Stock loans in incomplete markets
aa r X i v : . [ q -f i n . P M ] O c t Stock loans in incomplete markets
Matheus R Grasselli ∗ Cesar G. Velez † September 17, 2018
Abstract
A stock loan is a contract whereby a stockholder uses shares as collateral to borrow money froma bank or financial institution. In Xia and Zhou (2007), this contract is modeled as a perpetualAmerican option with a time varying strike and analyzed in detail within a risk–neutral framework.In this paper, we extend the valuation of such loans to an incomplete market setting, which takesinto account the natural trading restrictions faced by the client. When the maturity of the loan isinfinite, we use a time–homogeneous utility maximization problem to obtain an exact formula for thevalue of the loan fee to be charged by the bank. For loans of finite maturity, we characterize the feeusing variational inequality techniques. In both cases we show analytically how the fee varies withthe model parameters and illustrate the results numerically.
Keywords:
Stock loans, indifference pricing, illiquid assets, incomplete markets.
A stock loan is a contract between two parties: the lender, usually a bank or other financial institutionproviding a loan, and the borrower, represented by a client who owns one share of a stock used as collateralfor the loan. Several reasons might motivate the client to get into such a deal. For example he might notwant to sell his stock or even face selling restrictions, while at the same time being in need of availablefunds to attend to another financial operation.Our main task consists of determining the fair values of the parameters of the loan, particularly thevalue of the fee that the bank charges for the service along with the interest rate to be charge over theamount borrowed, taking into account the stock price at the moment of taking the loan. In addition,we take into account the fact that the bank typically collects any dividends paid by the stock for theduration of the loan. Finally, whereas the client can recover the stock at any time by paying the loanprincipal plus interest, he is not obliged to do so, even if the stock price falls down, and this optionalityalso needs to be accounted in the valuation of the loan.In [6], a stock loan is modeled as a perpetual American option with a time varying strike and analyzedin detail using probabilistic methods within the Black-Scholes framework. Assuming that the risk neutraldynamics of the stock follows a geometric Brownian motion, they obtained explicit formulas for the bank’sfee in terms of the amount lent and the stock price at the moment of signing the loan. Implicit in theiruse of the risk neutral paradigm is the assumption that the option can be replicated by trading in theunderlying stock and the money market. Whereas this is certainly plausible from the bank’s point ofview, we argue that neither type of trade is readily available for the client, who presumably does nothave unrestricted access to the money market (hence the need to post collateral in the form of a stock)nor can freely trade in the stock (otherwise he would simply sell the stock instead of take the loan).Moreover, while risk neutral valuation yields the fair price at which the option itself can be traded inthe market without introducing arbitrage opportunities, a stock loan typically cannot be sold or boughtin a secondary market once it is initiated. In other words, the client does not operate in the frictionlessmarket that is assumed by the Black–Scholes framework.Accordingly, we treat a stock loan as an option in an incomplete market. We assume that the clientcannot trade directly in the underlying stock, but is allowed to trade in a portfolio of assets that is ∗ McMaster University † Universidad Nacional de Colombia
