Paper: arXiv 2407.14728

Authors: Minh-Quan Nguyen, Nhat-Tan Le, Khuong Nguyen-An, Duc-Thi Luu

Abstract

This paper examines the pricing issue of margin-call stock loans with finite maturities under the Black-Scholes-Merton framework. In particular, using a Fourier Sine transform method, we reduce the partial differential equation governing the price of a margin-call stock loan into an ordinary differential equation, the solution of which can be easily found (in the Fourier Sine space) and analytically inverted into the original space. As a result, we obtain an integral representation of the value of the stock loan in terms of the unknown optimal exit prices, which are, in turn, governed by a Volterra integral equation. We thus can break the pricing problem of margin-call stock loans into two steps: 1) finding the optimal exit prices by solving numerically the governing Volterra integral equation and 2) calculating the values of margin-call stock loans based on the obtained optimal exit prices. By validating and comparing with other available numerical methods, we show that our proposed numerical scheme offers a reliable and efficient way to calculate the service fee of a margin-call stock loan contract, track the contract value over time, and compute the level of stock price above which it is optimal to exit the contract. The effects of the margin-call feature on the loan contract are also examined and quantified.

Complexity vs Empirical Score

  • Math Complexity: 8.0/10
  • Empirical Rigor: 3.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous analytical framework using Fourier Sine Transforms and Volterra integral equations to price finite-maturity margin-call stock loans under Black-Scholes-Merton. While mathematically sound and clear in its derivation, the work is primarily theoretical with validation limited to synthetic numerical comparisons rather than empirical market data backtesting.

Research Flowchart

  flowchart TD
    A[Goal: Value Finite-Maturity Margin-Call Stock Loans] --> B[BSM Model & Contract Inputs: Interest, Dividends, Volatility, LTV]
    B --> C[Apply Fourier Sine Transform to Governing PDE]
    C --> D[Solve ODE & Invert to Derive Volterra Integral Equation]
    D --> E[Step 1: Numerically Solve Integral Equation for Optimal Exit Boundary]
    E --> F[Step 2: Calculate Stock Loan Value & Service Fees]
    F --> G[Outcomes: Validated Efficient Scheme & Quantified Margin-Call Impact]

Cited References (182)

