DOI 10.1007/s11147-007-9007-5
Valuation of vulnerable American options
with correlated credit risk
Lung-Fu Chang · Mao-Wei Hung
Published online: 14 August 2007
© Springer Science+Business Media, LLC 2007
Abstract This article evaluates vulnerable American options based on the two-point Geske and Johnson method. In accordance with the Martingale approach, we provide analytical pricing formulas for European and multi-exercisable options under risk-neutral measures. Employing Richardson’s extrapolation gets the values of vulnerable American options. To demonstrate the accuracy of our proposed method, we use numerical examples to compare the values of vulnerable American options from our proposed method with the benchmark values from the least-square Monte Carlo simu-lation method. We also perform sensitivity analyses for vulnerable American options and show how the prices of vulnerable American options vary with the correlation between the underlying assets and the option writer’s assets.
Keywords American options· Derivatives · Default · Credit risk · Multi-exercisable · Martingale
JEL Classification G13
1 Introduction
Over-the-counter markets have grown rapidly in recent years. Most financial deri-vatives in the over-the-counter market have American-style properties. Compared with exchange-listed options markets, there is no organizing exchange, such as the Options Clearing Corporation, in over-the-counter markets requiring that options positions need to be resettled daily and sufficient collateral need to be posted. Thus, it
L.-F. Chang· M.-W. Hung (
B
)College of Management, National Taiwan University, No. 50, Lane 144, Keelung Rd. Sec. 4, Taipei, 100, Taiwan
is unreasonable to assume that holders of over-the-counter options are not exposed to the possible risk of their counterparty defaulting. This kind of risk, known as counter-party risk, is not factored into standard option pricing formulas derived byBlack and Scholes(1973) orMerton(1973). Therefore, several studies concentrate on pricing vulnerable options, which take into account the issue of counterparty risk, for example, Johnson and Stulz(1987),Hull and White(1995),Jarrow and Turnbull(1995),Klein (1996),Klein and Inglis(1999,2001) andHung and Liu(2005).
Johnson and Stulz(1987) assume that when the counterparty cannot make the pro-mised payment, the option holders will receive all their assets from the options writer and that the option is the option writer’s only obligation. But these assumptions are not consistent with real business situations. By extending Johnson and Stulz’s (1987) model, Hull and White(1995) allow the counterparty to have other equal-ranking liabilities. However, they oversimplify the situation by assuming that the payout ra-tio is exogenous. The opra-tion holders only receive the proporra-tion of the opra-tion’s no-default value when the counterparty no-defaults. Their model not only values vulnerable European options but also evaluates vulnerable American options. They orthogona-lize the two stochastic processes to eliminate the correlation between the two state variables and then construct a three-dimensional recombined lattice model with two new transformed state variables. Converting the transformed new state variables to the original form at each node and carrying out backward induction through the lattice acquire the values of the vulnerable European options and vulnerable American opti-ons. Furthermore,Jarrow and Turnbull(1995) develop a risk-neutral pricing procedure to evaluate derivative securities, which are subject to default risk. But they do not show any numerical examples for vulnerable options.
Klein(1996) argues that the assumptions of previous studies are unrealistic in many business situations. He allows other liabilities to exist in the capital structure of an option writer and assumes that the proportional nominal claims paid out when a coun-terparty defaults is endogenous and based on the ultimate value of the option writer’s assets.Klein(1996) also takes the correlation between the option’s underlying asset and the option writer’s asset into consideration and provides a closed form formula for vulnerable European options. In a recent paper,Hung and Liu(2005) extend the framework ofKlein(1996) to price vulnerable options when the market is incomplete. This paper will follow the framework ofKlein(1996) to evaluate vulnerable American options.Klein and Inglis(1999) extend the framework ofKlein(1996) to derive an analytical formula for European options that are subject to financial distress and inte-rest rate risk.Klein and Inglis(2001) also expand on the results of pricing vulnerable European options from the Johnson andJohnson and Stulz(1987) andKlein(1996) models. They assume the default boundary is related to the payoff of the option and the counterparty has other liabilities in her/his capital structure.
Up to now, most of the previous literature has focused on vulnerable European options. However, in over-the-counter markets, most financial derivatives have American-style properties. Klein’s models do not take the American-style options into consideration. For that reason, this article adopts Klein’s (1996) framework to evaluate vulnerable American options by using numerical methods, such as Longstaff and Schwartz’s (2001) least-square Monte Carlo simulation, and provide a more nu-merically efficient analytical formula.
Generally speaking, the numerical methods for evaluating American options are categorized by the following three approaches: lattice model, Monte Carlo simulation and the finite difference method. Longstaff and Schwartz1 (2001) provide a least-square Monte Carlo simulation to evaluate American options. This approach compares the value of exercising the option immediately with the conditional expectation value of continuation. The conditional expectation of continuation for each path at each point is obtained from the linear regression of the discount value of the cash flow by a set of basis functions of the state variables. Compared with numerical approaches,Geske and Johnson(1984) provide analytical formulas for American options by evaluating multi-exercisable options and utilizing Richardson’s extrapolation. In view of computation time, Geske and Johnson’s (1984) model is more numerically efficient than numerical methods such as the Monte Carlo simulation and lattice model. Due to the improved accuracy of numerical methods, computation time is increasing in the wake of the increase in the simulation paths or the time steps of the lattice model.2For this reason, we also apply Geske and Johnson’s method to derive analytical solutions for vulnerable American options.
The purpose of this article is to provide analytical formulas and the least-square Monte Carlo simulation to evaluate vulnerable American options under suitable assumptions that are closer to the real-life business situation. The assumptions of this article are similar to those in Klein’s model (1996). On the one hand, we provide an analytical formula to evaluate vulnerable American options by pricing European and twice-exercisable options and employing Richardson’s extrapolation which is cal-led the two-point Geske and Johnson method;3and on the other hand, the least-square Monte Carlo simulation suggested byLongstaff and Schwartz(2001) is used to com-pare with our proposed method. Based on the benchmark values of the least-square Monte Carlo simulation, the following numerical examples report that the error in analytical approximation is almost smaller than one percent. Our proposed method is more numerically efficient than the least-square Monte Carlo simulation ofLongstaff and Schwartz(2001) or the three-dimensional lattice model ofHull and White(1995). For example, it takes about 20 min to estimate the value of a vulnerable American option, using the least-square Monte Carlo simulation method with 20,000 (10,000 plus 10,000 antithetic) paths. In contrast, the computation time of our proposed me-thod is a few seconds at most. Numerical evaluations show the values of vulnerable American options from our proposed method and the least-square Monte Carlo simu-lation. We also illustrate the impact of counterparty risk and the correlation between the underlying assets and the assets of the option writer on the prices of vulnerable American options by the following numerical examples.
1 Longstaff and Schwartz(2001) survey recent articles that address the pricing of American options by
simulation.
2 For the sake of enhancing computational accuracy, numerical methods have to increase the simulation
paths or the time steps, which results in taking more computation time than analytical solutions. For example, it takes more than 4 min to price a plain vanilla American option with a 5,000-step binomial tree. In contrast, the computation time of Geske and Johnson’s method is within a few seconds at most running on a Pentium 4 2.00 GHz PC.