We consider a market consisting of two correlated assets S and V with discounted prices given by dS t = ( µ − r ) S t dt + σ S t dW t dV t = ( µ − r ) V t dt + σ V t ( ρdW t + p − ρ dW t ) , (1)for t ≤ t ≤ T ≤ ∞ , where W = ( W , W ) is a standard two–dimensional Brownian motion. Wesuppose further that the client can trade dynamically by holding H t units of the asset S t and investingthe remaining of his wealth in a bank account with normalized value B t = e r ( t − t ) for a constant interestrate r . It follows that the discounted value of the corresponding self–financing portfolio satisfies dX πt = π t ( µ − r ) dt + π t σ dW t , t ≤ t ≤ T, (2)where π t = H t S t .At a given time t , the client borrows an amount L from a bank leaving the asset with value V t asa collateral. We assume that the bank collects the dividends paid by the underlying asset V at a rate δ for the duration of the loan. In addition, the bank charges the client a fee c and stipulates an interestrate α to be charged on the loan amount L , so that the client can redeem the asset with value e r ( t − t ) V t at time t ≤ t ≤ T by paying an amount e α ( t − t ) L . At the maturity time T , we assume that the clientneeds to decide between repaying the loan or forfeiting the underlying asset indefinitely.In other words, at the beginning of the loan the client gives the bank an asset worth V t and receivesa net amount ( L − c ) plus the option to buy back an asset with market price e r ( t − t ) V t for an amount e α ( t − t ) L . Denoting the cost of this option for the bank by C t , the loan parameters are related by c = L + C t − V t (3) Let us first assume that T = ∞ and that α = r . Given an exponential utility U ( x ) = − e − γx , we considera client trying to maximize the expected utility of discounted wealth. We assume that, upon repayingthe loan at time τ , the borrower adds the discounted payoff ( V τ − L ) to his discounted wealth X πτ andcontinues to invest optimally. Accordingly, having taken the loan at time t , the borrower needs to solvethe following optimization problem: G ( x, v ) = sup ( τ,π ) ∈A E (cid:2) − e ( µ − r )22 σ τ e − γ ( X πτ +( V τ − L ) + ) (cid:12)(cid:12) X πt = x, V t = v (cid:3) . (4)Here A is a set of admissible pairs ( τ, π ), where τ ∈ [0 , ∞ ] is a stopping time and π is a portfolio process.Observe that the factor e ( µ − r )22 σ τ leads to a horizon unbiased optimization problem (see Appendix 1 of[1] for details), while the choice α = r removes the dependence on time from the factor e ( α − r )( t − t ) L atthe repayment date. Combined with the infinite maturity assumption, this allows us to deduce that the2orrower should decide to pay back the loan at the first time that V reaches a stationary threshold V ∗ ,that is τ ∗ = inf { s ≥ t : V s = V ∗ } . (5)We follow [2] and define the indifference value for the option to pay back the loan as the amount p ( v )satisfying G ( x,
0) = G ( x − p ( v ) , v ) . (6)The following proposition summarizes the resulst in [1] regarding the value function G ( x, v ), thethreshold V ∗ and the indifference value p ( v ). Proposition 1 (Henderson, 2007) . The function G ( x, v ) solves the following non-linear HJB equation ( µ − r ) σ G + σ v G vv + ( µ − r ) vG v − ( ρσ σ vG xv + ( µ − r ) G x ) σ G xx = 0 (7) subject to the following boundary, value matching and smooth pasting conditions: G ( x,
