stochstic calculus and binary options

Public Interest Argument

The option pricing problem is an of import topic of finance field. Now, nosotros mainly focus on the report of the option pricing problem (for example, the pricing of the vulnerable binary option in this paper), the option pricing formula of the price process of assets which is more reasonable in practice, and some respective real data assay.

1. Introduction

Credit risk is ane of the major financial risk, it has go a major challenge facing the earth's financial markets. On the exchange transaction market, because of the margin requirements, futures exchanging and options exchange derivatives nearly no credit gamble. Therefore, when option pricing happened in the exchange, we can assume credit run a risk does non be. Notwithstanding, On OTC market, counter-party credit hazard is a factor that must be considered. For derivatives on OTC market, since there is no guarantee, option bulls exposure to the credit risk and market risk. The assumption of no default risk no longer valid. Therefore, how to properly consider the counter-party risk factors and establish choice pricing model containing credit risk are of import to investors.

At that place are some articles about option pricing take credit risk theory into consideration. Firm value model is 1 of the virtually important model most vulnerable choice pricing put forward by Merton (1974). Blackness, Fisher, and Cox (1976) studied the risk of bankrupt toll pricing of corporate bonds, improved Merton's thought about events of default. Johnson and Stulz (1987) extended Merton default model, discussing the selection pricing problem under the structural model that has credit take chances, putting forrard the concept of vulnerable option. The seminal piece of work that discussed option pricing containing credit risk is from Johnson and Stulz (1987). They used the term vulnerable option to ascertain those options containing counter-party default risk, and pointed out a lot of features most them. Hull and White (1995) supposed the underlying nugget and the counter-party firm's assets are independent of one another, obtained vulnerable options pricing formula. Jarrow and Tunbull (1995) supposed default fourth dimension obey the strength of $\mathit{\lambda }$ (non-negative constant) independent homogeneous poisson process, used discrete method to create a uncomplicated model of credit risk. Klein (1996) supposed alienation of contract happened with a certain probability at any time, considered the pricing model of null-coupon bonds can be liquidated at initial time cipher.

In recent years, there is a considerable interest in the counter-party risk factors and establish selection 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-political party liability case. They discussed the trouble of option pricing with credit risk and getting its pricing formula. However, people always assume that stock price obey standard brown motility, witch is not reality and has its limitation. Lin, Wang, and Feng (2000) proposed the O-U process model that stock price obeys, fugitive the limitations of standard brown motion. In the previous option pricing model, we assume interest rate is a abiding. However, in reality, involvement rate changes randomly. A lot of scholars conducted all-encompassing research in this field. Zhou, Xinyu, and Gao (2011) gave European option pricing model nether stochastic interest rate. Xue (2000) gave convertible bond pricing formula based on Hull-White Model which satisfy fractional Brownian movement drive.

This article mainly research the trouble of vulnerable binary choice pricing under the O-U process. Based on the firm value model, we presume the stock toll, counter-party business firm's assets and counter-party firm'due south debts obey O-U procedure, interest charge per unit obeys Hull-White Model. By using martingale arroyo nosotros derive an analytical pricing formula for such vulnerable binary option. This article is structured as follows. Section 2 describes business firm value model and its supposition. Section 3 presents the utilise of the martingale approach for vulnerable binary option pricing. Moreover,we obtain the pricing formula of vulnerable binary option. Section 4 contains some conclusion.

2. Model and its assumptions

two.1. Firm value model

Because continuous-time fiscal market, let $(\mathrm{\Omega },\mathcal{F},\{{\mathcal{F}}_t{\}}_{0\le t\le T},P)$ be a filtered probability space such that the filtration satisfies the usual conditions. Based on house value model past Merton (1974), assume Ten(T) is the counter party firm promise to pay at the time T, 10 is European way option, if the counter party firm always have the ability of pay earlier the due maturity engagement, and so the owner of this option will get X(T) at the fourth dimension T, otherwise the payoff depend on the ratio of company's assets and its liability. If we assume Five(T) is the counter party firm'south assets, D(T) is counter party firm'due south liability, when the counter party firm bankrupt, the payoff is $X(T)\times [\mathrm{Five}(T)/D(T)]$ , and then the owner of this vulnerable option'due south yield to maturity is $Y(T)=X(T)I_{\{{\mathit{\delta }}_{(T)}\ge \mathrm{one}\}}+{\mathit{\delta }}_{(T)}\mathrm{10}(T)I_{\{{\mathit{\delta }}_{(T)}<\mathrm{one}\}}$ , where $I_A$ is indicator function of the set up A, $\mathit{\delta }(T)$ is the ratio of compensation, $\mathit{\delta }(T)=V(T)/D(T)$ . Co-ordinate to martingale theory of option pricing, the price of this option at the time $0\le t\le T$ can be represented every bit beneath nether the equivalent martingale measurement $\tilde{P}$ , (1) $$\begin{array}{c}\hfill C(t)=B(t)\tilde{E}\left[\frac{X(T)}{B(T)}(I_{\{{\mathit{\delta }}_{(T)}\ge \mathrm{ane}\}}+{\mathit{\delta }}_{(T)}{I}_{\{{\mathit{\delta }}_{(T)}<1\}})\left|\right.{\mathcal{F}}_t\right],\end{array}$$ (ane)

where B(t) is the price of default-free bonds, satisfy differential equation $dB(t)=r(t)B(t)dt$ , r(t) is risk-free interest charge per unit, ${\mathcal{F}}_t$ is data sets before the time t, $\widetilde{\mathrm{Eastward}}(·)$ is the mathematical expectation under equivalent martingale measurement $\tilde{P}$ .

2.2. Model edifice

Assume stock price $S_t$ , company's assets $V_t$ , company's liability $D_t$ all follow O-U procedure as follows (two) $$\begin{array}{cc}\hfill \text{d}S(t)& =({\mathit{\mu }}_{\mathrm{South}}(t)-{\mathit{\alpha }}_{\mathrm{South}}(t)lnS(t))\mathrm{Due\; south}(t)\text{d}t+{\mathit{\sigma }}_{\mathrm{South}}(t)S(t)\text{d}{B}_{\mathrm{Due\; south}}(t),\hfill \\ \hfill \text{d}V(t)& =({\mathit{\mu }}_5(t)-{\mathit{\alpha }}_5(t)ln\mathrm{Five}(t))V(t)\text{d}t+{\mathit{\sigma }}_V(t)V(t)\text{d}{B}_V(t),\hfill \\ \hfill \text{d}D(t)& =({\mathit{\mu }}_D(t)-{\mathit{\alpha }}_D(t)lnD(t))D(t)\text{d}t+{\mathit{\sigma }}_D(t)D(t)\text{d}{B}_D(t),\hfill \end{array}$$ (2)

where $B_S(t),B_V(t),B_D(t)$ is a standard 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 return charge per unit and volatility respectively, which are continuous functions nigh t.

O-U process is a Dark-brown move considering expected return charge per unit depends on underlying assets'southward price, ${\mathit{\alpha }}_{\mathit{\xi }}(\mathit{\xi }=S,5,D)$ can subtract underlying assets's cost when the price up to a sure height. Assume $\mathbf{B}=(B_S(t),B_V(t),B_D(t))$ is a 3-dimensional correlated ${\mathcal{F}}_t$ -Dark-brown motion with the correlation matrix C given by (3) $$\begin{array}{c}\hfill \mathbf{C}=\left(\begin{array}{ccc}\mathrm{ane}& {\mathit{\rho }}_{SV}& {\mathit{\rho }}_{SD}\\ {\mathit{\rho }}_{SV}& 1& {\mathit{\rho }}_{\mathrm{Five}D}\\ {\mathit{\rho }}_{\mathrm{Due\; south}D}& {\mathit{\rho }}_{VD}& \mathrm{one}\end{array}\right)\end{array}$$ (3)

In the following, nosotros shall introduce Girsanov's theorem and martingale representation theorem about multidimensional correlated Brown motion B.

