Stochstic Calculus And Binary Options

Public Interest Argument

The pick pricing problem is an important topic of finance field. At present, we mainly focus on the study of the selection pricing problem (for example, the pricing of the vulnerable binary option in this paper), the selection pricing formula of the price process of assets which is more reasonable in practise, and some corresponding real data assay.

1. Introduction

Credit risk is one of the major financial adventure, it has become a major claiming facing the earth's financial markets. On the exchange transaction market, because of the margin requirements, futures exchanging and options exchange derivatives virtually no credit risk. Therefore, when option pricing happened in the substitution, we tin can assume credit chance does not be. However, On OTC market place, counter-party credit risk is a gene that must be considered. For derivatives on OTC market, since at that place is no guarantee, selection bulls exposure to the credit risk and marketplace chance. The assumption of no default chance no longer valid. Therefore, how to properly consider the counter-political party run a risk factors and establish choice pricing model containing credit risk are important to investors.

At that place are some articles about option pricing have credit hazard theory into consideration. Firm value model is one of the nearly important model nearly vulnerable option pricing put forrard past Merton (1974). Black, Fisher, and Cox (1976) studied the risk of bankrupt cost pricing of corporate bonds, improved Merton's idea about events of default. Johnson and Stulz (1987) extended Merton default model, discussing the option pricing problem under the structural model that has credit adventure, putting forward the concept of vulnerable option. The seminal work that discussed pick pricing containing credit risk is from Johnson and Stulz (1987). They used the term vulnerable option to define those options containing counter-political party default risk, and pointed out a lot of features nearly them. Hull and White (1995) supposed the underlying asset and the counter-party firm's avails are independent of one another, obtained vulnerable options pricing formula. Jarrow and Tunbull (1995) supposed default time obey the forcefulness of $\mathit{\lambda }$ (non-negative constant) independent homogeneous poisson process, used discrete method to create a simple model of credit gamble. Klein (1996) supposed breach of contract happened with a certain probability at whatever fourth dimension, considered the pricing model of zero-coupon bonds tin can be liquidated at initial fourth dimension cypher.

In recent years, there is a considerable involvement in the counter-party risk factors and establish option pricing model containing credit risk. Wang and Li (2003), Fu and Zhang (2002) and Xu and Li (2005), researched the problem of derivative pricing which have default risk. Deng and Kaleong (2007) and Wu, Lv, and Min (2007) consider stochastic interest rate and random counter-party liability instance. They discussed the trouble of choice pricing with credit risk and getting its pricing formula. However, people always assume that stock toll obey standard chocolate-brown motion, witch is non reality and has its limitation. Lin, Wang, and Feng (2000) proposed the O-U process model that stock price obeys, avoiding the limitations of standard dark-brown motion. In the previous selection pricing model, we assume involvement rate is a constant. However, in reality, involvement charge per unit changes randomly. A lot of scholars conducted all-encompassing research in this field. Zhou, Xinyu, and Gao (2011) gave European option pricing model under stochastic interest rate. Xue (2000) gave convertible bond pricing formula based on Hull-White Model which satisfy fractional Brownian motion drive.

This commodity mainly enquiry the problem of vulnerable binary choice pricing under the O-U process. Based on the business firm value model, nosotros presume the stock toll, counter-party firm'due south assets and counter-party firm'due south debts obey O-U procedure, interest rate obeys Hull-White Model. By using martingale approach we derive an analytical pricing formula for such vulnerable binary option. This commodity is structured as follows. Section ii describes firm value model and its supposition. Section iii presents the utilise of the martingale approach for vulnerable binary pick pricing. Moreover,we obtain the pricing formula of vulnerable binary pick. Section four contains some conclusion.

2. Model and its assumptions

2.one. Firm value model

Considering continuous-time financial market, allow $(\mathrm{\Omega },\mathcal{F},\{{\mathcal{F}}_t{\}}_{0\le t\le T},P)$ be a filtered probability space such that the filtration satisfies the usual conditions. Based on firm value model by Merton (1974), assume 10(T) is the counter party house promise to pay at the time T, Ten is European fashion pick, if the counter party firm always take the ability of pay before the due maturity date, then the owner of this selection volition get Ten(T) at the time T, otherwise the payoff depend on the ratio of company's avails and its liability. If nosotros assume V(T) is the counter party firm's assets, D(T) is counter party firm's liability, when the counter party firm bankrupt, the payoff is $\mathrm{10}(T)\times [V(T)/D(T)]$ , so the owner of this vulnerable choice's yield to maturity is $Y(T)=X(T)I_{\{{\mathit{\delta }}_{(T)}\ge 1\}}+{\mathit{\delta }}_{(T)}X(T)I_{\{{\mathit{\delta }}_{(T)}<1\}}$ , where $I_A$ is indicator function of the fix A, $\mathit{\delta }(T)$ is the ratio of compensation, $\mathit{\delta }(T)=V(T)/D(T)$ . According to martingale theory of pick pricing, the cost of this option at the fourth dimension $0\le t\le T$ tin can be represented equally below nether the equivalent martingale measurement $\tilde{P}$ , (one) $$\begin{array}{c}\hfill C(t)=B(t)\widetilde{\mathrm{Due\; east}}\left[\frac{\mathrm{10}(T)}{B(T)}(I_{\{{\mathit{\delta }}_{(T)}\ge 1\}}+{\mathit{\delta }}_{(T)}{I}_{\{{\mathit{\delta }}_{(T)}<1\}})\left|\right.{\mathcal{F}}_t\right],\end{array}$$ (i)

where B(t) is the price of default-free bonds, satisfy differential equation $dB(t)=r(t)B(t)dt$ , r(t) is chance-gratuitous involvement rate, ${\mathcal{F}}_t$ is information sets before the time t, $\widetilde{\mathrm{East}}(·)$ is the mathematical expectation under equivalent martingale measurement $\tilde{P}$ .

2.two. Model building

Presume stock cost $S_t$ , company'southward assets $V_t$ , company's liability $D_t$ all follow O-U process equally follows (ii) $$\begin{array}{cc}\hfill \text{d}S(t)& =({\mathit{\mu }}_S(t)-{\mathit{\alpha }}_S(t)ln\mathrm{Due\; south}(t))S(t)\text{d}t+{\mathit{\sigma }}_{\mathrm{Due\; south}}(t)S(t)\text{d}{B}_S(t),\hfill \\ \hfill \text{d}V(t)& =({\mathit{\mu }}_{\mathrm{Five}}(t)-{\mathit{\alpha }}_5(t)lnV(t))\mathrm{Five}(t)\text{d}t+{\mathit{\sigma }}_V(t)5(t)\text{d}{B}_V(t),\hfill \\ \hfill \text{d}D(t)& =({\mathit{\mu }}_D(t)-{\mathit{\alpha }}_D(t)lnD(t))D(t)\text{d}t+{\mathit{\sigma }}_D(t)D(t)\text{d}{B}_D(t),\hfill \end{array}$$ (2)

where $B_{\mathrm{Southward}}(t),B_V(t),B_D(t)$ is a standard Chocolate-brown motion in probability space $(\mathrm{\Omega },\mathcal{F},{\mathcal{F}}_{0\le t\le T},P)$ , ${\mathit{\mu }}_{\mathit{\xi }}(t),{\mathit{\sigma }}_{\mathit{\xi }}(t),(\mathit{\xi }=S,V,D)$ is the corresponding expected render rate and volatility respectively, which are continuous functions about t.

O-U process is a Brownish motion considering expected render rate depends on underlying assets's price, ${\mathit{\alpha }}_{\mathit{\xi }}(\mathit{\xi }=S,\mathrm{Five},D)$ tin can decrease underlying avails'southward price when the price upward to a certain height. Assume $\mathbf{B}=(B_{\mathrm{South}}(t),B_5(t),B_D(t))$ is a 3-dimensional correlated ${\mathcal{F}}_t$ -Dark-brown motion with the correlation matrix C given by (3) $$\begin{array}{c}\hfill \mathbf{C}=\left(\begin{array}{ccc}1& {\mathit{\rho }}_{S5}& {\mathit{\rho }}_{\mathrm{Due\; south}D}\\ {\mathit{\rho }}_{SV}& 1& {\mathit{\rho }}_{VD}\\ {\mathit{\rho }}_{\mathrm{Southward}D}& {\mathit{\rho }}_{\mathrm{Five}D}& \mathrm{one}\end{array}\right)\end{array}$$ (three)

In the post-obit, we shall innovate Girsanov's theorem and martingale representation theorem about multidimensional correlated Brown motion B.

