The Pricing of Power Options on Two Assets

Gerald H. L. Cheang¹, Semyon Chernykh¹¹School of Mathematical Sciences, Adelaide University, Adelaide, SA 5000, Australia

Abstract. Employees and board members are often given bonus stock options as performance incentives. Such options can often be evaluated as regular stock-or-nothing options if the KPI criterion triggering the bonus is basic enough. However, for more complicated criteria, the equivalent payoff at the exercise time could take the form of a power stock-or-nothing option, where the trigger also depends on another stock, e.g.

𝑆_{1, 𝑇}^{𝛾} 1_{{𝑆_{1, 𝑇}^{𝛼} > 𝑆_{2, 𝑇}^{𝛽}}}

. We price the value of such options, as well as the stock-or-nothing option with the payoff

𝑆_{2, 𝑇}^{𝛾} 1_{{𝑆_{1, 𝑇}^{𝛼} > 𝑆_{2, 𝑇}^{𝛽}}}

. The expressions for the prices of these two stock-or-nothing options also enable us to easily price the power exchange option proposed by Blenman and Clark [1].

Keywords: power options; stock-or-nothing options; exchange options; Black-Scholes model; Esscher transforms; executive compensation

1. Introduction

In this paper, we derive the price of power stock-or-nothing options based on two different stocks. These options are often used for employee and board member bonus stock options as performance incentives. Such options can often be evaluated as regular stock-or-nothing options if the KPI criterion triggering the bonus are basic enough. However, for more complicated criteria, the equivalent payoff at the exercise time could take the form of a power stock-or-nothing option, where the trigger also depends on another stock. Additionally, these options simplify the pricing of power exchange options.

Tompkins [2] discusses the use of power options to hedge nonlinear risks, focusing on their application in managing implied volatility. He utilises the property that a power straddle’s sensitivity to volatility is invariant to the underlying asset price, which allows the use of the Black-Scholes model (Black and Scholes [3]). Tompkins critiques the prior derivations of power options for being incomplete and derives the right formulas for pricing several power options.

Esser [4] generalises the pricing framework for stochastic volatility using Esscher transforms. She demonstrates that a stock raised to a power cannot serve as a traded asset, and uses martingale properties and change-of-measure technique to develop a pricing framework for power options. While Esser [4] primarily focuses on the stochastic volatility model for pricing power options, the change-of-measure technique described in the text simplifies the calculations for the Black-Scholes model.

Blenman and Clark [1] use the change-of-measure technique developed by Esser [4] to price power exchange options in the Black-Scholes setting. Furthermore, they provide a formula for hedging such options using the underlying stocks.

In this paper, we generalise and price stock-or-nothing options in the Black-Scholes setting using the change-of-measure technique developed by Esser [4]. The payoff is the equivalent amount of the power of the value of one stock, and the trigger for the payoff depends on a scenario involving the powers of two different stocks. One power stock-or-nothing option that we price in this paper has a payoff of the form

𝑆_{1, 𝑇}^{𝛾} 1_{{𝑆_{1, 𝑇}^{𝛼} > 𝑆_{2, 𝑇}^{𝛽}}} .

(1)

Another stock-or-nothing option that we price in this paper has a payoff of the form

𝑆_{2, 𝑇}^{𝛾} 1_{{𝑆_{1, 𝑇}^{𝛼} > 𝑆_{2, 𝑇}^{𝛽}}} .

(2)

The expressions for the prices of these two stock-or-nothing options also allow us to easily price the power exchange option of Blenman and Clark [1] with the payoff function ${(𝑆_{1, 𝑇}^{𝛼} - 𝑆_{2, 𝑇}^{𝛽})}^{+}$ , from which the price of the standard exchange option of Fisher [5] and Margrabe [6] is a special case.

This paper is organised as follows. Section 2 introduces the dynamics of the stock prices. In Section 3, we define a Radon-Nikodým derivative that transforms the dynamics of the two stock prices from the market measure to the risk-neutral measure. We then show how the parameters of this Radon-Nikodým derivative, which induces the measure transformation, can be interpreted as the market prices of risk for the individual stocks. The main results and their proofs are presented in Section 4, followed by examples of applications in Section 5. Finally, Section 6 concludes the paper.

2. The Model

Let $(Ω, ℱ︀, {{ℱ︀}_{𝑡}}, ℙ)$ be a probability measure space. We are only interested in the filtration over $0 \leq 𝑡 \leq 𝑇$ for some fixed $𝑇$ , the expiry time of the option, that is ${{ℱ︀}_{𝑡}}_{𝑡 \leq 𝑇}$ . Let $𝑾_{𝑡} = {(𝑊_{1, 𝑡}, 𝑊_{2, 𝑡})}^{⊤}$ be a bivariate correlated Brownian motion process which is adapted to the filtration. We define the matrix

Σ = (\begin{matrix} 1 & 𝜌 \\ 𝜌 & 1 \end{matrix}),

(3)

where $𝑑 𝑊_{1, 𝑡} 𝑑 𝑊_{2, 𝑡} = 𝜌 𝑑 𝑡$ , with $𝜌$ the constant instantaneous correlation between the two Brownian motion components.

Throughout this paper, we assume that $𝑆_{1, 𝑡}$ and $𝑆_{2, 𝑡}$ are traded financial assets with return dynamics under the market measure $ℙ$ given by

\frac{𝑑 𝑆_{𝑖, 𝑡}}{𝑆_{𝑖, 𝑡}} = 𝜇_{𝑖} 𝑑 𝑡 + 𝜎_{𝑖} 𝑑 𝑊_{𝑖, 𝑡}

(4)

for $𝑖 = 1, 2$ , and where for asset $𝑖$ , $𝜇_{𝑖}$ is the constant instantaneous expected return per unit time, and $𝜎_{𝑖}$ is the constant instantaneous volatility per unit time. We also assume that asset $𝑖$ pays a constant continuous dividend yield at rate $𝛿_{𝑖}$ . There is also a money market account $𝑀_{𝑡} = 𝑒^{𝑟 𝑡}$ , where $𝑟$ is the constant instantaneous risk-free interest rate. We choose constant parameters for our model for convenience. The results in this paper can easily be extended to the case where the parameters are non-stochastic time varying, subject to the usual $𝑙_{1}$ -integrability conditions for the drifts and risk-free rate, and the $𝑙_{2}$ -integrability condition for the volatility.

3. Transformation of Measures

In option pricing problems, the main goal is a suitable risk-neutral evaluation of the final payoff conditional on information about the underlying asset prices up to the current time. In our model given by Equation 4, randomness in the model is driven by the correlated bivariate Wiener process $𝑾_{𝑡}$ . Options are priced as discounted conditional expected payoffs under the risk-neutral measure $ℚ$ , the measure that corresponds to the money market account $𝑀_{𝑡}$ as the numéraire. Thus, if $𝐶_{𝑡} (𝑆_{1, 𝑡}, 𝑆_{2, 𝑡})$ is the price of an option at time $𝑡$ where the payoff at maturity is $𝐶_{𝑇} (𝑆_{1, 𝑇}, 𝑆_{2, 𝑇})$ , then the discounted option price process $\frac{𝐶_{𝑡}}{𝑀_{𝑡}}$ is a $ℚ$ -martingale. Similarly, we also required the discounted stock yield processes $\frac{𝑆_{1, 𝑡} 𝑒^{𝛿_{1} 𝑡}}{𝑀_{𝑡}}$ and $\frac{𝑆_{2, 𝑡} 𝑒^{𝛿_{2} 𝑡}}{𝑀_{𝑡}}$ to also be $ℚ$ -martingales.