Allow $\mathrm{\Lambda }(t)=({\mathit{\lambda }}_{\mathrm{South}}(t),{\mathit{\lambda }}_V(t),{\mathit{\lambda }}_D(t))$ is a 3-dimensional ${\mathcal{F}}_t$ -adapted process, where ${\mathit{\lambda }}_{\mathit{\xi }}(t)=[{\mathit{\mu }}_{\mathit{\xi }}-{\mathit{\alpha }}_{\mathit{\xi }}ln{\mathit{\xi }}_t-r(t)]/{\mathit{\sigma }}_{\mathit{\xi }},(\mathit{\xi }=\mathrm{South},V,D)$ , now ascertain $\mathit{\lambda }(t)={(\mathrm{\Lambda }(t)\mathbf{C}{\mathrm{\Lambda }}^{\ast }(t))}^{\frac{1}{2}},\forall 0\le t\le T$ , satisfy Novikov status, where ${\mathrm{\Lambda }}^{\ast }(t)$ is the transposition of $\mathrm{\Lambda }(t)$ , C is the matrix defined by formula (5). Define (4) $$\begin{array}{c}\hfill Z_{\mathit{\lambda }}(t)=exp\left\{-\underset{0}{\overset{t}{\int }}\mathrm{\Lambda }(\mathrm{southward})d\mathbf{B}(s)-\frac{1}{\mathrm{two}}\underset{0}{\overset{t}{\int }}{\mathrm{\Lambda }}^{\mathrm{two}}(s)d\mathrm{southward}\right.\},\forall 0\le t\le T,\end{array}$$ (4)

then $Z_{\mathit{\lambda }}(t)$ is a ${\mathcal{F}}_t$ -martingale. We further assume that for each $0\le t\le T$ , ${\mathit{\lambda }}_{\mathit{\xi }}(t)\ne 0$ , and there exists a abiding ${\mathit{\rho }}_{\mathit{\lambda }\mathit{\xi }},\mathit{\xi }=S,5,D$ such that (5) $$\begin{array}{cc}\hfill {\mathit{\rho }}_{\mathit{\lambda }S}\mathit{\lambda }(t)& ={\mathit{\lambda }}_S(t)+{\mathit{\rho }}_{\mathrm{South}V}{\mathit{\lambda }}_V(t)+{\mathit{\rho }}_{SD}{\mathit{\lambda }}_D(t),\hfill \end{array}$$ (five) (6) $$\begin{array}{cc}\hfill {\mathit{\rho }}_{\mathit{\lambda }V}\mathit{\lambda }(t)& ={\mathit{\rho }}_{S5}{\mathit{\lambda }}_{\mathrm{South}}(t)+{\mathit{\lambda }}_V(t)+{\mathit{\rho }}_{5D}{\mathit{\lambda }}_D(t),\hfill \end{array}$$ (6) (7) $$\begin{array}{cc}\hfill {\mathit{\rho }}_{\mathit{\lambda }D}\mathit{\lambda }(t)& ={\mathit{\rho }}_{\mathrm{South}D}{\mathit{\lambda }}_{\mathrm{Due\; south}}(t)+{\mathit{\rho }}_{VD}{\mathit{\lambda }}_V(t)+{\mathit{\lambda }}_D(t).\hfill \end{array}$$ (7)

Theorem one

On the measurable space $(\mathrm{\Omega },{\mathcal{F}}_T)$ , nosotros ascertain new probability mensurate $\tilde{P}$ , and for each $A\in {\mathcal{F}}_t$ , $\tilde{P}(A)={\mathrm{Eastward}}_P[I_{A}Z(T)]$ , then $\tilde{P}$ is the martingale measure equivalent to P, such that (viii) $$\begin{array}{cc}\hfill {\tilde{B}}_S(t)& =B_S(t)+\underset{0}{\overset{t}{\int }}{\mathit{\rho }}_{\mathit{\lambda }S}\mathit{\lambda }(\mathrm{south})ds,\hfill \end{array}$$ (8) (9) $$\begin{array}{cc}\hfill {\tilde{B}}_V(t)& =B_V(t)+\underset{0}{\overset{t}{\int }}{\mathit{\rho }}_{\mathit{\lambda }\mathrm{Five}}\mathit{\lambda }(s)ds,\hfill \end{array}$$ (nine) (10) $$\begin{array}{cc}\hfill {\tilde{B}}_D(t)& =B_D(t)+\underset{0}{\overset{t}{\int }}{\mathit{\rho }}_{\mathit{\lambda }D}\mathit{\lambda }(s)d\mathrm{south},\hfill \end{array}$$ (10)

for $0\le t\le T,{\tilde{B}}_S(t),{\tilde{B}}_V(t),{\tilde{B}}_D(t)$ are all ${\mathcal{F}}_t$ standard Brownish 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 nether the advisable measure P and $\tilde{P}$ .

This result is due to Doob (1953), and its proof can be found in Deng and Kaleong (2007) (Theorem two.2).

Lemma 1

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

where ${\mathit{\beta }}_1(t),{\mathit{\beta }}_{\mathrm{two}}(t)$ are deterministic continuous function. And so B(t) is a standard ${\mathcal{F}}_t$ -Brown motion under the probability P.

three. The martingale approach for vulnerable binary option pricing

Now we volition discuss vulnerable binary option pricing under stochastic interest rate and liability.

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

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

For cash or cypher put choice, payoff office on maturity engagement is $X_T=R{I}_{\{\mathrm{South}(T)<K\}}$ ;

For asset or nothing phone call selection, payoff function on maturity date is $X_T=S(T)I_{\{\mathrm{South}(T)\ge K\}}$ ;

For asset or null put choice, payoff role on maturity appointment is $X_T=S(T)I_{\{\mathrm{South}(T)<\mathrm{1000}\}}$ .

3.i. Model solution

Presume the 3-dimensional ${\mathcal{F}}_t$ -adapted procedure $\mathrm{\Lambda }(t)=({\mathit{\lambda }}_S(t),{\mathit{\lambda }}_{\mathrm{Five}}(t),{\mathit{\lambda }}_D(t))$ satisfy the following SDEs: (12) $$\begin{array}{cc}\hfill {\mathit{\sigma }}_S(t)[{\mathit{\rho }}_{\mathrm{Due\; south}5}{\mathit{\lambda }}_V(t)+{\mathit{\rho }}_{SD}{\mathit{\lambda }}_D(t)]& =0,\hfill \end{array}$$ (12) (13) $$\begin{array}{cc}\hfill {\mathit{\sigma }}_V(t)[{\mathit{\rho }}_{\mathrm{Due\; south}V}{\mathit{\lambda }}_S(t)+{\mathit{\rho }}_{\mathrm{Five}D}{\mathit{\lambda }}_D(t)]& =0,\hfill \end{array}$$ (13) (14) $$\begin{array}{cc}\hfill {\mathit{\sigma }}_D(t)[{\mathit{\rho }}_{SD}{\mathit{\lambda }}_{\mathrm{Due\; south}}(t)+{\mathit{\rho }}_{VD}{\mathit{\lambda }}_{\mathrm{Five}}(t)]& =0.\hfill \end{array}$$ (14)

Let $Z_{\mathit{\lambda }}(t)$ is the ${\mathcal{F}}_t$ -martingale divers past Equation (six), and then by the Theorem i, $\tilde{\mathbf{B}}(t)=({\tilde{B}}_S(t),{\tilde{B}}_5(t),{\tilde{B}}_D(t))$ is a 3-dimensional correlated standard $({\mathcal{F}}_t,\tilde{P})$ - Brown motion, correlation matrix is C defined by formula (v).