0) = − e − γx G ( x, V ∗ ) = − e − γ ( x + V ∗ − L ) G v ( x, V ∗ ) = γe − γ ( x + V ∗ − L ) . Let β = 1 − σ (cid:16) µ − rσ − ρ µ − rσ (cid:17) . If β > , the threshold V ∗ > L is the unique solution to V ∗ − L = 1 γ (1 − ρ ) log (cid:20) γ (1 − ρ ) V ∗ β (cid:21) (8) and the solution to (7) and associated conditions is given by G ( x, v ) = − e − γx (cid:20) − (1 − e − γ ( V ∗ − L )(1 − ρ ) ) (cid:0) vV ∗ (cid:1) β (cid:21) − ρ , if v < V ∗ − e γx e − γ ( v − L ) , if v ≥ V ∗ . (9) In this case, the indifference value p ( ρ,γ ) is given by p ( v ) = − γ (1 − ρ ) log (cid:20) ( e − γ ( V ∗ − L )(1 − ρ ) − (cid:0) vV ∗ (cid:1) β + 1 (cid:21) , if v < V ∗ ( v − L ) , if v ≥ V ∗ . (10) Alternatively, if β ≤ , then the smooth pasting fails and there is no solution to (7) and associatedconditions. In this case, V ∗ = ∞ and the option to repay the loan is never exercised. Let us assume from now on that S is the discounted price of the market portfolio, so that theequilibrium rate of return µ on the asset V satisfies the CAPM condition µ − rσ = ρ µ − rσ . (11)The dividend rate paid by V is then δ = µ − µ , and we have that β = 1 − σ (cid:18) µ − rσ − ρ µ − rσ (cid:19) = 1 + 2 δσ > . (12)Because the bank is well–diversified and can hedge in the financial market by directly trading theasset V , the cost C t = C ( V t ) of granting the repayment option is given by the complete market price ofa perpetual barier–type call option on e r ( t − t ) V t with strike e α ( t − t ) L exercised at the borrower’s optimalexercise boundary obtained in Proposition 1. In other words, denoting by Q the unique risk–neutralmeasure for the complete market consisting of S and V , we have the following result.3 roposition 2. Assuming that the borrower exercises the repayment option optimally according to Propo-sition 1, the cost of this option for the bank is given by C ( v ) = ( ( V ∗ − L ) E Q (cid:2) { τ ∗ < ∞} (cid:3) , if v < V ∗ v − L, if v ≥ V ∗ = ( V ∗ − L ) (cid:16) vV ∗ (cid:17) β if v < V ∗ v − L, if v ≥ V ∗ . (13) Proof.
Observe that the risk–neutral dynamics for V is dV t = − δV t dt + σ V t dW Qt , (14)where W Q is a Brownian motion. Therefore E Q (cid:2) { τ ∗ < ∞} (cid:3) corresponds to the risk–neutral probabilitythat the geometric Brownian motion V t started at V t = v will cross the barrier V ∗ in a finite time. Theresult then follows from standard Laplace transform techniques.We can now use (3) and (13) to establish that c = L + C t − V t = L + ( V ∗ − L ) (cid:18) V t V ∗ (cid:19) β − V t , if V t < V ∗ , if V t ≥ V ∗ . (15)As shown in Proposition 3.5 of [1], one can find by direct differentiation of expressions (8) and (10)that both the threshold V ∗ and the indifference value p ( v ) are increasing in ρ . In other words, all thingsbeing equal, a higher degree of market incompleteness, expressed as a smaller absolute value for thecorrelation between the stock and the market portfolio, lead the client to exercise the option to repaythe loan earlier than in the complete market case, resulting in a smaller indifference value for the option.Similarly both V ∗ and p ( v ) are decreasing in γ , meaning that a higher degree of risk aversion has a similareffect in decreasing the value of the option to repay the loan. These properties carry over to the loan feeobtained in expression (15), as we establish in the next proposition. Proposition 3.