In order to facilitate the change of measure in our analyses, we state the following lemma for the convenience of the reader. The proof is relatively straightforward using the moment generating function of multivariate normally distributed random variables. In the statement of the lemma and in the rest of the paper, the expression ${ℰ︀}_{ℙ} (𝑋_{𝑡})$ denotes the Doléans-Dade [7] exponential of a semi-martingale process ${𝑋_{𝑡}}$ under the measure $ℙ$ .

Lemma 1. Let

𝐿_{𝑡} = {ℰ︀}_{ℙ} (- 𝜽^{⊤} 𝑾_{𝑡}) = exp [- 𝜽^{⊤} 𝑾_{𝑡} - \frac{1}{2} 𝜽^{⊤} Σ 𝜽 𝑡] .

(5)

where $𝜽 = {(𝜃_{1}, 𝜃_{2})}^{⊤}$ . Then $𝐿_{𝑡}$ is a Radon-Nikodým derivative that induces the measure transformation from $ℙ$ to $\hat{ℙ}$ , where under $\hat{ℙ}$ , $𝑾_{𝑡}$ is distributed as multivariate normal MVN $(- Σ 𝜽 𝑡, Σ 𝑡)$ and

\begin{aligned} 𝑑 𝑊_{1, 𝑡} & = - (𝜃_{1} + 𝜌 𝜃_{2}) 𝑑 𝑡 + 𝑑 {\hat{𝑊}}_{1, 𝑡}, \\ 𝑑 𝑊_{2, 𝑡} & = - (𝜌 𝜃_{1} + 𝜃_{2}) 𝑑 𝑡 + 𝑑 {\hat{𝑊}}_{2, 𝑡} . \end{aligned}

(6)

where ${\hat{𝑾}}_{𝑡} = {({\hat{𝑊}}_{1, 𝑡}, {\hat{𝑊}}_{2, 𝑡})}^{⊤}$ is correlated Brownian motion without drift and with correlation matrix $Σ$ under $\hat{ℙ}$ .

The proof of Lemma 1 is just a standard observation that the moment generating function of $𝑾_{𝑡}$ under $\hat{ℙ}$ is $𝔼_{\hat{ℙ}} [𝑒^{𝒖^{⊤} 𝑾_{𝑡}}] = 𝔼_{ℙ} [𝐿_{𝑡} 𝑒^{𝒖^{⊤} 𝑾_{𝑡}}]$ . The expression for Radon-Nikodým derivative in Equation Equation 5 is a type of Esscher transform similar to those in Gerber and Shiu [8].

In what follows in the rest of the paper, all the options will be priced as expected values of the final payoffs under the risk-neutral measure $ℚ$ . It is a requirement that the discounted process for each stock must be a martingale under $ℚ$ . Thus, for $𝑖 = 1, 2$ , the dynamics for the discounted yield process for each stock is

𝑑 (\frac{𝑆_{𝑖, 𝑡} 𝑒^{𝛿_{𝑖} 𝑡}}{𝑀_{𝑡}}) = \frac{𝑆_{𝑖, 𝑡} 𝑒^{𝛿_{𝑖} 𝑡}}{𝑀_{𝑡}} ((𝜇_{𝑖} + 𝛿_{𝑖} - 𝑟) 𝑑 𝑡 + 𝜎_{𝑖} 𝑑 𝑊_{𝑖, 𝑡})

(7)

by the application of Itô’s Lemma (for examples, see Shreve [9]). Since each stock discounted stock yield process are $ℚ$ -martigales, we must be able to express their dynamics as

𝑑 (\frac{𝑆_{𝑖, 𝑡} 𝑒^{𝛿_{𝑖} 𝑡}}{𝑀_{𝑡}}) = \frac{𝑆_{𝑖, 𝑡} 𝑒^{𝛿_{𝑖} 𝑡}}{𝑀_{𝑡}} 𝜎_{𝑖} 𝑑 {\tilde{𝑊}}_{𝑖, 𝑡},

(8)

where ${\tilde{𝑾}}_{𝑡} = {({\tilde{𝑊}}_{1, 𝑡}, {\tilde{𝑊}}_{2, 𝑡})}^{⊤}$ is correlated Brownian motion without drift under $ℚ$ and $𝑑 {\tilde{𝑊}}_{1, 𝑡} . 𝑑 {\tilde{𝑊}}_{2, 𝑡} = 𝜌 𝑑 𝑡$ .

In going from Equation 7 to Equation 8, we have

𝑑 {\tilde{𝑊}}_{𝑖, 𝑡} = 𝜃_{𝑖} 𝑑 𝑡 + 𝑑 𝑊_{𝑖, 𝑡}

(9)

for $𝑖 = 1, 2$ , and

𝜃_{𝑖} = \frac{𝜇_{𝑖} + 𝛿_{𝑖} - 𝑟}{𝜎_{𝑖}}

(10)

is the market price of risk for stock $𝑖$ . It is an easy application of Lemma 1 to show that the Radon-Nikodým that induces the measure change from $ℙ$ to the risk-neutral measure $ℚ$ is given by

\frac{𝑑 ℚ}{𝑑 ℙ} |_{𝑡} = {ℰ︀}_{ℙ} (- {(Σ^{- 1} 𝜽)}^{⊤} 𝑾_{𝑡}) = exp [- {(Σ^{- 1} 𝜽)}^{⊤} 𝑾_{𝑡} - \frac{1}{2} 𝜽^{⊤} Σ^{- 1} 𝜽 𝑡],

(11)

where the components of $𝜽$ are given by Equation 10 for $𝑖 = 1, 2$ .

The integrated form of Equation 8 is

\frac{𝑆_{𝑖, 𝑡} 𝑒^{𝛿_{𝑖} 𝑡}}{𝑀_{𝑡}} = 𝑆_{𝑖, 0} exp {- \frac{1}{2} 𝜎_{𝑖}^{2} 𝑡 + 𝜎_{𝑖} {\tilde{𝑊}}_{𝑖, 𝑡}} = 𝑆_{𝑖, 0} {ℰ︀}_{ℚ} (𝜎_{𝑖} {\tilde{𝑊}}_{𝑖, 𝑡}),

(12)

which gives us the stock price dynamics under the risk-neutral measure $ℚ$ as

𝑆_{𝑖, 𝑡} = 𝑆_{𝑖, 0} exp {(𝑟 - 𝛿_{𝑖} - \frac{1}{2} 𝜎_{𝑖}^{2}) 𝑡 + 𝜎_{𝑖} {\tilde{𝑊}}_{𝑖, 𝑡}} = 𝑆_{𝑖, 0} 𝑒^{(𝑟 - 𝛿_{𝑖}) 𝑡} {ℰ︀}_{ℚ} (𝜎_{𝑖} {\tilde{𝑊}}_{𝑖, 𝑡}) .

(13)

In Equation 13, we explicitly see that under the risk-neutral measure $ℚ$ , the cost of carry for each stock is the risk-free rate less its dividend yield. In the analyses that follows in Section 4, we apply the expression for the stock price in Equation 13 when we work under the risk-neutral measure $ℚ$ .

