With the development of globalization and cross-market integration, multi-asset options are created. In this section, we suppose that the asset price follows uncertain jump processes whose right-continuous pure jump part is determined by an uncertain renewal reward process. Let *r* denote the risk-free interest rate, *e* be the log drift, *σ* be the log diffusion, and *C*
_{
t
} be a canonical Liu process. \(R_{t}=\sum _{i=1}^{N_{t}}Y_{i}\) is an uncertain renewal reward process where *N*
_{
t
} is the number of renewals in (0,*t*] with independent and identically distributed (*i*.*i*.*d*) uncertain interarrival times *X*
_{1},*X*
_{2},⋯ and *Y*
_{1},*Y*
_{2},⋯ are *i*.*i*.*d* uncertain variables. We assume that *X*
_{1},*Y*
_{1},*X*
_{2},*Y*
_{2},⋯ are also independent. Then, the underlying uncertain model of asset price is given by

$$ \mathrm{d} Z_{t}=eZ_{t}\mathrm{d} t+\sigma Z_{t}\mathrm{d} C_{t}+Z_{t-}\mathrm{d} R_{t}. $$

((20))

Let *Y*
_{
i
}>−1,*i*=1,2,⋯. This assumption guarantees that the asset price can jump down but it cannot jump from positive to negative or to zero. The solution to (20) is

$$ Z_{t}=Z_{0}\text{exp}\{et+\sigma C_{t}\}\prod\limits_{i=1}^{N_{t}}(1+Y_{i}). $$

((21))

To check if (21) satisfies the uncertain differential Eq. (20), we define the continuous part

$$ {Z_{t}^{c}}=Z_{0}\text{exp}\{et+\sigma C_{t}\} $$

((22))

and the pure jump part

$$ J_{t}=\prod_{i=1}^{N_{t}}(1+Y_{i}). $$

((23))

Then, \(Z_{t}={Z_{t}^{c}}J_{t}\). By using Theorem 2, we have

$$ \mathrm{d} {Z_{t}^{c}}=e{Z_{t}^{c}}\mathrm{d} t+\sigma {Z_{t}^{c}}\mathrm{d} C_{t}. $$

((24))

And the jump size of *J*
_{
t
} at time *t* is

$$ \Delta J_{t}=J_{t}-J_{t-}=J_{t-}(1+Y_{i})-J_{t}=J_{t-}\Delta R_{t}. $$

((25))

By using Theorem 5 and the above equations, we obtain

$$\begin{array}{@{}rcl@{}} Z_{t}&=&{Z_{t}^{c}}J_{t}\\ &=&{Z_{0}^{c}}J_{0}+{\int_{0}^{t}}e{Z_{s}^{c}}J_{s}\mathrm{d} s+{\int_{0}^{t}}\sigma {Z_{s}^{c}}J_{s}\mathrm{d} C_{s}+\sum_{0<s\leq t}\left[Z_{s}^{c}J_{s}-Z_{s-}^{c}J_{s-}\right]\\ &=&{Z_{0}^{c}}J_{0}+{\int_{0}^{t}}e{Z_{s}^{c}}J_{s}\mathrm{d} s+{\int_{0}^{t}}\sigma {Z_{s}^{c}}J_{s}\mathrm{d} C_{s}+\sum_{0<s\leq t}Z_{s-}^{c}J_{s-}\Delta R_{s}\\ &=&Z_{0}+{\int_{0}^{t}}eZ_{s}\mathrm{d} s+{\int_{0}^{t}}\sigma Z_{s}\mathrm{d} C_{s}+\sum_{0<s\leq t}Z_{s-}\Delta R_{s}. \end{array} $$

((26))

This verifies that (21) is the solution to (20).

###
**Theorem**
**6**.

Assume *C*
_{
t
} has distribution *Φ*
_{
t
}(*x*), *Y*
_{1},*Y*
_{2},⋯ and *X*
_{1},*X*
_{2},⋯ have distributions *F*(*x*)and *H*(*x*), respectively. Then, *Z*
_{
t
} has an uncertainty distribution

$$\begin{array}{@{}rcl@{}} \Psi_{t}(x)&=&\sup_{x_{1}x_{2}={x}/{Z_{0}}}\left\{\Phi_{t}\left(\frac{\ln x_{1}-et}{\sigma}\right)\right.\\ &&\left.\wedge\left(\max_{k\geq 0}\left(1-H\left(\frac{t}{k+1}\right)\right)\wedge F\left(\exp\left(\frac{x_{2}}{k}\right)-1\right)\right)\right\}. \end{array} $$

((27))