From Equations (7)–(9), (xiv)-(16), the definition of ${\mathit{\lambda }}_{\mathit{\xi }}(t)(\mathit{\xi }=S,V,D)$ and the Theorem 1, we can get counter party firm's assets V(t), counter political party firm's liability D(T), stock cost S(t) satisfy the following SDEs: (15) $$\begin{array}{cc}\hfill \text{d}S(t)& =r(t)\mathrm{Southward}(t)\text{d}t+{\mathit{\sigma }}_S(t)S(t)\text{d}{\tilde{B}}_S(t),\hfill \end{array}$$ (xv) (16) $$\begin{array}{cc}\hfill \text{d}\mathrm{Five}(t)& =r(t)V(t)\text{d}t+{\mathit{\sigma }}_V(t)5(t)\text{d}{\tilde{B}}_V(t),\hfill \end{array}$$ (sixteen) (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)

Nosotros assume brusque-term interest rate follows Hull-White model under the probability $\tilde{P}$ as 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: $$\begin{array}{c}\hfill \underset{t}{\overset{T}{\int }}r(u)du=K(t,T,r(t))+\underset{t}{\overset{T}{\int }}{\mathit{\sigma }}_r(u)m(u,T)d{\tilde{B}}_r(u),\end{array}$$

where $\mathrm{Grand}(t,T,r(t))=r(t)m(t,T)+{\int }_t^{T}a(u)\mathrm{thou}(u,T)du$ , $k(u,v)={\int }_u^{5}e\mathrm{10}p(\mathit{\eta }(u)-\mathit{\eta }(\mathrm{due\; south}))ds$ , $\mathit{\eta }(\mathrm{south})={\int }_0^{s}b(u)du$ .

Nether the probability $\tilde{P}$ , applying Ito formulas to (17)–(xix) nosotros can get: $$\begin{array}{c}\hfill \mathit{\xi }(T)=\mathit{\xi }(t)exp\left\{\underset{t}{\overset{T}{\int }}(r(u)-{\mathit{\sigma }}^2(\mathit{\xi })/2)du+\underset{t}{\overset{T}{\int }}{\mathit{\sigma }}_{\mathit{\xi }}(u)d{\tilde{B}}_{\mathit{\xi }}(u)\right\},\mathit{\xi }=S,V,D.\end{array}$$

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 Brownish motion under probability $\tilde{P}$ , the correlation matrix is (19) $$\begin{array}{c}\hfill \tilde{\mathbf{c}}=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& {\mathit{\rho }}_{\mathrm{Due\; south}V}& {\mathit{\rho }}_{SD}\\ 0& {\mathit{\rho }}_{\mathrm{Due\; south}5}& \mathrm{one}& {\mathit{\rho }}_{VD}\\ 0& {\mathit{\rho }}_{\mathrm{South}D}& {\mathit{\rho }}_{VD}& \mathrm{one}\end{array}\right).\end{array}$$ (19)

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

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

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

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

where (23) $$\begin{array}{cc}& d{\tilde{B}}_{\mathit{\sigma }}(t)=\frac{{\mathit{\sigma }}_V(t)d{\tilde{B}}_V(t)-{\mathit{\sigma }}_D(t)d{\tilde{B}}_D(t)}{{\mathrm{\Delta }}_2},\hfill \\ \hfill & {\mathrm{\Delta }}_{\mathrm{ii}}=\sqrt{{\mathit{\sigma }}_5^{\mathrm{ii}}(t)+{\mathit{\sigma }}_D^{\mathrm{two}}(t)-\mathrm{two}{\mathit{\sigma }}_{\mathrm{Five}D}(t)},{\mathit{\sigma }}_{VD}(t)={\mathit{\rho }}_{5D}(t){\mathit{\sigma }}_V(t){\mathit{\sigma }}_D(t).\hfill \end{array}$$ (23)

By Lemma 1 once again, we have that ${\tilde{B}}_{\mathit{\sigma }}(t)$ is a standard brown motion nether the probability $\tilde{P}$ .

3.ii. The pricing formula of vulnerable binary option

Theorem two

Under the firm value model, assume underlying nugget price is ${\mathrm{Southward}}_t$ at the time t, exercise cost is K, the ratio of bounty is $\mathit{\delta }(T)=V(T)/D(T)$ , the price of vulnerable cash or nothing call option at the time $t(0\le t\le T)$ is: (24) $$\begin{array}{c}\hfill C(t)=\mathrm{1000}RN(d_1,d_{\mathrm{ii}};\mathit{\rho })+\mathrm{One\; thousand}K\mathit{\delta }(t)b_2\mathrm{Due\; north}(d_1+a_1,-(d_2+\sqrt{{\int }_t^T{\mathrm{\Delta }}_2^{2}du});-\mathit{\rho })).\end{array}$$ (24)

where $$\begin{array}{cc}\hfill d_1& =\frac{ln\frac{\mathrm{Southward}(t)}{K}+G(t,T,r(t))-\underset{t}{\overset{T}{\int }}{\mathit{\sigma }}_S^2(u)/\mathrm{two}du}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_{\mathrm{i}}^{2}du}},\hfill \\ \hfill d_{\mathrm{two}}& =\frac{ln\mathit{\delta }(t)+{\int }_t^T\frac{1}{2}({\mathit{\sigma }}_D^2(u)-{\mathit{\sigma }}_V^2(u))du}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_{\mathrm{ii}}^{\mathrm{two}}du}},\hfill \\ \hfill a_{\mathrm{one}}& =\frac{{\int }_t^T({\mathit{\sigma }}_{\mathrm{Due\; south}5}(u)-{\mathit{\sigma }}_{\mathrm{Five}D}(u))du}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_1^{\mathrm{ii}}du}},\hfill \\ \hfill b_2& =exp\left\{\underset{t}{\overset{T}{\int }}({\mathit{\sigma }}_D^2(u)-{\mathit{\sigma }}_{VD}(u))du\right\},\hfill \\ \hfill G& =exp\left\{-G(t,T,r(t))+\frac{1}{2}\underset{t}{\overset{T}{\int }}{\mathit{\sigma }}_r^{\mathrm{two}}(u)m^2(u,T)du\right\},\hfill \\ \hfill \mathit{\rho }& =\frac{\underset{t}{\overset{T}{\int }}{\mathrm{\Delta }}_1{\mathrm{\Delta }}_{\mathrm{ii}}{\mathit{\rho }}_{{\tilde{B}}_{\overline{S}}{\tilde{B}}_{\mathit{\sigma }}}du}{\sqrt{\underset{t}{\overset{T}{\int }}{\mathrm{\Delta }}_{\mathrm{ane}}^{2}du}\sqrt{\underset{t}{\overset{T}{\int }}{\mathrm{\Delta }}_2^{2}du}},\hfill \\ \hfill {\mathit{\rho }}_{{\tilde{B}}_{\overline{S}}{\tilde{B}}_{\mathit{\sigma }}}& =\frac{({\mathit{\sigma }}_{\mathrm{South}}(t){\mathit{\sigma }}_V(t){\mathit{\rho }}_{SV}+{\mathit{\sigma }}_S(t){\mathit{\sigma }}_D(t){\mathit{\rho }}_{SD})}{\sqrt{({\mathit{\sigma }}_r^{\mathrm{two}}(t)m^2(t,T)+{\mathit{\sigma }}_{\mathrm{Due\; south}}^{\mathrm{two}}(t))}\sqrt{({\mathit{\sigma }}_r^2(t){\mathrm{grand}}^2(t,T)+2{\mathit{\sigma }}_V(t){\mathit{\sigma }}_D(t)g(t,T){\mathit{\rho }}_{VD})}},\hfill \end{array}$$ ${\mathrm{\Delta }}_1$ , ${\mathrm{\Delta }}_2$ is defined by Equations (23)and (25).