The loan fee:1. decreases as the risk aversion γ increases;2. decreases as the dividend rate δ increases;3. increases as ρ increases.Moreover, its limiting values either as ρ → or γ → coincide and are given by c = L + ( e V − L ) (cid:18) V t e V (cid:19) β − V t , if V t < V ∗ , if V t ≥ V ∗ . (16) where e V = ββ − L = (cid:16) σ δ (cid:17) L .Proof. The first part of the proposition follows by explicit differentiation of expressions (15), (8) and(12). For the second part, observe that it follows from our equilibrium condition (11) that both limitingthresholds in Proposition 3.5 of [1] are given by e V = ββ − L . Substituting expression (12) for β thencompletes the proof.We conclude this section by observing that the limiting threshold e V corresponds to the completemarket threshold a found in [6] using risk-neutral valuation arguments. Consequently, provided α = r ,the complete market, risk–neutral setting can be recovered as a special case of our results.4 Finite maturity
Consider
T < ∞ and define the value function (see [4]) M ( t, x ) = sup π ∈A [ t,T ] E [ − e − γX πT | X πt = x ] = − e − γx e − ( µ − r )22 σ ( T − t ) , (17)for t ≤ t ≤ T , where X πt follows the dynamics (2) and A [ t,T ] is the set of admissible investment policieson the interval [ t, T ], which we take to be progressively measurable processes satisfying the integrabilitycondition E "Z Tt π s ds < ∞ . As before, let the repayment time by a stopping time τ and assume that the borrower will add thediscounted payoff ( V τ − e ( α − r )( τ − t ) L ) + to his discounted wealth X πτ at time τ and then invest optimallyuntil time T . Accordingly, having taken the stock loan at time t , the borrower consider the followingoptimization problem: u ( t , x, v ) = sup τ ∈T [ t ,T ] sup π ∈A [ t,τ ] E [ M ( τ, X πτ + ( V τ − e ( α − r )( τ − t ) L ) + ) | X πt = x, V t = v ] , (18)where T [ t , T ] denotes the set of stopping times in the interval [ t , T ]. The indifference value for therepayment option is then given by the amount p satisfying M ( t , x ) = u ( t , x − p, v ) . (19)It follows from the dynamic programming principle that the value function u solves the free boundaryproblem ∂u∂t + sup π L π u ≤ ,u ( t, x, v ) ≥ Λ( t, x, v ) , (cid:18) ∂u∂t + sup π L π u (cid:19) · ( u − Λ) = 0 , (20)for ( t, x, v ) ∈ [ t , T ) × R × (0 , ∞ ), where L π = ( µ − r ) v ∂∂v + σ v ∂ ∂v + π ( µ − r ) ∂∂x + ρπσ σ v ∂ ∂x∂v + π σ ∂ ∂x is the infinitesimal generator of ( X π , V ) andΛ( t, x, v ) = M ( t, x + ( v − e ( α − r )( t − t ) L ) + )is the utility obtained from exercising the repayment option at time t . The boundary conditions forProblem (20) are u ( T, x, v ) = − e − γ [ x +( v − e ( α − r )( T − t L ) + ] u ( t, x,
0) = − e − γx e − ( µ − r )22 σ ( T − t ) . (21)Using the factorization u ( t, x, v ) = M ( t, x ) F ( t, v ) − ρ , (22)we find that the corresponding free boundary problem for F becomes ∂F∂t + L F ≥ ,F ( t, v ) ≤ κ ( t, v ) , (cid:18) ∂F∂t + L F (cid:19) · ( F − κ ) = 0 , (23)5or ( t, v ) ∈ [ t , T ) × (0 , ∞ ), where L = (cid:20) µ − r − ρ µ − rσ σ (cid:21) v ∂∂v + σ v ∂ ∂v (24)and κ ( t, v ) = e − γ (1 − ρ )( v − e ( α − r )( t − t L ) + . (25)The boundary conditions for Problem (23) are F ( T, v ) = e − γ (1 − ρ )( v − e ( α − r )( T − t L ) + F ( t,