4. Option Pricing

In this section, we price two different power stock-or-nothing options and show that these can also be used to price the power exchange option found in Blenman and Clark [1].

It must be noted that powers of stocks are not traded assets (Esser [4]). However, a market consisting of two stocks under the Margrabe [6] correlated geometric Brownian motion dynamics is a complete market. Consequently, all discounted option prices based on these two stocks are $ℚ$ -martingales. In the proofs presented in this section, we apply the change-of-measure technique described by Esser [4] in order to evaluate the payoff functions of these assets.

The first stock-or-nothing option based on two stocks has a final payoff of the form $𝐶_{1, 𝛾, 𝑇} = 𝑆_{1, 𝑇}^{𝛾} 1_{{𝑆_{1, 𝑇}^{𝛼} > 𝑆_{2, 𝑇}^{𝛽}}}$ at maturity time $𝑇$ , where the parameters are $𝛼, 𝛽, 𝛾 > 0$ .

Theorem 1. The price $𝐶_{1, 𝛾, 𝑡}$ at time $𝑡$ of a stock-or-nothing option with final payoff $𝐶_{1, 𝛾, 𝑇} = 𝑆_{1, 𝑇}^{𝛾} 1_{{𝑆_{1, 𝑇}^{𝛼} > 𝑆_{2, 𝑇}^{𝛽}}}$ is

𝐶_{1, 𝛾, 𝑡} = 𝑆_{1, 𝑡}^{𝛾} exp {((𝛾 - 1) 𝑟 - 𝛾 𝛿_{1} - \frac{1}{2} 𝛾 (1 - 𝛾) 𝜎_{1}^{2}) (𝑇 - 𝑡)} Φ (𝑑_{1, 𝛾, 𝑡}),

(14)

where $Φ (\cdot)$ is the cumulative distribution function of the standard normal distribution, and

\begin{matrix} 𝑑_{1, 𝛾, 𝑡} = \frac{1}{𝜎 \sqrt{𝑇 - 𝑡}} [ln \frac{𝑆_{1, 𝑡}^{𝛼}}{𝑆_{2, 𝑡}^{𝛽}} + (𝛼 (𝑟 - 𝛿_{1}) - 𝛽 (𝑟 - 𝛿_{2}) - \frac{1}{2} 𝛼 (1 - 2 𝛾 + 𝛼) 𝜎_{1}^{2} \\ + \frac{1}{2} 𝛽 (1 - 𝛽) 𝜎_{2}^{2} - 𝜌 𝛽 (𝛾 - 𝛼) 𝜎_{1} 𝜎_{2} + \frac{1}{2} 𝜎^{2}) (𝑇 - 𝑡)] \end{matrix}

(15)

with the variance

𝜎^{2} = {(𝛼 𝜎_{1})}^{2} - 2 𝜌 (𝛼 𝜎_{1}) (𝛽 𝜎_{2}) + {(𝛽 𝜎_{2})}^{2} .

(16)

Proof of Theorem 1. Without loss of generality, we price the option at time $𝑡 = 0$ , since the price at time $𝑡$ is easily obtained by changing the time to maturity in the expression that we derive to $𝑇 - 𝑡$ .

Since the payoff function is a martingale under the risk-neutral measure $ℚ$ , the option price at time $0$ is given by

𝐶_{1, 𝛾, 0} = 𝔼_{ℚ} [\frac{𝐶_{1, 𝛾, 𝑇}}{𝑀_{𝑇}}] = 𝔼_{ℚ} [\frac{𝑆_{1, 𝑇}^{𝛾} 1_{{𝑆_{1, 𝑇}^{𝛼} > 𝑆_{2, 𝑇}^{𝛽}}}}{𝑀_{𝑇}}] .

(17)

Recall the dynamics of the stock price under the risk-neutral measure $ℚ$ from Equation 13, and evaluate the discounted stock prices raised to the power of $𝛾$

\begin{aligned} \frac{𝑆_{1, 𝑇}^{𝛾}}{𝑀_{𝑇}} = \frac{𝑆_{1, 𝑇}^{𝛾}}{𝑒^{𝑟 𝑇}} & = 𝑆_{1, 0}^{𝛾} exp {𝛾 (𝑟 - 𝛿_{1} - \frac{1}{2} 𝜎_{1}^{2}) 𝑇 - 𝑟 𝑇 + 𝛾 𝜎_{1} {\tilde{𝑊}}_{1, 𝑇}} \\ = 𝑆_{1, 0}^{𝛾} exp {((𝛾 - 1) 𝑟 - 𝛾 𝛿_{1} - \frac{1}{2} 𝛾 𝜎_{1}^{2}) 𝑇 + 𝛾 𝜎_{1} {\tilde{𝑊}}_{1, 𝑇}} \\ = 𝑆_{1, 0}^{𝛾} exp {((𝛾 - 1) 𝑟 - 𝛾 𝛿_{1} - \frac{1}{2} 𝛾 𝜎_{1}^{2}) 𝑇 + \frac{1}{2} 𝛾^{2} 𝜎_{1}^{2} 𝑇 - \frac{1}{2} 𝛾^{2} 𝜎_{1}^{2} 𝑇 + 𝛾 𝜎_{1} {\tilde{𝑊}}_{1, 𝑇}} \\ = 𝑆_{1, 0}^{𝛾} exp {((𝛾 - 1) 𝑟 - 𝛾 𝛿_{1} - \frac{1}{2} 𝛾 (1 - 𝛾) 𝜎_{1}^{2}) 𝑇} exp {- \frac{1}{2} 𝛾^{2} 𝜎_{1}^{2} 𝑇 + 𝛾 𝜎_{1} {\tilde{𝑊}}_{1, 𝑇}} . \end{aligned}

(18)

Observe that the left exponent in Equation 18 is deterministic, while the right exponent takes the form of the Radon-Nikodým derivative. Thus, we can define a new measure $\hat{ℚ}$ such that

\frac{𝑑 \hat{ℚ}}{𝑑 ℚ} |_{𝑡} = {ℰ︀}_{ℚ} (𝛾 𝜎_{1} {\tilde{𝑊}}_{1, 𝑡}) = exp {- \frac{1}{2} 𝛾^{2} 𝜎_{1}^{2} 𝑡 + 𝛾 𝜎_{1} {\tilde{𝑊}}_{1, 𝑡}} .