3 Bunch and Johnson(1992),Ho et al.(1997), andChung(2002) use a two-point Geske and Johnson
The remainder of this article is organized as follows: Section 2 outlines the valuation model used in this paper. Section 3 describes the closed-form solution for vulnerable European options and the analytical formula for twice-exercisable options and applies Richardson’s extrapolation to evaluate vulnerable American options. The results of numerical examples are reported in Section4. Finally, Section5presents the conclusion.
2 The model 2.1 Assumptions
The basic assumptions of our valuation model for vulnerable European and American options are as follows:
Assumption 1 Let S(t) be the market value of the underlying asset of the option at time t and q be the continuously compounded dividend yields of S(t). The dynamics of S(t) are given by
d S(t)/S(t) = µSdt+ σSd ZS(t) (1)
whereµSis the instantaneous expected return of S(t), σSis the instantaneous constant
volatility of the return, and ZS(t) is a standard Brownian motion on probability space
(,F, P).
Assumption 2 Let V(t) denote the market value of the underlying asset of the option writer at time t. The dynamics of V(t) are given by
d V(t)/V (t) = µVdt+ σVd ZV(t) (2)
whereµV is the instantaneous expected return of V(t), σVis the instantaneous
volati-lity of the return (assumed to be constant), and ZV(t) is a standard Brownian motion on
probability space (, F, P). The instantaneous correlation between ZS(t) and ZV(t)
isρSV.
Assumption 3 Let B(t) represent the money market account which is the wealth accumulated by an initial $1 investment at the spot interest rate in the subsequent period. This risk-free security is assumed to continuously compounded in value at the constant interest rate r . Therefore,
d B(t) = r B(t)dt (3)
By convention, we take B(0) = 1.
Assumption 4 We suppose markets are frictionless and perfect. Securities trade in continuous time and there are no taxes or transaction costs.
Assumption 5 Assume European options and twice-exercisable options4 have the same maturity (T ). Default occurs at maturity, T, for both European and twice-exercisable options, and default also happens at half maturity, T/2, for twice-exercisable options, only if the value of the option writer’s assets is less than a threshold value, D∗, which represents the value of the option writer’s critical liabilities. Assumption 6 In the event of default, then the option holder only obtains (1− ω) proportion of the nominal claim, whereω symbolizes the percentage write-down of the nominal claim.
Assumption 7 The percentage write-down on the nominal claim isω = 1 − (1 − α)
V(t)
D∗
∀t = T/2 or T where α represents the deadweight costs of the default event.
2.2 Two-point Geske and Johnson Method
Geske and Johnson(1984) make use of a three-point Richardson extrapolation me-thod to evaluate American options. However, Bunch and Johnson5 (1992) demon-strate how a two-point Richardson method can provide computational accuracy aside from deep-in-the-money options. This paper also adopts the two-point Richardson approach that we substitute for the name of the two-point Geske and Johnson method to price vulnerable American options. For example, we can apply vulnerable Euro-pean option, V EC(t), and vulnerable twice-exercisable option, V ECT (t), to value vulnerable American option, V AC(t), by utilizing a particular form of the Richardson extrapolation that is suggested by Geske and Johnson6(1984).
V AC(t) = V ECT (t) + (V ECT (t) − V EC(t)) (4)
2.3 Risk-neutral valuation
Under the Martingale approach, the value at time 0 of the contingent claim is given by the conditional expectation under the risk-neutral probability measure. For example: The value of the vulnerable European call option at time 0, V EC, with the promised payoff V EC(T ) = (S(T ) − K )+and actual payoff V EC(T ) = (1 − α)
4 For twice-exercisable options, the maturity, T, or the half maturity, T/2, are the only times the options
can be exercised.
5 By utilizing different extrapolation, the Geske and Johnson method has been expanded in a series of
articles byOmberg(1987),Bunch and Johnson(1992), andHo et al.(1994).
6 Geske and Johnson(1984) show that arbitrary accuracy can be achieved by considering American options
which can only be exercised at a few discrete dates and then using their prices to extrapolate the price of the American option, which can be exercised at any date. The sum of a series of compound options is the solution to an optimal stopping time problem, and the American option can be thought of as an infinite series of contingent payoffs. They demonstrate their proposed method for the American problem is exact in the limit.
(V (T )/D∗) (S(T ) − K )+in the event of a default is given by V EC = B(0)E0Q B−1(T )(S(T ) − K )+(1{V (T )≥D∗}) + B(0)EQ 0 B−1(T )(S(T ) − K )+(1 − α) V(T ) D∗ (1{V (T )<D∗})
where E0Q(.) is the expectation at time 0 under risk-neutral probability measure Q. K represents the strike price, and T denotes the maturity of the option. 1{•}is the indicator function.
The price of the vulnerable European put option at time 0, VEP, with the promised payoff VEP(T ) = (K − S(T ))+and actual payoff V E P(T ) = (1 − α)(V (T )/D∗)
(K − S(T ))+in the event of default is given by
V E P = B(0)E0Q B−1(T )(K − S(T ))+(1{V (T )≥D∗}) + B(0)EQ 0 B−1(T )(K − S(T ))+(1 − α) V(T ) D∗ (1{V (T )<D∗})
The value of the vulnerable twice-exercisable call option at time 0, V EC T , with the promised payoff V EC T(t) = (S(t)− K )+and actual payoff V EC T(t) = (1− α)
(V (t)/D∗) (S(t) − K )+ in the event of default where t equals to T or T/2 is given by V EC T = B(0)E0Q B−1 T 2 S T 2 − K 1 ST2≥S∗2T,VT2≥D∗ + B(0)EQ 0 B−1 T 2 S T 2 − K (1 − α) V T2 D∗ × 1 S T 2 ≥S∗∗T 2 ,VT 2 <D∗ + B(0)EQ 0 B−1(T )(S(T ) − K ) 1 ST2<S∗T2,S(T )≥K,V (T )≥D∗ + B(0)EQ 0 B−1(T )(S(T ) − K )(1 − α) V(T ) D∗ × 1 S T 2 <S∗∗T 2 ,S(T )≥K,V (T )<D∗
where the critical stock prices,S∗(T/2) and S∗∗(T/2) are derived as follows: The critical stock price S∗(T/2) is obtained by
S∗ T 2 = K + V EC 0; S, K, V, D∗, r,T 2, σS, σV,ρSV, q = S
As the market value of the underlying asset of the option writer exceeds the value of the threshold liability, D∗, of the option writer on that intermediate date, T/2, if default occurs, the critical stock price S∗∗(T/2) is obtained by
S∗∗ T 2 = K +V EC 0; S, K, V, D∗, r,T2, σS, σV,ρSV, q 1− ω = S
The price of the vulnerable twice-exercisable put option at time 0, V E P T , with the promised payoff V E P T(t) = (K − S(t))+and actual payoff V E P T(t) = (1 − α)
(V (t)/D∗)(K − S(t))+in the event of default where t equals to T or T/2 is given by
V E P T = B(0)E0Q B−1 T 2 K− S T 2 1 S T 2 ≤S∗T 2 ,VT 2 ≥D∗ + B (0) EQ 0 B−1 T 2 K− S T 2 (1 − α) V T2 D∗ × 1 ST2≤S∗∗T2,VT2<D∗ + B(0)EQ 0 B−1(T )(K − S(T )) 1 S T 2 >S∗T 2 ,S(T )≤K,V (T )≥D∗ + B(0)EQ 0 B−1(T )(K − S(T ))(1 − α) V(T ) D∗ × 1{ST 2 >S∗∗T 2 ,S(T )≤K,V (T )<D∗}
where the critical stock prices,S∗(T/2) and S∗∗(T/2) are exhibited as follows: If the market value of the underlying asset of the option writer exceeds the value of the threshold liability, D∗, of the option writer on that intermediate date, T/2, the critical stock price S∗(T/2) is acquired by
S∗ T 2 = K − V E P 0; S, K, V, D∗, r,T 2, σS, σV,ρSV, q = S
In the event default occurs, the critical stock price S∗∗(T/2) is acquired by
S∗∗ T 2 = K −V E P 0; S, K, V, D∗, r,T2, σS, σV,ρSV, q 1− ω = S
In order to implement the model, we can make use of the numerical method7to obtain the critical stock prices S∗(T/2) and S∗∗(T/2). In the following section, we will evaluate V EC (or V E P) and V ECT (or V E PT ) firstly and then use Eq. (4) to price vulnerable American call and put options.