Proof

Denote $A_1=(\mathrm{Southward}(T)>K,\mathit{\delta }(T)\ge 1),A_2=(\mathrm{Southward}(T)>K,\mathit{\delta }(T)<\mathrm{ane})$ , according to Equation (13), under the risk neutral probability, the price of vulnerable cash or nil call option is: $$\begin{array}{cc}\hfill C(t)& =\tilde{E}\left[exp\left\{-\underset{t}{\overset{T}{\int }}r(u)du\right\}(X(T)I_{\{{\mathit{\delta }}_{(T)}\ge \mathrm{ane}\}}+\mathit{\delta }(T)X(T)I_{\{{\mathit{\delta }}_{(T)}<1\}})|{\mathcal{F}}_t\right]\hfill \\ \hfill & =\widetilde{\mathrm{East}}\left[exp\left\{-\underset{t}{\overset{T}{\int }}r(u)du\right\}(R{I}_{A_1}+\mathit{\delta }(T)R{I}_{A_2})|{\mathcal{F}}_t\right]\hfill \\ \hfill & =\tilde{E}\left[exp\left\{-G(t,T,r(t))-\underset{t}{\overset{T}{\int }}{\mathit{\sigma }}_r(u)k(u,T)d{\tilde{B}}_r(u)\right\}(R{I}_{A_{\mathrm{ane}}}+\mathit{\delta }(T)R{I}_{A_2})|{\mathcal{F}}_t\right]\hfill \\ \hfill & =\widetilde{\mathrm{East}}\left[exp\left\{-K(t,T,r(t))+\frac{1}{2}\underset{t}{\overset{T}{\int }}{\mathit{\sigma }}_r^{\mathrm{two}}(u)m^2(u,T)du\right\}(R{I}_{A_{\mathrm{i}}}+\mathit{\delta }(T)R{I}_{A_2})|{\mathcal{F}}_t\right]\hfill \\ \hfill & =G\tilde{E}\left[(R{I}_{A_{\mathrm{ane}}}+\mathit{\delta }(T)R{I}_{A_2})|{\mathcal{F}}_t\right]\hfill \\ \hfill & =G\tilde{E}\left[R{I}_{A_{\mathrm{one}}}|{\mathcal{F}}_t]+\mathrm{Grand}\widetilde{\mathrm{Due\; east}}[\mathit{\delta }(T)R{I}_{A_2}|{\mathcal{F}}_t\right]\hfill \\ \hfill & =D_1+D_{\mathrm{two}},\hfill \end{array}$$

where $G=exp\left\{-G(t,T,r(t))+\frac{1}{2}{\int }_t^T{\mathit{\sigma }}_r^2(u){\mathrm{chiliad}}^2(u,T)du\right\},D_{\mathrm{i}}=\mathrm{1000}\tilde{E}[R{I}_{A_1}|{\mathcal{F}}_t],D_2=\mathrm{Grand}\widetilde{\mathrm{Due\; east}}[\mathit{\delta }(T)R{I}_{A_2}|{\mathcal{F}}_t]$ .

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

Evaluation of term $D_{\mathrm{i}}$ : $$\begin{array}{c}\hfill D_1=\mathrm{One\; thousand}R\tilde{P}(A_1|{\mathcal{F}}_t)=\mathrm{Yard}R\tilde{P}((S(T)>\mathrm{Grand},\mathit{\delta }(T)\ge 1)|{\mathcal{F}}_t),\end{array}$$

co-ordinate to $S(T)>K$ and South(T) follows from (22) that $$\begin{array}{c}\hfill S(T)=S(t)exp\left\{G(t,T,r(t))-\underset{t}{\overset{T}{\int }}{\mathit{\sigma }}_S^2(u)/2du+\underset{t}{\overset{T}{\int }}{\mathrm{\Delta }}_{1}d{\tilde{B}}_{\overline{S}}(u)\right\}>\mathrm{1000}\end{array}$$

then, $$\begin{array}{c}\hfill \mathrm{Yard}(t,T,r(t))-\underset{t}{\overset{T}{\int }}{\mathit{\sigma }}_{\mathrm{Southward}}^2(u)/2du+\underset{t}{\overset{T}{\int }}{\mathrm{\Delta }}_{1}d{\tilde{B}}_{\overline{S}}(u)>ln\frac{K}{\mathrm{Due\; south}(t)}\end{array}$$

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

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

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

Substituting this into $D_{\mathrm{ane}}$ , we obtain $$\begin{array}{cc}\hfill D_{\mathrm{ane}}& =GR\tilde{P}((S(T)>\mathrm{Yard},\mathit{\delta }(T)\ge 1)|{\mathcal{F}}_t)\hfill \\ \hfill & =GR\tilde{P}\left[-\frac{\underset{t}{\overset{T}{\int }}{\mathrm{\Delta }}_{\mathrm{one}}d{\tilde{B}}_{\overline{S}}(u)}{\sqrt{\underset{t}{\overset{T}{\int }}{\mathrm{\Delta }}_1^{2}du}}\right.\hfill \\ \hfill & <\frac{ln\frac{S(t)}{K}+G(t,T,r(t))-\underset{t}{\overset{T}{\int }}{\mathit{\sigma }}_S^{\mathrm{two}}(u)/\mathrm{two}du}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_{\mathrm{ane}}^{2}du}},\hfill \\ \hfill & \left.-\frac{{\int }_t^T{\mathrm{\Delta }}_{2}d{\tilde{B}}_{\mathit{\sigma }}(u)}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_2^{\mathrm{ii}}du}}\le \frac{ln\mathit{\delta }(t)+\underset{t}{\overset{T}{\int }}\frac{\mathrm{ane}}{2}({\mathit{\sigma }}_D^2(u)-{\mathit{\sigma }}_V^2(u))du}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_2^{2}du}}|{\mathcal{F}}_t\right]\hfill \\ \hfill & =KRN(d_1,d_{\mathrm{two}};\mathit{\rho }),\hfill \end{array}$$

