catalyze industrialization.
  - Jeffrey Sachs (2005, as discussed by Easterly (2006)) advocated scaling financial resources to LICs; criticisms include "information" and "incentive" problems favoring piecemeal interventions.
- This paper resuscitates the "big push" using an options framework:
  - Introduces a hybrid between planning and randomness: planning chooses strategies likely to yield complementarity and implements them close in time; randomness (optionality) retains flexibility to modify course based on outcomes.
  - Repeated experimentation yields lesson values (e.g., skill development) that can eventually produce successful outcomes.
- Related work: Dihle (2015) uses a real options framework to examine macro implications of microeconomic irreversibility and idiosyncratic uncertainty in a simple growth model; principles of abandon and modification in the options framework that mitigate irreversibility are discussed in the policy section.

### Volatility Ratio (Statistical Measure to Measure Drift)
- Volatility of economic growth measured by standard deviation:
  - V(g) = √[ ∑_{i=1}^{n} (g_t − g_m)^2 / (n−1) ]
  - V(g) lumps positive and negative deviations from the mean.
- Proposed ratio isolating positive to negative deviations:
  - V(g+) / V(g−) = VR
  - VR = √[ ∑_{i=1}^{n} (g_(t+) − g_m)^2 / (n−1) ] / √[ ∑_{i=1}^{n} (g_(t−) − g_m)^2 / (n−1) ]
- Hypothesis: over long stretches of time, VR < 1 for LICs, meaning negative deviations overtake positive ones, producing a negative drift.
- Two important considerations for VR:
  - Balance: a few large negative deviations could produce large negative volatility overtaking smaller but more numerous positive deviations. Ensure sufficient balance in the number of observations between numerator and denominator.
  - Sustainability: VR alone does not indicate sustainability. In combination with population growth and per capita GDP growth, VR indicates whether growth path avoids drift. Overall growth must exceed population growth sufficiently, and have large enough positive deviations to overcome negative pulls.
- Therefore, per capita GDP growth is used when measuring VR for groups.

### Empirical Ratios (Table 1: Volatility Ratios for Selected Groups (1980-2018), PPP Real GDP Growth)
- Table 1 results (1980-2018):
  - SSA: Ratio 0.72 ; Balance* 0.95
  - SSA LIC: Ratio 0.93 ; Balance* 0.86
  - SSA Fragile Countries: Ratio 0.77 ; Balance* 0.70
  - SSA Middle Income: Ratio 0.60 ; Balance* 0.70
  - WAEMU: Ratio 1.01 ; Balance* 0.95
  - Emerging LICs: Ratio 2.75 ; Balance* 1.17
- Notes:
  - Source: WEO and author's calculations.
  - 1/ Country groups are obtained from IMF SSA REO.
  - *Balance represents the ratio of number of positive to number of negative deviations regardless of the size of the deviation.
- Observation:
  - SSA middle-income countries show a ratio significantly below 1, consistent with a middle income trap.