Let $\mathrm{\Lambda }(t)=({\mathit{\lambda }}_S(t),{\mathit{\lambda }}_V(t),{\mathit{\lambda }}_D(t))$ is a iii-dimensional ${\mathcal{F}}_t$ -adapted process, where ${\mathit{\lambda }}_{\mathit{\xi }}(t)=[{\mathit{\mu }}_{\mathit{\xi }}-{\mathit{\alpha }}_{\mathit{\xi }}ln{\mathit{\xi }}_t-r(t)]/{\mathit{\sigma }}_{\mathit{\xi }},(\mathit{\xi }=\mathrm{Due\; south},V,D)$ , now define $\mathit{\lambda }(t)={(\mathrm{\Lambda }(t)\mathbf{C}{\mathrm{\Lambda }}^{\ast }(t))}^{\frac{1}{\mathrm{ii}}},\forall 0\le t\le T$ , satisfy Novikov condition, where ${\mathrm{\Lambda }}^{\ast }(t)$ is the transposition of $\mathrm{\Lambda }(t)$ , C is the matrix defined past formula (5). Define (4) $$\begin{array}{c}\hfill Z_{\mathit{\lambda }}(t)=exp\left\{-\underset{0}{\overset{t}{\int }}\mathrm{\Lambda }(s)d\mathbf{B}(s)-\frac{\mathrm{i}}{\mathrm{ii}}\underset{0}{\overset{t}{\int }}{\mathrm{\Lambda }}^{\mathrm{two}}(s)d\mathrm{southward}\right.\},\forall 0\le t\le T,\end{array}$$ (4)

then $Z_{\mathit{\lambda }}(t)$ is a ${\mathcal{F}}_t$ -martingale. We further presume that for each $0\le t\le T$ , ${\mathit{\lambda }}_{\mathit{\xi }}(t)\ne 0$ , and there exists a constant ${\mathit{\rho }}_{\mathit{\lambda }\mathit{\xi }},\mathit{\xi }=S,5,D$ such that (5) $$\begin{array}{cc}\hfill {\mathit{\rho }}_{\mathit{\lambda }S}\mathit{\lambda }(t)& ={\mathit{\lambda }}_S(t)+{\mathit{\rho }}_{SV}{\mathit{\lambda }}_V(t)+{\mathit{\rho }}_{\mathrm{Southward}D}{\mathit{\lambda }}_D(t),\hfill \end{array}$$ (five) (half dozen) $$\begin{array}{cc}\hfill {\mathit{\rho }}_{\mathit{\lambda }\mathrm{Five}}\mathit{\lambda }(t)& ={\mathit{\rho }}_{SV}{\mathit{\lambda }}_S(t)+{\mathit{\lambda }}_V(t)+{\mathit{\rho }}_{VD}{\mathit{\lambda }}_D(t),\hfill \end{array}$$ (6) (7) $$\begin{array}{cc}\hfill {\mathit{\rho }}_{\mathit{\lambda }D}\mathit{\lambda }(t)& ={\mathit{\rho }}_{SD}{\mathit{\lambda }}_S(t)+{\mathit{\rho }}_{VD}{\mathit{\lambda }}_V(t)+{\mathit{\lambda }}_D(t).\hfill \end{array}$$ (7)

Theorem 1

On the measurable space $(\mathrm{\Omega },{\mathcal{F}}_T)$ , we define new probability measure $\tilde{P}$ , and for each $A\in {\mathcal{F}}_t$ , $\tilde{P}(A)=E_P[I_{A}Z(T)]$ , then $\tilde{P}$ is the martingale measure out equivalent to P, such that (8) $$\begin{array}{cc}\hfill {\tilde{B}}_S(t)& =B_{\mathrm{South}}(t)+\underset{0}{\overset{t}{\int }}{\mathit{\rho }}_{\mathit{\lambda }S}\mathit{\lambda }(s)ds,\hfill \end{array}$$ (8) (9) $$\begin{array}{cc}\hfill {\tilde{B}}_V(t)& =B_{\mathrm{Five}}(t)+\underset{0}{\overset{t}{\int }}{\mathit{\rho }}_{\mathit{\lambda }V}\mathit{\lambda }(\mathrm{southward})ds,\hfill \end{array}$$ (9) (10) $$\begin{array}{cc}\hfill {\tilde{B}}_D(t)& =B_D(t)+\underset{0}{\overset{t}{\int }}{\mathit{\rho }}_{\mathit{\lambda }D}\mathit{\lambda }(s)ds,\hfill \end{array}$$ (ten)

for $0\le t\le T,{\tilde{B}}_S(t),{\tilde{B}}_5(t),{\tilde{B}}_D(t)$ are all ${\mathcal{F}}_t$ standard Chocolate-brown motion, and for each $\mathit{\xi },\mathit{\zeta }=S,V,D$ $$\begin{array}{c}\hfill ⟨{\tilde{B}}_{\mathit{\xi }},{\tilde{B}}_{\mathit{\zeta }}⟩(t)=⟨B_{\mathit{\xi }},B_{\mathit{\zeta }}⟩(t)={\mathit{\rho }}_{\mathit{\xi }\mathit{\zeta }}t,\end{array}$$

where the cross-variations are computed under the appropriate measure P and $\tilde{P}$ .

This upshot is due to Doob (1953), and its proof can be establish in Deng and Kaleong (2007) (Theorem 2.2).

Lemma 1

(Steven, 1987)    Assume $\{B_1(t),t\ge 0\}$ , $\{B_2(t),t\ge 0\}$ are ${\mathcal{F}}_t$ -standard Brown motions nether the probability P, correlation coefficient is ${\mathit{\rho }}_{12}$ , and covariance is $⟨B_1,B_2⟩(t)={\mathit{\rho }}_{12}t$ . Announce $$\begin{array}{cc}\hfill B(t)& =\frac{{\mathit{\beta }}_{\mathrm{one}}(t)B_1(t)-{\mathit{\beta }}_2(t)B_2(t)}{\mathrm{\Delta }(t)},\hfill \\ \hfill \mathrm{\Delta }(t)& =\sqrt{{\mathit{\beta }}_1^{\mathrm{two}}(t)+{\mathit{\beta }}_{\mathrm{ii}}^{\mathrm{two}}(t)-2{\mathit{\rho }}_{12}{\mathit{\beta }}_{\mathrm{one}}(t){\mathit{\beta }}_2(t)},\hfill \end{array}$$

where ${\mathit{\beta }}_{\mathrm{i}}(t),{\mathit{\beta }}_2(t)$ are deterministic continuous function. Then B(t) is a standard ${\mathcal{F}}_t$ -Chocolate-brown move under the probability P.

three. The martingale approach for vulnerable binary option pricing

At present we will discuss vulnerable binary option pricing under stochastic interest rate and liability.

Under the risk-neutral measure $\tilde{P}$ , from Equation (1) we can get vulnerable binary option price tin can express as follow at the fourth dimension $0\le t\le T,$ (11) $$\begin{array}{c}\hfill C(t)=\tilde{E}\left[exp\left\{-\underset{t}{\overset{T}{\int }}r(u)du\right\}(X(T)I_{\{{\mathit{\delta }}_{(T)}\ge 1\}}+\mathit{\delta }(T)X(T)I_{\{{\mathit{\delta }}_{(T)}<1\}})|{\mathcal{F}}_t\right].\end{array}$$ (11)

For cash or nothing call option, payoff role on maturity date is $X_T=R{I}_{\{S(T)\ge \mathrm{Yard}\}}$ ;

For greenbacks or nothing put option, payoff part on maturity date is $X_T=R{I}_{\{\mathrm{Southward}(T)<K\}}$ ;

For asset or nothing call pick, payoff function on maturity date is ${\mathrm{Ten}}_T=\mathrm{Due\; south}(T)I_{\{S(T)\ge K\}}$ ;

For asset or nothing put option, payoff function on maturity date is $X_T=S(T)I_{\{S(T)<\mathrm{1000}\}}$ .

3.1. Model solution

Assume the three-dimensional ${\mathcal{F}}_t$ -adapted process $\mathrm{\Lambda }(t)=({\mathit{\lambda }}_S(t),{\mathit{\lambda }}_V(t),{\mathit{\lambda }}_D(t))$ satisfy the post-obit SDEs: (12) $$\begin{array}{cc}\hfill {\mathit{\sigma }}_S(t)[{\mathit{\rho }}_{SV}{\mathit{\lambda }}_{\mathrm{Five}}(t)+{\mathit{\rho }}_{SD}{\mathit{\lambda }}_D(t)]& =0,\hfill \end{array}$$ (12) (xiii) $$\begin{array}{cc}\hfill {\mathit{\sigma }}_{\mathrm{Five}}(t)[{\mathit{\rho }}_{S\mathrm{Five}}{\mathit{\lambda }}_S(t)+{\mathit{\rho }}_{VD}{\mathit{\lambda }}_D(t)]& =0,\hfill \end{array}$$ (13) (14) $$\begin{array}{cc}\hfill {\mathit{\sigma }}_D(t)[{\mathit{\rho }}_{SD}{\mathit{\lambda }}_{\mathrm{South}}(t)+{\mathit{\rho }}_{VD}{\mathit{\lambda }}_{\mathrm{Five}}(t)]& =0.\hfill \end{array}$$ (14)

Let $Z_{\mathit{\lambda }}(t)$ is the ${\mathcal{F}}_t$ -martingale defined by Equation (6), and so by the Theorem ane, $\tilde{\mathbf{B}}(t)=({\tilde{B}}_S(t),{\tilde{B}}_V(t),{\tilde{B}}_D(t))$ is a 3-dimensional correlated standard $({\mathcal{F}}_t,\tilde{P})$ - Brown motion, correlation matrix is C divers by formula (five).

From Equations (7)–(9), (14)-(16), the definition of ${\mathit{\lambda }}_{\mathit{\xi }}(t)(\mathit{\xi }=S,V,D)$ and the Theorem one, we tin go counter political party business firm'due south assets 5(t), counter party firm's liability D(T), stock price S(t) satisfy the following SDEs: (fifteen) $$\begin{array}{cc}\hfill \text{d}S(t)& =r(t)S(t)\text{d}t+{\mathit{\sigma }}_S(t)S(t)\text{d}{\tilde{B}}_{\mathrm{Due\; south}}(t),\hfill \end{array}$$ (xv) (16) $$\begin{array}{cc}\hfill \text{d}5(t)& =r(t)V(t)\text{d}t+{\mathit{\sigma }}_5(t)\mathrm{Five}(t)\text{d}{\tilde{B}}_V(t),\hfill \end{array}$$ (16) (17) $$\begin{array}{cc}\hfill \text{d}D(t)& =r(t)D(t)\text{d}t+{\mathit{\sigma }}_D(t)D(t)\text{d}{\tilde{B}}_D(t).\hfill \end{array}$$ (17)

We assume short-term interest rate follows Hull-White model under the probability $\tilde{P}$ every bit below: (18) $$\begin{array}{c}\hfill \text{d}r(t)=(a(t)-b(t)r(t))\text{d}t+{\mathit{\sigma }}_r(t)\text{d}{\tilde{B}}_r(t).\end{array}$$ (18)

Under the probability $\tilde{P}$ , from Ito formulas we can go: $$\begin{array}{c}\hfill \underset{t}{\overset{T}{\int }}r(u)du=G(t,T,r(t))+\underset{t}{\overset{T}{\int }}{\mathit{\sigma }}_r(u)m(u,T)d{\tilde{B}}_r(u),\end{array}$$

where $\mathrm{Grand}(t,T,r(t))=r(t)\mathrm{grand}(t,T)+{\int }_t^{T}a(u)m(u,T)du$ , $m(u,v)={\int }_u^{v}e\mathrm{10}p(\mathit{\eta }(u)-\mathit{\eta }(s))d\mathrm{due\; south}$ , $\mathit{\eta }(s)={\int }_0^{s}b(u)du$ .