where $$\begin{array}{cc}\hfill d_{\mathrm{one}}& =\frac{ln\frac{S(t)}{\mathrm{Thousand}}+G(t,T,r(t))-{\int }_t^T{\mathit{\sigma }}_S^{\mathrm{two}}(u)/\mathrm{ii}du}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_1^{\mathrm{two}}du}},\hfill \\ \hfill d_2& =\frac{ln\mathit{\delta }(t)+{\int }_t^T\frac{\mathrm{one}}{2}({\mathit{\sigma }}_D^2(u)-{\mathit{\sigma }}_{\mathrm{Five}}^2(u))du}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_2^{2}du}}|{\mathcal{F}}_t],\hfill \\ \hfill \mathit{\rho }& =\text{Corr}\left(-\frac{{\int }_t^T{\mathrm{\Delta }}_{\mathrm{ane}}d{\tilde{B}}_{\overline{S}}(u)}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_{\mathrm{one}}^{2}du}},-\frac{{\int }_t^T{\mathrm{\Delta }}_{2}d{\tilde{B}}_{\mathit{\sigma }}(u)}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_{\mathrm{two}}^{2}du}}\right)\hfill \\ \hfill & =\frac{{\int }_t^T{\mathrm{\Delta }}_{\mathrm{i}}{\mathrm{\Delta }}_{\mathrm{two}}{\mathit{\rho }}_{{\tilde{B}}_{\overline{\mathrm{Due\; south}}}{\tilde{B}}_{\mathit{\sigma }}}du}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_1^{2}du}\sqrt{{\int }_t^T{\mathrm{\Delta }}_2^{2}du}},\hfill \\ \hfill {\mathit{\rho }}_{{\tilde{B}}_{\overline{\mathrm{Due\; south}}}{\tilde{B}}_{\mathit{\sigma }}}& =\frac{({\mathit{\sigma }}_{\mathrm{Due\; south}}(t){\mathit{\sigma }}_{\mathrm{Five}}(t){\mathit{\rho }}_{SV}+{\mathit{\sigma }}_S(t){\mathit{\sigma }}_D(t){\mathit{\rho }}_{SD})}{\sqrt{({\mathit{\sigma }}_r^{\mathrm{ii}}(t)m^2(t,T)+{\mathit{\sigma }}_S^{\mathrm{ii}}(t))}\sqrt{({\mathit{\sigma }}_r^2(t)m^{\mathrm{two}}(t,T)+2{\mathit{\sigma }}_V(t){\mathit{\sigma }}_D(t)m(t,T){\mathit{\rho }}_{VD})}}.\hfill \end{array}$$

Evaluation of term $D_2$ : firstly nosotros define a new probability measure, since ${\tilde{B}}_{\mathit{\sigma }}(t),{\tilde{B}}_{\overline{\mathrm{Due\; south}}}(t)$ are standard chocolate-brown motions under the measure $\tilde{P}$ . Ascertain Rodon-Nikodym derivative: $$\begin{array}{c}\hfill \frac{\text{d}\widehat{P}}{\text{d}\tilde{P}}|{\mathcal{F}}_t=exp\left\{\underset{t}{\overset{T}{\int }}{\mathrm{\Delta }}_{\mathrm{ii}}d{\tilde{B}}_{\overline{S}}(u)-\frac{1}{\mathrm{ii}}\underset{t}{\overset{T}{\int }}{\mathrm{\Delta }}_2^{2}du\right\}{Z}_{\mathrm{one}}(t).\end{array}$$

Permit $$\begin{array}{cc}& d{\widehat{B}}_{\mathit{\sigma }}(t)=-{\mathrm{\Delta }}_{\mathrm{two}}dt+d{\tilde{B}}_{\mathit{\sigma }}(t),\hfill \\ \hfill & d{\widehat{B}}_{\overline{S}}=-{\mathit{\rho }}_{\overline{\mathrm{Due\; south}}\mathit{\sigma }}{\mathrm{\Delta }}_{\mathrm{two}}dt+d{\tilde{B}}_{\overline{S}}(t)=-\frac{{\mathit{\sigma }}_{\mathrm{Due\; south}V}(t)-{\mathit{\sigma }}_{SD}(t)}{{\mathrm{\Delta }}_1}dt+d{\tilde{B}}_{\overline{S}}(t),\hfill \end{array}$$

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

According to the Lemma one, we see that ${\widehat{B}}_{\mathit{\sigma }}(t),{\widehat{B}}_{\overline{\mathrm{Due\; south}}}(t)$ are standard brown motions nether the measure $\widehat{P}$ .

Under the mensurate $\widehat{P}$ , we have $$\begin{array}{cc}\hfill \mathrm{South}(T)& =S(t)exp\left\{K(t,T,r(t))-\underset{t}{\overset{T}{\int }}({\mathit{\sigma }}_{\mathrm{South}}^2(u)/2-{\mathit{\sigma }}_{SV}(u)\right.\hfill \\ \hfill & \left.+{\mathit{\sigma }}_{SD}(u))du+\underset{t}{\overset{T}{\int }}{\mathrm{\Delta }}_{\mathrm{i}}d{\widehat{B}}_{\overline{S}}(t)\right\},\hfill \\ \hfill \mathit{\delta }(T)& =\mathit{\delta }(t)exp\left\{\underset{t}{\overset{T}{\int }}\left[\frac{1}{2}({\mathit{\sigma }}_D^2(u)-{\mathit{\sigma }}_{\mathrm{Five}}^2(u))+{\mathrm{\Delta }}_2^2\right]du+\underset{t}{\overset{T}{\int }}{\mathrm{\Delta }}_{2}d{\widehat{B}}_{\mathit{\sigma }}(u)\right\}.\hfill \end{array}$$

Substituting this into $D_2$ , nosotros get $$\begin{array}{cc}\hfill D_2& =\mathrm{Chiliad}\tilde{E}(\mathit{\delta }(T)R{I}_{A_{\mathrm{two}}}|{\mathcal{F}}_t)\hfill \\ \hfill & =GR\widetilde{\mathrm{Eastward}}\left[\mathit{\delta }(t)exp\left\{\underset{t}{\overset{T}{\int }}\left[\frac{1}{2}({\mathit{\sigma }}_D^{\mathrm{two}}(u)-{\mathit{\sigma }}_V^2(u))+{\mathrm{\Delta }}_2^{\mathrm{two}}\right]du+\underset{t}{\overset{T}{\int }}{\mathrm{\Delta }}_{\mathrm{ii}}d{\widehat{B}}_{\mathit{\sigma }}(u)\right\}{I}_{A_2}|{\mathcal{F}}_t\right]\hfill \\ \hfill & =GR\mathit{\delta }(t)exp\left\{\underset{t}{\overset{T}{\int }}\left[\frac{1}{2}({\mathit{\sigma }}_D^{\mathrm{two}}(u)-{\mathit{\sigma }}_{\mathrm{Five}}^2(u))+{\mathrm{\Delta }}_2^2\right]du-\frac{1}{2}\underset{t}{\overset{T}{\int }}{\mathrm{\Delta }}_{\mathrm{two}}^{2}du\right\}\tilde{E}[Z_{\mathrm{one}}(t)Z_{\mathrm{i}}^{-1}(t)I_{A_{\mathrm{two}}}|{\mathcal{F}}_t]\hfill \\ \hfill & =\mathrm{Grand}R\mathit{\delta }(t)exp\left\{\underset{t}{\overset{T}{\int }}({\mathit{\sigma }}_D^{\mathrm{two}}(u)-{\mathit{\sigma }}_{5D}(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{E}[S(T)>K,\mathit{\delta }(T)<\mathrm{i}|{\mathcal{F}}_t]\hfill \\ \hfill & =GR\mathit{\delta }(t)\widehat{P}\left[\left(-\frac{{\int }_t^T{\mathrm{\Delta }}_{1}d{\widehat{B}}_{\overline{S}}(u)}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_1^{\mathrm{ii}}du}}<\frac{ln\frac{S(t)}{K}+G(t,T,r(t))-\underset{t}{\overset{T}{\int }}({\mathit{\sigma }}_{\mathrm{South}}^2(u)/2-{\mathit{\sigma }}_{SV}(u)+{\mathit{\sigma }}_{\mathrm{Southward}D}(u))du}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_1^{2}du}},\right.\right.\hfill \\ \hfill & \left.-\frac{{\int }_t^T{\mathrm{\Delta }}_{2}d{\widehat{B}}_{\mathit{\sigma }}(u)}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_2^{\mathrm{two}}du}}>\frac{ln\mathit{\delta }(t)+{\int }_t^T\left[\frac{1}{\mathrm{ii}}({\mathit{\sigma }}_D^2(u)-{\mathit{\sigma }}_V^{\mathrm{ii}}(u))+{\mathrm{\Delta }}_2^2\right]du}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_2^{2}du}})|{\mathcal{F}}_t\right]\hfill \\ \hfill & =\mathrm{Thousand}R\mathit{\delta }(t)b_{\mathrm{two}}N\left(d_1+a_1,-\left(d_2+\sqrt{{\int }_t^T{\mathrm{\Delta }}_2^{2}du};-\mathit{\rho }\right)\right),\hfill \end{array}$$