### The Model
A. Growth as a Function of Decisions
- Economy comprised of private sector agents p_i and government G.
- Economic production results from accumulation of a series of decisions and their interactions: dp_i, dG, dp_i dp_ii, and dG dp_i leading to continuous output:
  - Y_t = f(dp_i, dG, dp_i dp_ii, dG dp_i)   (2)
- Two fundamental decision types:
  - Routine decisions (D1): sustain same production structure for an agent.
  - Structural and non-linear decisions (D2): change production structure and involve considerably more risk.
- Two parts of growth:
  - Linear component driven by D1 (higher production of same products).
  - Transformative component (speculative premium) driven by D2 (new products, higher value products, transformation beyond productivity in Solow model).
  - The set of D2 that results in the speculative premium involves purchase and exercise of options.
- Time treated as continuous given a multiplicity of agents and interactions.
- Probabilistic assumptions:
  - D1 ~ N(0,1) (normal distribution with zero mean and variance 1).
  - D2 has two embedded probabilities:
    - Perceived probability of success if decision taken: P(D2S).
    - Probability decision is taken (purchase of option): P(D2I). P(D2I) is conditional on P(D2S).
  - Bayes relation:
    - P(D2I / D2S) = P(D2I and D2S) / P(D2S)   (3)
  - P(D2S)_tk = f_t(PS, Σ_{k=1}^{n} D2k)   (4)
    - D2 for agent k ∈ {0,1}; PS stands for factors linked to success of the positive shock.
  - PD2I governed by binomial distribution.
- Key inference:
  - Absent PS, as number of agents increases for a given t, the joint probability of these agents to purchase options for a significant impact on P(D2S) tends to zero:
    - ∏_{k=1}^{n} P_t(D2I) → 0 as n → a   (5)
    - where a represents a sufficiently large number such that it has significant impact on P(D2S).
  - Conclusion: absent exogenous intervention, probability of meaningful impact from these decisions tends to zero, reverting growth path to equilibrium defined by negative drift (formalization of path dependence).
- Implication of D2 structure:
  - Two embedded probabilities (decision and success) imply no stable middle ground: either a virtuous cycle or reversion to negative drift.
- Expected economic production for a given agent at time ti:
  - EP(ti) = Pdensity * qt(dil) + P * dsnl * Psuccess * qt(dsnl)
    - qt(dil): function of D1 (linear decisions).
    - qt(dsnl): function of D2 (non-linear decisions).
- Aggregate expected GDP:
  - E(GDP) = Σ_{n=1}^{nag} (EP(tn))   (6)
- Aggregate option value:
  - Σ_{n=1}^{k} (C_n) = C(gdp)   (7)
    - C_n is a call option available to agent n inherent in any D2 decision.
    - C(gdp) is the sum of all call options available to agents, representing the overall call option value on the economy.
  - D2 is an implicit argument of C(gdp); isolating D2 allows calculation of option value associated with gdp.
- Tradeability and aggregation:
  - Overall production is tradeable internally and externally; real options embedded are tradeable through trade in goods and services associated with D2 decisions.
  - Agents' implicit direction and volatility bets aggregate to determine aggregate direction and volatility of the economy.
  - The aggregate option C(gdp) can be valued using standard option valuation methods.

B. Model Objectives
- Primary objective 1: Determine the minimum expected overall economic growth rate, ES, at which agents find it profitable to purchase real options available to them — the rate enabling more risk taking toward transformation and beating the speed trap.
- Primary objective 2: Propose a framework to value aggregation of individual options at the expected growth rate, yielding a hypothetical overall option on the economy representing the speculative premium.
  - Estimation purposes:
    - Provide an estimate of the speculative premium for country-level relative ranking.
    - Provide a minimum threshold for the overall size of the positive shock: the aggregate option value (sum of individual premiums) indicates the size of subsidy or support needed to induce option purchase.
  - Policy interpretation:
    - The cost of the positive shock can be represented as a generic subsidy for option purchase; the gap between aggregate premium and what the premium should be for profitability represents additional cost of the positive shock.
    - Example instrument: a pool of funds saved in good times to take advantage of opportunities (not direct purchase of each agent's option by planners, but bridging the profitability threshold).

C. General Assumptions (to apply Black-Scholes)
- Baseline: use Black-Scholes model to find closed-form solution for ES. Assumptions:
  1. Agents face two decision types at any time t: D1 (routine, normally distributed via N(0,1)) and D2 (purchase of "real" options in riskier, transformative projects).
  2. Strike price K is current per capita GDP, g, factored by propensity c (that takes out consumption) that the agent is willing to pay for the project underlying the option at exercise.
  3. Current value of project underlying the option, S_t, is K (as defined) expected to grow at economic growth rate g, discounted to present value by the prevailing real interest rate.
  4. Agent splits net resources (after consumption) between linear activity (investment in prevailing economy) and purchase of the option (riskier bet on higher growth rate).
  5. Given assumption 3, unless implied return of the option relative to current income outweighs the return on prevailing growth rate, the agent will expend all post-consumption resources on routine decision-type behavior (i.e., will not buy the option).

