Question

A so called binary option is a claim which pays a certain amount if the stock price at a certain date falls within some prespecified interval. Otherwise nothing will be paid out. Consider a binary option which pays \(K\) SEK to the holder at date \(T\) if the stock price at time \(T\) is in the inerval \([\alpha, \beta]\). Determine the arbitrage free price. The pricing formula will involve the standard Gaussian cumulative distribution function \(N\).

Short answer

Answer: The arbitrage-free price for the binary option is given by the following formula: $$ \Pi = K\cdot e^{-rT}\left[N\left(\frac{\ln(\beta)-\mu T}{\sigma\sqrt{T}}\right) - N\left(\frac{\ln(\alpha)-\mu T}{\sigma\sqrt{T}}\right)\right] $$ where - \(K\) is the payout amount if the stock price falls within the interval, - \(T\) is the specific time, - \([\alpha, \beta]\) is the given interval, - \(r\) is the risk-free interest rate, - \(\mu\) is the mean of the log-return process, - \(\sigma\) is the standard deviation of the log-return process, and - \(N(x)\) is the standard Gaussian cumulative distribution function.

Step-by-step solution

Define the payoff function for the binary option

The binary option pays out \(K\) SEK if the stock price at time \(T\) is within the interval \([\alpha, \beta]\), and nothing otherwise. Therefore, we can represent the payoff function $$V(S_T)$$ at time \(T\) as: $$ V(S_T) = \begin{cases} K & \mbox{if } \alpha \leq S_T \leq \beta \\ 0 & \mbox{otherwise } \end{cases} $$

Identify the risk-neutral probability

Under the risk-neutral probability measure, we need to find the probability that the stock price will be within the given interval \([\alpha, \beta]\) at time \(T\). Let \(x = \frac{S_T - \mu}{\sigma\sqrt{T}}\), where \(\mu\) and \(\sigma^2\) are the mean and variance of the log-return process, respectively. Now, the probability of the stock price falling within the interval is given by: $$ P(\alpha \leq S_T \leq \beta) = N\left(\frac{\ln(\beta)-\mu T}{\sigma\sqrt{T}}\right) - N\left(\frac{\ln(\alpha)-\mu T}{\sigma\sqrt{T}}\right) $$

Calculate the expected value of the binary option

To calculate the expected value of the binary option's payoff, we multiply the payoff by the probability and take the integral over the possible values: $$ E(V(S_T)) = \int_{\alpha}^{\beta} K\cdot p(S_T) dS_T = K\cdot P(\alpha \leq S_T \leq \beta) $$ By substituting the expression for \(P(\alpha \leq S_T \leq \beta)\) from Step 2, we get: $$ E(V(S_T)) = K\cdot\left[N\left(\frac{\ln(\beta)-\mu T}{\sigma\sqrt{T}}\right) - N\left(\frac{\ln(\alpha)-\mu T}{\sigma\sqrt{T}}\right)\right] $$

Price the binary option using the arbitrage-free principle

The arbitrage-free price of the binary option is the present value of the expected payoff under the risk-neutral probability measure: $$ \Pi = e^{-rT} \cdot E(V(S_T)) $$ Substituting the expression for \(E(V(S_T))\) from Step 3, we obtain the arbitrage-free price for the binary option: $$ \Pi = K\cdot e^{-rT}\left[N\left(\frac{\ln(\beta)-\mu T}{\sigma\sqrt{T}}\right) - N\left(\frac{\ln(\alpha)-\mu T}{\sigma\sqrt{T}}\right)\right] $$ This is the final expression for the arbitrage-free price for the binary option that pays \(K\) SEK if the stock price at time \(T\) is within the interval \([\alpha, \beta]\).

Key concepts

Binary Option

A binary option is an exciting type of financial derivative. Unlike standard options, a binary option has a simple payoff structure. It pays out a fixed amount, say \(K\) SEK, if the stock price on a specified future date falls within a predetermined range, \([\alpha, \beta]\). Otherwise, it pays nothing. Imagine a scenario where you're betting on the stock price of a tech company. If it lands within your chosen range, you earn a tidy sum; if not, you walk away empty-handed.

This all-or-nothing format makes binary options straightforward yet unique. They're a tool often used for speculative purposes or hedging. But to accurately assess them, it's vital to understand their pricing under the Arbitrage Pricing Theory, which requires more advanced concepts like risk-neutral measures.

Risk-Neutral Measure

Risk-neutral measure is a mathematical framework used for pricing derivatives such as binary options. In this measure, we modify the original probability distribution of stock prices to simplify calculations. Here, investors are assumed to be indifferent to risk, meaning they don't require a risk premium. Thus, we adjust the probability of future events happening in a way that matches the market's pricing of these events. This makes pricing derivatives easier and more consistent.

By transforming into a risk-neutral world, we calculate the present value of expected future payoffs by discounting them at the risk-free rate. This approach helps us determine the arbitrage-free price, without considering individual investor risk tolerances. It's like using a "neutral" lens to see the future of stock prices, focusing purely on probability rather than risk perception.

Gaussian Distribution

A Gaussian distribution, also known as the normal distribution, is a bell-shaped curve widely used in probability and statistics. When it comes to stock prices and financial derivatives, it's assumed that the log returns of stock prices follow this distribution. In simple terms, most stock price changes are average, but you may get occasional extreme movements.

For binary options, the Gaussian cumulative distribution function \(N\) helps estimate the probability that a stock price will fall within a specific range \([\alpha, \beta]\). Using the properties of this distribution allows us to determine the likelihood of different stock price outcomes in a risk-neutral world. This concept underpins the calculation of the binary option's expected payoff.

Pricing Formula

The pricing formula for a binary option ties together all the concepts we've discussed. To find the arbitrage-free price of a binary option, we calculate the expected payoff in a risk-neutral measure and then discount it to its present value.

The steps are as follows:

• Identify the payoffs: The option pays \(K\) SEK if the stock price falls in the interval \([\alpha, \beta]\) at time \(T\).
• Use Gaussian distribution to find the probability of the stock ending within this range.
• Calculate the expected payoff using this probability and the fixed payoff \(K\).
• Discount this expected payoff using the risk-free rate \(r\) to find the present value, known as the arbitrage-free price.

This formula, given by \[ \Pi = K \cdot e^{-rT} \left[ N \left( \frac{\ln(\beta) - \mu T}{\sigma \sqrt{T}} \right) - N \left( \frac{\ln(\alpha) - \mu T}{\sigma \sqrt{T}} \right) \right], \] lets us evaluate the option's value today based on future expectations, risk-neutral settings, and the statistical behavior of stock prices.