Under the probability $\tilde{P}$ , applying Ito formulas to (17)–(nineteen) we can go: $$\begin{array}{c}\hfill \mathit{\xi }(T)=\mathit{\xi }(t)exp\left\{\underset{t}{\overset{T}{\int }}(r(u)-{\mathit{\sigma }}^2(\mathit{\xi })/2)du+\underset{t}{\overset{T}{\int }}{\mathit{\sigma }}_{\mathit{\xi }}(u)d{\tilde{B}}_{\mathit{\xi }}(u)\right\},\mathit{\xi }=\mathrm{Due\; south},5,D.\end{array}$$

Assume ${\tilde{B}}_r(t)$ and ${\tilde{B}}_S(t),{\tilde{B}}_V(t),{\tilde{B}}_D(t)$ are uncorrelated, then $({\tilde{B}}_r(t),{\tilde{B}}_S(t),{\tilde{B}}_V(t),{\tilde{B}}_D(t))$ is a 4-dimensional correlated standard Brown motion nether probability $\tilde{P}$ , the correlation matrix is (19) $$\begin{array}{c}\hfill \tilde{\mathbf{c}}=\left(\begin{array}{cccc}\mathrm{one}& 0& 0& 0\\ 0& 1& {\mathit{\rho }}_{SV}& {\mathit{\rho }}_{\mathrm{Due\; south}D}\\ 0& {\mathit{\rho }}_{\mathrm{Due\; south}V}& \mathrm{i}& {\mathit{\rho }}_{VD}\\ 0& {\mathit{\rho }}_{SD}& {\mathit{\rho }}_{VD}& \mathrm{i}\end{array}\right).\end{array}$$ (19)

For the convenience of calculating, we transform $\mathrm{Due\; south}(T),\mathit{\delta }(T)$ every bit follows: (20) $$\begin{array}{cc}\hfill \mathrm{Due\; south}(T)& =S(t)exp\left\{\underset{t}{\overset{T}{\int }}(r(u)-{\mathit{\sigma }}_S^{\mathrm{two}}(u)/2)du+\underset{t}{\overset{T}{\int }}{\mathit{\sigma }}_S(u)d{\tilde{B}}_S(u)\right\}\hfill \\ \hfill & =S(t)exp\left\{\mathrm{Thousand}(t,T,r(t))-\underset{t}{\overset{T}{\int }}{\mathit{\sigma }}_S^2(u)/\mathrm{ii}du\right.\hfill \\ \hfill & \left.+\underset{t}{\overset{T}{\int }}({\mathit{\sigma }}_r(u)m(u,T)d{\tilde{B}}_r(u)+{\mathit{\sigma }}_S(u)d{\tilde{B}}_S(u))\right\}\hfill \\ \hfill & =S(t)exp\left\{K(t,T,r(t))-\underset{t}{\overset{T}{\int }}\frac{{\mathit{\sigma }}_{\mathrm{Southward}}^2(u)}{2}du+\underset{t}{\overset{T}{\int }}{\mathrm{\Delta }}_{1}d{\tilde{B}}_{\overline{S}}(u)\right\},\hfill \end{array}$$ (20)

where (21) $$\begin{array}{cc}\hfill \text{d}{\tilde{B}}_{\overline{\mathrm{Due\; south}}}(t)& =\frac{{\mathit{\sigma }}_r(t)\mathrm{1000}(t,T)\text{d}{\tilde{B}}_r(t)+{\mathit{\sigma }}_S(t)\text{d}{\tilde{B}}_{\mathrm{Southward}}(t)}{{\mathrm{\Delta }}_{\mathrm{i}}},\hfill \\ \hfill {\mathrm{\Delta }}_1& =\sqrt{{\mathit{\sigma }}_r^2(t)g^2(t,T)+{\mathit{\sigma }}_{\mathrm{Southward}}^{\mathrm{ii}}(t)}.\hfill \end{array}$$ (21)

From Lemma 1 we see that ${\tilde{B}}_{\overline{S}}(t)$ is a standard brown movement under the probability $\tilde{P}$ .

Under the probability $\tilde{P}$ , let $\mathit{\delta }(t)=V(t)/D(t)$ , co-ordinate to It $\widehat{o}$ 's formula we take (22) $$\begin{array}{cc}\hfill \mathit{\delta }(T)& =\mathit{\delta }(t)exp\left\{\underset{t}{\overset{T}{\int }}\frac{1}{\mathrm{ii}}({\mathit{\sigma }}_D^2(u)-{\mathit{\sigma }}_V^{\mathrm{ii}}(u))du\right.\hfill \\ \hfill & \left.+\underset{t}{\overset{T}{\int }}({\mathit{\sigma }}_V(u)d{\tilde{B}}_V(u)-{\mathit{\sigma }}_D(u)d{\tilde{B}}_D(u))\right\}\hfill \\ \hfill & =\mathit{\delta }(t)exp\left\{\underset{t}{\overset{T}{\int }}\frac{1}{2}({\mathit{\sigma }}_D^2(u)-{\mathit{\sigma }}_{\mathrm{Five}}^{\mathrm{two}}(u))du+\underset{t}{\overset{T}{\int }}{\mathrm{\Delta }}_{2}d{\tilde{B}}_{\mathit{\sigma }}(u)\right\},\hfill \end{array}$$ (22)

where (23) $$\begin{array}{cc}& d{\tilde{B}}_{\mathit{\sigma }}(t)=\frac{{\mathit{\sigma }}_{\mathrm{Five}}(t)d{\tilde{B}}_V(t)-{\mathit{\sigma }}_D(t)d{\tilde{B}}_D(t)}{{\mathrm{\Delta }}_2},\hfill \\ \hfill & {\mathrm{\Delta }}_2=\sqrt{{\mathit{\sigma }}_V^{\mathrm{ii}}(t)+{\mathit{\sigma }}_D^{\mathrm{two}}(t)-2{\mathit{\sigma }}_{5D}(t)},{\mathit{\sigma }}_{VD}(t)={\mathit{\rho }}_{\mathrm{Five}D}(t){\mathit{\sigma }}_V(t){\mathit{\sigma }}_D(t).\hfill \end{array}$$ (23)

By Lemma ane again, nosotros have that ${\tilde{B}}_{\mathit{\sigma }}(t)$ is a standard dark-brown move under the probability $\tilde{P}$ .

iii.2. The pricing formula of vulnerable binary option

Theorem two

Under the firm value model, assume underlying asset price is $S_t$ at the time t, exercise price is 1000, the ratio of compensation is $\mathit{\delta }(T)=5(T)/D(T)$ , the price of vulnerable cash or nothing call option at the fourth dimension $t(0\le t\le T)$ is: (24) $$\begin{array}{c}\hfill C(t)=\mathrm{Thou}R\mathrm{North}(d_{\mathrm{ane}},d_{\mathrm{two}};\mathit{\rho })+\mathrm{Yard}\mathrm{Yard}\mathit{\delta }(t)b_{\mathrm{ii}}N(d_1+a_1,-(d_{\mathrm{two}}+\sqrt{{\int }_t^T{\mathrm{\Delta }}_2^{\mathrm{ii}}du});-\mathit{\rho })).\end{array}$$ (24)

where $$\begin{array}{cc}\hfill d_1& =\frac{ln\frac{S(t)}{K}+G(t,T,r(t))-\underset{t}{\overset{T}{\int }}{\mathit{\sigma }}_S^{\mathrm{two}}(u)/\mathrm{ii}du}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_{\mathrm{ane}}^{2}du}},\hfill \\ \hfill d_{\mathrm{two}}& =\frac{ln\mathit{\delta }(t)+{\int }_t^T\frac{1}{2}({\mathit{\sigma }}_D^{\mathrm{ii}}(u)-{\mathit{\sigma }}_V^2(u))du}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_2^{2}du}},\hfill \\ \hfill a_1& =\frac{{\int }_t^T({\mathit{\sigma }}_{SV}(u)-{\mathit{\sigma }}_{VD}(u))du}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_1^{2}du}},\hfill \\ \hfill b_{\mathrm{two}}& =exp\left\{\underset{t}{\overset{T}{\int }}({\mathit{\sigma }}_D^{\mathrm{ii}}(u)-{\mathit{\sigma }}_{\mathrm{Five}D}(u))du\right\},\hfill \\ \hfill \mathrm{Yard}& =exp\left\{-G(t,T,r(t))+\frac{1}{\mathrm{two}}\underset{t}{\overset{T}{\int }}{\mathit{\sigma }}_r^2(u)m^2(u,T)du\right\},\hfill \\ \hfill \mathit{\rho }& =\frac{\underset{t}{\overset{T}{\int }}{\mathrm{\Delta }}_{\mathrm{i}}{\mathrm{\Delta }}_{\mathrm{two}}{\mathit{\rho }}_{{\tilde{B}}_{\overline{\mathrm{South}}}{\tilde{B}}_{\mathit{\sigma }}}du}{\sqrt{\underset{t}{\overset{T}{\int }}{\mathrm{\Delta }}_1^{2}du}\sqrt{\underset{t}{\overset{T}{\int }}{\mathrm{\Delta }}_2^{2}du}},\hfill \\ \hfill {\mathit{\rho }}_{{\tilde{B}}_{\overline{S}}{\tilde{B}}_{\mathit{\sigma }}}& =\frac{({\mathit{\sigma }}_{\mathrm{Due\; south}}(t){\mathit{\sigma }}_V(t){\mathit{\rho }}_{SV}+{\mathit{\sigma }}_{\mathrm{Due\; south}}(t){\mathit{\sigma }}_D(t){\mathit{\rho }}_{\mathrm{South}D})}{\sqrt{({\mathit{\sigma }}_r^2(t){\mathrm{thousand}}^2(t,T)+{\mathit{\sigma }}_S^{\mathrm{two}}(t))}\sqrt{({\mathit{\sigma }}_r^{\mathrm{two}}(t)m^2(t,T)+\mathrm{ii}{\mathit{\sigma }}_V(t){\mathit{\sigma }}_D(t)\mathrm{yard}(t,T){\mathit{\rho }}_{VD})}},\hfill \end{array}$$ ${\mathrm{\Delta }}_1$ , ${\mathrm{\Delta }}_{\mathrm{ii}}$ is defined past Equations (23)and (25).