where $$\begin{array}{cc}\hfill b_2& =exp\left\{\underset{t}{\overset{T}{\int }}({\mathit{\sigma }}_D^{\mathrm{ii}}(u)-{\mathit{\sigma }}_{VD}(u))du\right\};\hfill \\ \hfill a_1& =\frac{{\int }_t^T({\mathit{\sigma }}_{SV}(u)-{\mathit{\sigma }}_{SD}(u))du}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_1^{\mathrm{two}}du}},\hfill \\ \hfill \mathit{\rho }& =\frac{{\int }_t^T{\mathrm{\Delta }}_{\mathrm{ane}}{\mathrm{\Delta }}_2{\mathit{\rho }}_{{\tilde{B}}_{\overline{S}}{\tilde{B}}_{\mathit{\sigma }}}du}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_1^{2}du}\sqrt{{\int }_t^T{\mathrm{\Delta }}_2^{2}du}}\hfill \\ \hfill & =\text{Corr}\left(-\frac{{\int }_t^T{\mathrm{\Delta }}_{\mathrm{i}}d{\widehat{B}}_{\overline{S}}(u)}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_{\mathrm{one}}^{2}du}},-\frac{{\int }_t^T{\mathrm{\Delta }}_{2}d{\widehat{B}}_{\mathit{\sigma }}(u)}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_{\mathrm{ii}}^{2}du}}\right).\hfill \end{array}$$

And so, past the result of $D_1,D_{\mathrm{ii}}$ , 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 ${\mathrm{Due\; south}}_t$ , do price is K, the ratio of bounty is $\mathit{\delta }(T)=V(T)/D(T)$ , the toll of vulnerable cash or zero put selection at the fourth dimension $t(0\le t\le T)$ is every bit follows: (25) $$\begin{array}{c}\hfill C(t)=GRN(-d_{\mathrm{one}},d_{\mathrm{two}};-\mathit{\rho })+\mathrm{1000}R\mathit{\sigma }(t)b_{2}N\left(-(d_1+a_{\mathrm{ane}}),-\left(d_{\mathrm{ii}}+\sqrt{{\int }_t^T{\mathrm{\Delta }}_2^{2}du}\right);\mathit{\rho })\right),\end{array}$$ (25)

where $G,d_{\mathrm{ane}},d_{\mathrm{ii}},a_1,b_2,\mathit{\rho }$ are given in the Theorem ii.

Theorem 3

Under the business firm value model, assume the cost of underlying asset at time t is ${\mathrm{South}}_t$ , practise price is K, the ratio of bounty is $\mathit{\delta }(T)=V(T)/D(T)$ , the price of vulnerable asset or naught call selection at the fourth dimension $t(0\le t\le T)$ is as follows: (26) $$\begin{array}{cc}\hfill C(t)& =S(t)N(d_1+\sqrt{{\int }_t^T{\mathrm{\Delta }}_{\mathrm{ane}}^{\mathrm{two}}du},d_{\mathrm{ii}}+a_2;\mathit{\rho }))\hfill \\ \hfill & +S(t)\mathit{\delta }(t)b_{\mathrm{one}}N\left(d_1+a_{\mathrm{ane}}+2\sqrt{{\int }_t^T{\mathrm{\Delta }}_{\mathrm{i}}^{\mathrm{two}}du},-(d_2+\mathrm{two}{a}_2+\sqrt{{\int }_t^T{\mathrm{\Delta }}_2^{\mathrm{ii}}du});-\mathit{\rho }\right).\hfill \end{array}$$ (26)

where $$\begin{array}{cc}\hfill a_2& =\frac{{\int }_t^T({\mathit{\sigma }}_{\mathrm{Southward}5}(u)-{\mathit{\sigma }}_{SD}(u))du}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_2^{\mathrm{two}}du}},\hfill \\ \hfill b_{\mathrm{one}}& =exp\underset{t}{\overset{T}{\int }}(-{\mathit{\sigma }}_{\mathrm{Five}}^2(u)-{\mathit{\sigma }}_S^2(u)+{\mathit{\sigma }}_{VD}(u)+{\mathit{\sigma }}_{\mathrm{South}\mathrm{Five}}(u)-{\mathit{\sigma }}_{\mathrm{Southward}D}(u))du,\hfill \end{array}$$ $a_{\mathrm{i}},d_{\mathrm{ane}},d_{\mathrm{two}},\mathit{\rho }$ are given in the theorem 2.

Proof

Permit $A_{\mathrm{i}}=(S(T)>K,\mathit{\delta }(T)\ge \mathrm{one}),A_{\mathrm{ii}}=(S(T)>K,\mathit{\delta }(T)<1)$ , according to (13), under the run a risk neutral probability, the price of vulnerable greenbacks or zilch telephone call pick is as follows: $$\begin{array}{cc}\hfill C(t)& =\tilde{E}\left[exp\left\{-\underset{t}{\overset{T}{\int }}r(u)du\right\}(X(T)I_{\{{\mathit{\delta }}_{(T)}\ge 1\}}+\mathit{\sigma }(T)\mathrm{Ten}(T)I_{\{{\mathit{\delta }}_{(T)}<1\}})|{\mathcal{F}}_t\right]\hfill \\ \hfill & =\tilde{E}\left[exp\left\{-\underset{t}{\overset{T}{\int }}r(u)du\right\}(S(T)I_{A_1}+\mathit{\delta }(T)\mathrm{Due\; south}(T)I_{A_2})|{\mathcal{F}}_t\right]\hfill \\ \hfill & =\tilde{E}\left[exp\left\{-\mathrm{1000}(t,T,r(t))-\underset{t}{\overset{T}{\int }}{\mathit{\sigma }}_r(u)m(u,T)d{\tilde{B}}_r(u)\right\}(S(T)I_{A_{\mathrm{ane}}}+\mathit{\delta }(T)S(T)I_{A_2})|{\mathcal{F}}_t\right]\hfill \\ \hfill & =\widetilde{\mathrm{East}}\left[exp\left\{-\mathrm{Thou}(t,T,r(t))+\frac{1}{\mathrm{two}}{\int }_t^T{\mathit{\sigma }}_r^{\mathrm{two}}(u)m^{\mathrm{two}}(u,T)du\right\}(S(T)I_{A_{\mathrm{i}}}+\mathit{\delta }(T)S(T)I_{A_2})|{\mathcal{F}}_t\right]\hfill \\ \hfill & =G\widetilde{\mathrm{East}}[(\mathrm{South}(T)I_{A_1}+\mathit{\delta }(T)S(T)I_{A_2})|{\mathcal{F}}_t]\hfill \\ \hfill & =G\tilde{E}[\mathrm{Due\; south}(T)I_{A_1}|{\mathcal{F}}_t]+\mathrm{1000}\tilde{E}[\mathit{\delta }(T)\mathrm{Due\; south}(T)I_{A_{\mathrm{ii}}}|{\mathcal{F}}_t]\hfill \\ \hfill & =5_1+V_2,\hfill \end{array}$$

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

Evaluation of $5_{\mathrm{i}}$ : we tin can define a new probability measure equivalent to $\overline{P}$ .