3 The pricing formulas
3.1 Vulnerable European Call and Put Options
Proposition 18The price of the vulnerable European call option at time 0 is given by
V EC = Se−qTN2(d1, d8, ρSV) − K e−rTN2(d2, d6, ρSV) + (1 − α)Se(r−q+ρSVσSσV)TV D N2(d3, −d7, −ρSV) − (1 − α)K V D N2(d4, −d5, −ρSV) (5) where d1= ln S K + r− q + 12σS2T σS √ T d2= ln S K + r− q − 12σS2T σS √ T d3= ln S K + r− q + 12σS2+ ρSVσSσV T σS √ T d4= ln S K + r− q − 12σS2+ ρSVσSσV T σS √ T d5= ln V D∗ + r+12σV2T σV √ T d6= ln V D∗ + r−12σV2T σV √ T d7= ln V D∗ + r+12σV2+ ρSVσSσV T σV √ T d8= ln V D∗ + r−12σV2+ ρSVσSσV T σV √ T
N2(•) is the bivariate normal cumulative distribution function.
Proposition 29The price of the vulnerable European put option at time 0 is given as follows: V E P = K e−rTN2(−d2, d6, −ρSV) − Se−qTN2(−d1, d8, −ρSV) − (1 − α)S V D e(r−q+ρSVσSσV)TN2(−d3, −d7, ρSV) + (1 − α)K V D N2(−d4, −d5, ρSV) (6)
8 Without paying dividends (i.e., q= 0), our solution of VEC reduces to the one of Klein’s model (1996). 9 Without paying dividends (i.e., q= 0), VEP is similar to Klein’s pricing formula (1996).
If no default can occur, then D∗ equals zero, then lim
D∗→0d6 = d8 = ∞ and
lim
D∗→0−d5 = −d7 = −∞. Therefore, N2(x, ∞, ρSV) and N2(x, ∞, −ρSV) will
degenerate to the marginal distribution of x which is denoted by N1(x). N2(x, −∞,
ρSV) and N2(x, −∞, −ρSV) will equal zero. Hence, Eqs. (5) and (6) can be
rewritten as
V EC= Se−qTN1(d1) − K e−rTN1(d2) (7)
V E P = K e−rTN1(−d2) − Se−qTN1(−d1) (8)
Therefore, our valuation formula of V EC and V E P will reduce to the option pricing formulas of Merton’s (1973)10model.
3.2 Vulnerable twice-exercisable call and put options
Proposition 311The price of the vulnerable twice-exercisable call option at time 0 is
V EC T = Se−qT2N2 d1 S, S∗ T 2 ,T 2 , d8 V, D∗,T 2 , ρSV − K e−rT2N2 d2 S, S∗ T 2 ,T 2 , d6 V, D∗,T 2 , ρSV + (1 − α) V D Se(r−q+ρSVσSσV)T2N2 d3 S, S∗∗ T 2 ,T 2 , − d7 V, D∗,T 2 , −ρSV − (1 − α)K V D∗ N2 d4 S, S∗∗ T 2 ,T 2 , − d5 V, D∗,T 2 , −ρSV
10 Without paying dividends (that is q= 0), VEC and VEP also reduce to the option pricing formula of
Black and Shotes (1973).
+ Se−qTN3 −d10 S, S∗ T 2 ,T 2 , d1(S, K, T ), d8(V, D∗, T ), −√1 2, −ρSV, ρSV − K e−rTN3 −d2 S, S∗ T 2 ,T 2 , d2(S, K, T ), d6(V, D∗, T ), −√1 2, −ρSV, ρSV + (1 − α) V D∗ Se(r−q+ρSVσSσV)TN3 −d12 S, S∗∗ T 2 ,T 2 , ×d3(S, K, T ), −d7(V, D∗, T ), −√1 2, ρSV, −ρSV + (1 − α) V D∗ K N3 −d4 S, S∗∗ T 2 ,T 2 , d4(S, K, T ), − d5(V, D∗, T ), −√1 2, ρSV, −ρSV (9)
where N3(•) is the trivariate normal cumulative distribution function.12
d1 S, S∗ T 2 ,T 2 = ln S S∗ T 2 + r− q + 12σS2T2 σS T 2 d1(S, K, T ) = ln S K + r− q +12σS2T σS √ T d2 S, S∗ T 2 ,T 2 = ln S S∗T2+ r− q − 12σS2T2 σS T 2 d2(S, K, T ) = ln S K + r− q −12σS2T σS √ T
12 To implement the trivariate normal cumulative distribution function, we utilize the method ofCurnow
and Dunnett(1962) and the numerical integration method, which is the extended trapezoidal rule with the fifth order closed Newton–Cotes formula (refer toPress et al. 1994).