Proof

Denote $A_1=(\mathrm{Due\; south}(T)>K,\mathit{\delta }(T)\ge 1),A_2=(S(T)>K,\mathit{\delta }(T)<1)$ , according to Equation (xiii), under the risk neutral probability, the price of vulnerable cash or nothing call option is: $$\begin{array}{cc}\hfill C(t)& =\widetilde{\mathrm{Due\; east}}\left[exp\left\{-\underset{t}{\overset{T}{\int }}r(u)du\right\}(X(T)I_{\{{\mathit{\delta }}_{(T)}\ge \mathrm{one}\}}+\mathit{\delta }(T)X(T)I_{\{{\mathit{\delta }}_{(T)}<1\}})|{\mathcal{F}}_t\right]\hfill \\ \hfill & =\tilde{E}\left[exp\left\{-\underset{t}{\overset{T}{\int }}r(u)du\right\}(R{I}_{A_{\mathrm{ane}}}+\mathit{\delta }(T)R{I}_{A_2})|{\mathcal{F}}_t\right]\hfill \\ \hfill & =\widetilde{\mathrm{East}}\left[exp\left\{-G(t,T,r(t))-\underset{t}{\overset{T}{\int }}{\mathit{\sigma }}_r(u)m(u,T)d{\tilde{B}}_r(u)\right\}(R{I}_{A_{\mathrm{one}}}+\mathit{\delta }(T)R{I}_{A_2})|{\mathcal{F}}_t\right]\hfill \\ \hfill & =\widetilde{\mathrm{Eastward}}\left[exp\left\{-\mathrm{Thou}(t,T,r(t))+\frac{\mathrm{one}}{\mathrm{ii}}\underset{t}{\overset{T}{\int }}{\mathit{\sigma }}_r^2(u){\mathrm{grand}}^2(u,T)du\right\}(R{I}_{A_1}+\mathit{\delta }(T)R{I}_{A_{\mathrm{two}}})|{\mathcal{F}}_t\right]\hfill \\ \hfill & =M\tilde{E}\left[(R{I}_{A_1}+\mathit{\delta }(T)R{I}_{A_2})|{\mathcal{F}}_t\right]\hfill \\ \hfill & =G\tilde{E}\left[R{I}_{A_1}|{\mathcal{F}}_t]+G\tilde{E}[\mathit{\delta }(T)R{I}_{A_2}|{\mathcal{F}}_t\right]\hfill \\ \hfill & =D_{\mathrm{ane}}+D_2,\hfill \end{array}$$

where $\mathrm{Chiliad}=exp\left\{-\mathrm{1000}(t,T,r(t))+\frac{1}{\mathrm{ii}}{\int }_t^T{\mathit{\sigma }}_r^2(u)m^2(u,T)du\right\},D_1=\mathrm{Yard}\tilde{E}[R{I}_{A_1}|{\mathcal{F}}_t],D_{\mathrm{ii}}=G\widetilde{\mathrm{Due\; east}}[\mathit{\delta }(T)R{I}_{A_2}|{\mathcal{F}}_t]$ .

Denote the joint distribution function of two dimensional standardized normal random vector is: $$\begin{array}{cc}& N(z_{\mathrm{one}},z_2;\mathit{\rho })=\underset{z_1}{\overset{-\infty }{\int }}\underset{z_2}{\overset{-\infty }{\int }}f(x_1,{\mathrm{10}}_2;\mathit{\rho })d{x}_{1}d{x}_2,(\forall -\infty <z_{\mathrm{one}},z_2<+\infty ),\hfill \\ \hfill & f({\mathrm{ten}}_1,x_2;\mathit{\rho })=\frac{\mathrm{one}}{2\mathit{\pi }\sqrt{\mathrm{ane}-{\mathit{\rho }}^2}}exp\left\{-\frac{\mathrm{ane}}{2(\mathrm{i}-{\mathit{\rho }}^{\mathrm{ii}})}({\mathrm{10}}_1^2-2\mathit{\rho }{\mathrm{10}}_{\mathrm{one}}{x}_{\mathrm{ii}}+{\mathrm{10}}_{\mathrm{two}}^{\mathrm{ii}})\right\},(\forall -\infty <{\mathrm{ten}}_{\mathrm{one}},{\mathrm{ten}}_{\mathrm{two}}<+\infty ,1-{\mathit{\rho }}^{\mathrm{ii}}\ne 0).\hfill \end{array}$$

Evaluation of term $D_1$ : $$\begin{array}{c}\hfill D_1=GR\tilde{P}(A_{\mathrm{i}}|{\mathcal{F}}_t)=\mathrm{Thou}R\tilde{P}((S(T)>\mathrm{Thou},\mathit{\delta }(T)\ge \mathrm{i})|{\mathcal{F}}_t),\end{array}$$

according to $S(T)>K$ and Southward(T) follows from (22) that $$\begin{array}{c}\hfill \mathrm{Due\; south}(T)=\mathrm{Due\; south}(t)exp\left\{K(t,T,r(t))-\underset{t}{\overset{T}{\int }}{\mathit{\sigma }}_{\mathrm{Southward}}^2(u)/2du+\underset{t}{\overset{T}{\int }}{\mathrm{\Delta }}_{\mathrm{one}}d{\tilde{B}}_{\overline{S}}(u)\right\}>M\end{array}$$

then, $$\begin{array}{c}\hfill G(t,T,r(t))-\underset{t}{\overset{T}{\int }}{\mathit{\sigma }}_S^2(u)/\mathrm{two}du+\underset{t}{\overset{T}{\int }}{\mathrm{\Delta }}_{\mathrm{i}}d{\tilde{B}}_{\overline{\mathrm{Southward}}}(u)>ln\frac{K}{S(t)}\end{array}$$

therefore, $$\begin{array}{c}\hfill \frac{-{\int }_t^T{\mathrm{\Delta }}_{1}d{\tilde{B}}_{\overline{S}}(u)}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_{\mathrm{one}}^{2}du}}<\frac{ln\frac{S(t)}{\mathrm{Thou}}+M(t,T,r(t))-{\int }_t^T\frac{{\mathit{\sigma }}_S^{\mathrm{two}}(u)}{2}du}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_1^{2}du}}.\end{array}$$

Past (24) and $\mathit{\delta }(T)\ge 1$ , we have $$\begin{array}{c}\hfill \mathit{\delta }(T)=\mathit{\delta }(t)exp\left\{\underset{t}{\overset{T}{\int }}\frac{\mathrm{one}}{2}({\mathit{\sigma }}_D^{\mathrm{two}}(u)-{\mathit{\sigma }}_V^{\mathrm{two}}(u))du+\underset{t}{\overset{T}{\int }}{\mathrm{\Delta }}_{2}d{\tilde{B}}_{\mathit{\sigma }}(u)\right\}\ge \mathrm{i}.\end{array}$$

And so $$\begin{array}{c}\hfill -\frac{{\int }_t^T{\mathrm{\Delta }}_{2}d{\tilde{B}}_{\mathit{\sigma }}(u)}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_{\mathrm{ii}}^{2}du}}\le \frac{ln{\mathit{\delta }}_t+{\int }_t^T\frac{1}{\mathrm{two}}({\mathit{\sigma }}_D^2(u)-{\mathit{\sigma }}_V^2(u))du}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_2^{2}du}}.\end{array}$$