0) = 1 . (26)Since Problem (23) is independent of X and S , we define the borrower’s optimal exercise boundaryas the function V ∗ ( t ) = inf { v ≥ F ( t, v ) = κ ( t, v ) } (27)and the optimal repayment time as τ ∗ = inf { t ≤ t ≤ T : V t = V ⋆ ( t ) } . (28)It follows from the definition (19) and the factorization (22) that the indifference value for the repay-ment option is given by p = p ( t , V t ) where p ( t, v ) = − γ (1 − ρ ) log F ( t, v ) . (29)Therefore, the original free boundary problem can be rewritten in terms of the indifference value as ∂p∂t + L p − γ (1 − ρ ) σ v (cid:18) ∂p∂v (cid:19) ≤ ,p ( t, v ) ≥ (cid:16) v − e ( α − r )( t − t ) L (cid:17) + , " ∂p∂t + L p − γ (1 − ρ ) σ v (cid:18) ∂p∂v (cid:19) · ( p − ( v − e ( α − r )( t − t ) L ) + ) = 0 , (30)Similarly, the optimal exercise time τ ∗ can be expressed in terms of p as follows: τ ∗ = inf n t ≤ t ≤ T : p ( t, V t ) = ( V t − e ( α − r )( t − t ) L ) + o (31)Once we find the optimal exercise boundary V ∗ ( t ), say by solving problem (23) numerically, we cancalculate the bank’s cost of granting the repayment option as the risk–neutral value of a barier–type calloption on e r ( t − t ) V t with strike e α ( t − t ) L and maturity T exercised at the barrier e r ( t − t ) V ∗ ( t ). In otherwords C t = C ( t , v ) = E Q (cid:20) e − r ( τ − t ) (cid:16) e r ( τ − t ) V ∗ ( t ) − e α ( τ − t ) L (cid:17) + { τ ∗ < ∞} (cid:12)(cid:12)(cid:12)(cid:12) V t = v (cid:21) (32)= E Q (cid:20) e − b r ( τ − t ) (cid:16) e ( r − α )( τ − t ) V ∗ ( t ) − L (cid:17) + { τ ∗ < ∞} (cid:12)(cid:12)(cid:12)(cid:12) V t = v (cid:21) (33)= E Q (cid:20) e − b r ( τ − t ) (cid:16) b V ∗ ( t ) − L (cid:17) + { τ ∗ < ∞} (cid:12)(cid:12)(cid:12)(cid:12) V t = v (cid:21) (34)where b r = r − α and b V ( t ) = e b r ( τ − t ) V ∗ ( t ). Denoting b V t = e ( r − α )( τ − t ) V t , it is trivial to see that τ ∗ defined in (28) can be written as τ ∗ = inf { t ≤ t ≤ T : V t = V ⋆ ( t ) } = inf n t ≤ t ≤ T : b V t = b V ⋆ ( t ) o (35)6herefore, since the risk–neutral dynamics for the process b V t is d b V t = ( b r − δ ) V t dt + σ V t dW Qt , (36)we have that the function C ( t, v ) satisfies the Black–Scholes PDE ∂C∂t + ( r − α − δ ) v ∂C∂v + σ v ∂ C∂v = ( r − α ) C (37)over the domain D = { ( t, v ) : t ≤ t ≤ T, ≤ v ≤ V ∗ ( t ) } , subject to the boundary conditions C ( t,
0) = 0 , t ≤ t ≤ T,C ( t, b V ∗ ( t )) = ( b V ∗ ( t ) − L ) + , t ≤ t ≤ T,C ( T, v ) = ( v − L ) + , ≤ v ≤ b V ∗ ( T ) (38)As before, once we calculate the cost C t , the fee to be charged for the loan is given by (3). It is easyto see that the cost C t , and consequently the fee c , increase if the optimal exercise boundary V ∗ ( t ) isshifted upward and decrease otherwise. In this section we investigate how the loan fee to be charged by the bank depends on the underlyingparameters. We will always assume that the interest rate r , the expected return µ and volatility σ forthe market portfolio S , the loan interest rate α , and loan amount L are fixed. On the other hand, wetreat the risk aversion γ , the dividend rate δ , the correlation ρ , and the underlying asset volatility σ asvariable parameters. We then perform comparative statics, that is, we change each of these parameterswhile keeping the others constant and analyze the corresponding behavior of the loan fee.Observe that for each choice of values for δ , σ and ρ the expected return µ is automatically deter-mined by the assumption that asset prices are in equilibrium. For simplicity, we continue to assume that S is the discounted price of the market portfolio, so that the CAPM condition (11) holds and we havethat µ = ρ µ − rσ σ + r − δ. (39)The behavior of the loan fee with respect to the underlying parameters is established in the nextproposition, which we prove using the same technique as in [3], but adapted the problem at hand. Proposition 4.