*Source: wpiea2023152-print-pdf - 3. Volatility*

### 6. If the return on the value of the option is greater than the return on the prevailing economy, she would

### 6. If the return on the value of the option is greater than the return on the prevailing economy, she would 

### D. Solution to the model: Black-Scholes Formula Applied
- Black-Scholes formula for a given agent at time t:
  - C(St, t) = N(d1) St – N(d2) PV(K) (8)
  - C represents the value of the option; St is the current stock price; K is the strike price; PV is present value; N is the cumulative density function for the normal distribution.
- In the framework:
  - St = [(i c l) (1+g*)] e^{-r (T-t)} (9)
    - i = GDP per capita
    - c = consumption factor less than 1
    - l = routine behavior factor less than 1
    - g = expected economic growth rate
    - r = risk-free real interest rate
    - T-t = time to maturity of the real option
  - K = i c l (10)
- Re-formulated option value with assumptions:
  - C(St, t) = N(d1) [(i c l) (1+g*)] e^{-r (T-t)} - N(d2) (i c l) e^{-r (T-t)} (11)
- Option attractive condition (must be ≥ present value of prevailing-return investment):
  - N(d1) [(i c l) (1+g*)] e^{-r (T-t)} - N(d2) (i c l) e^{-r (T-t)} ≥ [(i c l) (1+g)] e^{-r (T-t)} (12)
- Standard Black-Scholes d1 and d2:
  - d1 = [1 / (σ (√T-t))] [ln (St / K) + (r + σ^2 /2) (T -t)] (13)
  - d2 = d1 – σ (√T-t) (14)
  - σ represents volatility.

### Solving for the minimum g* (growth needed for option to be attractive)
- Intermediate algebraic expression for g*:
  - g* = [ ((1+g) + N(d2)) / N(d1)] -1 (15)
- N(d1) expanded via series (cumulative normal expansion):
  - N(d1) = [( 1 / (√2π ) (d1 – (1/6) d1^3 + (1/40)d1^5 – (1/336) d1^7 ....) ) ] (16)
- One-year option life assumption (T-t = 1) for interpretability and alignment with LIC/fragile contexts (footnote justification).
  - Under one-year horizon: d2 = d1 - σ (17)
  - d1 rewritten as: [1 / σ] [ln (St / K) + (r + σ^2 /2)] and substituting St and K leads to expanded N(d1) and N(d2) expressions (18)-(21).
- Final algebra ends with g* expressed by substitution (22) resulting in an equation that reduces to a quadratic involving key variables listed below.

### Key model variables (quadratic outcome variables)
1) icl — resources available to the agent net of consumption and routine decisions.  
2) r — real “risk-free” interest rate on a government paper (one-year security minus average inflation).  
3) g — “prevailing” economic growth rate (latest realized growth rate).  
4) σ — standard deviation of growth in the past 20 years to represent a realistic expectation for the agent.  
5) g* — the variable solved for: the growth rate needed so the option premium is valuable enough to beat the prevailing growth rate; ES.  
6) The difference in valuation of the option using g* and g from existing nominal GDP base indicates the additional “premium” per agent (potential overall cost of the positive shock).

### Numerical simulation approach and a robustness note
- Simulations used Mathematica and Newton’s method to approximate g* across combinations of the above variables for selected countries, regions, and groupings.
- Simulations varied icl; results were orthogonal to levels of icl (resources available), i.e., outcomes did not materially depend on the nominal level of icl in the model runs — noted as a potential weakness or interpretation nuance of the model.
- Baseline σ uses 20 years of growth data. Footnote: using shorter-horizon volatility increases the gap between ES, g* and prevailing growth rate by 13 percent for fragile countries in SSA when positive volatility decreased in the last 10 years.

### Simulation results and key statistics (grouping-level outcomes)
- For LICs in general:
  - ES is nearly 8.3 percent higher than the prevailing growth rate or more than 0.4 percentage points.
- For LICs in SSA:
  - Differential rises to 68 percent or 2.8 percentage points (potentially reflecting fragility).
- For fragile economies in SSA:
  - ES is nearly 88 percent higher than the prevailing growth rate or 2.1 percentage points, confirming the need for a “shock” effect.
- Longer-term option (9-year maturity) results:
  - For SSA LICs, growth differential rises to 16.8 percentage points over the longer horizon.
  - For fragile economies in SSA, differential is 8.5 percentage points vs. prevailing medium-term growth assumption.
- Interpretation: longer maturities increase required g* (higher premium) because more uncertainties accumulate over time.

### VI. Policy Implications — Convergence vs Divergence and incentives
- Two possible macro outcomes when more options are purchased:
  - Convergence to a critical mass: options end up in the money, increasing overall option premium and raising next-period option values.
  - Failure to reach critical mass: options end up out of the money; continuation of linear path or convergence back to status-quo.
- Without an initial coordinated push, probability of sufficient transformative decisions (D2) converging to critical mass is practically zero.
- To incentivize agents to purchase riskier options and empower D2, policy must provide:
  - i) conditions conducive to more risk-taking for transformative micro-level behaviors;
  - ii) an initial macro-level push via strategic risks with prioritization and asymmetric concentration (avoid equally spreading resources — the “Christmas tree approach”);
  - iii) project credibility, specificity, and speed of reform to match short-term private-agent horizons (agents bet on both direction and speed of reforms);
  - iv) simultaneity — complementary projects and measures implemented relatively close in time to maximize synergy.