###
*Proof*.

Because *Y*
_{
i
} has distribution *F*(*x*), ln(1+*Y*
_{
i
}) has distribution *F*(exp(*x*)−1). \(\prod _{i=1}^{N_{t}}(1+Y_{i})\) has an uncertainty distribution

$$\begin{array}{@{}rcl@{}} \Upsilon_{t}(x)&=&\mathscr{M}\left\{\prod_{i=1}^{N_{t}}(1+Y_{i})\leq x\right\}\\ &=&\mathscr{M}\left\{\sum_{i=1}^{N_{t}}\ln(1+Y_{i})\leq \ln x\right\}\\ &=&\mathscr{M}\left\{\bigcup_{k=0}^{\infty}(N_{t}=k)\cap\sum_{i=1}^{k}\ln(1+Y_{i})\leq x\right\}\\ &=&\mathscr{M}\left\{\bigcup_{k=0}^{\infty}(N_{t}=k)\cap\left(\ln(1+Y_{1})\leq \frac{x}{k}\right)\right\}\\ &=&\max_{k\geq 0}\mathscr{M}\left\{(N_{t}\leq k)\cap\left(\ln(1+Y_{1})\leq \frac{x}{k}\right)\right\}\\ &=&\max_{k\geq 0}\mathscr{M}\left\{(N_{t}\leq k)\right\}\wedge\mathscr{M}\left\{\left(\ln(1+Y_{1})\leq \frac{x}{k}\right)\right\}\\ &=&\max_{k\geq 0}\left(1-H\left(\frac{t}{k+1}\right)\right)\wedge F\left(\exp\left(\frac{x}{k}\right)-1\right). \end{array} $$

((28))

On the other hand, exp{*e*
*t*+*σ*
*C*
_{
t
}} has uncertainty distribution \(\Phi \left (\frac {\ln x-et}{\sigma }\right)\). By the operational law of uncertain variables, *Z*
_{
t
} has an uncertainty distribution

$$\begin{array}{@{}rcl@{}} \Psi_{t}(x)&=&\mathscr{M}\{Z_{t}\leq x\}\\ &=&\mathscr{M}\left\{{Z_{0}\text{exp}\{et+\sigma C_{t}\}\prod_{i=1}^{N_{t}}(1+Y_{i})\leq x}\right\}\\ &=&\sup_{x_{1}x_{2}={x}/{Z_{0}}}\mathscr{M}\left\{\left(\text{exp}\{et+\sigma C_{t}\}\leq x_{1}\right)\cap\left(\prod_{i=1}^{N_{t}}(1+Y_{i})\leq x_{2}\right)\right\}\\ &=&\sup_{x_{1}x_{2}={x}/{Z_{0}}}\mathscr{M}\left\{\left(\text{exp}\{et+\sigma C_{t}\}\leq x_{1}\right)\right\}\wedge\mathscr{M}\left\{\left(\prod_{i=1}^{N_{t}}(1+Y_{i})\leq x_{2}\right)\right\}\\ &=&\sup_{x_{1}x_{2}={x}/{Z_{0}}}\left\{\Phi_{t}\left(\frac{\ln x_{1}-et}{\sigma}\right)\wedge\left(\max_{k\geq 0}\left(1-H\left(\frac{t}{k+1}\right)\right)\right.\right.\\&&\left.\left.\wedge F\left(\exp\left(\frac{x_{2}}{k}\right)-1\right)\right)\right\}. \end{array} $$

The theorem is proved. □

### Dual-Strike Options

A dual-strike option is an option which has two strike prices written on two underlying financial assets. In an uncertain environment, suppose the two assets \(Z_{t}^{(1)}\) and \(Z_{t}^{(2)}\) both follow the uncertain jump process given by (20) and are independent. Then, the payoff of a European dual-strike option is \(\max \left \{\omega _{1}\left (Z_{1}^{(1)}-K_{1}\right), \omega _{2}\left (Z_{1}^{(2)}-K_{2}\right), 0\right \}\) where *K*
_{1} and *K*
_{2} are the strike prices, *ω*
_{1} and *ω*
_{2} are 1 for a call option and −1 for a put option. By discounting the expected payoff at expiration date *T*, the price of the dual-strike option is

$$ f_{ds}=\text{exp}(-rT)E\left[\max\left\{\omega_{1}\left(Z_{t}^{(1)}-K_{1}\right), \omega_{2}\left(Z_{t}^{(2)}-K_{2}\right), 0\right\}\right]. $$

((29))

In the following, we derive the pricing formula for dual-strike option with *ω*
_{1}=*ω*
_{2}=1.