The loan fee c :1. decreases as the risk aversion γ increases;2. decreases as the dividend rate δ increases;3. increases as ρ increases;Proof. Observe first that for fixed values of
L, α and r , it follows from (31) that a smaller indifferencevalue leads to a smaller optimal exercise time, which in turns implies a lower optimal exercise boundaryand consequently a smaller loan fee. To establish how the indifference value changes with the underlyingparameters, we use the comparison principle for the variational inequalitymin ( − ∂p∂t − L p + 12 γ (1 − ρ ) σ v (cid:18) ∂p∂v (cid:19) , p ( t, v ) − (cid:16) v − e ( α − r )( t − t ) L (cid:17) + ) = 0 (40)which is known to be equivalent to (30).For item (1), observe that the variational inequality (40) depends on γ only through the nonlinearterm 12 γ (1 − ρ ) σ v (cid:18) ∂p∂v (cid:19) . (41)7ince this is increasing in γ , it follows that p is decreasing in γ .For item (2), observe first that ∂p∂v ≥
0, because u ( , t, x, v ) defined in (18) (and consequently p ( t, v )) isan increasing function v . Next, recalling the definition of L in (24), we see that the variational inequality(40) depends on δ through the term − (cid:20) µ − r − ρ µ − rσ σ (cid:21) ∂p∂v = δ ∂p∂v , on account of (39). Since this is increasing is δ , we have that p is decreasing in δ .Similarly for item (3), using (24) we see that the variational inequality (40) depends on ρ through theterm − (cid:20) µ − r − ρ µ − rσ σ (cid:21) ∂p∂v + 12 γ (1 − ρ ) σ v (cid:18) ∂p∂v (cid:19) . By virtue of (39), we then see that the dependence on ρ reduces to the nonlinear term (41). Thereforethe indifference price is a symmetric function of ρ , and increases as ρ increases from 0 to 1.Notice that the variational inequality (40) depends on σ through the term − σ v ∂ p∂v + 12 γ (1 − ρ ) σ v (cid:18) ∂p∂v (cid:19) . (42)Since this is not necessarily monotone in σ , we cannot expect the indifference value, and consequentlythe loan fee c , to be monotone function of the underlying stock volatility.Regarding the behavior of the loan fee with respect to the maturity length of the loan, one intuitivelyexpects that a longer maturity increases the optionality of the repayment and should contribute to higherfee. As establish in the next proposition, this is indeed the case provided we can ignore the effects ofinterest rates. Proposition 5. If α = r , the loan fee is an increasing function of the maturity T .Proof. The solution to problem (23) admits a probabilistic representation (see [5]) of the form F ( t, v ) = inf τ ∈T [ t,T ] E [ κ ( τ, V τ ) | V t = v ] , where E [ · ] denotes the expectation operator under the minimal martingale measure Q defined by dQ dP = e − µ − rσ W T −
12 ( µ − r )2 σ T . (43)When α = r , we can use the fact that V t is a time–homogeneous diffusion to obtain that F ( t, v ) = inf τ ∈T [ t,T ] E [ e − γ (1 − ρ )( V τ − L ) + | V t = v ]= inf τ ∈T t ,T − t + t E [ e − γ (1 − ρ )( V τ − L ) + | V t = v ] . For any s ≤ t we have that T [ t , T − t + t ] ⊂ T [ t , T − s + t ], so F ( s, v ) ≤ F ( t, v ). Now fix v > s, v ), that is, F ( s, v ) = k ( s, v ). Using the fact that F isincreasing in time (as we just established), we have that e − γ (1 − ρ )( v − L ) + = k ( s, v ) = F ( s, v ) ≤ F ( t, v ) ≤ k ( t, v ) = e − γ (1 − ρ )( v − L ) + , so that F ( t, v ) = k ( t, v ), which implies that it is also optimal to exercise at ( t, v ). This means that, foreach fixed T , the optimal exercise boundary V ∗ ( t ) is a decreasing function of time, and consequently anincreasing function of the time–to–maturity parameter ( T − t ). Therefore, as we modify the problem byincreasing the maturity T , the optimal exercise boundary shits upwards, leading to a higher cost for thebank and a higher loan fee. 8able 1: Loan fee c as for different loan amounts L (infinite maturity) L
50 60 70 80 90 100 110 120Case 1 c 50 60 70 80 90 100 110 120Case 2 c 31.0528 39.5086 48.1242 56.8653 65.7084 74.6363 83.6361 92.6978 V ∗ c a V ∗ The only numerical step involved in this case consists of finding the value of the threshold V ∗ by solvingthe nonlinear equation (8) for given parameter values. We can then use expression (15) to find the loanfee c . For comparison, we calculate the corresponding loan fee in a complete market scenario using theformulas found in [6]. Observe that we always need to use r = α as explained in Section 3 in order tomaintain time-homogeneity.We start by calculating the value of the loan fee c for a range of loan amounts L and four differentsets of model parameters. The results are summarized in Table 1 and correspond to the following cases:1. Complete market with σ = 0 . , δ = 0 , r = 0 . , α = 0 .