(19)

Using Lemma 1, we define the new correlated Brownian motion ${\hat{𝑾}}_{𝑡} = {({\hat{𝑊}}_{1, 𝑡}, {\hat{𝑊}}_{1, 𝑡})}^{⊤}$ under the $\hat{ℚ}$ measure

\begin{aligned} 𝑑 {\hat{𝑊}}_{1, 𝑡} & = - 𝛾 𝜎_{1} 𝑑 𝑡 + 𝑑 {\tilde{𝑊}}_{1, 𝑡}, \\ 𝑑 {\hat{𝑊}}_{2, 𝑡} & = - 𝜌 𝛾 𝜎_{1} 𝑑 𝑡 + 𝑑 {\tilde{𝑊}}_{2, 𝑡} . \end{aligned}

(20)

We substitute the Brownian motion ${\hat{𝑾}}_{𝑡}$ into the equation Equation 13 to get the stock price dynamics under the new measure $\hat{ℚ}$

𝑆_{1, 𝑇} = 𝑆_{1, 0} exp {(𝑟 - 𝛿_{1} - \frac{1}{2} 𝜎_{1}^{2} + 𝛾 𝜎_{1}^{2}) 𝑇 + 𝜎_{1} {\hat{𝑊}}_{1, 𝑇}},

(21)

𝑆_{2, 𝑇} = 𝑆_{2, 0} exp {(𝑟 - 𝛿_{2} - \frac{1}{2} 𝜎_{2}^{2} + 𝜌 𝛾 𝜎_{1} 𝜎_{2}) 𝑇 + 𝜎_{2} {\hat{𝑊}}_{2, 𝑇}},

(22)

which will be used to evaluate the inequality $𝑆_{1, 𝑇}^{𝛼} > 𝑆_{2, 𝑇}^{𝛽}$ under the $\hat{ℚ}$ measure.

Substituting the discounted stock price Equation 18 back into the option price Equation 17, we can take the deterministic term outside the expectation to get

\begin{aligned} 𝐶_{1, 𝛾, 0} & = 𝑆_{1, 0}^{𝛾} exp {((𝛾 - 1) 𝑟 - 𝛾 𝛿_{1} - \frac{1}{2} 𝛾 (1 - 𝛾) 𝜎_{1}^{2}) 𝑇} 𝔼_{ℚ} [\frac{𝑑 \hat{ℚ}}{𝑑 ℚ} |_{𝑇} 1_{{𝑆_{1, 𝑇}^{𝛼} > 𝑆_{2, 𝑇}^{𝛽}}}] \\ = 𝑆_{1, 0}^{𝛾} exp {((𝛾 - 1) 𝑟 - 𝛾 𝛿_{1} - \frac{1}{2} 𝛾 (1 - 𝛾) 𝜎_{1}^{2}) 𝑇} 𝔼_{\hat{ℚ}} [1_{{𝑆_{1, 𝑇}^{𝛼} > 𝑆_{2, 𝑇}^{𝛽}}}] \\ = 𝑆_{1, 0}^{𝛾} exp {((𝛾 - 1) 𝑟 - 𝛾 𝛿_{1} - \frac{1}{2} 𝛾 (1 - 𝛾) 𝜎_{1}^{2}) 𝑇} \hat{ℚ} {𝑆_{1, 𝑇}^{𝛼} > 𝑆_{2, 𝑇}^{𝛽}} \end{aligned}

(23)

To evaluate $\hat{ℚ} {𝑆_{1, 𝑇}^{𝛼} > 𝑆_{2, 𝑇}^{𝛽}}$ , we take the natural logarithm of both sides of the inequality

ln 𝑆_{1, 𝑇}^{𝛼} > ln 𝑆_{2, 𝑇}^{𝛽},

(24)

and substitute the stock price dynamics from Equation 21 and Equation 22 into Equation 24, to yield

\begin{matrix} ln 𝑆_{1, 0}^{𝛼} + 𝛼 (𝑟 - 𝛿_{1} - \frac{1}{2} 𝜎_{1}^{2} + 𝛾 𝜎_{1}^{2}) 𝑇 + 𝛼 𝜎_{1} {\hat{𝑊}}_{1, 𝑇} \\ > ln 𝑆_{2, 0}^{𝛽} + 𝛽 (𝑟 - 𝛿_{2} - \frac{1}{2} 𝜎_{2}^{2} + 𝜌 𝛾 𝜎_{1} 𝜎_{2}) 𝑇 + 𝛽 𝜎_{2} {\hat{𝑊}}_{2, 𝑇}, \end{matrix}

(25)

and finally

\begin{matrix} ln \frac{𝑆_{1, 0}^{𝛼}}{𝑆_{2, 0}^{𝛽}} + (𝛼 (𝑟 - 𝛿_{1}) - 𝛽 (𝑟 - 𝛿_{2}) - \frac{1}{2} 𝛼 (1 - 2 𝛾) 𝜎_{1}^{2} + \frac{1}{2} 𝛽 𝜎_{2}^{2} - 𝜌 𝛽 𝛾 𝜎_{1} 𝜎_{2}) 𝑇 \\ > 𝛽 𝜎_{2} {\hat{𝑊}}_{2, 𝑇} - 𝛼 𝜎_{1} {\hat{𝑊}}_{1, 𝑇} . \end{matrix}

(26)

The term on the right-hand side in Equation 26 is a sum of correlated Brownian motions, which is itself a Brownian motion ${\hat{𝑊}}_{𝑇}$ with mean 0 and variance

𝜎^{2} = {(𝛼 𝜎_{1})}^{2} - 2 𝜌 (𝛼 𝜎_{1}) (𝛽 𝜎_{2}) + {(𝛽 𝜎_{2})}^{2} .

(27)

We now rewrite the inequality Equation 26 as

\begin{matrix} 𝜎 \sqrt{𝑇} \hat{𝑍} = 𝜎 {\hat{𝑊}}_{𝑇} \\ < ln \frac{𝑆_{1, 0}^{𝛼}}{𝑆_{2, 0}^{𝛽}} + (𝛼 (𝑟 - 𝛿_{1}) - 𝛽 (𝑟 - 𝛿_{2}) - \frac{1}{2} 𝛼 (1 - 2 𝛾) 𝜎_{1}^{2} + \frac{1}{2} 𝛽 𝜎_{2}^{2} - 𝜌 𝛽 𝛾 𝜎_{1} 𝜎_{2}) 𝑇 \end{matrix}

(28)

where $\hat{𝑍}$ is a standard normal random variable under the $\hat{ℚ}$ measure. Isolate $\hat{𝑍}$ on the left-hand side and rearrange the right-hand side to yield

\begin{matrix} \hat{𝑍} < \frac{1}{𝜎 \sqrt{𝑇}} [ln \frac{𝑆_{1, 0}^{𝛼}}{𝑆_{2, 0}^{𝛽}} + (𝛼 (𝑟 - 𝛿_{1}) - 𝛽 (𝑟 - 𝛿_{2}) - \frac{1}{2} 𝛼 (1 - 2 𝛾 + 𝛼) 𝜎_{1}^{2} \\ + \frac{1}{2} 𝛽 (1 - 𝛽) 𝜎_{2}^{2} - 𝜌 𝛽 (𝛾 - 𝛼) 𝜎_{1} 𝜎_{2} + \frac{1}{2} ({(𝛼 𝜎_{1})}^{2} - 2 𝜌 (𝛼 𝜎_{1}) (𝛽 𝜎_{2}) + {(𝛽 𝜎_{2})}^{2})) 𝑇] \\ = \frac{1}{𝜎 \sqrt{𝑇}} [ln \frac{𝑆_{1, 0}^{𝛼}}{𝑆_{2, 0}^{𝛽}} + (𝛼 (𝑟 - 𝛿_{1}) - 𝛽 (𝑟 - 𝛿_{2}) - \frac{1}{2} 𝛼 (1 - 2 𝛾 + 𝛼) 𝜎_{1}^{2} \\ + \frac{1}{2} 𝛽 (1 - 𝛽) 𝜎_{2}^{2} - 𝜌 𝛽 (𝛾 - 𝛼) 𝜎_{1} 𝜎_{2} + \frac{1}{2} 𝜎^{2}) 𝑇] \\ = 𝑑_{1, 𝛾, 0} . \end{matrix}

(29)

Thus,

\hat{ℚ} {𝑆_{1, 𝑇}^{𝛼} > 𝑆_{2, 𝑇}^{𝛽}} = \hat{ℚ} {\hat{𝑍} < 𝑑_{1, 𝛾, 0}} = Φ (𝑑_{1, 𝛾, 0})

(30)

where $Φ (\cdot)$ is the cumulative distribution function of the standard normal distribution.

