The martingale approach for vulnerable binary option pricing under stochastic interest rate

Abstract: We consider the vulnerable option pricing problem when the stochastic interest rate is driven by a Hull-White model. Based on the firm value model, we suppose that the stock prices, assets and liabilities of a company follow the relevant O-U processes. We adopt the martingale approach to determine the equivalent martingale measure for pricing the vulnerable binary option, the analytical pricing formula of the vulnerable binary options is derived.


Introduction
Credit risk is one of the major financial risk, it has become a major challenge facing the world's financial markets. On the exchange transaction market, because of the margin requirements, futures exchanging and options exchange derivatives almost no credit risk. Therefore, when option pricing happened in the exchange, we can assume credit risk does not exist. However, On OTC market, counter-party credit risk 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

PUBLIC INTEREST STATEMENT
The option pricing problem is an important topic of finance field. At present, we mainly focus on the study 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 corresponding real data analysis.
default risk no longer valid. Therefore, how to properly consider the counter-party risk factors and establish option pricing model containing credit risk are important to investors.
There are some articles about option pricing take credit risk theory into consideration. Firm value model is one of the most important model about vulnerable option pricing put forward by 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 risk, putting forward the concept of vulnerable option. The seminal work that discussed option pricing containing credit risk is from Johnson and Stulz (1987). They used the term vulnerable option to define those options containing counter-party default risk, and pointed out a lot of features about them. Hull and White (1995) supposed the underlying asset and the counter-party firm's assets are independent of one another, obtained vulnerable options pricing formula. Jarrow and Tunbull (1995) supposed default time obey the strength of (non-negative constant) independent homogeneous poisson process, used discrete method to create a simple model of credit risk. Klein (1996) supposed breach of contract happened with a certain probability at any time, considered the pricing model of zerocoupon bonds can be liquidated at initial time zero.
In recent years, there is a considerable interest 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 case. They discussed the problem of option pricing with credit risk and getting its pricing formula. However, people always assume that stock price obey standard brown motion, witch is not 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 brown motion. In the previous option pricing model, we assume interest rate is a constant. However, in reality, interest rate changes randomly. A lot of scholars conducted extensive 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 article mainly research the problem of vulnerable binary option pricing under the O-U process. Based on the firm value model, we assume the stock price, counter-party firm's assets and counterparty firm's debts obey O-U process, interest rate obeys Hull-White Model. By using martingale approach we derive an analytical pricing formula for such vulnerable binary option. This article is structured as follows. Section 2 describes firm value model and its assumption. Section 3 presents the use of the martingale approach for vulnerable binary option pricing. Moreover, we obtain the pricing formula of vulnerable binary option. Section 4 contains some conclusion.

Firm value model
Considering continuous-time financial market, let (Ω,  , { t } 0≤t≤T , P) be a filtered probability space such that the filtration satisfies the usual conditions. Based on firm value model by Merton (1974), assume X(T) is the counter party firm promise to pay at the time T, X is European style option, if the counter party firm always have the ability of pay before the due maturity date, then the owner of this option will get X(T) at the time T, otherwise the payoff depend on the ratio of company's assets and its liability. 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 rate,  t is information sets before the time t, Ẽ (⋅) is the mathematical expectation under equivalent martingale measurement P .

Model building
Assume stock price S t , company's assets V t , company's liability D t all follow O-U process as follows O-U process is a Brown motion considering expected return rate depends on underlying assets's price, ( = S, V, D) can decrease underlying assets's price when the price up to a certain height.
In the following, we shall introduce Girsanov's theorem and martingale representation theorem about multidimensional correlated Brown motion B. Let Define then Z (t) is a  t -martingale. We further assume that for each 0 ≤ t ≤ T, (t) ≠ 0, and there exists a constant , = S, V, D such that Theorem 1 On the measurable space (Ω,  T ), we define new probability measure P , and for each A ∈  t , P (A) = E P [I A Z(T)], then P is the martingale measure equivalent to P, such that (1) (4) https://doi.org/10.1080/23311835.2017.1340073 This result is due to Doob (1953), and its proof can be found in Deng and Kaleong (2007, Theorem 2.2).
Lemma 1 (Steven, 1997) where 1 (t), 2 (t) are deterministic continuous function. Then B(t) is a standard  t -Brown motion under the probability P.

The martingale approach for vulnerable binary option pricing
Now we will discuss vulnerable binary option pricing under stochastic interest rate and liability.
Under the risk-neutral measure P , from Equation (1)

Model solution
Assume the 3-dimensional  t -adapted process Λ(t) = ( S (t), V (t), D (t)) satisfy the following SDEs: Let Z (t) is the  t -martingale defined by Equation (6), then by the Theorem 1, is a 3-dimensional correlated standard ( t ,P)-Brown motion, correlation matrix is C defined by formula (5).
From Equations (7)- (9), (14)- (16), the definition of (t)( = S, V, D) and the Theorem 1, we can get counter party firm's assets V(t), counter party firm's liability D(T), stock price S(t) satisfy the following SDEs: We assume short-term interest rate follows Hull-White model under the probability P as below: Under the probability P , from Ito formulas we can get: where Under the probability P , applying Ito formulas to (17)-(19) we can get:

t)) is a 4-dimensional correlated standard Brown motion under probability P , the correlation matrix is
For the convenience of computing, we transform S(T), (T) as follows: where From Lemma 1 we see that B̄S (t) is a standard brown motion under the probability P .
Under the probability P , let (t) = V(t)∕D(t), according to Itô's formula we have where By Lemma 1 again, we have that B (t) is a standard brown motion under the probability P .

The pricing formula of vulnerable binary option
Theorem 2 Under the firm value model, assume underlying asset price is S t at the time t, exercise price is K, the ratio of compensation is (T) = V(T)∕D(T), the price of vulnerable cash or nothing call option at the time t(0 ≤ t ≤ T) is: where Δ 1 , Δ 2 is defined by Equations (23)and (25).
Proof Denote A 1 = (S(T) > K, (T) ≥ 1), A 2 = (S(T) > K, (T) < 1), according to Equation (13), under the risk neutral probability, the price of vulnerable cash or nothing call option is: Denote the joint distribution function of two dimensional standardized normal random vector is: Evaluation of term D 1 : according to S(T) > K and S(T) follows from (22) Substituting this into D 1 , we obtain where Evaluation of term D 2 : firstly we define a new probability measure, since B (t),B̄S(t) are standard brown motions under the measure P . Define Rodon-Nikodym derivative: According to the Lemma 1, we see that B (t),B̄S(t) are standard brown motions under the measure P .
Under the measure P , we have Substituting this into D 2 , we get where Evaluation of V 1 : we can define a new probability measure equivalent to P .
Under the measure P , we obtain Substituting thus into V 1 , we have

S(T) = S(t) exp
where Evaluation of V 2 , under the measure P , we get