05 and V t = 100. This corresponds to case(a) of Theorem 3.1 in [6], that is, δ = 0 and α − r = 0 < σ /
2. Because the stock pays no dividendand the excess interest rate on the loan is small, it follows that the option to repay the loan has thesame value as the stock itself (that is C t = V t ), which leads to c = L . In other words, the bankhas no incentive to provide the loan and charges a fee exactly equal to the loan amount. In effect,the client gives away the stock and receives a perpetual American option with an infinite exercisethreshold.2. Incomplete market with σ = 0 . , δ = 0 , r = 0 . , α = 0 . , V t = 100 , ρ = 0 . γ = 0 . V ∗ even when the stock pays no dividend and theexcess interest rate on the loan is small. In other words, the option to repay the loan is exercisedsooner, and consequently has a smaller value for the client, than in the complete market case. As aconsequence, the bank has a smaller cost for providing the loan and can charge a reduced fee c < L .3. Complete market with σ = 0 . , δ = 0 . , r = 0 . , α = 0 .
05 and V t = 100. This correspondsto case (b) of Theorem 3.1 in [6], since δ >
0. To calculate c , we first find the exercise threshold a as in page 314 of [6] for each value of L . If a ≤ V t = 100 (which happens for low enough L )then c = 0, meaning that the client receives L in exchange of V t at no cost, and then immediatelyexercises the option to repay. In other words, there is no incentive for the client to seek the loan.On the other hand, if a > V t , we calculate the fee c using the formula at the end of page 316 of[6].4. Incomplete market with σ = 0 . , δ = 0 . , r = 0 . , α = 0 . , V t = 100 , ρ = 0 . γ = 0 . V ∗ < a ,leading to a smaller fee c charged by the bank.Next in Figure 1 we illustrate the dependence of the loan fee upon the other model parameters. Inparticular, each curve on the top left plot represents c as a function of γ for a particular value of thecorrelation ρ . Similarly, each curve on the top right plot the loan fee value c as a function of ρ for aparticular value of the risk aversion γ . Finally the curve on the bottom plot represents the loan fee value c as a function of the dividend rate δ . When not explicitly shown in the figure, the values of the remainingparameters are σ =0.15, δ =0.05, r = α = 0 . L =90, V = 100, ρ = 0 . γ = 0 . Risk aversion γ Loan f ee ρ =0.92 ρ =0.9 ρ =0.88 −1 −0.5 0 0.5 11.821.831.841.851.861.871.881.891.91.911.92 Correlation ρ Loan f ee γ =0.008 γ =0.01 γ =0.0120 0.05 0.1 0.15 0.2010203040506070 Dividend rate δ Loan ´ s f ee c Figure 1: Dependence on model parameters for infinite maturity10able 2: Loan fee c for different loan amounts L (finite maturity) L
50 60 70 80 90 100 110 120 c c is decreasing in γ and δ and increasing in ρ . Moreover, the limits as γ → ρ → ± L = 90, namely c = 1 . δ →
0, but convergesto the value c = 65 . <
90 as obtained in Case 2 of Table 1 for L = 90. As mentioned in Section 4.1, the numerical procedures in this case are slightly more involved. First weuse finite differences with projected successive–over–relaxation (PSOR) to solve the linear free boundaryproblem (23). This yields a threshold function V ∗ ( t ), which we then use to solve equation (37) subjectto the boundary conditions (38), again by finite differences.To start with, Table 2 shows the loan fee c for different loan amounts L , with the following parametervalues: σ = 0 . , ρ = 0 . , γ = 0 . , δ = 0 . , r = 0 . , α = 0 . , V t = 100 and T = 5 (in years). Observethat we do not need to restrict ourselves to the case r = α as we did before, since the time–homogeneityproperty is not used in the finite–maturity case.Next in Figure 2, we illustrate in detail the dependence upon the model parameters analyzed inPropositions 4 and 5. We use T = 5, L = 80, σ = 0 . r = 0 . α = 0 . δ = 0 .
