Stock Loan In A Jump-Diffusion Model

Stock loan enables the client to get a loan from the bank with mortgage. Theclient can regain the stock with repaying the loan at maturity, or surrender the stock instead of repaying the loan.A stock loan provided many advantages for investor. First of all, it creates liquidity while overcoming the barrier of large block sales, such as triggering tax events or control restrictions on sales of stocks. Second, it serves as a hedge against a market downturn. The client receives the loan at the initial time. If the stock price goes down, the client may simply forfeit the stock instead of repaying the loan. Consequently, the maximum downside risk is capped at the initial price of the stock minus the loan principal. On the other hand, with the above controlled downside risk, the client has unlimited upside potential. If the stock price goes up, the client keeps all the upside by repaying the principal plus interest. Finally, the personal liability with the loan is limited. There is no recourse and no margin requirement with most of the currently available stock loan products in the market. A natural problem arises for both the client and the bank: what are the fair values of the principal, the loan interest, and the fee charged by the bank for providing the service?In this paper we extend results in [3] , written by JM,Xia and XY,Zhou, to a positive exponential jump-diffusion model. In our model, the stock price follows the following stochastic process under the martingale measure where e^x = S₀, W = {W_t,t≥0} is a Wiener process , N = {N_t,t≥0} is a Poisson process with intensityλ> 0, and Y≡{Y_k,k≥1} is a sequence of independent nonnegative random variables , with identified distribution F, F is exponential withparameterη> 0, that is, F - exp{η}.The stock loan in our paper has the following specifications:(1) At time 0, a client borrows amount K (K > 0) from a bank with one share of the stock as collateral,·whereas the bank charges amount c (0≤c≤K) for this service. Then the client gets amount (K - c) from the bank.(2) The continuously compounding loan interest rate isγ> 0. The client may regain the stock by repaying amount Ke^γt to the bank at any time t≥0. The dividends of the stock accrued are collected by the bank until the client regain the stock, and the paid dividends are not credited to the client.(3) The client is not obliged to regain the stock.(4) No deadline is required for repayment of stock loan. The By [5], the value of the underlying stock loan iswhere (?) denotes all (F*_t)_t≥0-stopping times.In this paper, the closed form solution of the above model is given from the following lemmas and theorems.Lemma 1 Suppose g(x) = {e^x-K)₊, X_t = x+σW_t+ (?) + ct, c∈R,{Y_k}_k≥1is independent identical random variables with identical distribution F(y) {W_t}_t≥0 isBrownian motion, {N(t)}_t≥0 is a Poisson process with intensityλ> 0. If there exist x₀∈R and a nonnegative function u(x) such that(1) u(x) is C₂ function defined on R\{x₀}, convex and u"(x₀^-),u"(x₀⁺) existing;(2) ((?)u)(x) -αu(x) =0, (?)_x?x₀; (3) ((?)u){x) -αu(x)≤0, (?)_x≠x₀ ;(4) u(x)≥g(x)≥0, (?)_x∈R;(5) u(X_(?)^*) = g(X_(?)^*), Q-a.s.;(6) (?)t≥0, stopping time (?), there exists a random variableZ, E \ Z \<∞, such where ((?)u)(x) = (?)σ²u"(x) + cu'(x) +λ∫_R[u(x + y) - u{x)]dF(y), c∈R, (?)* = inf {t≥0;X_t≥x₀},where (?), c∈R, (?)*=inf{t≥0;X_t≥x₀},then (u(x), (?)*)is the solution for the optimal stopping problem supE[e^-α(?) g(X_?)], thatisLemma 2 Suppose (?), under the assumption of Lemma 1, {Y_k} follows exponential distribution with parameterη> 0. Then for (?) < 0,η> 1Lemma 3 Let (?) whereμ= (?), (?)< 0,σ,λ> 0,η> 1 are constants. Then h(x) = 0 has two roots on R, and h(x) = 0 have only two roots when (?)=0, and -1 is two-multi-roots.2 \Theorem 1 Under the assumption of Lemma 2, for (?),η> 1, is the solution of the optimal problem supwhereβ₁,β2 are two roots of the following equation Lemma 4 Suppose U_t = (?) = inf{t≤0; U_t < 0},infφ= +∞, M_Y(·) is moment generating function, R is the positive root and satisfyThenTheorem 2 The value of the stock loan considered in this papercan be written in detail...