Finally, the price of the option at time 0 is

𝐶_{1, 𝛾, 0} = 𝑆_{1, 0}^{𝛾} exp {((𝛾 - 1) 𝑟 - 𝛾 𝛿_{1} - \frac{1}{2} 𝛾 (1 - 𝛾) 𝜎_{1}^{2}) 𝑇} Φ (𝑑_{1, 𝛾, 0}),

(31)

or similarly at time $𝑡$

𝐶_{1, 𝛾, 𝑇} = 𝑆_{1, 𝑡}^{𝛾} exp {((𝛾 - 1) 𝑟 - 𝛾 𝛿_{1} - \frac{1}{2} 𝛾 (1 - 𝛾) 𝜎_{1}^{2}) (𝑇 - 𝑡)} Φ (𝑑_{1, 𝛾, 𝑡}),

(32)

where

\begin{matrix} 𝑑_{1, 𝛾, 𝑡} = \frac{1}{𝜎 \sqrt{𝑇 - 𝑡}} [ln \frac{𝑆_{1, 𝑡}^{𝛼}}{𝑆_{2, 𝑡}^{𝛽}} + (𝛼 (𝑟 - 𝛿_{1}) - 𝛽 (𝑟 - 𝛿_{2}) - \frac{1}{2} 𝛼 (1 - 2 𝛾 + 𝛼) 𝜎_{1}^{2} \\ + \frac{1}{2} 𝛽 (1 - 𝛽) 𝜎_{2}^{2} - 𝜌 𝛽 (𝛾 - 𝛼) 𝜎_{1} 𝜎_{2} + \frac{1}{2} 𝜎^{2}) (𝑇 - 𝑡)] \end{matrix}

(33)

$□$

The second stock-or-nothing option based on two stocks has a final payoff of the form $𝐶_{2, 𝛾, 𝑇} = 𝑆_{2, 𝑇}^{𝛾} 1_{{𝑆_{1, 𝑇}^{𝛼} > 𝑆_{2, 𝑇}^{𝛽}}}$ at maturity time $𝑇$ , where the parameters are $𝛼, 𝛽, 𝛾 > 0$ .

Theorem 2. The price $𝐶_{2, 𝛾, 𝑡}$ at time $𝑡$ of a stock-or-nothing option with final payoff $𝐶_{2, 𝛾, 𝑇} = 𝑆_{2, 𝑇}^{𝛾} 1_{{𝑆_{1, 𝑇}^{𝛼} > 𝑆_{2, 𝑇}^{𝛽}}}$ is

𝐶_{2, 𝛾, 𝑡} = 𝑆_{2, 𝑡}^{𝛾} exp {((𝛾 - 1) 𝑟 - 𝛾 𝛿_{2} - \frac{1}{2} 𝛾 (1 - 𝛾) 𝜎_{2}^{2}) (𝑇 - 𝑡)} Φ (𝑑_{2, 𝛾, 𝑡}),

(34)

where $Φ (\cdot)$ is the cumulative distribution function of the standard normal distribution, and

\begin{matrix} 𝑑_{2, 𝛾, 𝑡} = \frac{1}{𝜎 \sqrt{𝑇 - 𝑡}} [ln \frac{𝑆_{1, 𝑡}^{𝛼}}{𝑆_{2, 𝑡}^{𝛽}} + (𝛼 (𝑟 - 𝛿_{1}) - 𝛽 (𝑟 - 𝛿_{2}) - \frac{1}{2} 𝛼 (1 - 𝛼) 𝜎_{1}^{2} \\ + \frac{1}{2} 𝛽 (1 - 2 𝛾 + 𝛽) 𝜎_{2}^{2} + 𝜌 𝛼 (𝛾 - 𝛽) 𝜎_{1} 𝜎_{2} - \frac{1}{2} 𝜎^{2}) (𝑇 - 𝑡)] \end{matrix}

(35)

with the variance

𝜎^{2} = {(𝛼 𝜎_{1})}^{2} - 2 𝜌 (𝛼 𝜎_{1}) (𝛽 𝜎_{2}) + {(𝛽 𝜎_{2})}^{2} .

(36)

Proof of Theorem 2. Without loss of generality, we price the option at time $𝑡 = 0$ , since the price at time $𝑡$ is easily obtained by changing the time to maturity in the expression that we derive to $𝑇 - 𝑡$ . We will take advantage of our work in Theorem 1 by using the following identity

1 = 1_{{𝑆_{1, 𝑇}^{𝛼} > 𝑆_{2, 𝑇}^{𝛽}}} + 1_{{𝑆_{2, 𝑇}^{𝛽} \geq 𝑆_{1, 𝑇}^{𝛼}}} .

(37)

Multiplying through by $𝑆_{2, 𝑇}^{𝛾}$ , we have

𝑆_{2, 𝑇}^{𝛾} = 𝑆_{2, 𝑇}^{𝛾} 1_{{𝑆_{1, 𝑇}^{𝛼} > 𝑆_{2, 𝑇}^{𝛽}}} + 𝑆_{2, 𝑇}^{𝛾} 1_{{𝑆_{2, 𝑇}^{𝛽} \geq 𝑆_{1, 𝑇}^{𝛼}}},

(38)

and rearranging for the payoff of the second stock-or-nothing option gives

𝑆_{2, 𝑇}^{𝛾} 1_{{𝑆_{1, 𝑇}^{𝛼} > 𝑆_{2, 𝑇}^{𝛽}}} = 𝑆_{2, 𝑇}^{𝛾} - 𝑆_{2, 𝑇}^{𝛾} 1_{{𝑆_{2, 𝑇}^{𝛽} \geq 𝑆_{1, 𝑇}^{𝛼}}} .

(39)

Since the event ${𝑆_{2, 𝑇}^{𝛽} = 𝑆_{1, 𝑇}^{𝛼}}$ has $ℚ$ probability 0, we can rewrite Equation 39 as

𝑆_{2, 𝑇}^{𝛾} 1_{{𝑆_{1, 𝑇}^{𝛼} > 𝑆_{2, 𝑇}^{𝛽}}} = 𝑆_{2, 𝑇}^{𝛾} - 𝑆_{2, 𝑇}^{𝛾} 1_{{𝑆_{2, 𝑇}^{𝛽} > 𝑆_{1, 𝑇}^{𝛼}}} a.s. ℚ .