05, and ρ = 0 . V ∗ ( t )for V = 100 , whereas each curve on the right side represent the loan fee c as function of V , for theparticular set of parameter values described below:1. For the top row, we use γ = 0 . , . , .
08 and find that both the optimal exercise boundary andthe loan fee decrease as risk aversion increases, in agreement with item 1 of Proposition 4.2. For the second row, we use δ = 0 . , . , .
15 and find that both the optimal exercise boundary andthe loan fee decrease as the dividend rate increases, in agreement with item 2 of Proposition 4.3. For the third row, we use ρ = 0 . , . , . α = r = 0 .
05 and find that the optimal exercise boundary is strictlydecreasing with respect to time-to-maturity ( T − t ), in agreement with Proposition 5 In this paper we have extended the analysis of [6] for stock loans in incomplete markets. This allows usto consider the realistic situation when the borrower faces trading restrictions and cannot use replicationarguments to find the unique arbitrage–free value for the repayment option embedded in such loans. Weshowed how an explicit expression for the loan fee can still be found in the infinite–horizon case providedthe loan interest rate is set to be equal to the risk–free rate. In the finite–horizon case we characterize theloan fee in terms of a free–boundary problem and show how to calculate it numerically. In both cases,we analyzed how the loan fee depends on the underlying model parameters.Based on the dependence on correlation and risk–aversion, we find that the complete–market, risk–neutral valuation of a stock loan provides an upper bound for the fee to be charged by the bank. Thisshows that by following our model a bank can quantify the effects of the restrictions faced by the clientthereby charging a smaller fee for the loan, presumably increasing its competitiveness.11 t V * ( t ) γ =0.01 γ =0.05 γ =0.08 0 20 40 60 80 100 120 14001020304050607080 V c ( V ) γ =0.01 γ =0.05 γ =0.080 1 2 3 4 59095100105110115120125130135 t V * ( t ) δ =0.05 δ =0.1 δ =0.15 60 70 80 90 100 110 120 130 1400510152025303540 V c ( V ) δ =0.05 δ =0.1 δ =0.150 1 2 3 4 59092949698100102 t V * ( t ) ρ =0.9 ρ =0.4 ρ =0.0 93 94 95 96 97 98 99 100 101 102123456 V c ( V ) ρ =0.9 ρ =0.4 ρ =0.00 1 2 3 4 59092949698100102 t V * ( t ) V c ( V ) Figure 2: Dependence on model parameters for finite maturity12 eferences [1] V. Henderson. Valuing the option to invest in an incomplete market.
Math. Financ. Econ. , 1(2):103–128, 2007.[2] S. D. Hodges and A. Neuberger. Optimal replication of contingent claims under transaction costs.
Rev. Fut. Markets , 8:222–239, 1989.[3] T. S. Leung and R. Sircar. Accounting for Risk Aversion, Vesting, Job Termination Risk and MultipleExercises in Valuation of Employee Stock Options.
Mathematical Finance , 19(1):99–128, January2009.[4] R. C. Merton. Lifetime portfolio selection under uncertainty: the continuous–time model.
Rev.Econom. Statist. , 51:247–257, 1969.[5] A. Oberman and T. Zariphopoulou. Pricing early exercise contracts in incomplete markets.
Compu-tational Management Science , 1(1):75–107, December 2003.[6] J. Xia and X. Y. Zhou. Stock Loans.