d3 S, S∗∗ T 2 ,T 2 = ln S S∗∗T2+ r− q +12σS2+ ρSVσSσV T 2 σS T 2 d3(S, K, T ) = ln S K + r− q + 12σS2+ ρSVσSσV T σS √ T d4 S, S∗∗ T 2 ,T 2 = ln S S∗∗T2+ r− q −12σS2+ ρSVσSσV T 2 σS T 2 d4(S, K, T ) = ln S K + r− q − 12σS2+ ρSVσSσV T σS √ T d5 V, D∗,T 2 = ln V D∗ + r+12σV2T2 σV T 2 d5(V, D∗, T ) = ln V D∗ + r+12σV2T σV √ T d6 V, D∗,T 2 = ln V D∗ + r−12σV2T2 σV T 2 d6(V, D∗, T ) = ln V D∗ + r−12σV2T σV √ T d7 V, D∗,T 2 = ln V D∗ + r+12σV2 + ρSVσSσV T 2 σV T 2 d7(V, D∗, T ) = ln V D∗ + r+12σV2 + ρSVσSσV T σV √ T
d8 V, D∗,T 2 = ln V D∗ + r−12σV2 + ρSVσSσV T 2 σV T 2 d8(V, D∗, T ) = ln V D∗ + r−12σV2 + ρSVσSσV T σV √ T d10 S, S∗ T 2 ,T 2 = ln S S∗ T 2 +r− q − 12σS2+√1 2σ 2 S T 2 σS T 2 d12 S, S∗∗ T 2 ,T 2 = ln S S∗∗ T 2 +r− q −12σS2+√1 2σ 2 S+ ρSVσSσV T 2 σS T 2
Proposition 413The value of the vulnerable twice-exercisable put option at time 0 is given by V E P T = K e−rT2N2 −d2 S, S∗ T 2 ,T 2 , d6 V, D∗,T 2 , −ρSV − Se−qT2N2 −d1 S, S∗ T 2 ,T 2 , d8 V, D∗,T 2 , −ρSV − (1 − α)Se(r−q+ρSVσSσV)T2 V D∗ N2 −d3 S, S∗∗ T 2 ,T 2 , − d7 V, D∗,T 2 , ρSV + (1 − α)K V D∗ N2 −d4 S, S∗∗ T 2 ,T 2 , − d5 V, D∗,T 2 , ρSV − Se−qTN3 d10 S, S∗ T 2 ,T 2 , −d1(S, K, T ), d8(V, D∗, T ), −√1 2, ρSV, −ρSV
+ K e−rTN3 d2 S, S∗ T 2 ,T 2 , −d2(S, K, T ), d6(V, D∗, T ), −√1 2, ρSV, −ρSV − (1 − α)Se(r−q+ρSVσSσV)T V D∗ N3 d12 S, S∗∗ T 2 ,T 2 , − d3(S, K, T ), −d7(V, D∗, T ), −√1 2, −ρSV, ρSV + (1 − α)K V D∗ N3 d4 S, S∗∗ T 2 ,T 2 , − d4(S, K, T ), −d5(V, D∗, T ), −√1 2, −ρSV, ρSV (10) If lim
D∗→0d6 = d8 = ∞ and limD∗→0−d5 = −d7 = −∞, then on the one hand, N3(x, y, ∞, −1/
√
2, −ρSV, ρSV) and N3(x, y, ∞, −1/
√
2, ρSV, −ρSV) will
dege-nerate to the bivariate normal distribution function N2(x, y, −1/√2). N2(x, ∞, ρSV)
and N2(x, ∞, −ρSV) will reduce to the marginal distribution of x which is distributed
as N1(x); on the other hand N2(x, −∞, −ρSV), N2(x, −∞, ρSV), N3(x, y, −∞, 1/
√
2, −ρSV, ρSV) and N3(x, y, −∞, 1/
√
2, ρSV, −ρSV) will equal zero. Hence, our
valuation formula will be similar to the twice-exercisable analytical solution from Geske and Johnson(1984).
After obtaining a closed-form solution for vulnerable European options and an analytical formula for vulnerable twice-exercisable options, we can combine V EC (or
V E P) and V EC T (or V E P T ) with a particular form of Richardson’s extrapolation,
such as Eq. (4), to value vulnerable American call and put options. We also show the numerical results of our proposed method and the benchmark values of least-square Monte Carlo simulation in the following section.
4 Numerical examples
In order to confirm the accuracy of our proposed method, we compare the least-square Monte Carlo simulation ofLongstaff and Schwartz(2001) with our proposed analytical formula.Longstaff and Schwartz(2001) provide a least-square Monte Carlo approach to value American options. This approach compares the value of immediate exercise with the conditional expectation value of continuation. The conditional expectation of continuation for each path at each point is obtained from the regression of the discount value of the cash flow by a set of basis functions of the state variables.
In this article, the basis functions consist of the first three powers of the option and the option’s writer underlying asset. The product of two variables is in the form of
SV , SV2and S2V . This results in a total of nine basis functions.14Thus, we regress
14 Numerical tests show that using different basis functions has little effect on the results. This finding is
discounted early exercise values on a constant and nonlinear basis functions of the stock price and the option writer’s underlying asset to estimate the continuation value of a vulnerable American option. For every vulnerable American option, we use 20 least-square Monte Carlo simulation estimates with each least-square Monte Carlo simulation, which is based on 20,000 (10,000 plus 10,000 antithetic) paths using 24 exercise points per year.15 The mean and standard deviation of the 20 least-square Monte Carlo simulation estimates for vulnerable American options are listed in the following tables.
We use the following parameters’ values to evaluate vulnerable and non-vulnerable options. The numerical evaluations of options are reported in Table1and2. All option values in Table1are calculated by the following parameters: S= K = 40, V = 100,
D∗ = 90, r = 0.05, q = 0, T − t = 0.3333, σS = 0.2, σV = 0.2, α = 0.25, ρSV = 0. The prices of vulnerable European and American call options in Table
1 are based on the above propositions and Eq. (4). The values of non-vulnerable American call options in Table1 are according to Trigeorgis’ (1991) method with 5,000 time steps. We adopt the definition of error in analytical approximation from Klein and Inglis(2001), which is defined as the difference between a least-square Monte Carlo simulation estimate (i.e., numerical solution) and ours (i.e., analytical solution) divided by a least-square Monte Carlo simulation estimate. The relative error between the analytical approximation and default risk from the counterparty is defined as the difference between a least-square Monte Carlo simulation estimate and ours as a percentage of the difference between the values of the non-vulnerable option and the vulnerable option obtained by least-square Monte Carlo simulation.
Table 1 shows that the prices of vulnerable American options are smaller than those of non-vulnerable American options due to the default risk from the counter-party. The effect of counterparty risk on vulnerable American options is less than that on vulnerable European options because early exercise will eliminate the influence of counterparty risk. These results are expected and consistent with previous studies. Table1reveals that the values of vulnerable American call options from our proposed method are very close to those from least-square Monte Carlo simulation. The diffe-rences between least-square Monte Carlo simulation estimates and ours are generally small. The error in analytical approximation between our proposed method and least-square Monte Carlo simulation is almost one percent in most cases. From the results of Table1, these facts demonstrate that our proposed method can provide accurate estimations for vulnerable American options. In view of the computation time, our proposed method is more numerically efficient than least-square Monte Carlo simu-lation. For example, the computation time of least-square Monte Carlo simulation to obtain a price of the vulnerable American option is about 20 min, but the computation time of our proposed method to obtain a price of the vulnerable American option is under four seconds. When the simulation paths increase to improve computational accuracy, the advantage of our model is even more apparent.
Since there would be a longer time period during which the possibility of fi-nancial distress could affect the value of an option, we provide numerical
examp-15 It takes about 20 min to obtain 20 least-square Monte Carlo simulation estimates for each vulnerable
les representing a longer period under a broader range of assumptions. All option values in Table2are evaluated by the following parameters: S= K = 40, V = 100,
D∗ = 90, r = 0.05, q = 0, T − t = 1, σS = 0.2, σV = 0.2, α = 0.25, ρSV = 0.
Table2presents a number of numerical examples for vulnerable and non-vulnerable call options.