(40)

The payoff function is a martingale under the risk-neutral measure $ℚ$ , thus, we express the option price at time $0$ as expectation and substitute the identity Equation 40 to get

𝐶_{2, 𝛾, 0} = 𝔼_{ℚ} [\frac{𝐶_{2, 𝛾, 𝑇}}{𝑀_{𝑇}}] = 𝔼_{ℚ} [\frac{𝑆_{2, 𝑇}^{𝛾} 1_{{𝑆_{1, 𝑇}^{𝛼} > 𝑆_{2, 𝑇}^{𝛽}}}}{𝑀_{𝑇}}] = 𝔼_{ℚ} [\frac{𝑆_{2, 𝑇}^{𝛾}}{𝑀_{𝑇}}] - 𝔼_{ℚ} [\frac{𝑆_{2, 𝑇}^{𝛾} 1_{{𝑆_{2, 𝑇}^{𝛽} > 𝑆_{1, 𝑇}^{𝛼}}}}{𝑀_{𝑇}}] .

(41)

Recall the dynamics of the stock price under the risk-neutral measure $ℚ$ from Equation 13, and evaluate the discounted stock price raised to the power of $𝛾$ (the steps are the same as those leading to Equation Equation 18) to yield

\begin{aligned} \frac{𝑆_{2, 𝑇}^{𝛾}}{𝑀_{𝑡}} & = \frac{𝑆_{2, 𝑇}^{𝛾}}{𝑒^{𝑟 𝑇}} \\ = 𝑆_{2, 0}^{𝛾} exp {((𝛾 - 1) 𝑟 - 𝛾 𝛿_{2} - \frac{1}{2} 𝛾 (1 - 𝛾) 𝜎_{2}^{2}) 𝑇} exp {- \frac{1}{2} 𝛾^{2} 𝜎_{2}^{2} 𝑇 + 𝛾 𝜎_{2} {\tilde{𝑊}}_{2, 𝑇}} . \\ = 𝑆_{2, 0}^{𝛾} exp {((𝛾 - 1) 𝑟 - 𝛾 𝛿_{2} - \frac{1}{2} 𝛾 (1 - 𝛾) 𝜎_{2}^{2}) 𝑇} {ℰ︀}_{ℚ} (𝛾 𝜎_{2} {\tilde{𝑊}}_{2, 𝑇}) . \end{aligned}

(42)

Observe that the left exponent in Equation 42 is deterministic and can be taken outside of the expectation; while the right exponent is a Doléans-Dade [7] exponential, which has expected value of 1 under $ℚ$ . Thus, the first expectation in Equation 41 is

𝔼_{ℚ} [\frac{𝑆_{2, 𝑇}^{𝛾}}{𝑀_{𝑇}}] = 𝑆_{2, 0}^{𝛾} exp {((𝛾 - 1) 𝑟 - 𝛾 𝛿_{2} - \frac{1}{2} 𝛾 (1 - 𝛾) 𝜎_{2}^{2}) 𝑇} .

(43)

The second expectation in Equation 41 is the payoff of the first stock-or-nothing option (see Equation Equation 17) with the stocks 1 and 2, and the parameters $𝛼$ and $𝛽$ switched, while $𝛾$ remains the same. Thus, from Theorem 1, we have

𝔼_{ℚ} [\frac{𝑆_{2, 𝑇}^{𝛾} 1_{{𝑆_{2, 𝑇}^{𝛽} > 𝑆_{1, 𝑇}^{𝛼}}}}{𝑀_{𝑇}}] = 𝑆_{2, 0}^{𝛾} exp {((𝛾 - 1) 𝑟 - 𝛾 𝛿_{2} - \frac{1}{2} 𝛾 (1 - 𝛾) 𝜎_{2}^{2}) 𝑇} Φ (- 𝑑_{2, 𝛾, 0}),

(44)

where

\begin{matrix} - 𝑑_{2, 𝛾, 0} = \frac{1}{𝜎 \sqrt{𝑇}} [ln \frac{𝑆_{2, 0}^{𝛽}}{𝑆_{1, 0}^{𝛼}} + (𝛽 (𝑟 - 𝛿_{2}) - 𝛼 (𝑟 - 𝛿_{1}) - \frac{1}{2} 𝛽 (1 - 2 𝛾 + 𝛽) 𝜎_{2}^{2} \\ + \frac{1}{2} 𝛼 (1 - 𝛼) 𝜎_{1}^{2} - 𝜌 𝛼 (𝛾 - 𝛽) 𝜎_{1} 𝜎_{2} + \frac{1}{2} 𝜎^{2}) 𝑇], \end{matrix}

(45)

\begin{matrix} 𝑑_{2, 𝛾, 0} = \frac{1}{𝜎 \sqrt{𝑇}} [ln \frac{𝑆_{1, 0}^{𝛼}}{𝑆_{2, 0}^{𝛽}} + (𝛼 (𝑟 - 𝛿_{1}) - 𝛽 (𝑟 - 𝛿_{2}) - \frac{1}{2} 𝛼 (1 - 𝛼) 𝜎_{1}^{2} \\ + \frac{1}{2} 𝛽 (1 - 2 𝛾 + 𝛽) 𝜎_{2}^{2} + 𝜌 𝛼 (𝛾 - 𝛽) 𝜎_{1} 𝜎_{2} - \frac{1}{2} 𝜎^{2}) 𝑇], \end{matrix}

(46)

and the variance

𝜎^{2} = {(𝛼 𝜎_{1})}^{2} - 2 𝜌 (𝛼 𝜎_{1}) (𝛽 𝜎_{2}) + {(𝛽 𝜎_{2})}^{2} .

(47)

Substituting the expectations Equation 43 and Equation 44 back into Equation Equation 41, we get the price of the option at time 0

\begin{aligned} 𝐶_{2, 𝛾, 0} & = 𝑆_{2, 0}^{𝛾} exp {((𝛾 - 1) 𝑟 - 𝛾 𝛿_{2} - \frac{1}{2} 𝛾 (1 - 𝛾) 𝜎_{2}^{2}) 𝑇} (1 - Φ (- 𝑑_{2, 𝛾, 0})) \\ = 𝑆_{2, 0}^{𝛾} exp {((𝛾 - 1) 𝑟 - 𝛾 𝛿_{2} - \frac{1}{2} 𝛾 (1 - 𝛾) 𝜎_{2}^{2}) 𝑇} Φ (𝑑_{2, 𝛾, 0}), \end{aligned}

(48)

and similarly at time $𝑡$

𝐶_{2, 𝛾, 𝑡} = 𝑆_{2, 𝑡}^{𝛾} exp {((𝛾 - 1) 𝑟 - 𝛾 𝛿_{2} - \frac{1}{2} 𝛾 (1 - 𝛾) 𝜎_{2}^{2}) (𝑇 - 𝑡)} Φ (𝑑_{2, 𝛾, 𝑡}),

(49)

where

\begin{matrix} 𝑑_{2, 𝛾, 𝑡} = \frac{1}{𝜎 \sqrt{𝑇 - 𝑡}} [ln \frac{𝑆_{1, 𝑡}^{𝛼}}{𝑆_{2, 𝑡}^{𝛽}} + (𝛼 (𝑟 - 𝛿_{1}) - 𝛽 (𝑟 - 𝛿_{2}) - \frac{1}{2} 𝛼 (1 - 𝛼) 𝜎_{1}^{2} \\ + \frac{1}{2} 𝛽 (1 - 2 𝛾 + 𝛽) 𝜎_{2}^{2} + 𝜌 𝛼 (𝛾 - 𝛽) 𝜎_{1} 𝜎_{2} - \frac{1}{2} 𝜎^{2}) (𝑇 - 𝑡)] . \end{matrix}

(50)

$□$

Next, we derive the pricing formula for a power exchange option where the final payoff takes the form $𝐶_{𝑇} = {(𝑆_{1, 𝑇}^{𝛼} - 𝑆_{2, 𝑇}^{𝛽})}^{+}$ at maturity time $𝑇$ .