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

Define the Rodon-Nikodym derivative: $$\begin{array}{c}\hfill \frac{\text{d}\overline{P}}{\text{d}\tilde{P}}|{\mathcal{F}}_t=exp\left\{\underset{t}{\overset{T}{\int }}{\mathrm{\Delta }}_1\text{d}{\overline{B}}_{\overline{\mathrm{South}}}(u)-\frac{1}{2}\underset{t}{\overset{T}{\int }}{\mathrm{\Delta }}_1^{\mathrm{two}}\text{d}u\right\}{Z}_{\mathrm{ii}}(t).\end{array}$$

Let $\text{d}{\overline{B}}_{\overline{S}}(t)=-{\mathrm{\Delta }}_1\text{d}t+\text{d}{\tilde{B}}_{\overline{S}}(t)$ , $\text{d}{\overline{B}}_{\mathit{\sigma }}(t)=-{\mathit{\rho }}_{\overline{\mathrm{Southward}}\mathit{\sigma }}{\mathrm{\Delta }}_1\text{d}t+\text{d}{\overline{B}}_{\mathit{\sigma }}(t)=-\frac{{\mathit{\sigma }}_{SV}(t)-{\mathit{\sigma }}_{SD}(t)}{{\mathrm{\Delta }}_2}\text{d}t+\text{d}{\overline{B}}_{\mathit{\sigma }}(t)$ , co-ordinate to the Lemma i, we see that ${\overline{B}}_{\overline{\mathrm{Due\; south}}}(t),{\overline{B}}_{\mathit{\sigma }}(t)$ are standard chocolate-brown motions under the measure out $\overline{P}$ .

Under the measure $\overline{P}$ , we obtain $$\begin{array}{cc}\hfill S(T)& =S(t)exp\left\{G(t,T,r(t))-\underset{t}{\overset{T}{\int }}({\mathit{\sigma }}_{\mathrm{Southward}}^2(u)/\mathrm{two}-{\mathrm{\Delta }}_1^2)du+\underset{t}{\overset{T}{\int }}{\mathrm{\Delta }}_{\mathrm{one}}d{\overline{B}}_{\overline{S}}(u)\right\},\hfill \\ \hfill \mathit{\delta }(T)& =\mathit{\delta }(t)exp\left\{\underset{t}{\overset{T}{\int }}\left[\frac{1}{\mathrm{ii}}({\mathit{\sigma }}_D^{\mathrm{ii}}(u)-{\mathit{\sigma }}_5^2(u))+({\mathit{\sigma }}_{S\mathrm{Five}}(u)+{\mathit{\sigma }}_{SD}(u))\right]du+\underset{t}{\overset{T}{\int }}{\mathrm{\Delta }}_{2}d{\overline{B}}_{\mathit{\sigma }}(u)\right\}.\hfill \end{array}$$

Substituting thus into $V_{\mathrm{i}}$ , nosotros have $$\begin{array}{cc}\hfill {\mathrm{Five}}_1& =G\tilde{E}(S(T)I_{A_{\mathrm{ii}}}|{\mathcal{F}}_t)\hfill \\ \hfill & =\mathrm{Chiliad}\widetilde{\mathrm{Eastward}}\left[S(t)exp\left\{\mathrm{Yard}(t,T,r(t))-\underset{t}{\overset{T}{\int }}(\frac{{\mathit{\sigma }}_S^{\mathrm{two}}(u)}{2}du-{\mathrm{\Delta }}_{\mathrm{ane}}^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 & =GS(t)exp\left\{\mathrm{1000}(t,T,r(t))-\underset{t}{\overset{T}{\int }}\frac{{\mathit{\sigma }}_S^2(u)}{2}du\right.\hfill \\ \hfill & \left.+\frac{\mathrm{ane}}{2}\underset{t}{\overset{T}{\int }}{\mathrm{\Delta }}_{\mathrm{one}}^{\mathrm{two}}du\right\}\tilde{E}[Z_2(t)Z_2^{-1}(t)I_{A_1}|{\mathcal{F}}_t]\hfill \\ \hfill & =\mathrm{South}(t)\overline{E}[I_{A_1}|{\mathcal{F}}_t]\hfill \\ \hfill & =\mathrm{South}(t)\overline{P}[(S(T)>K,\mathit{\delta }(T)\ge 1)|{\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^{2}du}}<\frac{ln\frac{S(t)}{K}+G(t,T,r(t))-{\int }_t^T({\mathit{\sigma }}_{\mathrm{South}}^{\mathrm{two}}(u)/2-{\mathrm{\Delta }}_1^2)du}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_{\mathrm{ane}}^{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.\right.\hfill \\ \hfill & \left.<\frac{ln\mathit{\delta }(t)+{\int }_t^T\left[\frac{1}{2}({\mathit{\sigma }}_D^{\mathrm{ii}}(u)-{\mathit{\sigma }}_V^2(u))+{\int }_t^T({\mathit{\sigma }}_{\mathrm{Southward}5}(u)-{\mathit{\sigma }}_{SD}(u))du\right]}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_{\mathrm{ii}}^{2}du}})|{\mathcal{F}}_t\right]\hfill \\ \hfill & =\mathrm{Southward}(t)\mathrm{Northward}\left(d_{\mathrm{one}}+\sqrt{{\int }_t^T{\mathrm{\Delta }}_1^{2}du},d_2+a_{\mathrm{ii}};\mathit{\rho }\right),\hfill \end{array}$$

where $$\begin{array}{cc}\hfill a_2& =\frac{{\int }_t^T({\mathit{\sigma }}_{\mathrm{Southward}5}(u)-{\mathit{\sigma }}_{SD}(u))du}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_2^{\mathrm{ii}}du}},\hfill \\ \hfill \mathit{\rho }& =\frac{{\int }_t^T{\mathrm{\Delta }}_{\mathrm{i}}{\mathrm{\Delta }}_2{\mathit{\rho }}_{{\tilde{B}}_{\overline{S}}{\tilde{B}}_{\mathit{\sigma }}}du}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_{\mathrm{ane}}^{2}du}\sqrt{{\int }_t^T{\mathrm{\Delta }}_2^{\mathrm{two}}du}}\hfill \\ \hfill & =\text{Corr}\left(-\frac{{\int }_t^T{\mathrm{\Delta }}_{1}d{\overline{B}}_{\overline{\mathrm{Due\; south}}}(u)}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_1^{2}du}},-\frac{{\int }_t^T{\mathrm{\Delta }}_{\mathrm{two}}d{\overline{B}}_{\mathit{\sigma }}(u)}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_2^{2}du}}\right).\hfill \end{array}$$

Evaluation of $V_2$ , under the measure $\tilde{P}$ , we get $$\begin{array}{cc}\hfill \mathit{\delta }(T)S(T)& =\mathrm{Southward}(t)\mathit{\delta }(t)exp\left\{G(t,T,r(t))+\underset{t}{\overset{T}{\int }}\left[\frac{1}{2}({\mathit{\sigma }}_D^{\mathrm{two}}(u)-{\mathit{\sigma }}_V^2(u)-{\mathit{\sigma }}_{\mathrm{Due\; south}}^2(u))\right]du\right.\hfill \\ \hfill & \left.+\underset{t}{\overset{T}{\int }}({\mathrm{\Delta }}_{1}d{\tilde{B}}_{\overline{S}}(u)+\underset{t}{\overset{T}{\int }}{\mathrm{\Delta }}_{2}d{\tilde{B}}_{\mathit{\sigma }}(u))\right\}\hfill \\ \hfill & =S(t)\mathit{\delta }(t)exp\left\{\mathrm{One\; thousand}(t,T,r(t))+\underset{t}{\overset{T}{\int }}\left[\frac{\mathrm{ane}}{2}({\mathit{\sigma }}_D^2(u)-{\mathit{\sigma }}_V^{\mathrm{ii}}(u)-{\mathit{\sigma }}_S^{\mathrm{two}}(u))\right]du\right.\hfill \\ \hfill & \left.+\underset{t}{\overset{T}{\int }}{\mathrm{\Delta }}_{3}d\tilde{B}(u)\right\},\hfill \end{array}$$

where $\text{d}\tilde{B}(t)=\frac{{\mathrm{\Delta }}_1\text{d}{\tilde{B}}_{\overline{S}}(t)+{\mathrm{\Delta }}_2\text{d}{\tilde{B}}_{\mathit{\sigma }}(t)}{{\mathrm{\Delta }}_3},{\mathrm{\Delta }}_{\mathrm{three}}=\sqrt{{\mathrm{\Delta }}_1^{\mathrm{two}}+{\mathrm{\Delta }}_2^{\mathrm{two}}-2{\mathit{\rho }}_{\overline{S}\mathit{\sigma }}{\mathrm{\Delta }}_1{\mathrm{\Delta }}_2}$ , then according to Lemma 1, nosotros meet that $\tilde{B}(t)$ is standard brown motion under the measure $\tilde{P}$ .