As in the results of Table1, Table2reveals that the values of vulnerable American options are smaller than the ones of non-vulnerable American options, which results from the default risk of the counterparty. Table2presents that the prices of vulnerable American call options obtained from our proposed method are close to those from least-square Monte Carlo simulation. The results in Table2 are similar to those in Table1. The differences between least-square Monte Carlo simulation estimates and ours are generally small. Based on the results of Tables1and2, these findings show that our proposed method can still give accurate price estimations for pricing vulne-rable American options. As expected, increasing interest rates increases the values of vulnerable calls. Increasing the volatility of the returns of the underlying assets of the options increases the value of vulnerable calls. Also, as expected, increasing the volatility of the returns of the writer’s assets decreases the value of vulnerable calls. Finally, increasing the costs of bankruptcy also decreases the value of vulnerable calls. The numerical results of Figs.1–5illustrate the sensitivity analysis of vulnerable American options. All option values in the following figures are calculated by these parameters: S = K = 40, V = 100, D∗ = 90, r = 0.05, q = 0, T = 1, σS = 0.2, σV = 0.2, α = 0.25, ρSV = 0.
Figures1and2show how the ratio of underlying value to strike price, S/K , and debt ratio, D/V , affect the values of vulnerable and non-vulnerable American op-tions. As in the standard option pricing result, the value of a vulnerable American call increases as the moneyness of the option increases. Figure1shows the value of both non-vulnerable and vulnerable American calls as a function of the moneyness of the option and exhibits how the default risk from the counterparty weakens the advantage of in-the-money options. The impact of default risk on the price of Ame-rican options is even more obvious as the ratio of the underlying value to strike price increases.
Figure 2also displays that the impact of default risk on the value of American option is more apparent if the debt ratio approaches to one. This means that if default becomes more and more possible, the decrease in the value of vulnerable American options rises more rapidly. The value of the vulnerable American option decreases as the writer is leveraged over a 55% debt ratio. The greater the default risk, the greater the reduction in the vulnerable call’s value.
Figures3and4illustrate the effect of varying degrees of correlation between the returns on the underlying assets and the assets of the option writer. In Figure3, we show the reduction in the vulnerable American call’s value as a function of the option’s moneyness.
Ta b le 1 V alues of vulnerable and non-vulnerable call options based o n p arameter v alues: S = K = 40 , V = 100 , D ∗= 90 , r = 0. 05, q = 0, T − t = 0. 3333, σS = 0. 2, σV = 0. 2, α = 0. 25, ρSV = 0 Analytical solution o f V u lnerable American option LSMC of
vulnerable American option Standard deviation of
LSMC
(S.E)
Error
in
analytical approxi- mation (%)
Closed
form
solution
o
f
vulnerable European option Numerical solution o f non-vulnerable American option Percentage reduction (%) Relati v e error b etween approximation and d ef ault risk Base case 2 .0771 2.0774 0.0113 0.01 2.0744 2.1757 4.52 0.0031 ρ = 0. 5 2 .1513 2.1556 0.0176 0.20 2.1512 2.1757 0.92 0.2139 ρ =− 0. 5 1 .9733 1.9548 0.0146 0.95 1.9542 2.1757 10.15 0.0837 σS = 0. 15 1.6455 1.6472 0.0087 0.10 1.6455 1.7258 4.55 0.0216 σS = 0. 25 2.5305 2.5098 0.0164 0.82 2.5055 2.6278 4.49 0.1754 σv = 0. 15 2.1238 2.1239 0.015 0.00 2.1236 2.1757 2.38 0.0019 σv = 0. 25 2.1081 2.066 0.0112 2.04 2.0286 2.1757 5.04 0.3838 T − t= 0. 083 0.9964 0.9971 0.0132 0.07 0.9964 1.0046 0.75 0.0933 T − t= 0. 583 2.8635 2.829 0.0242 1.22 2.8212 3.0184 6.27 0.1822 S = 35 0.3322 0.3332 0.0147 0.30 0.3322 0.3484 4.36 0.0658 S = 45 5.8242 5.7881 0.0149 0.62 5.6764 5.9538 2.78 0.2179 V = 105 2.1273 2.1331 0.0142 0.27 2.1273 2.1757 1.96 0.1362 V = 110 2.1549 2.1567 0.0137 0.08 2.1549 2.1757 0.87 0.0947 α = 0 2 .156 2.1562 0.0134 0.01 2.1559 2.1757 0.90 0.0103 α = 0. 5 2 .0141 2.0601 0.0122 2.23 1.993 2.1757 5.31 0.3979 r = 0. 03 2.0169 1.9758 0.0113 2.08 1.9343 2.0381 3.06 0.6597 r = 0. 07 2.2200 2.2192 0.0181 0.04 2.2200 2.3184 4.28 0.0081 The v alues of vulnerable A merican and E uropean options are acquired b y our pre v ious proposed analytical formulas. L east-square Monte C arlo simulat ion is b ased on 20,000 (10,000 plus 10,000 antithetic) paths u sing 24 ex ercise points p er year . T he standard errors of the simulation estimates are g iv en in the columns of (S .E). The v alues of non-vulnerable A merican-type options are computed by the T rigeor g is ( 1991 ) m ethod with 5,000 time steps. The error in analytical approximation is d efined as the dif ference between a least-square Monte C arlo simulation estimate and ours d iv ided by a least-square Monte C arlo simulation estimate. The relati v e error b etwe en the analytical approximation and def ault risk from the counterparty is d efined as the dif ference between a least-square Monte C arlo simulation estimate and ours as a percentage o f the dif ference between the v alues of the non-vulnerable option and the vulnerable option obtained by least-square Monte C arlo simulation
Ta b le 2 V alues of vulnerable and non-vulnerable call options based o n p arameter v alues: S = K = 40 , V = 100 , D ∗= 90 , r = 0. 05, q = 0, T − t = 1, σS = 0. 2, σV = 0. 2, α = 0. 25, ρSV = 0 Analytical solution o f V u lnerable American option LSMC of
vulnerable American option Standard deviation of
LSMC
(S.E)
Error
in
analytical approxi- mation (%)
Closed form solution o f vulnerable European option Numerical solution o f non-vulnerable American option Percentage reduction (%) Relati v e error between appro-ximation and def ault risk Base case 3 .8509 3.8853 0.0295 0.89 3.8351 4.1801 7.05 0.1167 ρ = 0. 5 4 .0947 4.0824 0.0301 0.30 4.073 4.1801 2.34 0.1259 ρ =− 0. 5 3 .6907 3.6071 0.0245 2.32 3.4994 4.1801 13.71 0.1459 σS = 0. 15 3.1683 3.1735 0.0189 0.16 3.153 3.4366 7.66 0.0198 σS = 0. 25 4.5608 4.5565 0.034 0.09 4.527 4.9342 7.65 0.0114 σv = 0. 15 3.9675 3.9907 0.0316 0.58 3.9648 4.1801 4.53 0.1225 σv = 0. 25 3.727 3.7413 0.0232 0.38 3.7191 4.1801 10.50 0.0326 T − t= 0. 5 2 .6156 2.6247 0.013 0.35 2.5895 2.7554 4.74 0.0696 T − t= 1. 5 5 .0409 4.9593 0.0315 1.65 4.8701 5.377 7.77 0.1954 S = 35 1.5076 1.5054 0.0271 0.15 1.499 1.634 7.87 0.0171 S = 45 7.7645 7.6585 0.0217 1.38 7.2262 7.8764 2.77 0.4865 V = 105 3.9411 3.9783 0.025 0.94 3.9382 4.1801 4.83 0.1843 V = 110 4.0147 4.0273 0.0225 0.31 4.0147 4.1801 3.66 0.0825 α = 0 4 .0675 4.0588 0.0207 0.21 4.0674 4.1801 2.90 0.0717 α = 0. 5 3 .6332 3.7075 0.0291 2.00 3.6028 4.1801 11.31 0.1572 r = 0. 03 3.4582 3.4949 0.0265 1.05 3.4103 3.7653 7.18 0.1357 r = 0. 07 4.2958 4.3064 0.0326 0.25 4.2853 4.6164 6.72 0.0342 See footnote o f T able 1
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 0 5 10 15 20 25 30 35 40 45