Corollary 1. The price at time $𝑡$ for the power exchange option with final payoff $𝐶_{𝑇} = {(𝑆_{1, 𝑇}^{𝛼} - 𝑆_{2, 𝑇}^{𝛽})}^{+}$ is

\begin{matrix} 𝐶_{𝑡} = 𝑆_{1, 𝑡}^{𝛼} exp {((𝛼 - 1) 𝑟 - 𝛼 𝛿_{1} - \frac{1}{2} 𝛼 (1 - 𝛼) 𝜎_{1}^{2}) (𝑇 - 𝑡)} Φ (𝑑_{1, 𝑡}) \\ - 𝑆_{2, 𝑡}^{𝛽} exp {((𝛽 - 1) 𝑟 - 𝛽 𝛿_{2} - \frac{1}{2} 𝛽 (1 - 𝛽) 𝜎_{2}^{2}) (𝑇 - 𝑡)} Φ (𝑑_{2, 𝑡}), \end{matrix}

(51)

where

\begin{aligned} 𝑑_{1, 𝑡} & = \frac{ln \frac{𝑆_{1, 𝑡}^{𝛼}}{𝑆_{2, 𝑡}^{𝛽}} + (𝛼 (𝑟 - 𝛿_{1}) - 𝛽 (𝑟 - 𝛿_{2}) - \frac{1}{2} 𝛼 (1 - 𝛼) 𝜎_{1}^{2} + \frac{1}{2} 𝛽 (1 - 𝛽) 𝜎_{2}^{2} + \frac{1}{2} 𝜎^{2}) (𝑇 - 𝑡)}{𝜎 \sqrt{𝑇 - 𝑡}}, \\ 𝑑_{2, 𝑡} & = \frac{ln \frac{𝑆_{1, 𝑡}^{𝛼}}{𝑆_{2, 𝑡}^{𝛽}} + (𝛼 (𝑟 - 𝛿_{1}) - 𝛽 (𝑟 - 𝛿_{2}) - \frac{1}{2} 𝛼 (1 - 𝛼) 𝜎_{1}^{2} + \frac{1}{2} 𝛽 (1 - 𝛽) 𝜎_{2}^{2} - \frac{1}{2} 𝜎^{2}) (𝑇 - 𝑡)}{𝜎 \sqrt{𝑇 - 𝑡}}, \end{aligned}

(52)

and the variance

𝜎^{2} = {(𝛼 𝜎_{1})}^{2} - 2 𝜌 (𝛼 𝜎_{1}) (𝛽 𝜎_{2}) + {(𝛽 𝜎_{2})}^{2} .

(53)

Remark 1. We note that

𝑑_{2, 𝑡} = 𝑑_{1, 𝑡} - 𝜎 \sqrt{𝑇 - 𝑡},

(54)

which is analogous to the relationship between $𝑑_{1}$ and $𝑑_{2}$ in the usual Black-Scholes formula.

Proof of Corollary 1. Since the final payoff function is a martingale under the risk-neutral measure $ℚ$ , the option price at time $𝑡$ is given by

\begin{aligned} 𝐶_{𝑡} & = 𝑀_{𝑡} 𝔼_{ℚ} [\frac{𝐶_{𝑇}}{𝑀_{𝑇}}] = 𝑀_{𝑡} 𝔼_{ℚ} [\frac{{(𝑆_{1, 𝑇}^{𝛼} - 𝑆_{2, 𝑇}^{𝛽})}^{+}}{𝑀_{𝑇}}] \\ = 𝑀_{𝑡} 𝔼_{ℚ} [\frac{𝑆_{1, 𝑇}^{𝛼} - 𝑆_{2, 𝑇}^{𝛽}}{𝑀_{𝑇}} 1_{{𝑆_{1, 𝑇}^{𝛼} > 𝑆_{2, 𝑇}^{𝛽}}}] \\ = 𝑀_{𝑡} 𝔼_{ℚ} [\frac{𝑆_{1, 𝑇}^{𝛼}}{𝑀_{𝑇}} 1_{{𝑆_{1, 𝑇}^{𝛼} > 𝑆_{2, 𝑇}^{𝛽}}}] - 𝑀_{𝑡} 𝔼_{ℚ} [\frac{𝑆_{2, 𝑇}^{𝛽}}{𝑀_{𝑇}} 1_{{𝑆_{1, 𝑇}^{𝛼} > 𝑆_{2, 𝑇}^{𝛽}}}] \\ = 𝐶_{1, 𝛼, 𝑡} - 𝐶_{2, 𝛽, 𝑡} . \end{aligned}

(55)

We know the payoff function $𝐶_{1, 𝛼, 𝑡}$ from Theorem 1

𝐶_{1, 𝛼, 𝑡} = 𝑆_{1, 𝑡}^{𝛼} exp {((𝛼 - 1) 𝑟 - 𝛼 𝛿_{1} - \frac{1}{2} 𝛼 (1 - 𝛼) 𝜎_{1}^{2}) (𝑇 - 𝑡)} Φ (𝑑_{1, 𝛼, 𝑡}),

(56)

where

\begin{matrix} 𝑑_{1, 𝛼, 𝑡} = \frac{1}{𝜎 \sqrt{𝑇 - 𝑡}} [ln \frac{𝑆_{1, 𝑡}^{𝛼}}{𝑆_{2, 𝑡}^{𝛽}} + (𝛼 (𝑟 - 𝛿_{1}) - 𝛽 (𝑟 - 𝛿_{2}) - \frac{1}{2} 𝛼 (1 - 2 𝛼 + 𝛼) 𝜎_{1}^{2} \\ + \frac{1}{2} 𝛽 (1 - 𝛽) 𝜎_{2}^{2} - 𝜌 𝛽 (𝛼 - 𝛼) 𝜎_{1} 𝜎_{2} + \frac{1}{2} 𝜎^{2}) (𝑇 - 𝑡)] \\ = \frac{ln \frac{𝑆_{1, 𝑡}^{𝛼}}{𝑆_{2, 𝑡}^{𝛽}} + (𝛼 (𝑟 - 𝛿_{1}) - 𝛽 (𝑟 - 𝛿_{2}) - \frac{1}{2} 𝛼 (1 - 𝛼) 𝜎_{1}^{2} + \frac{1}{2} 𝛽 (1 - 𝛽) 𝜎_{2}^{2} + \frac{1}{2} 𝜎^{2}) (𝑇 - 𝑡)}{𝜎 \sqrt{𝑇 - 𝑡}} \\ = 𝑑_{1, 𝑡} . \end{matrix}

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Similarly, we know the payoff function $𝐶_{2, 𝛽, 𝑡}$ from Theorem 2

𝐶_{2, 𝛽, 𝑡} = 𝑆_{2, 𝑡}^{𝛽} exp {((𝛽 - 1) 𝑟 - 𝛽 𝛿_{2} - \frac{1}{2} 𝛽 (1 - 𝛽) 𝜎_{2}^{2}) (𝑇 - 𝑡)} Φ (𝑑_{2, 𝛽, 𝑡}),