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

Let $\text{d}\check{B}(t)=-{\mathrm{\Delta }}_3\text{d}t+\text{d}\tilde{B}(t),\text{d}{\check{B}}_{\overline{S}}(t)=-\frac{{\mathit{\sigma }}_{\mathrm{Southward}V}(t)-{\mathit{\sigma }}_{SD}(t)+{\mathrm{\Delta }}_1^2}{{\mathrm{\Delta }}_1}\text{d}t+\text{d}{\overline{B}}_{\overline{\mathrm{Southward}}}(t),\text{d}{\check{B}}_{\mathit{\sigma }}(t)=-\frac{{\mathit{\sigma }}_{\mathrm{Due\; south}\mathrm{Five}}(t)-{\mathit{\sigma }}_{SD}(t)+{\mathrm{\Delta }}_2^2}{{\mathrm{\Delta }}_2}\text{d}t+\text{d}{\overline{B}}_{\mathit{\sigma }}(t)$ , according to the Lemma 1, nosotros can become that $(\check{B}(t),{\check{B}}_{\overline{S}}(t),{\check{B}}_{\mathit{\sigma }}(t))$ is standard chocolate-brown motion under the mensurate $\check{P}$ .

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

Substituting this into $V_2$ , we have $$\begin{array}{cc}\hfill V_{\mathrm{two}}& =G\tilde{E}(\mathrm{Southward}(T)\mathit{\delta }(T)I_{A_{\mathrm{ii}}}|{\mathcal{F}}_t)\hfill \\ \hfill & =G\tilde{E}[S(t)\mathit{\delta }(t)exp\left\{G(t,T,r(t))-\underset{t}{\overset{T}{\int }}\frac{1}{2}({\mathit{\sigma }}_D^{\mathrm{two}}(u)-{\mathit{\sigma }}_5^{\mathrm{two}}(u)\right.\hfill \\ \hfill & \left.\left.-{\mathit{\sigma }}_{\mathrm{Southward}}^{\mathrm{two}}(u))du+\underset{t}{\overset{T}{\int }}{\mathrm{\Delta }}_{\mathrm{three}}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 }}_V^2(u)-{\mathit{\sigma }}_S^2(u)+{\mathit{\sigma }}_{\mathrm{Five}D}(u)-{\mathit{\sigma }}_{SV}(u)\right.\hfill \\ \hfill & \left.+{\mathit{\sigma }}_{SD}(u))du\right\}\tilde{E}[Z_3(t)Z_3^{-\mathrm{one}}(t)I_{A_2}|{\mathcal{F}}_t]\hfill \\ \hfill & =\mathrm{Southward}(t)\mathit{\delta }(t)b_1\check{\mathrm{Due\; east}}[I_{A_2}|{\mathcal{F}}_t]\hfill \\ \hfill & =S(t)\mathit{\delta }(t)b_{\mathrm{one}}\check{P}[(\mathrm{Southward}(T)>K,\mathit{\delta }(T)<\mathrm{i})|{\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 }}_{\mathrm{i}}^{2}du}}<\frac{ln\frac{\mathrm{South}(t)}{K}}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_1^{2}du}}\right.\right.\hfill \\ \hfill & +\frac{\mathrm{Yard}(t,T,r(t))-\underset{t}{\overset{T}{\int }}\left(\frac{{\mathit{\sigma }}_S^2(u)}{\mathrm{two}}-{\mathit{\sigma }}_{\mathrm{South}V}(u)-{\mathit{\sigma }}_{SD}(u)-2{\mathrm{\Delta }}_1^2\right)du}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_1^{2}du}}-\frac{{\int }_t^T{\mathrm{\Delta }}_{\mathrm{ii}}d{\check{B}}_{\mathit{\sigma }}(u)}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_{\mathrm{ii}}^{2}du}}\hfill \\ \hfill & \left.>\frac{ln\mathit{\delta }(t)+{\int }_t^T\left[\frac{1}{2}({\mathit{\sigma }}_D^2(u)-{\mathit{\sigma }}_V^2(u)+\mathrm{ii}{\mathit{\sigma }}_{SV}(u)-2{\mathit{\sigma }}_{SD}(u)+{\mathrm{\Delta }}_2^{\mathrm{ii}})du\right]}{\sqrt{{\int }_t^T{\mathrm{\Delta }}_{\mathrm{ii}}^{2}du}})|{\mathcal{F}}_t\right]\hfill \\ \hfill & =S(t)\mathit{\delta }(t)b_1\mathrm{North}\left(d_{\mathrm{ane}}+a_1+2\sqrt{{\int }_t^T{\mathrm{\Delta }}_1^{2}du},-\left(d_2+\mathrm{two}{a}_2+\sqrt{{\int }_t^T{\mathrm{\Delta }}_2^{\mathrm{ii}}du}\right);-\mathit{\rho }\right),\hfill \end{array}$$

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

Then by the consequence of $D_1,D_2$ , we consummate the proof of Theorem 3.

Corollary 2

Nether the firm value model,assume the cost of underlying nugget at time t is $S_t$ , exercise price is G, the ratio of bounty is $\mathit{\delta }(T)=V(T)/D(T)$ , the price of vulnerable asset or nothing put choice at the time $t(0\le t\le T)$ is as follows: (27) $$\begin{array}{cc}\hfill C(t)& =-S(t)N\left(-\left(d_{\mathrm{i}}+\sqrt{{\int }_t^T{\mathrm{\Delta }}_{\mathrm{one}}^{2}du}\right),d_2+a_2;-\mathit{\rho }\right)+S(t)\mathit{\delta }(t)b_1\hfill \\ \hfill & N\left(-\left(d_1+a_1+2\sqrt{{\int }_t^T{\mathrm{\Delta }}_{\mathrm{ane}}^{\mathrm{two}}du}\right),-\left(d_{\mathrm{two}}+2a_2+\sqrt{{\int }_t^T{\mathrm{\Delta }}_2^{2}du}\right);\mathit{\rho }\right),\hfill \end{array}$$ (27)

where $b_1,a_{\mathrm{two}},\mathit{\rho }$ are given in the Theorem 3.

four. Conclusion

The analytical pricing formula of the vulnerable binary options is derived in this paper past 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 charge per unit follows a Hull-White model. Comparing with the models of the pricing vulnerable European pick 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 abiding, the geometric Brownian move is corrected to be an O-U process. And also the involvement rate is assumed to be random ane following the Hull-White model rather than only a function with respect to fourth dimension in Ding and Chan(2007). All the corrections make the model more than realistic.

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

Posted by: hughesthomed.blogspot.com

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