Values of Vulnerable and Non-Vulnerable American Call Options
Values of Call Options
Ratio of Underlying Asset Value to Strike Price (S/K) VAC:Vulnerable American Call Option
NVAC:Non-Vulnerable American Call Option
Fig. 1 Values of vulnerable and non-vulnerable American options: showing how the ratio of underlying
value to strike price, S/K , affects the values of vulnerable and non-vulnerable American options. The values of vulnerable and non-vulnerable options in Figure 1 are based on the following parameter values:
S= K = 40, V = 100, D∗ = 90, r = 0.05, q = 0, T = 1, σS = 0.2 = σV,α = 0.25, ρSV = 0. The values of vulnerable American options are computed by our previous proposed method. The values of non-vulnerable American options are also acquired by theTrigeorgis(1991) method with 5,000 time steps
In the case of a positive correlation between the counterparty’s assets and the option’s underlying asset, if the counterparty’s asset increases, there is a tendency for the option’s underlying asset to increase and for the value of the vulnerable Ameri-can call option to increase. Therefore, a positive correlation between the counterparty’s assets and the option’s underlying asset implies a small reduction in the vulnerable American call’s value. Moving toward a negative correlation makes it more important to take this effect into account. The effect of negative correlation is more pronounced for out-of-money options.
Figure4plots the reduction as a function of the option writer’s debt ratio. If the underlying assets and the assets of the option writer are positively correlated, then there is little reduction in the value of the vulnerable American option. Note that the effect of negative correlation is much more significant for highly leveraged option writers.
For the sake of understanding the impact of counterparty risk and the correlation between the underlying assets and the assets of the option writer on the prices of vulnerable American options, Figure5illustrates how the price of vulnerable American options are affected by the correlation between the underlying asset of the option and the option writer and the ratio of underlying value to strike price. The values of vulnerable American options will go up as the correlation between the underlying asset of the option and the writer of option,ρSV, or the ratio of underlying value to
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 3.5 3.6 3.7 3.8 3.9 4 4.1 4.2 4.3
Values of Vulnerable and Non-Vulnerable American Call Options
Values of Call Options
Debt Ratio (D/V) VAC : Vulnerable American Call Option NVAC : Non-Vulnerable American Call Option
Fig. 2 Values of vulnerable and non-vulnerable American options: the debt ratio, D/V , influences the
values of vulnerable and non-vulnerable American options. The values of vulnerable and non-vulnerable options in Fig.2are computed by the following parameter values: S= K = 40, V = 100, D∗ = 90,
r = 0.05, q = 0, T = 1, σS = 0.2 = σV,α = 0.25, ρSV = 0. The values of vulnerable American options are based on our proposed pricing formula. The values of non-vulnerable American options are also obtained by theTrigeorgis(1991) method with 5,000 time steps
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 0 10 20 30 40 50 60
Percentage Reduction inValue of Vulnerable Call for Different Values of Rho
Percetage Reduction (%)
Ratio of Underlying Asset Value to Strike Price (S/K) PR0 : Percentage Reducation in Case of Rho=0 PR1 : Percentage Reducation in Case of Rho=0.5 PR2 : Percentage Reducation in Case of Rho=0.5
Fig. 3 Percentage reduction of vulnerable American options: the effect of varying degrees of correlation
between the returns on the underlying assets and the assets of the option writer. In Fig.3, we show the reduction in the vulnerable American call’s value as a function of the option’s moneyness. The values of vulnerable options in Figure3are calculated by the following parameter values: S= K = 40, V = 100,
D∗ = 90, r = 0.05, q = 0, T = 1, σS = 0.2 = σV,α = 0.25, ρSV = 0. The values of vulnerable American options are computed by our proposed method
0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 0 5 10 15 20 25 30
Percentage Reducation in Value of Vulnerable Call for Different Values of Rho
Percentage Reduction (%)
Debt Ratio (D/V) PR0 : Percentage Reduction in Case of Rho=0 PR1 : Percentage Reduction in Case of Rho=0.5 PR2 : Percentage Reduction in Case of Rho=0.5
Fig. 4 Percentage reduction of vulnerable American options: exhibits the effect of varying degrees of
correlation between the returns on the underlying assets and the assets of the option writer. Figure4 presents the reduction as a function of the option writer’s debt ratio. The values of vulnerable options in Fig.4are computed by the following parameter values: S = K = 40, V = 100, D∗ = 90, r = 0.05,
q = 0, T = 1, σS = 0.2 = σV,α = 0.25, ρSV = 0. The values of vulnerable American options are computed by our proposed method
-0.8 -0.6 -0.4-0.2 0 0.2 0.4 0.6 0.8 0 0.5 1 1.5 2 0 5 10 15 20 25 30 35 40 rho Values of Vulnerable American Call Option
S/K
Values of Call Option
Fig. 5 Values of vulnerable American Options: illustrating how the price of vulnerable American options are affected by the correlation between the underlying asset of the option and the writer of option and the ratio of underlying value to strike price. Values of vulnerable American options in Fig.5are based on the following parameter values: S= K = 40, V = 100, D∗= 90, r = 0.05, q = 0, T = 1, σS= 0.2 = σV,
α = 0.25, ρSV = 0. The values of vulnerable American options are obtained by our proposed pricing method
5 Conclusion
This article use the two-point Geske and Johnson method and least-square Monte Carlo simulation method suggested byLongstaff and Schwartz(2001) to evaluate vulnerable American options under suitable assumptions that are close to real business situations. We provide analytical pricing formulas for vulnerable European and multi-exercisable options. Based on a particular form of Richardson’s extrapolation, we employ our analytical pricing formulas to evaluate vulnerable American options.
To confirm the accuracy of our proposed method, we compare the estimates of our proposed method with those calculated by the least-square Monte Carlo simulation. The values of vulnerable American options from ours are close to those from the least-squares Monte Carlo simulation. The error in analytical approximation between our proposed method and the least-squares Monte Carlo simulation is generally smaller than 1%, which demonstrates that our proposed method can give accurate price estima-tes for vulnerable American option. Our analytical formula for vulnerable American options is more numerically efficient than the least-squares Monte Carlo simulation as the simulation paths increase to improve the accuracy of the numerical method. Finally, we conduct the numerical evaluations of sensitivity analysis for vulnerable American options. In future research, we hope to extend our model to take the interest risk into account.