Substituting this into $D_{\mathrm{i}}$ , we obtain $$\begin{array}{cc}\hfill D_1& =GR\tilde{P}((S(T)>\mathrm{1000},\mathit{\delta }(T)\ge \mathrm{i})|{\mathcal{F}}_t)\hfill \\ \hfill & =\mathrm{1000}R\tilde{P}\left[-\frac{\underset{t}{\overset{T}{\int }}{\mathrm{\Delta }}_{1}d{\tilde{B}}_{\overline{S}}(u)}{\sqrt{\underset{t}{\overset{T}{\int }}{\mathrm{\Delta }}_{\mathrm{i}}^{2}du}}\right.\hfill \\ \hfill & <\frac{ln\frac{S(t)}{K}+\mathrm{Thousand}(t,T,r(t))-\underset{t}{\overset{T}{\int }}{\mathit{\sigma }}_{\mathrm{Due\; south}}^2(u)/\mathrm{two}du}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_{\mathrm{i}}^{2}du}},\hfill \\ \hfill & \left.-\frac{{\int }_t^T{\mathrm{\Delta }}_{2}d{\tilde{B}}_{\mathit{\sigma }}(u)}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_{\mathrm{two}}^{2}du}}\le \frac{ln\mathit{\delta }(t)+\underset{t}{\overset{T}{\int }}\frac{\mathrm{one}}{\mathrm{two}}({\mathit{\sigma }}_D^2(u)-{\mathit{\sigma }}_5^2(u))du}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_2^{2}du}}|{\mathcal{F}}_t\right]\hfill \\ \hfill & =\mathrm{Thou}RN(d_1,d_2;\mathit{\rho }),\hfill \end{array}$$

where $$\begin{array}{cc}\hfill d_1& =\frac{ln\frac{\mathrm{South}(t)}{K}+\mathrm{Thou}(t,T,r(t))-{\int }_t^T{\mathit{\sigma }}_S^2(u)/2du}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_1^{2}du}},\hfill \\ \hfill d_{\mathrm{two}}& =\frac{ln\mathit{\delta }(t)+{\int }_t^T\frac{1}{\mathrm{ii}}({\mathit{\sigma }}_D^2(u)-{\mathit{\sigma }}_V^2(u))du}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_2^{2}du}}|{\mathcal{F}}_t],\hfill \\ \hfill \mathit{\rho }& =\text{Corr}\left(-\frac{{\int }_t^T{\mathrm{\Delta }}_{\mathrm{one}}d{\tilde{B}}_{\overline{S}}(u)}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_1^{2}du}},-\frac{{\int }_t^T{\mathrm{\Delta }}_{2}d{\tilde{B}}_{\mathit{\sigma }}(u)}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_2^{2}du}}\right)\hfill \\ \hfill & =\frac{{\int }_t^T{\mathrm{\Delta }}_{\mathrm{i}}{\mathrm{\Delta }}_2{\mathit{\rho }}_{{\tilde{B}}_{\overline{S}}{\tilde{B}}_{\mathit{\sigma }}}du}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_1^{\mathrm{two}}du}\sqrt{{\int }_t^T{\mathrm{\Delta }}_2^{\mathrm{two}}du}},\hfill \\ \hfill {\mathit{\rho }}_{{\tilde{B}}_{\overline{S}}{\tilde{B}}_{\mathit{\sigma }}}& =\frac{({\mathit{\sigma }}_{\mathrm{Due\; south}}(t){\mathit{\sigma }}_{\mathrm{Five}}(t){\mathit{\rho }}_{SV}+{\mathit{\sigma }}_{\mathrm{South}}(t){\mathit{\sigma }}_D(t){\mathit{\rho }}_{SD})}{\sqrt{({\mathit{\sigma }}_r^2(t){\mathrm{grand}}^{\mathrm{ii}}(t,T)+{\mathit{\sigma }}_S^{\mathrm{two}}(t))}\sqrt{({\mathit{\sigma }}_r^{\mathrm{two}}(t)m^{\mathrm{two}}(t,T)+2{\mathit{\sigma }}_V(t){\mathit{\sigma }}_D(t)\mathrm{yard}(t,T){\mathit{\rho }}_{5D})}}.\hfill \end{array}$$

Evaluation of term $D_{\mathrm{two}}$ : firstly we define a new probability measure, since ${\tilde{B}}_{\mathit{\sigma }}(t),{\tilde{B}}_{\overline{S}}(t)$ are standard brown motions under the measure $\tilde{P}$ . Define Rodon-Nikodym derivative: $$\begin{array}{c}\hfill \frac{\text{d}\widehat{P}}{\text{d}\tilde{P}}|{\mathcal{F}}_t=exp\left\{\underset{t}{\overset{T}{\int }}{\mathrm{\Delta }}_{2}d{\tilde{B}}_{\overline{S}}(u)-\frac{\mathrm{i}}{2}\underset{t}{\overset{T}{\int }}{\mathrm{\Delta }}_{\mathrm{two}}^{2}du\right\}{Z}_1(t).\end{array}$$

Let $$\begin{array}{cc}& d{\widehat{B}}_{\mathit{\sigma }}(t)=-{\mathrm{\Delta }}_{\mathrm{two}}dt+d{\tilde{B}}_{\mathit{\sigma }}(t),\hfill \\ \hfill & d{\widehat{B}}_{\overline{S}}=-{\mathit{\rho }}_{\overline{\mathrm{South}}\mathit{\sigma }}{\mathrm{\Delta }}_{2}dt+d{\tilde{B}}_{\overline{S}}(t)=-\frac{{\mathit{\sigma }}_{SV}(t)-{\mathit{\sigma }}_{SD}(t)}{{\mathrm{\Delta }}_1}dt+d{\tilde{B}}_{\overline{\mathrm{Southward}}}(t),\hfill \end{array}$$

where ${\mathit{\rho }}_{\overline{S}\mathit{\sigma }}=\frac{{\mathit{\sigma }}_{\mathrm{Due\; south}V}(t)-{\mathit{\sigma }}_{SD}(t)}{{\mathrm{\Delta }}_{\mathrm{ane}}{\mathrm{\Delta }}_2},{\mathit{\sigma }}_{\mathrm{Due\; south}5}(t)={\mathit{\rho }}_{\mathrm{Due\; south}V}{\mathit{\sigma }}_{\mathrm{Due\; south}}(t){\mathit{\sigma }}_V(t),{\mathit{\sigma }}_{SD}(t)={\mathit{\rho }}_{SD}{\mathit{\sigma }}_S(t){\mathit{\sigma }}_V(t)$ .

According to the Lemma ane, we see that ${\widehat{B}}_{\mathit{\sigma }}(t),{\widehat{B}}_{\overline{S}}(t)$ are standard brown motions nether the mensurate $\widehat{P}$ .

Under the mensurate $\widehat{P}$ , nosotros have $$\begin{array}{cc}\hfill \mathrm{South}(T)& =S(t)exp\left\{G(t,T,r(t))-\underset{t}{\overset{T}{\int }}({\mathit{\sigma }}_S^2(u)/2-{\mathit{\sigma }}_{\mathrm{Southward}5}(u)\right.\hfill \\ \hfill & \left.+{\mathit{\sigma }}_{\mathrm{Southward}D}(u))du+\underset{t}{\overset{T}{\int }}{\mathrm{\Delta }}_{1}d{\widehat{B}}_{\overline{S}}(t)\right\},\hfill \\ \hfill \mathit{\delta }(T)& =\mathit{\delta }(t)exp\left\{\underset{t}{\overset{T}{\int }}\left[\frac{1}{2}({\mathit{\sigma }}_D^2(u)-{\mathit{\sigma }}_V^2(u))+{\mathrm{\Delta }}_{\mathrm{two}}^2\right]du+\underset{t}{\overset{T}{\int }}{\mathrm{\Delta }}_{\mathrm{two}}d{\widehat{B}}_{\mathit{\sigma }}(u)\right\}.\hfill \end{array}$$

Substituting this into $D_{\mathrm{ii}}$ , we become $$\begin{array}{cc}\hfill D_2& =M\widetilde{\mathrm{Due\; east}}(\mathit{\delta }(T)R{I}_{A_2}|{\mathcal{F}}_t)\hfill \\ \hfill & =\mathrm{1000}R\tilde{E}\left[\mathit{\delta }(t)exp\left\{\underset{t}{\overset{T}{\int }}\left[\frac{1}{2}({\mathit{\sigma }}_D^2(u)-{\mathit{\sigma }}_{\mathrm{Five}}^2(u))+{\mathrm{\Delta }}_2^2\right]du+\underset{t}{\overset{T}{\int }}{\mathrm{\Delta }}_{2}d{\widehat{B}}_{\mathit{\sigma }}(u)\right\}{I}_{A_{\mathrm{two}}}|{\mathcal{F}}_t\right]\hfill \\ \hfill & =\mathrm{Chiliad}R\mathit{\delta }(t)exp\left\{\underset{t}{\overset{T}{\int }}\left[\frac{1}{\mathrm{ii}}({\mathit{\sigma }}_D^2(u)-{\mathit{\sigma }}_5^{\mathrm{two}}(u))+{\mathrm{\Delta }}_{\mathrm{ii}}^{\mathrm{two}}\right]du-\frac{1}{\mathrm{ii}}\underset{t}{\overset{T}{\int }}{\mathrm{\Delta }}_2^{2}du\right\}\tilde{E}[Z_1(t)Z_1^{-\mathrm{one}}(t)I_{A_2}|{\mathcal{F}}_t]\hfill \\ \hfill & =GR\mathit{\delta }(t)exp\left\{\underset{t}{\overset{T}{\int }}({\mathit{\sigma }}_D^2(u)-{\mathit{\sigma }}_{VD}(u))\right\}\widehat{E}[I_{A_2}|{\mathcal{F}}_t]\hfill \\ \hfill & =GR\mathit{\delta }(t)exp\left\{\underset{t}{\overset{T}{\int }}({\mathit{\sigma }}_D^2(u)-{\mathit{\sigma }}_{VD}(u))\right\}\widehat{\mathrm{Due\; east}}[\mathrm{South}(T)>\mathrm{Thousand},\mathit{\delta }(T)<\mathrm{i}|{\mathcal{F}}_t]\hfill \\ \hfill & =GR\mathit{\delta }(t)\widehat{P}\left[\left(-\frac{{\int }_t^T{\mathrm{\Delta }}_{\mathrm{one}}d{\widehat{B}}_{\overline{\mathrm{Due\; south}}}(u)}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_1^{2}du}}<\frac{ln\frac{\mathrm{Southward}(t)}{\mathrm{Chiliad}}+G(t,T,r(t))-\underset{t}{\overset{T}{\int }}({\mathit{\sigma }}_{\mathrm{Due\; south}}^2(u)/2-{\mathit{\sigma }}_{SV}(u)+{\mathit{\sigma }}_{\mathrm{Due\; south}D}(u))du}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_1^{\mathrm{two}}du}},\right.\right.\hfill \\ \hfill & \left.-\frac{{\int }_t^T{\mathrm{\Delta }}_{\mathrm{ii}}d{\widehat{B}}_{\mathit{\sigma }}(u)}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_2^{2}du}}>\frac{ln\mathit{\delta }(t)+{\int }_t^T\left[\frac{1}{2}({\mathit{\sigma }}_D^2(u)-{\mathit{\sigma }}_{\mathrm{Five}}^{\mathrm{ii}}(u))+{\mathrm{\Delta }}_2^2\right]du}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_2^{\mathrm{two}}du}})|{\mathcal{F}}_t\right]\hfill \\ \hfill & =GR\mathit{\delta }(t)b_{2}N\left(d_{\mathrm{i}}+a_{\mathrm{i}},-\left(d_2+\sqrt{{\int }_t^T{\mathrm{\Delta }}_2^{\mathrm{two}}du};-\mathit{\rho }\right)\right),\hfill \end{array}$$