  1. On stock-based loans (2022) — link
  2. Stock loan valuation under a stochastic interest rate model (2015) — link
  3. Controlling shareholder share pledging and the cost of equity capital: Evidence from China (2022) — link
  4. Stock pledge, risk of losing control and corporate innovation (2020) — link
  5. Performance and risk of energy industrial firms with stock pledge in China (2022) — link
  6. Stock pledge restrictions and investment efficiency (2022) — link
  7. Share pledges and margin call pressure (2018) — link
  8. Shareholder Wealth Consequences of Insider Pledging of Company Stock as Collateral for Personal Loans (2019)
and 174 more
  1. Options, futures, and other derivatives (2016)
  2. The Pricing of Options and Corporate Liabilities (1973)
  3. Collateral equilibrium. I: A basic framework (2014) — link
  4. Collateral Unchained: Rehypothecation networks, concentration and systemic effects (2021) — link
  5. Asset scarcity and collateral rehypothecation (2022) — link
  6. Securitized banking and the run on repo (2012) — link
  7. Securities market theory: Possession, repo and rehypothecation (2012) — link
  8. Procyclicality, Collateral Values and Financial Stability (2006) — link
  9. A new integral equation formulation for American put options (2018)
  10. An integral equation approach for the valuation of American-style down-and-out calls with rebates (2016)
  11. The Analytic Valuation of American Options (1990)
  12. PENALTY AMERICAN OPTIONS (2019)
  13. Price dynamics and error correction in stock index and stock index futures markets: A cointegration approach (1993)
  14. Intraday price discovery and volatility transmission in stock index and stock index futures markets: Evidence from China (2011)
  15. Do individual index futures investors destabilize the underlying spot market? (2011)
  16. Do futures lead price discovery in electronic foreign exchange markets? (2009)
  17. Tick sizes and relative rates of price discovery in stock, futures, and options markets: Evidence from the Taiwan stock exchange (2009)
  18. An Empirical Investigation of the Lead-Lag Relations of Returns and Volatilities among the KOSPI200 Spot, Futures and Options Markets and their Explanations (2006)
  19. Intraday return dynamics between the cash and the futures markets in Japan (1996)
  20. Price discovery in the hang seng index markets: Index, futures, and the tracker fund (2004)
  21. Lead-lag relationship between spot index and futures price of the nikkei stock average (1995)
  22. The Lead-lag Relationship Between Stock Indices and Stock Index Futures Contracts: Further Australian Evidence (1999)
  23. A trading strategy based on the lead-��lag relationship between the spot index and futures contract for the FTSE 100 (2001)
  24. Return and volatility dynamics in the FTSE 100 stock index and stock index futures markets (1995)
  25. A Further Analysis of the Lead-��Lag Relationship Between the Cash Market and Stock Index Futures Market (1992)
  26. The Dynamics of Stock Index and Stock Index Futures Returns (1990)
  27. Price discovery in spot and futures markets: a reconsideration (2012)
  28. Price discovery and volatility spillovers in index futures markets: Some evidence from Mexico (2004)
  29. The Lead-Lag Relationship Between Cash and Stock Index Futures in a New Market (2008)
  30. The Temporal Price Relationship Between S & P500 Futures and the S & P500 Index (1987)
  31. Problems and Solutions in Mathematical Finance: Stochastic Calculus (2014)
  32. Theory of Rational Option Pricing (1973)
  33. Stock loans (2007)
  34. Capped stock loans (2010) — link
  35. Valuation of Stock Loans with Regime Switching (2009)
  36. Valuation of stock loans using exponential phase-type Levy models (2013)
  37. Stochastic volatility asymptotics of stock loans: Valuation and optimal stopping (2012)
  38. Valuation of stock loans with jump risk (2014) — link
  39. Valuation of stock loan under uncertain stock model with floating interest rate (2020) — link
  40. On buybacks, dilutions, dividends, and the pricing of stock-based claims (2022) — link
  41. Real options approach for fashionable and perishable products using stock loan with regime switching (2017) — link
  42. Stock loan with automatic termination clause, cap and margin (2010) — link
  43. Finite maturity margin call stock loans (2016) — link
  44. Stochastic calculus for finance II: Continuous-time models (2004)
  45. A decomposition approach via Fourier sine transform for valuing American knock-out options with rebates (2017) — link
  46. Semi-analytic valuation of stock loans with finite maturity (2015)
  47. Stock loans in incomplete markets (2013)
  48. Stock loan valuation under a regime-switching model with mean-reverting and finite maturity (2010)
  49. Optimal redeeming strategy of stock loans with finite maturity (2011)
  50. An integral equation approach for the valuation of American-style down-and-out calls with rebates (2016) — link
  51. Partial differential equations (2013)
  52. Parisian Option Pricing: A Recursive Solution for the Density of the Parisian Stopping Time (2013) — link
  53. Perturbed Brownian motion and its application to Parisian option pricing (2010) — link
  54. Brownian Excursions and Parisian Barrier Options (1997) — link
  55. Pricing P arisian options (1998)
  56. M athematics of D erivarive S ecurities (1997)
  57. Brownian excursion and P arisian barrier options (1997)
  58. Pricing algorithms for options with exotic path-dependence (2001)
  59. Edokko options: A New Framework of Barrier Options (2003)
  60. A new procedure for pricing P arisian options (2005)
  61. M athematical M odels of F inancial D erivatives, S econd E dition (2008)
  62. P artial D ifferential E quations (1997)
  63. D ifferential E quations: L inear, N onlinear, O rdinary, P artial (2003)
  64. G reen’s F unctions with A pplications (2001)
  65. Numerical Integration (1998)
  66. The F eynman- K ac formula and pricing occupation times derivatives (1999)
  67. Approximate inversion of the Laplace transform (1994)
  68. An algorithm for the numerical inversion of L aplace transforms (1989)
  69. Pricing P arisian options by L aplace inversion (2002)
  70. Brownian excusions and P arisian barrier options: a note (2003)
  71. F inite D ifference S chemes and P artial D ifferential E quations (1989)
  72. P ricing F inancial I nstruments, T he F inite D ifference M ethod (2000)
  73. I ntroduction to N umerical A nalysis (1956)
  74. Laplace transform tables and theorems (1965)
  75. Semigroups of L inear O perators and A pplications to P artial D ifferential E quations (1983)
  76. S pectral and H igh-order M ethods (2006)
  77. Partial D ifferential E quations T heorey and T echnique (1976)