Appendix
Under the Martingale approach, we should transform the standard Brownian motion on probability space (, F, P) to probability space (, F, Q). Using the money market account as the numeraire, all the asset prices normalized by money market account will follow martingales. Let Q be the corresponding risk-neutral probability measure in (, F), which is equivalent to P. The Radon–Nikodym derivative is
d Q d P = exp T 0 ηd Z(v) −1 2 T 0 |η|2 dv
whereη ∈ R is the market prices of risks corresponding to the sources of stochastic processes in the economy and satisfies
µS+ q − r + σSη1= 0
µV − r + σVη2= 0
By Girsanov’s theorem, Z∗S(t) and Z∗V(t) defined below is a standard Brownian motion under measure Q: Z∗S(t) = ZS(t) − t 0 η 1dv Z∗V(t) = ZV(t) − t 0 η2dv
The dynamics of the underlying asset of the option under risk-neutral probability measure Q is able to recover from the auxiliary process by Ito’s lemma under risk-neutral probability measure Q. For the dynamics of the option’sunderlying asset and the underlying asset of the option writer at time t under srisk-neutral probability measure Q is the following
d S(t)/S(t) = (r − q)dt + σSd Z∗S(t) d V(t)/V (t) = rdt + σVd Z∗V(t)
The instantaneous correlation between Z∗S(t) and ZV∗(t) is ρSV. Proof of proposition 3 V EC T = B(0)E0Q B−1 T 2 (ST 2 − K ) 1 S T 2 ≥S∗T 2 ,VT 2 ≥D∗ + (1 − α)B(0)EQ 0 B−1 T 2 V T2 D∗ S T 2 − K × 1 ST2≥S∗∗T2,VT2<D∗ + B (0) EQ 0 B−1(T ) (S (T ) − K ) 1 S T 2 <S∗T 2 ,S(T )≥K,V (T )≥D∗ + (1 − α)B(0)EQ 0 B−1(T )(S(T ) − K ) V(T ) D∗ × 1 ST2<S∗∗T2,S(T )≥K,V (T )<D∗ = e−rT2EQ 0 S T 2 1 S T 2 ≥S∗T 2 ,VT 2 ≥D∗ − e−rT2 ×K EQ 0 1 ST2≥S∗T2,VT2≥D∗ + (1 − α) e−rT2EQ 0 S T 2 V T2 D∗ 1 ST 2 ≥S∗∗T 2 ,VT 2 <D∗ − (1 − α)e−rT2K EQ 0 V T2 D∗ 1 S T 2 ≥S∗∗T 2 ,VT 2 <D∗ + e−rTE0Q S(T )1 ST2<S∗T2,S(T )≥K,V (T )≥D∗ − K e−rTE0Q 1 S T 2 <S∗T 2 ,S(T )≥K,V (T )≥D∗ + (1 − α)e−rTEQ 0 S(T ) V(T ) D∗ 1 ST2<S∗∗T2,S(T )≥K,V (T )<D∗
− (1 − α)e−rTK E0Q V(T ) D∗ 1 ST 2 <S∗∗T 2 ,S(T )≥K,V (T )<D∗ = J1− J2+ J3− J4+ J5− J6+ J7− J8 Let Xi(t) = Z∗i(t)/ √
t where i = S(t) or V (t), then Xi has the standard normal
distribution. J1= e−r T 2 ∞ −∞ ∞ −∞Se r−q−12σS2T2+σSXs T 2 T 21 S T 2 ≥S∗T 2 ,VT 2 ≥D∗ ∗ 1 2π 1− ρ2SV e− 1 2(1−ρ2SV )(X 2 s−2ρSVXsXV+X2V) d Xsd XV = Se−qT2 ∞ −∞ ∞ −∞1 S T 2 ≥S∗T 2 ,VT 2 ≥D∗ × 1 2π 1− ρ2SV e− 1 2(1−ρ2SV ) v2 S−2ρSVvsvV+vV2 d Xsd XV wherevS = XS− σS √ T/2 andvV = XV − ρSVσV √
T/2. We are able to define an
equivalent probability measure, d Q∗/d Q = exp(σSZ∗S(T/2) − 1/2σS2T/2) with ξ
and Z∗• vectors in R2andξ is defined to [σS, ρSVσS] for S and V, respectively. By
Girsanov’s theorem, d WS T 2 d WV T 2 = d Z∗S(T2) d Z∗V(T2) − ξdt
is a R2-valued standard Brownian motion under risk-neutral probability measureQ∗. Therefore, J1= Se−q T 2EQ∗ 0 1 ST2≥S∗T2,VT2≥D∗ = Se−qT2PQ∗ Se r−q+12σS2 T 2+σSW T 2 ≥ S∗ T 2 , V e r−12σV2+ρSVσSσS T 2+σVW T 2 ≥ D∗ = Se−qT 2N2 d1 S, S∗ T 2 ,T 2 , d8 V, D∗,T 2 , ρSV J2= K e−r T 2EQ 0 1 ST2≥S∗T2,VT2≥D∗ = K e−rT 2PQ Se r−q−1 2σS2 T 2+σSZ∗ T 2 ≥S∗T 2 , V e r−1 2σV2 T 2+σVZ∗ T 2 ≥D∗ = K e−rT2N2 d2 S, S∗ T 2 ,T 2 , d6 V, D∗,T 2 , ρSV
We are able to define an equivalent probability measure, d Qd Q = exp σSZ S T 2 − 1 2 σ2 S T 2+σVZV T 2 −1 2σ 2 V T 2−ρSVσSσV T 2
withξ and Z•∗vectors in R2andξ is defined to[σS+ ρSVσV, σV + ρSVσS] for S and V , respectively. By Girsanov’s theorem,
d WS T2 d WV T2 = d Z∗S T2 d ZV∗ T2 − ξdt
is a R2-valued standard Brownian motion under risk-neutral probability measureQ . Let ˜Xi(t) = Wi (t)/
√
t where i = S(t) or V (t), then ˜Xi has the standard normal
distribution and where ˜vS = ˜XS − (σS+ ρSVσV) √ T/2 and ˜vV = ˜XV − (σV + ρSVσS) √ T/2. Therefore, J3= (1 − α)e−r T 2 ∞ −∞ ∞ −∞Se r−q−12σ2 S T 2+σS˜Xs T 2 T 2 V D∗ ×e r−12σ2 V T 2+σV ˜XV T 2 T 21 S T 2 ≥ S∗∗ T 2 , V T 2 < D∗ × 1 2π 1− ρ2SV e− 1 2(1−ρ2SV ) ( ˜X2 S−2ρSV˜Xs˜XV+ ˜X2V) d ˜Xsd ˜XV = (1 − α) V D∗ Se(r−q+ρSVσSσV)T2 ∞ −∞ ∞ −∞1 S(T2)≥S∗∗(T2),V (T2)<D∗ × 1 2π 1− ρ2SV e− 1 2(1−ρ2SV )(˜v 2 S−2ρSV˜vs˜vV+˜v2V) d ˜Xsd ˜XV = (1 − α) V D∗ Se(r−q+ρSVσSσV)T2EQ 0 [1 ST2≥S∗∗T2,VT2<D∗] = (1 − α) V D∗ Se(r−q+ρSVσSσV)T2PQ Se(r−q+ 1 2σS2+ρSVσSσV)T2+σSW T 2 ≥ S∗∗T 2 , V e r+1 2σV2+ρSVσSσV T 2+σVW T 2 < D∗ = (1 − α) V D∗ Se(r−q+ρSVσSσV)T2N2 d3 S, S∗∗ T 2 ,T 2 , − d7 V, D∗,T 2 , −ρSV
We are able to define an equivalent probability measure, d ¯Q/d Q = exp(σVZV∗(T/2)−
1/2σV2T/2) with ξ and Z•∗vectors in R2andξ is defined to [ρSVσV, σV] for S and V , respectively. By Girsanov’s theorem,
d ¯WS T 2 d ¯WV T 2 = d Z∗S T2 d ZV∗ T2 − ξdt
is a R2-valued standard Brownian motion under risk-neutral probability measure ¯Q.