where $$\begin{array}{cc}\hfill b_2& =exp\left\{\underset{t}{\overset{T}{\int }}({\mathit{\sigma }}_D^{\mathrm{two}}(u)-{\mathit{\sigma }}_{VD}(u))du\right\};\hfill \\ \hfill a_{\mathrm{i}}& =\frac{{\int }_t^T({\mathit{\sigma }}_{SV}(u)-{\mathit{\sigma }}_{SD}(u))du}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_1^{2}du}},\hfill \\ \hfill \mathit{\rho }& =\frac{{\int }_t^T{\mathrm{\Delta }}_1{\mathrm{\Delta }}_2{\mathit{\rho }}_{{\tilde{B}}_{\overline{\mathrm{Southward}}}{\tilde{B}}_{\mathit{\sigma }}}du}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_1^{2}du}\sqrt{{\int }_t^T{\mathrm{\Delta }}_{\mathrm{ii}}^{2}du}}\hfill \\ \hfill & =\text{Corr}\left(-\frac{{\int }_t^T{\mathrm{\Delta }}_{1}d{\widehat{B}}_{\overline{S}}(u)}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_{\mathrm{ane}}^{\mathrm{two}}du}},-\frac{{\int }_t^T{\mathrm{\Delta }}_{2}d{\widehat{B}}_{\mathit{\sigma }}(u)}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_2^{2}du}}\right).\hfill \end{array}$$

And then, by the issue of $D_{\mathrm{ane}},D_{\mathrm{two}}$ , we complete the proof of the Theorem 2.

Corollary 1

Under the house value model, assume the price of underlying asset at time t is ${\mathrm{Due\; south}}_t$ , exercise price is Yard, the ratio of bounty is $\mathit{\delta }(T)=V(T)/D(T)$ , the toll of vulnerable cash or nothing put option at the fourth dimension $t(0\le t\le T)$ is as follows: (25) $$\begin{array}{c}\hfill C(t)=\mathrm{Yard}RN(-d_{\mathrm{i}},d_2;-\mathit{\rho })+\mathrm{Chiliad}R\mathit{\sigma }(t)b_2\mathrm{North}\left(-(d_{\mathrm{ane}}+a_1),-\left(d_2+\sqrt{{\int }_t^T{\mathrm{\Delta }}_2^{2}du}\right);\mathit{\rho })\right),\end{array}$$ (25)

where $\mathrm{Yard},d_1,d_2,a_1,b_2,\mathit{\rho }$ are given in the Theorem 2.

Theorem iii

Nether the firm value model, assume the price of underlying asset at fourth dimension t is $S_t$ , exercise cost is K, the ratio of compensation is $\mathit{\delta }(T)=V(T)/D(T)$ , the cost of vulnerable asset or cipher phone call choice at the time $t(0\le t\le T)$ is as follows: (26) $$\begin{array}{cc}\hfill C(t)& =S(t)\mathrm{North}(d_1+\sqrt{{\int }_t^T{\mathrm{\Delta }}_{\mathrm{ane}}^{2}du},d_2+a_2;\mathit{\rho }))\hfill \\ \hfill & +S(t)\mathit{\delta }(t)b_{1}N\left(d_1+a_1+2\sqrt{{\int }_t^T{\mathrm{\Delta }}_{\mathrm{ane}}^{2}du},-(d_2+2a_2+\sqrt{{\int }_t^T{\mathrm{\Delta }}_{\mathrm{two}}^{2}du});-\mathit{\rho }\right).\hfill \end{array}$$ (26)

where $$\begin{array}{cc}\hfill a_2& =\frac{{\int }_t^T({\mathit{\sigma }}_{SV}(u)-{\mathit{\sigma }}_{\mathrm{Due\; south}D}(u))du}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_2^{\mathrm{two}}du}},\hfill \\ \hfill b_1& =exp\underset{t}{\overset{T}{\int }}(-{\mathit{\sigma }}_V^2(u)-{\mathit{\sigma }}_S^2(u)+{\mathit{\sigma }}_{5D}(u)+{\mathit{\sigma }}_{\mathrm{South}V}(u)-{\mathit{\sigma }}_{\mathrm{Southward}D}(u))du,\hfill \end{array}$$ $a_1,d_{\mathrm{ane}},d_2,\mathit{\rho }$ are given in the theorem 2.

Proof

Let $A_1=(S(T)>\mathrm{Thou},\mathit{\delta }(T)\ge 1),A_2=(S(T)>\mathrm{Grand},\mathit{\delta }(T)<1)$ , according to (thirteen), nether the take a chance neutral probability, the price of vulnerable cash or zero call option is equally follows: $$\begin{array}{cc}\hfill C(t)& =\widetilde{\mathrm{East}}\left[exp\left\{-\underset{t}{\overset{T}{\int }}r(u)du\right\}(X(T)I_{\{{\mathit{\delta }}_{(T)}\ge 1\}}+\mathit{\sigma }(T)X(T)I_{\{{\mathit{\delta }}_{(T)}<1\}})|{\mathcal{F}}_t\right]\hfill \\ \hfill & =\widetilde{\mathrm{East}}\left[exp\left\{-\underset{t}{\overset{T}{\int }}r(u)du\right\}(S(T)I_{A_1}+\mathit{\delta }(T)S(T)I_{A_2})|{\mathcal{F}}_t\right]\hfill \\ \hfill & =\tilde{E}\left[exp\left\{-\mathrm{Chiliad}(t,T,r(t))-\underset{t}{\overset{T}{\int }}{\mathit{\sigma }}_r(u)\mathrm{thousand}(u,T)d{\tilde{B}}_r(u)\right\}(S(T)I_{A_1}+\mathit{\delta }(T)S(T)I_{A_2})|{\mathcal{F}}_t\right]\hfill \\ \hfill & =\tilde{E}\left[exp\left\{-G(t,T,r(t))+\frac{\mathrm{ane}}{2}{\int }_t^T{\mathit{\sigma }}_r^{\mathrm{ii}}(u)k^{\mathrm{ii}}(u,T)du\right\}(S(T)I_{A_1}+\mathit{\delta }(T)S(T)I_{A_2})|{\mathcal{F}}_t\right]\hfill \\ \hfill & =\mathrm{1000}\widetilde{\mathrm{Due\; east}}[(S(T)I_{A_1}+\mathit{\delta }(T)\mathrm{Southward}(T)I_{A_2})|{\mathcal{F}}_t]\hfill \\ \hfill & =G\tilde{E}[S(T)I_{A_{\mathrm{one}}}|{\mathcal{F}}_t]+G\widetilde{\mathrm{Due\; east}}[\mathit{\delta }(T)S(T)I_{A_{\mathrm{ii}}}|{\mathcal{F}}_t]\hfill \\ \hfill & =V_1+V_{\mathrm{ii}},\hfill \end{array}$$

where $G=exp\left\{-G(t,T,r(t))+\frac{1}{\mathrm{ii}}\underset{t}{\overset{T}{\int }}{\mathit{\sigma }}_r^2(u)k^2(u,T)du\right\}$ .

Evaluation of $V_{\mathrm{one}}$ : we can define a new probability measure equivalent to $\overline{P}$ .

${\tilde{B}}_{\overline{\mathrm{South}}}(t),{\tilde{B}}_{\mathit{\sigma }}(t)$ are standard brown motions nether the measure $\tilde{P}$ . Denote $\text{d}{\tilde{B}}_{\overline{S}}(t)\text{d}{\tilde{B}}_{\mathit{\sigma }}(t)={\mathit{\rho }}_{\overline{S}\mathit{\sigma }}\text{d}t$ .

Define the Rodon-Nikodym derivative: $$\begin{array}{c}\hfill \frac{\text{d}\overline{P}}{\text{d}\tilde{P}}|{\mathcal{F}}_t=exp\left\{\underset{t}{\overset{T}{\int }}{\mathrm{\Delta }}_1\text{d}{\overline{B}}_{\overline{S}}(u)-\frac{\mathrm{one}}{2}\underset{t}{\overset{T}{\int }}{\mathrm{\Delta }}_1^2\text{d}u\right\}{Z}_2(t).\end{array}$$

Let $\text{d}{\overline{B}}_{\overline{\mathrm{Due\; south}}}(t)=-{\mathrm{\Delta }}_1\text{d}t+\text{d}{\tilde{B}}_{\overline{S}}(t)$ , $\text{d}{\overline{B}}_{\mathit{\sigma }}(t)=-{\mathit{\rho }}_{\overline{\mathrm{Due\; south}}\mathit{\sigma }}{\mathrm{\Delta }}_{\mathrm{one}}\text{d}t+\text{d}{\overline{B}}_{\mathit{\sigma }}(t)=-\frac{{\mathit{\sigma }}_{\mathrm{Southward}V}(t)-{\mathit{\sigma }}_{SD}(t)}{{\mathrm{\Delta }}_{\mathrm{two}}}\text{d}t+\text{d}{\overline{B}}_{\mathit{\sigma }}(t)$ , according to the Lemma ane, we run into that ${\overline{B}}_{\overline{S}}(t),{\overline{B}}_{\mathit{\sigma }}(t)$ are standard brown motions under the measure $\overline{P}$ .

