Valuation of Stock Loans with Regime Switching

Abstract

Abstract. This paper is concerned with stock loan valuation in which the underlying stock price is dictated by geometric Brownian motion with regime switching. The stock loan pricing is quite different from that for standard American options because the associated variational inequalities may have infinitely many solutions. In addition, the optimal stopping time equals infinity with positive probability. Variational inequalities are used to establish values of stock loans and reasonable values of critical parameters such as loan sizes, loan rates and service fees in terms of certain algebraic equations. Numerical examples are included to illustrate the results.Key words. Stock loan, regime switching, variational inequalities, optimal stopping, smooth fit AMS subject classifications. 91B28, 60G35, 34B60Running head: Stock Loan Valuation 1. Introduction. This paper is concerned with valuation of stock loans. A stock loan involves two parties, a borrower and a lender. The borrower owns one share of a stock and obtains a loan from the lender with the share as collateral. The borrower may regain the stock in the future by repaying the lender the principal plus interest or alternatively surrender the stock. Stock loan valuation has been attracting attentions of both academic researchers and lending institutions. When a stock holder prefers not to sell his stock due to either capital gain tax consideration or restrictions on sales of his stock, a stock loan is a viable alternative for raising cash. In addition, the loan can help the stock holder hedge against a market downturn. For example, if the stock price goes down, the borrower may forfeit the stock instead of repaying the loan. On the other hand, if the stock price goes up, the borrower can repay the loan and regain the stock.In Xia and Zhou [16], the stock loan valuation is studied using a pure probabilistic approach with a classical geometric Brownian model. They pointed out that the variational inequality (VI) approach cannot be directly applied to stock loan pricing as in American option pricing because the associated VIs may have infinitely many solutions. In addition, the corresponding optimal stopping time equals infinity with positive probability. Nevertheless, the variational inequality approach is very useful for optimal stopping problems because it is associated with a set of sufficient conditions that are easy to verify. It is natural for models with regime switching, for instance. Moreover, the VI approach typically leads to partial differential equations that can be solved numerically.In this paper, we first use variational inequalities to solve the stock loan pricing problem considered in [16]. We overcome the difficulty of possibly infinitely many solutions to the VIs by pinning down the right solution which is identical to the value function. Then, we carry this approach over to models in which the underlying stock price follows a geometric Brownian motion with regime switching. The model with

A new efficient numerical method for solving American option under regime switching model

“…Furthermore, regime switching models are computationally inexpensive com- 15 pared to stochastic volatility jump diffusion models and have versatile applications in other fields, like electric markets [3], valuation of stock loans [35], forestry valuation [7], natural gas [8] and insurance [17].…”

“…Lattice methods [19,26] are popular for practitioners because they are easy to implement, but they have the drawback of the absence of numerical analysis and subsequent unreliability, because the lack of numerical analysis 35 may waste the best model. The penalty method [18,22,23,34] uses a coupling of the penalty term and the regime coupling terms.…”

“…As implicit method is employed for the numerical solution, the optimal stopping boundary is fully involved in the system, but has not an isolated expression like (33)- (35). The closed system of M + 2 equations (31)- (32) and (36) is solved by using the well know iterative Newton's method for every regime i = 1, .., I.…”

Pricing Multi-Asset Derivatives With Regime-Switching Volatilities

“…This means that each volatility of asset is influenced by a separated independent market uncertainty. We use a single jump regime-switching process which is used in Roh([4]), Zhang and Zhou( [7]). …”

Methods for Pricing American Options under Regime Switching

“…Recent research has shown that models based on stochastic volatility, jump diffusion, and regime switching processes produce better fits to market data. A nonexhaustive list of regime switching applications includes insurance [22], electricity markets [21,40], natural gas [12,2], optimal forestry management [11], trading strategies [15], valuation of stock loans [44], convertible bond pricing [3], and interest rate dynamics [27]. Regime switching models are intuitively appealing, and computationally inexpensive compared to a stochastic volatility jump diffusion model.…”