###
**Theorem**
**7**.

The price of a European dual-strike option is

$$ f_{ds}=\int_{0}^{+\infty}\left(1-\Psi_{T}^{(1)}\left(x+K_{1}\right)\right)\vee \left(1-\Psi_{T}^{(2)}\left(x+K_{2}\right)\right)\mathrm{d} x $$

((30))

where

$$\begin{array}{@{}rcl@{}} \Psi_{T}^{(i)}(x)&=&\sup_{x_{1}x_{2}={x}/{Z_{0}}}\left\{\Phi_{T}\left(\frac{\ln x_{1}-e_{i}t}{\sigma_{i}}\right)\wedge\left(\max_{k\geq 0}\left(1-H^{(i)}\left(\frac{t}{k+1}\right)\right)\right.\right.\\&&\left.\left.\wedge F^{(i)}\left(\exp\left(\frac{x_{2}}{k}\right)-1\right)\right)\right\}, \ i=1,2. \end{array} $$

###
*Proof*.

By the definition of expected value

$$\begin{array}{@{}rcl@{}} &&E\left[\max\left\{\left(Z_{T}^{(1)}-K_{1}\right), \left(Z_{T}^{(2)}-K_{2}\right), 0\right\}\right]\\ &&=\int_{0}^{+\infty}\mathscr{M}\left\{\left(Z_{T}^{(1)}-K_{1}\geq x\right)\cup \left(Z_{T}^{(2)}-K_{2}\geq x\right)\right\}\mathrm{d} x\\ &&=\int_{0}^{+\infty}\mathscr{M}\left\{Z_{T}^{(1)}\geq x+K_{1}\right\}\vee \mathscr{ M}\left\{Z_{T}^{(2)}\geq x+K_{2}\right\}\mathrm{d} x\\ &&=\int_{0}^{+\infty}\left(1-\mathscr{M}\left\{Z_{T}^{(1)}< x+K_{1}\right\}\right)\vee \left(1-\mathscr{ M}\left\{Z_{T}^{(2)}< x+K_{2}\right\}\right)\mathrm{d} x\\ &&=\int_{0}^{+\infty}\left(1-\Psi_{T}^{(1)}\left(x+K_{1}\right)\right)\vee \left(1-\Psi_{T}^{(2)}\left(x+K_{2}\right)\right)\mathrm{d} x. \end{array} $$

((31))

Using (27), we know that

$$\begin{array}{@{}rcl@{}} \Psi_{T}^{(i)}(x)&=&\sup_{x_{1}x_{2}={x}/{Z_{0}}}\left\{\Phi_{T}\left(\frac{\ln x_{1}-e_{i}t}{\sigma_{i}}\right)\wedge\left(\max_{k\geq 0}\left(1-H^{(i)}\left(\frac{t}{k+1}\right)\right) \right.\right.\\&&\wedge \left.\left.F^{(i)}\left(\exp\left(\frac{x_{2}}{k}\right)-1\right)\right)\right\}, \ i=1,2. \end{array} $$

Hence, the deal-strike option pricing formula is verified. □

### Product Options

A product option is an option which is written on the product of two financial assets. Let \(Z_{t}^{(1)}\) and \(Z_{t}^{(2)}\) be uncertain jump processes which satisfy (20) and (1). Then, the price of a European-style product option with strike price *K* and expiration date *T* is

$$ f_{p}=\text{exp}(-rT)E\left[\max\left\{\omega Z_{T}^{(1)}Z_{T}^{(2)}-\omega K, 0\right\}\right], $$

((32))

where *ω* is 1 for a call option and −1 for a put option. In the following, we consider the product call option.

###
**Theorem**
**8**.

The price of a European product option is

$$ f_{p}=\int_{0}^{+\infty}\left(1-\Psi_{p}(x+K)\right)\mathrm{d} x, $$

((33))

where

$$ \Psi_{p}(x)=\sup_{x_{1}x_{2}=x}\Psi_{T}^{(1)}(x_{1})\wedge\Psi_{T}^{(2)}(x_{2}), $$

((34))

$$\begin{array}{@{}rcl@{}} \Psi_{T}^{(i)}(x)&=&\sup_{x_{1}x_{2}={x}/{Z_{0}}}\left\{\Phi_{T}\left(\frac{\ln x_{1}-e_{i}t}{\sigma_{i}}\right)\wedge\left(\max_{k\geq 0}\left(1-H^{(i)}\left(\frac{t}{k+1}\right)\right)\right.\right.\\&&\left.\left.\wedge F^{(i)}\left(\exp\left(\frac{x_{2}}{k}\right)-1\right)\right)\right\}, \ i=1,2. \end{array} $$