Under the mensurate $\overline{P}$ , we obtain $$\begin{array}{cc}\hfill \mathrm{South}(T)& =\mathrm{South}(t)exp\left\{G(t,T,r(t))-\underset{t}{\overset{T}{\int }}({\mathit{\sigma }}_S^{\mathrm{ii}}(u)/2-{\mathrm{\Delta }}_1^2)du+\underset{t}{\overset{T}{\int }}{\mathrm{\Delta }}_{1}d{\overline{B}}_{\overline{S}}(u)\right\},\hfill \\ \hfill \mathit{\delta }(T)& =\mathit{\delta }(t)exp\left\{\underset{t}{\overset{T}{\int }}\left[\frac{\mathrm{one}}{2}({\mathit{\sigma }}_D^{\mathrm{two}}(u)-{\mathit{\sigma }}_{\mathrm{Five}}^2(u))+({\mathit{\sigma }}_{SV}(u)+{\mathit{\sigma }}_{\mathrm{Southward}D}(u))\right]du+\underset{t}{\overset{T}{\int }}{\mathrm{\Delta }}_{\mathrm{two}}d{\overline{B}}_{\mathit{\sigma }}(u)\right\}.\hfill \end{array}$$

Substituting thus into ${\mathrm{Five}}_1$ , we accept $$\begin{array}{cc}\hfill 5_1& =G\widetilde{\mathrm{Due\; east}}(\mathrm{South}(T)I_{A_{\mathrm{two}}}|{\mathcal{F}}_t)\hfill \\ \hfill & =G\tilde{E}\left[\mathrm{South}(t)exp\left\{\mathrm{Thousand}(t,T,r(t))-\underset{t}{\overset{T}{\int }}(\frac{{\mathit{\sigma }}_S^{\mathrm{two}}(u)}{\mathrm{ii}}du-{\mathrm{\Delta }}_{\mathrm{one}}^2)du\right.\right.\hfill \\ \hfill & \left.\left.+\underset{t}{\overset{T}{\int }}{\mathrm{\Delta }}_{1}d{\overline{B}}_{\overline{S}}(u)\right\}{I}_{A_1}|{\mathcal{F}}_t\right]\hfill \\ \hfill & =KS(t)exp\left\{G(t,T,r(t))-\underset{t}{\overset{T}{\int }}\frac{{\mathit{\sigma }}_{\mathrm{South}}^2(u)}{2}du\right.\hfill \\ \hfill & \left.+\frac{\mathrm{i}}{2}\underset{t}{\overset{T}{\int }}{\mathrm{\Delta }}_1^{\mathrm{ii}}du\right\}\tilde{E}[Z_{\mathrm{ii}}(t)Z_2^{-1}(t)I_{A_1}|{\mathcal{F}}_t]\hfill \\ \hfill & =S(t)\overline{\mathrm{Due\; east}}[I_{A_1}|{\mathcal{F}}_t]\hfill \\ \hfill & =S(t)\overline{P}[(S(T)>\mathrm{Chiliad},\mathit{\delta }(T)\ge \mathrm{one})|{\mathcal{F}}_t]\hfill \\ \hfill & =S(t)\overline{P}\left[\left(-\frac{{\int }_t^T{\mathrm{\Delta }}_{1}d{\overline{B}}_{\overline{S}}(u)}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_1^{\mathrm{two}}du}}<\frac{ln\frac{\mathrm{South}(t)}{K}+G(t,T,r(t))-{\int }_t^T({\mathit{\sigma }}_S^2(u)/\mathrm{ii}-{\mathrm{\Delta }}_1^{\mathrm{two}})du}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_1^{\mathrm{two}}du}},-\frac{{\int }_t^T{\mathrm{\Delta }}_{\mathrm{two}}d{\overline{B}}_{\mathit{\sigma }}(u)}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_2^{\mathrm{two}}du}}\right.\right.\hfill \\ \hfill & \left.<\frac{ln\mathit{\delta }(t)+{\int }_t^T\left[\frac{\mathrm{one}}{2}({\mathit{\sigma }}_D^2(u)-{\mathit{\sigma }}_V^2(u))+{\int }_t^T({\mathit{\sigma }}_{SV}(u)-{\mathit{\sigma }}_{SD}(u))du\right]}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_2^{2}du}})|{\mathcal{F}}_t\right]\hfill \\ \hfill & =\mathrm{Due\; south}(t)N\left(d_1+\sqrt{{\int }_t^T{\mathrm{\Delta }}_1^{2}du},d_2+a_2;\mathit{\rho }\right),\hfill \end{array}$$

where $$\begin{array}{cc}\hfill a_{\mathrm{two}}& =\frac{{\int }_t^T({\mathit{\sigma }}_{SV}(u)-{\mathit{\sigma }}_{SD}(u))du}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_2^{2}du}},\hfill \\ \hfill \mathit{\rho }& =\frac{{\int }_t^T{\mathrm{\Delta }}_1{\mathrm{\Delta }}_2{\mathit{\rho }}_{{\tilde{B}}_{\overline{\mathrm{South}}}{\tilde{B}}_{\mathit{\sigma }}}du}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_1^{2}du}\sqrt{{\int }_t^T{\mathrm{\Delta }}_2^{\mathrm{two}}du}}\hfill \\ \hfill & =\text{Corr}\left(-\frac{{\int }_t^T{\mathrm{\Delta }}_{\mathrm{i}}d{\overline{B}}_{\overline{\mathrm{Southward}}}(u)}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_1^{2}du}},-\frac{{\int }_t^T{\mathrm{\Delta }}_{2}d{\overline{B}}_{\mathit{\sigma }}(u)}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_2^{2}du}}\right).\hfill \end{array}$$

Evaluation of $5_{\mathrm{two}}$ , under the measure $\tilde{P}$ , we get $$\begin{array}{cc}\hfill \mathit{\delta }(T)\mathrm{South}(T)& =\mathrm{Southward}(t)\mathit{\delta }(t)exp\left\{K(t,T,r(t))+\underset{t}{\overset{T}{\int }}\left[\frac{\mathrm{i}}{2}({\mathit{\sigma }}_D^2(u)-{\mathit{\sigma }}_V^2(u)-{\mathit{\sigma }}_S^{\mathrm{ii}}(u))\right]du\right.\hfill \\ \hfill & \left.+\underset{t}{\overset{T}{\int }}({\mathrm{\Delta }}_{1}d{\tilde{B}}_{\overline{S}}(u)+\underset{t}{\overset{T}{\int }}{\mathrm{\Delta }}_{\mathrm{ii}}d{\tilde{B}}_{\mathit{\sigma }}(u))\right\}\hfill \\ \hfill & =\mathrm{South}(t)\mathit{\delta }(t)exp\left\{G(t,T,r(t))+\underset{t}{\overset{T}{\int }}\left[\frac{1}{2}({\mathit{\sigma }}_D^{\mathrm{ii}}(u)-{\mathit{\sigma }}_{\mathrm{Five}}^{\mathrm{two}}(u)-{\mathit{\sigma }}_{\mathrm{South}}^2(u))\right]du\right.\hfill \\ \hfill & \left.+\underset{t}{\overset{T}{\int }}{\mathrm{\Delta }}_{3}d\tilde{B}(u)\right\},\hfill \end{array}$$

where $\text{d}\tilde{B}(t)=\frac{{\mathrm{\Delta }}_1\text{d}{\tilde{B}}_{\overline{S}}(t)+{\mathrm{\Delta }}_2\text{d}{\tilde{B}}_{\mathit{\sigma }}(t)}{{\mathrm{\Delta }}_{\mathrm{iii}}},{\mathrm{\Delta }}_3=\sqrt{{\mathrm{\Delta }}_{\mathrm{ane}}^2+{\mathrm{\Delta }}_2^2-2{\mathit{\rho }}_{\overline{\mathrm{Due\; south}}\mathit{\sigma }}{\mathrm{\Delta }}_{\mathrm{i}}{\mathrm{\Delta }}_2}$ , and so co-ordinate to Lemma 1, we run across that $\tilde{B}(t)$ is standard chocolate-brown motion nether the mensurate $\tilde{P}$ .

Define the Rodon-Nikodym derivative: $$\begin{array}{c}\hfill \frac{\text{d}\check{P}}{\text{d}\tilde{P}}|{\mathcal{F}}_t=exp\left\{\underset{t}{\overset{T}{\int }}{\mathrm{\Delta }}_{3}d\tilde{B}(u)-\frac{1}{2}\underset{t}{\overset{T}{\int }}{\mathrm{\Delta }}_{\mathrm{three}}^{2}du\right\}{Z}_{\mathrm{three}}(t).\end{array}$$

Let $\text{d}\check{B}(t)=-{\mathrm{\Delta }}_3\text{d}t+\text{d}\tilde{B}(t),\text{d}{\check{B}}_{\overline{S}}(t)=-\frac{{\mathit{\sigma }}_{\mathrm{Due\; south}V}(t)-{\mathit{\sigma }}_{SD}(t)+{\mathrm{\Delta }}_{\mathrm{ane}}^{\mathrm{ii}}}{{\mathrm{\Delta }}_1}\text{d}t+\text{d}{\overline{B}}_{\overline{S}}(t),\text{d}{\check{B}}_{\mathit{\sigma }}(t)=-\frac{{\mathit{\sigma }}_{\mathrm{Southward}V}(t)-{\mathit{\sigma }}_{SD}(t)+{\mathrm{\Delta }}_2^2}{{\mathrm{\Delta }}_{\mathrm{ii}}}\text{d}t+\text{d}{\overline{B}}_{\mathit{\sigma }}(t)$ , co-ordinate to the Lemma 1, we can get that $(\check{B}(t),{\check{B}}_{\overline{S}}(t),{\check{B}}_{\mathit{\sigma }}(t))$ is standard dark-brown motion nether the measure $\check{P}$ .