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where

\begin{matrix} 𝑑_{2, 𝛽, 𝑡} = \frac{1}{𝜎 \sqrt{𝑇 - 𝑡}} [ln \frac{𝑆_{1, 𝑡}^{𝛼}}{𝑆_{2, 𝑡}^{𝛽}} + (𝛼 (𝑟 - 𝛿_{1}) - 𝛽 (𝑟 - 𝛿_{2}) - \frac{1}{2} 𝛼 (1 - 𝛼) 𝜎_{1}^{2} \\ + \frac{1}{2} 𝛽 (1 - 2 𝛽 + 𝛽) 𝜎_{2}^{2} - 𝜌 𝛼 (𝛽 - 𝛽) 𝜎_{1} 𝜎_{2} - \frac{1}{2} 𝜎^{2}) (𝑇 - 𝑡)] \\ = \frac{ln \frac{𝑆_{1, 𝑡}^{𝛼}}{𝑆_{2, 𝑡}^{𝛽}} + (𝛼 (𝑟 - 𝛿_{1}) - 𝛽 (𝑟 - 𝛿_{2}) - \frac{1}{2} 𝛼 (1 - 𝛼) 𝜎_{1}^{2} + \frac{1}{2} 𝛽 (1 - 𝛽) 𝜎_{2}^{2} - \frac{1}{2} 𝜎^{2}) (𝑇 - 𝑡)}{𝜎 \sqrt{𝑇 - 𝑡}} \\ = 𝑑_{2, 𝑡} . \end{matrix}

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Substituting $𝐶_{1, 𝛼, 𝑡}$ and $𝐶_{2, 𝛽, 𝑡}$ back into the option price Equation 55, we get

\begin{matrix} 𝐶_{𝑡} = 𝑆_{1, 𝑡}^{𝛼} exp {((𝛼 - 1) 𝑟 - 𝛼 𝛿_{1} - \frac{1}{2} 𝛼 (1 - 𝛼) 𝜎_{1}^{2}) (𝑇 - 𝑡)} Φ (𝑑_{1, 𝑡}) \\ - 𝑆_{2, 𝑡}^{𝛽} exp {((𝛽 - 1) 𝑟 - 𝛽 𝛿_{2} - \frac{1}{2} 𝛽 (1 - 𝛽) 𝜎_{2}^{2}) (𝑇 - 𝑡)} Φ (𝑑_{2, 𝑡}) . \end{matrix}

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$□$

5. Applications

The pricing formulas derived in Section 4 have practical implications in the design of incentive schemes, such as bonus stock options for employees and board members. A typical KPI is when the value of the company’s stock $𝑆_{1, 𝑇}$ at time $𝑇$ is greater than a benchmark $𝑆_{2, 𝑇}$ raised to some power $𝛽$ , denoted as $𝑆_{2, 𝑇}^{𝛽}$ , where $𝑆_{2, 𝑇}$ could be an industry index or the stock price of a rival company. Such an incentive can be priced using Theorem 1.

We list the following examples of payouts and their valuations.

Example 1. Under a constant payout bonus, we essentially have a cash-or-nothing reward. This is equivalent to an option with the payoff $𝐶_{𝑇} = 1_{{𝑆_{1, 𝑇} > 𝑆_{2, 𝑇}^{𝛽}}}$ . Using the pricing formula from Theorem 1 and setting $𝛼 = 1$ and $𝛾 = 0$ , the payoff reduces to a constant, yielding

𝐶_{𝑡} = exp {- 𝑟 (𝑇 - 𝑡)} Φ (𝑑_{1, 0, 𝑡}) .

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Example 2. When the bonus is paid out in units of the company’s stock, we have a standard stock-or-nothing option. The equivalent payoff for one unit of such a bonus is $𝐶_{𝑇} = 𝑆_{1, 𝑇} 1_{{𝑆_{1, 𝑇} > 𝑆_{2, 𝑇}^{𝛽}}}$ . By setting $𝛼 = 1$ and $𝛾 = 1$ in Theorem 1, the cost of this incentive becomes

𝐶_{𝑡} = 𝑆_{1, 𝑡} exp {- 𝛿_{1} (𝑇 - 𝑡)} Φ (𝑑_{1, 1, 𝑡}) .

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Example 3. The bonus can be designed for more complex criteria, where the reward can be amplified or reduced based on the price of the company’s stock at the time of the payout. Such a reward can be a nonlinear equity where the payout is tied to a power of the company’s stock price. In this scenario, the final payout takes the form $𝐶_{𝑇} = 𝑆_{1, 𝑇}^{𝛾} 1_{{𝑆_{1, 𝑇} > 𝑆_{2, 𝑇}^{𝛽}}}$ . This represents a power stock-or-nothing option, whose fair price is given by Theorem 1 with $𝛼 = 1$

𝐶_{𝑡} = 𝑆_{1, 𝑡}^{𝛾} exp {(𝑟 (𝛾 - 1) - 𝛾 𝛿_{1} - \frac{1}{2} 𝛾 (1 - 𝛾) 𝜎_{1}^{2}) (𝑇 - 𝑡)} Φ (𝑑_{1, 𝛾, 𝑡}) .

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In addition to these financial applications, the power stock-or-nothing options simplify the evaluation of other powered exotic options, such as the power exchange option of Blenman and Clark [1]. Their model generalises the standard exchange option of Margrabe [6] by allowing the holder to exchange one power of an asset for another. As shown in Corollary 1, the price of this power exchange option is simply the difference between two power stock-or-nothing options

𝐶_{𝑡} = 𝐶_{1, 𝛼, 𝑡} - 𝐶_{2, 𝛽, 𝑡} .

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Future research could continue with the framework and extend the model by incorporating jump-diffusion processes to better capture discontinuous market movements. Although only the Margrabe [6] model for two correlated stocks under the geometric Brownian motion model is considered in this paper for the pricing of all these power options, our various pricing models and change-of-measure techniques can easily be extended to the pricing of power options on two assets under the Cheang and Chiarella [10] jump-diffusion model.

6. Conclusion

In this paper, we derived closed-form pricing formulas for two types of power stock-or-nothing options on two assets. Using the change-of-measure technique described by Esser [4], we evaluated the risk-neutral expected payoffs for options with payoffs of the form $𝑆_{1, 𝑇}^{𝛾} 1_{{𝑆_{1, 𝑇}^{𝛼} > 𝑆_{2, 𝑇}^{𝛽}}}$ and $𝑆_{2, 𝑇}^{𝛾} 1_{{𝑆_{1, 𝑇}^{𝛼} > 𝑆_{2, 𝑇}^{𝛽}}}$ .

We demonstrated that these two formulas provide a straightforward method for pricing the power exchange option introduced by Blenman and Clark [1]. Specifically, the price of the power exchange option, with payoff ${(𝑆_{1, 𝑇}^{𝛼} - 𝑆_{2, 𝑇}^{𝛽})}^{+}$ , can be expressed simply as the difference between the prices of the two power stock-or-nothing options derived in this work.

Finally, we discussed the application of these results to executive compensation. The formulas allow for the valuation of incentive schemes where the payoff is tied to a power of the company’s stock price, conditional on performance relative to a benchmark.

References

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