  78. A F eynman- K ac formula for unbounded semigroups (2000)
  79. Using L aplace transform to price A merican puts (2012)
  80. Pricing double Parisian Options using Laplace transforms (2009)
  81. Convergence of Gauss-Laguerre quadrature formulae (2012)
  82. Error estimates for Gauss quadrature formulas for analytic functions (1968)
  83. Pricing P arisian down-and-in options (2014)
  84. The heat equation and reflected Brownian motion in time-dependent domains. II. Singularities of solutions (2003)
  85. The heat equation and reflected Brownian motion in time-dependent domains (2004)
  86. The heat equation in time dependent domains with insulated boundaries (2004)
  87. Pricing Parisian and Parasian options analytically (2013) — link
  88. Pricing Parisian down-and-in options (2015)
  89. The valuation of A merican barrier options using the decomposition technique (2000)
  90. Fast and accurate valuation of A merican barrier options (2003)
  91. A discrete time approach for E uropean and A merican barrier options (1995)
  92. The pricing of options and corporate liabilities (1973)
  93. Theory of rational option pricing (1973)
  94. The mathematical foundations of barrier option-pricing theory (1994)
  95. Breaking down the barriers (1991)
  96. Crossing barriers (1994)
  97. Partial barrier options (1994)
  98. Discrete partial barrier options with a moving barrier (1996)
  99. Two extensions to barrier option valuation (1995)
  100. Pricing Options With Curved Boundaries1 (1992)
  101. PRICING AND HEDGING DOUBLE-BARRIER OPTIONS: A PROBABILISTIC APPROACH (1996)
  102. Pricing and hedging more general double-barrier options (2002)
  103. The A merican put option valued analytically (1984)
  104. Efficient analytic approximation of A merican option values (1987)
  105. Analytic approximation for the A merican put option (1986)
  106. Randomization and the A merican put (1998)
  107. Alternative characterizations of A merican put options (1992)
  108. A free-boundary problem for the heat-equation arising from a problem of mathematical economics (1965)
  109. Heat Conduction (2012)
  110. The mathematics of financial derivatives: a student introduction (1995)
  111. A Survey of the Integral Representation of A merican Option Prices (2004)
  112. Evaluation of American option prices in a path integral framework using F ourier– H ermite series expansions (1999)
  113. Optimal stopping and the A merican put (1991)
  114. The analytic valuation of A merican options (1990)
  115. A new analytical approximation formula for the optimal exercise boundary of A merican put options (2006)
  116. The valuation of American put options (1977)
  117. The valuation of warrants: implementing a new approach (1977)
  118. A front-fixing finite difference method for the valuation of American options (1997)
  119. A new predictor–corrector scheme for valuing American puts (2011)
  120. Option pricing: A simplified approach (1979)
  121. A moving boundary approach to A merican option pricing (2008)
  122. Convergence and biases of Monte Carlo estimates of A merican option prices using a parametric exercise rule (2003)
  123. Valuing A merican options by simulation: A simple least-squares approach (2001)
  124. An exact and explicit solution for the valuation of A merican put options (2006)
  125. Calculating the early exercise boundary of A merican put options with an approximation formula (2007)
  126. A comparative study on time-efficient methods to price compound options in the H eston model (2014)
  127. Static H edging and P ricing A merican K nock- O ut O ptions (2013)
  128. Pricing P arisian and P arasian options analytically (2013)
  129. PDE methods for pricing barrier options (2000)
  130. Knock-in A merican options (2004)
  131. American P arisian options (2006)
  132. Static hedging and pricing A merican knock-in put options (2010)
  133. Pricing and hedging of A merican knock-in options (2004)
  134. The adaptive mesh model: a new approach to efficient option pricing (1999)
  135. A new procedure for pricing P arisian options (2005)
  136. Edokko options: A new framework of barrier options (2002)
  137. Monte C arlo methods in financial engineering (2004)
  138. A Hybrid Methodology for Valuing A merican P arisian O ptions (2009)
  139. Paul W ilmott on Q uantitative F inance, 3 Volume Set (2007)
  140. Paul W ilmott introduces Q uantitative F inance (2007)
  141. Quantitative Finance: A Simulation-Based Introduction Using E xcel (2014)
  142. An introduction to financial option valuation: mathematics, stochastics and computation (2004)
  143. Transforms and applications handbook (2009)
  144. Valuation of non-recourse stock loan using an integral equation approach (2020)
  145. A decomposition approach via Fourier sine transform for valuing American knock-out options with rebates (2017)
  146. Continuously Monitored Barrier Options Under M arkov Processes (2013)
  147. On pricing barrier options (1995)
  148. Pricing and hedging American options using approximations by Kim integral equations (2003)
  149. A new approach for solving nonlinear equations systems (2008)
  150. Methods for solving systems of nonlinear equations (1974)
  151. Option Pricing: Mathematical Models and Computation (1993)
  152. Convexity of the exercise boundary of the American put option on a zero dividend asset (2008)
  153. Boundary Evolution Equations For American Options (2014)
  154. Alternative characterizations of A merican put options (1992)
  155. Options, futures and other derivatives (2009)
  156. The concepts and practice of mathematical finance (2003)
  157. Real analysis: measure theory, integration, and Hilbert spaces (2009)
  158. Lebesgue integration on Euclidean space (2001)
  159. Stochastic calculus for finance (2012)
  160. Numerical methods for engineers (2010)
  161. An introduction to Laplace transforms and Fourier series (2001)
  162. Handbook of computational methods for integration (2004)
  163. Laplace transforms (1965)
  164. On Pricing Barrier Options (1995)
  165. Bumping up against the barrier with the binomial method (1994)
  166. Complex barrier options (1996)
  167. Pricing lookback and barrier options under the CEV process (1999)
  168. Derivative Securities and Difference Methods (2013)
  169. Adaptive and high-order methods for valuing A merican options (2011)
  170. Quadratic convergence for valuing A merican options using a penalty method (2002)
  171. Mathematical models of financial derivatives (2008)
  172. Handbook of Computational Methods for Integration (2014)
  173. American -style derivatives: Valuation and computation (2010)
  174. A decomposition approach via Fourier sine transform for valuing American knock-out options with rebates (2017)

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