Under the mensurate $\check{P}$ , we get $$\begin{array}{cc}\hfill S(T)& =\mathrm{Due\; south}(t)exp\left\{G(t,T,r(t))-\underset{t}{\overset{T}{\int }}({\mathit{\sigma }}_{\mathrm{Southward}}^2(u)/2-{\mathit{\sigma }}_{\mathrm{Due\; south}V}(u)\right.\hfill \\ \hfill & \left.+{\mathit{\sigma }}_{SD}(u)-2{\mathrm{\Delta }}_{\mathrm{one}}^2)du+\underset{t}{\overset{T}{\int }}{\mathrm{\Delta }}_{\mathrm{ane}}d{\check{B}}_{\overline{\mathrm{Due\; south}}}(u)\right\},\hfill \\ \hfill \mathit{\delta }(T)& =\mathit{\delta }(t)exp\left\{\underset{t}{\overset{T}{\int }}\left[\frac{1}{2}({\mathit{\sigma }}_D^2(u)-{\mathit{\sigma }}_V^2(u))+2({\mathit{\sigma }}_{SV}(u)\right.\right.\hfill \\ \hfill & \left.\left.-{\mathit{\sigma }}_{SD}(u)+{\mathrm{\Delta }}_2^2)\right]du+\underset{t}{\overset{T}{\int }}{\mathrm{\Delta }}_{2}d{\check{B}}_{\mathit{\sigma }}(u)\right\}.\hfill \end{array}$$

Substituting this into $V_2$ , we have $$\begin{array}{cc}\hfill {\mathrm{Five}}_{\mathrm{ii}}& =\mathrm{One\; thousand}\widetilde{\mathrm{Eastward}}(\mathrm{Due\; south}(T)\mathit{\delta }(T)I_{A_2}|{\mathcal{F}}_t)\hfill \\ \hfill & =G\widetilde{\mathrm{Eastward}}[S(t)\mathit{\delta }(t)exp\left\{\mathrm{Chiliad}(t,T,r(t))-\underset{t}{\overset{T}{\int }}\frac{\mathrm{one}}{2}({\mathit{\sigma }}_D^{\mathrm{ii}}(u)-{\mathit{\sigma }}_V^{\mathrm{two}}(u)\right.\hfill \\ \hfill & \left.\left.-{\mathit{\sigma }}_{\mathrm{Southward}}^{\mathrm{two}}(u))du+\underset{t}{\overset{T}{\int }}{\mathrm{\Delta }}_{3}d{\tilde{B}}_S(u)\right\}{I}_{A_2}|{\mathcal{F}}_t\right]\hfill \\ \hfill & =S(t)\mathit{\delta }(t)exp\left\{{\int }_t^T(-{\mathit{\sigma }}_{\mathrm{Five}}^{\mathrm{ii}}(u)-{\mathit{\sigma }}_{\mathrm{Southward}}^{\mathrm{two}}(u)+{\mathit{\sigma }}_{\mathrm{Five}D}(u)-{\mathit{\sigma }}_{SV}(u)\right.\hfill \\ \hfill & \left.+{\mathit{\sigma }}_{SD}(u))du\right\}\widetilde{\mathrm{East}}[Z_{\mathrm{three}}(t)Z_3^{-1}(t)I_{A_2}|{\mathcal{F}}_t]\hfill \\ \hfill & =S(t)\mathit{\delta }(t)b_1\check{E}[I_{A_{\mathrm{two}}}|{\mathcal{F}}_t]\hfill \\ \hfill & =S(t)\mathit{\delta }(t)b_{\mathrm{one}}\check{P}[(S(T)>K,\mathit{\delta }(T)<1)|{\mathcal{F}}_t]\hfill \\ \hfill & =S(t)\mathit{\delta }(t)b_1\check{P}\left[\left(-\frac{{\int }_t^T{\mathrm{\Delta }}_{1}d{\check{B}}_{\overline{S}}(u)}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_1^{2}du}}<\frac{ln\frac{\mathrm{Due\; south}(t)}{K}}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_1^{2}du}}\right.\right.\hfill \\ \hfill & +\frac{\mathrm{1000}(t,T,r(t))-\underset{t}{\overset{T}{\int }}\left(\frac{{\mathit{\sigma }}_{\mathrm{Due\; south}}^2(u)}{2}-{\mathit{\sigma }}_{SV}(u)-{\mathit{\sigma }}_{SD}(u)-2{\mathrm{\Delta }}_1^{\mathrm{ii}}\right)du}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_{\mathrm{ane}}^{2}du}}-\frac{{\int }_t^T{\mathrm{\Delta }}_{\mathrm{ii}}d{\check{B}}_{\mathit{\sigma }}(u)}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_2^{\mathrm{two}}du}}\hfill \\ \hfill & \left.>\frac{ln\mathit{\delta }(t)+{\int }_t^T\left[\frac{1}{\mathrm{ii}}({\mathit{\sigma }}_D^2(u)-{\mathit{\sigma }}_V^2(u)+2{\mathit{\sigma }}_{\mathrm{South}\mathrm{Five}}(u)-2{\mathit{\sigma }}_{SD}(u)+{\mathrm{\Delta }}_2^2)du\right]}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_2^{2}du}})|{\mathcal{F}}_t\right]\hfill \\ \hfill & =S(t)\mathit{\delta }(t)b_{1}N\left(d_1+a_1+2\sqrt{{\int }_t^T{\mathrm{\Delta }}_1^{2}du},-\left(d_2+\mathrm{ii}{a}_2+\sqrt{{\int }_t^T{\mathrm{\Delta }}_2^{2}du}\right);-\mathit{\rho }\right),\hfill \end{array}$$

where $$\begin{array}{cc}\hfill b_1& =exp\left\{\underset{t}{\overset{T}{\int }}(-{\mathit{\sigma }}_V^2(u)-{\mathit{\sigma }}_{\mathrm{Due\; south}}^{\mathrm{ii}}(u)+{\mathit{\sigma }}_{\mathrm{Five}D}(u)-{\mathit{\sigma }}_{S\mathrm{Five}}(u)+{\mathit{\sigma }}_{\mathrm{Southward}D}(u))du\right\},\hfill \\ \hfill \mathit{\rho }& =\frac{{\int }_t^T{\mathrm{\Delta }}_1{\mathrm{\Delta }}_{\mathrm{ii}}{\mathit{\rho }}_{{\tilde{B}}_{\overline{S}}{\tilde{B}}_{\mathit{\sigma }}}du}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_1^{\mathrm{two}}du}\sqrt{{\int }_t^T{\mathrm{\Delta }}_2^{2}du}}\hfill \\ \hfill & =\text{Corr}\left(-\frac{{\int }_t^T{\mathrm{\Delta }}_{\mathrm{one}}d{\check{B}}_{\overline{\mathrm{South}}}(u)}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_1^{2}du}},-\frac{{\int }_t^T{\mathrm{\Delta }}_{\mathrm{two}}d{\check{B}}_{\mathit{\sigma }}(u)}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_2^{2}du}}\right).\hfill \end{array}$$

And then by the issue of $D_1,D_2$ , we complete the proof of Theorem three.

Corollary two

Nether the firm value model,assume the toll of underlying asset at time t is $S_t$ , exercise toll is Thousand, the ratio of compensation is $\mathit{\delta }(T)=V(T)/D(T)$ , the price of vulnerable asset or nada put selection at the time $t(0\le t\le T)$ is as follows: (27) $$\begin{array}{cc}\hfill C(t)& =-S(t)N\left(-\left(d_1+\sqrt{{\int }_t^T{\mathrm{\Delta }}_1^{2}du}\right),d_{\mathrm{ii}}+a_2;-\mathit{\rho }\right)+\mathrm{Southward}(t)\mathit{\delta }(t)b_{\mathrm{i}}\hfill \\ \hfill & N\left(-\left(d_1+a_{\mathrm{i}}+2\sqrt{{\int }_t^T{\mathrm{\Delta }}_1^{\mathrm{ii}}du}\right),-\left(d_2+2a_{\mathrm{two}}+\sqrt{{\int }_t^T{\mathrm{\Delta }}_{\mathrm{two}}^{2}du}\right);\mathit{\rho }\right),\hfill \end{array}$$ (27)

where $b_1,a_2,\mathit{\rho }$ are given in the Theorem three.

4. Conclusion

The analytical pricing formula of the vulnerable binary options is derived in this paper by using the martingale method under the assumptions that the stock prices, assets and liabilities of a company follow the relevant O-U processes and the interest rate follows a Hull-White model. Comparing with the models of the pricing vulnerable European selection in Klein and Inglis (2001), Ammann (2002),and Ting and Deng (2007), the volatility is corrected to be a function with respect to time rather than a constant, the geometric Brownian movement is corrected to exist an O-U procedure. And also the interest charge per unit is assumed to exist random one following the Hull-White model rather than only a function with respect to time in Ding and Chan(2007). All the corrections make the model more realistic.

Stochstic Calculus And Binary Options,

Source: https://www.tandfonline.com/doi/abs/10.1080/23311835.2017.1340073

Posted by: faulknerhinger1958.blogspot.com

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