### VII. General Design Principles for a Positive Shock (real option theory applied)
- Three policy axes for designing the Positive Shock:
  1. Policy tools — range from policy incentives to governance reforms to lift private-sector constraints.
  2. Administrative actions — ad-hoc measures and direct support mechanisms (e.g., purchasing options for agents, training, grant competitions).
  3. Sectoral support — strategic selection of emerging sectors for targeted support.
- Project/measure selection criteria to maximize optionality:
  a. Abandon/modification quality — prefer measures easy to modify or abandon so lessons accrue and option value rises through exercisability.
  b. Lesson value and synergy — prefer measures that provide lessons and create synergies even if abandoned.
  c. Asymmetric concentration — do not split resources equally; choose low-hanging fruits plus selected strategic bets (one to improve an established industry toward export capacity; another to develop high-value sector). Invest in transferable technical skills (e.g., coding) and technology-enhancing skills.
  d. Simultaneity and speed — reforms must have a speed element and matrix of discrete actions within limited timeframe; pursue components simultaneously to preserve synergy and credibility to private agents. Speed increases the time-value component of the option and induces private purchases.

### VII. Conclusion — implications and summary findings
- Long-term development approach alone may not suffice to escape cycles of negative shocks. Options framework identifies substantial gaps between ES (required growth to escape negative drift) and actual prevailing growth rate:
  - Gap more than 88 percent for fragile SSA economies (footnote: subsequent work will test stochastic assumptions; agents’ actions affecting underlying asset price not explicitly modeled in current ES estimates).
- Options and optionality will tend to widen divergence: economies with more flexibility and space for risk-taking will better exploit profitable real options.
- Economies subject to routine negative shocks risk stagnation unless they can leverage positive shocks.
- Synergy between private agents and governments can accelerate non-linear decisions needed to achieve ES. Waiting for full institutional development may not be necessary; dynamic private-sector development could drive institution building.
- Broader implication: economies need a short-term growth strategy (positive shock) before transitioning to a longer-term virtuous cycle.

*Italic: Source — wpiea2023152-print-pdf (section content provided).*

### Annex I. Exploring a longer option expiration

### Annex I. Exploring a longer option expiration

### Objective and approach
- Aim: isolate g* from the relationship g* = [ ((1+g) + N(d2)) / N(d1) ] -1 when g* enters the definitions of d1 and d2 via the cumulative normal distribution N(·).
- Method: expand the cumulative normal distribution N(d1) and N(d2) via a series expansion and substitute expressions for d1 and d2 under the assumption of a hypothetical option life of 9 years.

### Series expansion of N(d1) and N(d2)
- Expansion used for N(d1):
  - N (d1) = [( 1 / (√2π) (d1 – (1/6) d1^3 + (1/40)d1^5 – (1/336) d1^7 ....) ) ]
- Expansion used for N(d2):
  - N (d2) = [(1 / (√2π) ((d1 - σ) – (1/6) (d1 - σ)^3) + (1/40) (d1 - σ)^5 – (1/336) (d1 - σ)^7 ....))]

### Definitions and substitutions for a 9-year option
- d2 = d1 - σ
- d1 expressed as:
  - [1 / σ] [ln (St / K) + (r + σ^2/2)]
- Substitution for the 9-year horizon and replacement of St and K leads to iterated expressions where time to maturity T - t = 9 appears throughout.
- Representative substituted forms (preserving original notation and numeric values):
  - N (d1) = [( 1 / (√2π ) (([1 / σ ] [ln (([(i c l) (1+g* /100) ] e^{-r (9)}) / (i c l)) + (r + σ^2/2) ] ) – ( (1/6) ([1 / σ ] [ln (([(i c l) (1+g* /100) ] e^{-r (9)}) / (i c l)) + (r + σ^2/2) ] )^3 + (1/40) ( (1/6) ([1 / σ ] [ln (([(i c l) (1+g* /100) ] e^{-r (9)}) / (i c l)) + (r + σ^2/2) ] )^5 – (1/336) ( (1/6) ([1 / σ ] [ln (([(i c l) (1+g* /100) ] e^{-r (9)}) / (i c l)) + (r + σ^2/2) ] )^7 ...)] )
  - N (d2) = [(1 / (√2π ) ((([1 / σ (√9)] [ln (St / K) + (r + σ^2/2) (T -t)] ) - σ) – (1/6) (([1 / σ (√9)] [ln (St / K) + (r + σ^2/2) (9)] ) - σ )^3) + (1/40) (([1 / σ (√9)] [ln (St / K) + (r + σ^2/2) (9)] ) - σ )^5 – (1/336) (([1 / σ (√9)] [ln (St / K) + (r + σ^2/2) (9)] ) - σ )^7 ....))]