Therefore, J4= (1 − α) K D∗e −rT 2EQ 0 V T 2 1 ST2≥S∗∗2T,VT2<D∗ = (1 − α)K V D∗ P¯Q Se r−q−12σS2+ρSVσSσV T 2+σSW¯ T 2 ≥ S∗∗ T 2 , V e r+12σ2 V T 2+σVW¯ T 2 < D∗ = (1 − α)K V D∗ N2 d4 S, S∗∗ T 2 ,T 2 , −d5 V, D∗,T 2 , −ρSV
We are able to introduce an equivalent probability measure, d ˆQ/d Q = exp(σSZ∗S(T )−
1/2σS2T) with Z∗· andγ vectors in R3andγ is defined to [σ
S, ρS(T )S(T/2)σS, ρSVσS]
whereρS(T )S(T 2)is 1/
√
2. It is easy to show by the uniqueness of the moment genera-ting function. By Girsanov’s theorem,
⎡ ⎣d ˆWS T 2 d ˆWS(T ) d ˆWV(T ) ⎤ ⎦ = ⎡ ⎣d Z ∗ S T 2 d Z∗S(T ) d Z∗V(T ) ⎤ ⎦ − γ dt
is a R3-valued standard Brownian motion under risk-neutral probability measure ˆQ.
Therefore, J5= e−rTE0Q S(T )1 ST 2 <S∗T 2 ,S(T )≥K,V (T )≥D∗ = Se−rTE0ˆQ 1 ST2<S∗T2,S(T )≥K,V (T )≥D∗ = Se−qTP ˆQ Se r−q−12σS2+√1 2σ 2 S T 2+σSWˆS T 2 < S∗T 2 , Se r−q+12σS2 T+σSWˆS(T ) ≥ K, V e r−12σV2+ρSVσSσS T+σVWˆV(T ) ≥ D∗ = Se−qTN3 −d10 S, S∗ T 2 ,T 2 , d1(S, K, T ), d8(V, D∗, T ), −√1 2, −ρSV, ρSV J6= K e−rTE0Q 1 ST2<S∗T2,S(T )≥K,V (T )≥D∗
= K e−rTPQ Se r−q−12σ2 S T 2+σSZs∗ T 2 < S∗ T 2 , Se r−q−12σ2 S T+σSZ∗S(T ) ≥ K, V e r−12σ2 V T+σVZ∗V(T )≥ D∗ = K e−rTN3 −d2 S, S∗ T 2 ,T 2 , d2(S, K, T ), d6(V, D∗, T ), −√1 2, −ρSV, ρSV J7= (1 − α)e−rTE0Q S(T ) V(T ) D∗ 1 ST 2 <S∗∗T 2 ,S(T )≥K,V (T )<D∗ = (1 − α) V D∗ Se(r−q+ρSVσSσV)TE0Q 1 ST2<S∗∗T2,S(T )≥K,V (T )<D∗ = (1−α) V D∗ Se(r−q+ρSVσSσV)TN3 −d12 S, S∗∗ T 2 ,T 2 , d3(S, K, T ), − d7(V, D∗, T ), −√1 2, ρSV, −ρSV J8= (1 − α)K ( V D∗)E ¯Q 0 1 {ST 2 <S∗∗T 2 ,S(T )≥K,V (T )<D∗} = (1 − α)K V D∗ P¯Q(Se r−q−12σS2+ρSVσSσV T 2+σS¯Zs T 2 < S∗∗ T 2 , Se r−q−12σ2 S+ρSVσSσV T+σS¯ZS(T )≥ K, V e r+12σ2 V T+σV¯ZV(T )< D∗) = (1 − α)K V D∗ N3 −d4 S, S∗∗ T 2 ,T 2 , d4(S, K, T ), − d5(V, D∗, T ), −√1 2, ρSV, −ρSV where d1 S, S∗ T 2 ,T 2 = ln S S∗T2+ r− q + 12σS2T2 σS T 2 d1(S, K, T ) = ln S K + r− q + 12σS2T σS √ T d2 S, S∗ T 2 ,T 2 = ln S S∗T2+ r− q − 12σS2T2 σS T 2
d2(S, K, T ) = ln S K + r− q −12σS2T σS √ T d3 S, S∗∗ T 2 ,T 2 = ln S S∗∗T2+ r− q +12σS2+ ρSVσSσV T 2 σS T 2 d3(S, K, T ) = ln S K + r− q + 12σS2+ ρSVσSσV T σS √ T d4 S, S∗∗ T 2 ,T 2 = ln S S∗∗T2+ r− q −12σS2+ ρSVσSσV T 2 σS T 2 d4(S, K, T ) = ln S K + r− q −12σS2+ ρSVσSσV T σS √ T d5 V, D∗,T 2 = ln V D∗ + r+12σV2T2 σV T 2 d5(V, D∗, T ) = ln V D∗ + r+12σV2T σV √ T d6 V, D∗,T 2 = ln V D∗ + r−12σV2T2 σV T 2 d6 V, D∗, T=ln V D∗ + r−12σV2T σV √ T d7 V, D∗,T 2 = ln V D∗ + r+12σV2 + ρSVσSσV T 2 σV T 2
d7 V, D∗, T=ln V D∗ + r+12σV2+ ρSVσSσV T σV √ T d8 V, D∗,T 2 = ln V D∗ + r−12σV2 + ρSVσSσV T 2 σV T 2 d8(V, D∗, T ) = ln V D∗ + r−12σV2 + ρSVσSσV T σV √ T d10 S, S∗ T 2 ,T 2 = ln S S∗T2+ r− q − 12σS2+√1 2σ 2 S T 2 σS T 2 d12 S, S∗∗ T 2 ,T 2 = ln S S∗∗ T 2 +r− q −12σS2+√1 2σ 2 S+ ρSVσSσV T 2 σS T 2 Q.E.D. The proof of the other propositions is similar to the Proof of the proposition 3.
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