###
*Proof*.

By the operational law, we can obtain the distribution of \(Z_{T}^{(1)}Z_{T}^{(2)}\)

$$\Psi_{p}(x)=\sup_{x_{1}x_{2}=x}\Psi_{T}^{(1)}(x_{1})\wedge\Psi_{T}^{(2)}(x_{2}), $$

where \(\Psi _{T}^{(1)}\) and \(\Psi _{T}^{(2)}\) are distribution functions of \(Z_{t}^{(1)}\) and \(Z_{t}^{(2)}\). Then, by the definition of expected value,

$$\begin{array}{@{}rcl@{}} E\left[\max\left\{Z_{T}^{(1)}Z_{T}^{(2)}-K, 0\right\}\right]&=&\int_{0}^{+\infty}\mathscr{M}\left\{\max\left\{Z_{T}^{(1)}Z_{T}^{(2)}-K, 0\right\}\geq x\right\}\mathrm{d} x\\ &=&\int_{0}^{+\infty}\mathscr{M}\left\{Z_{T}^{(1)}Z_{T}^{(2)}\geq x+K\right\}\mathrm{d} x\\ &=&\int_{0}^{+\infty}\left(1-\mathscr{M}\left\{Z_{T}^{(1)}Z_{T}^{(2)}<x+K\right\}\right)\mathrm{d} x\\ &=&\int_{0}^{+\infty}\left(1-\Psi_{p}(x+K)\right)\mathrm{d} x. \end{array} $$

Thus, the product option pricing formula is verified. □

### Quotient Options

A quotient option is an option which is written on the ratio of two financial assets. Let \(Z_{t}^{(1)}\) and \(Z_{t}^{(2)}\) be uncertain jump processes which satisfy (20) and (1). Then, the price of a European-style quotient option with strike price *K* and expiration date *T* is

$$ f_{q}=\text{exp}(-rT)E\left[\max\left\{\omega\frac{Z_{T}^{(1)}}{Z_{T}^{(2)}}-\omega K, 0\right\}\right], $$

((35))

where *ω* is 1 for a call option and −1 for a put option. In the following, we derive the pricing formula for quotient call option.

###
**Theorem**
**9**.

The price of a European quotient option is

$$ f_{q}=\int_{0}^{+\infty}\left(1-\Psi_{q}(x+K)\right)\mathrm{d} x, $$

((36))

where

$$ \Psi_{q}(x)=\sup_{y>0}\Psi_{T}^{(1)}(xy)\wedge \left(1-\Psi_{T}^{(2)}(y)\right), $$

((37))

$$\begin{array}{@{}rcl@{}} \Psi_{T}^{(i)}(x)&=&\sup_{x_{1}x_{2}={x}/{Z_{0}}}\left\{\Phi_{T}\left(\frac{\ln x_{1}-e_{i}t}{\sigma_{i}}\right)\wedge\left(\max_{k\geq 0}\left(1-H^{(i)}\left(\frac{t}{k+1}\right)\right)\right.\right.\\&&\left.\left.\wedge F^{(i)}\left(\exp\left(\frac{x_{2}}{k}\right)-1\right)\right)\right\}, i=1,2. \end{array} $$

###
*Proof*.

By the operational law, we can obtain the distribution of \({Z_{T}^{(1)}}/{Z_{T}^{(2)}}\),

$$\Psi_{q}(x)=\sup_{y>0}\Psi_{T}^{(1)}(xy)\wedge \left(1-\Psi_{T}^{(2)}(y)\right) $$

where \(\Psi _{T}^{(1)}\) and \(\Psi _{T}^{(2)}\) are distribution functions of \(Z_{t}^{(1)}\) and \(Z_{t}^{(2)}\). Then, by the definition of expected value,

$$\begin{array}{@{}rcl@{}} E\left[\max\left\{\frac{Z_{T}^{(1)}}{Z_{T}^{(2)}}-K, 0\right\}\right]&=&\int_{0}^{+\infty}\mathscr{M}\left\{\max\left\{\frac{Z_{T}^{(1)}}{Z_{T}^{(2)}}-K, 0\right\}\geq x\right\}\mathrm{d} x\\ &=&\int_{0}^{+\infty}\mathscr{M}\left\{\frac{Z_{T}^{(1)}}{Z_{T}^{(2)}}\geq x+K\right\}\mathrm{d} x\\ &=&\int_{0}^{+\infty}\left(1-\mathscr{M}\left\{\frac{Z_{T}^{(1)}}{Z_{T}^{(2)}}<x+K\right\}\right)\mathrm{d} x\\ &=&\int_{0}^{+\infty}\left(1-\Psi_{q}(x+K)\right)\mathrm{d} x. \end{array} $$

The quotient option pricing formula is verified. □