### Final algebraic expression for g* with 9-year maturity
- Core relationship:
  - g* = [ ((1+g) + N (d2)) / N (d1) ] -1
- Substituting the expanded N(d2) and N(d1) with the 9-year substitutions yields:
  - g* = [ ((1+g) + {[(1 / (√2π ) ((([1 / σ (9)] [ln ((St)/ (K)) + (r + σ^2/2) (9)] ) - σ) – (1/6) (([1 / σ (√9)] [ln ((St) / (K)) + (r + σ^2/2) (9)] ) - σ )^3) + (1/40) (([1 / σ (√9)] [ln ((St) / (K)) + (r + σ^2/2) (9)] ) - σ )^5 – (1/336) (([1 / σ (√9)] [ln ((St) / (K)) + (r + σ^2/2) (9)] ) - σ )^7 ....))]} / {[( 1 / (√2π ) (([1 / σ ] [ln (([(i c l) (1+g*) ] e^{-r (9)}) / (i c l)) + (r + σ^2/2) ] ) – ( (1/6) ([1 / σ ] [ln (([(i c l) (1+g*) ] e^{-r (9)}) / (i c l)) + (r + σ^2/2) ] )^3 + (1/40) ( (1/6) ([1 / σ ] [ln (([(i c l) (1+g*) ] e^{-r (9)}) / (i c l)) + (r + σ^2/2) ] )^5 – (1/336) ( (1/6) ([1 / σ ] [ln (([(i c l) (1+g*) ] e^{-r (9)}) / (i c l)) + (r + σ^2/2) ] )^7 ...]}] -1
- Equivalent form after final substitution for St and K (preserving the original algebraic structure and numeric values):
  - g* = [ ((1+g) + {[( 1 / (√2π ) ((([1 /9σ] [ln (([(i c l) (1+g*) ] e^{-9r} )/ (i c l)) + (r + σ^2/2)9]) - σ) – (1/6) (([1 / 3σ ] [ln (([(i c l) (1+g*) ] e^{-9r} ) / (i c l)) + (r + σ^2/2)9]) - σ )^3) + (1/40) (([1 / 3σ ] [ln (([(i c l) (1+g*) ] e^{-9r} ) / (i c l)) + (r + σ^2/2)9 ]) - σ )^5 – (1/336) (([1 / 3σ ] [ln (([(i c l) (1+g*) ] e^{-9r} ) / (i c l)) + (r + σ^2/2)9]) - σ )^7 ....) ) ]} / {[( 1 / (√2π ) (([1 / 3σ ] [ln (([(i c l) (1+g*) ] e^{-9r} ) / (i c l)) + (r + σ^2/2)9 ] ) – ( (1/6) ([1 / 3σ ] [ln (([(i c l) (1+g*) ] e^{-9r} ) / (i c l)) + (r + σ^2/2)9 ] )^3 + (1/40) ( (1/6) ([1 / 3σ ] [ln (([(i c l) (1+g*) ] e^{-9r} ) / (i c l)) + (r + σ^2/2) 9] )^5 – (1/336) ( (1/6) ([1 / 3σ ] [ln (([(i c l) (1+g*) ] e^{-9r} ) / (i c l)) + (r + σ^2/2)9 ] )^7 ...]}] -1

### Key technical points
- The derivation explicitly accounts for g* appearing inside N(d1) and N(d2), requiring expansion of the cumulative normal via polynomial terms up to odd powers (d1^3, d1^5, d1^7, ...).
- Time to maturity is set to 9 years throughout the substituted expressions (appearances of 9, e^{-9r}, and factors like [1 / 3σ] and [1 /9σ] stem from the √9 and scaling in d1).
- The final expression for g* remains implicit and nested due to g* appearing inside logarithmic and normal CDF expansions.

*International Monetary Fund — Annex I. Exploring a longer option expiration*

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_Source: https://www.imf.org/-/media/files/publications/wp/2023/english/wpiea2023152-print-pdf.pdf_