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Chapter 14: Problem 45

Calculate the implied volatility of soybean futures prices from the following information concerning a European put on soybean futures: $$\begin{array}{ll}\hline \text { Current futures price } & 525 \\\\\text { Exercise price } & 525 \\\\\text { Risk-free rate } & 6 \% \text { per annum } \\\\\text { Time to maturity } & 5 \text { months } \\ \text { Put price } & 20 \\\\\hline\end{array}$$

Short Answer

Expert verified
Implied volatility can be found by solving the Black-Scholes equation numerically for \( \sigma \), which requires trial and adjustment.

Step by step solution

01

Understand the problem

We need to find the implied volatility of a European put option on soybean futures with the given market data. Implied volatility is not directly observable and must be estimated from a model such as the Black-Scholes model.
02

Collect known parameters

Identify and list all the known parameters: \( F = 525 \) (futures price), \( K = 525 \) (exercise price), \( r = 0.06 \) (risk-free rate), \( T = \frac{5}{12} \) years (time to maturity), and \( P = 20 \) (put price).
03

Formulate the equation

Use the Black-Scholes model for pricing European options on futures. The formula for a European put option is:\[P = Ke^{-rT}N(-d_2) - FN(-d_1)\]where \(d_1\) and \(d_2\) depend on the implied volatility \(\sigma\).
04

Define \(d_1\) and \(d_2\)

The terms \(d_1\) and \(d_2\) are defined as:\[d_1 = \frac{\ln(\frac{F}{K}) + (\frac{\sigma^2}{2})T}{\sigma\sqrt{T}}\]\[d_2 = d_1 - \sigma\sqrt{T}\]
05

Solve iteratively for \(\sigma\)

Set up the equation \( 20 = 525e^{-0.06 \times (5/12)}N(-d_2) - 525N(-d_1) \) using trial and error or a numerical method like the Newton-Raphson method to find the implied volatility \( \sigma \) that satisfies this equation.
06

Verify solution

Check that substituting the found \(\sigma\) back into the Black-Scholes formula results in a put price close to 20. This confirms the accuracy of the implied volatility estimate.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

European options
European options are a type of derivative financial instrument that can only be exercised at the end of their life, or at maturity. Unlike American options, which can be exercised at any time before expiration, European options restrict the holder to a single exercise point. This simplification makes them easier to model mathematically, which is why they're often used when applying the Black-Scholes model.

Key characteristics include:
  • Exercise style: Can only be exercised on expiration.
  • Standardization: Typically follows stricter terms.
  • Models: Utilizes mathematical models like the Black-Scholes model efficiently.

    Because the value of a European option hinges on its expiration price, much of the option pricing theory for these options revolves around estimating future prices at this single point. This is often done using sophisticated financial models to account for volatility and other market factors.
Black-Scholes model
The Black-Scholes model is a critical tool in option pricing theory, especially for European options. It provides a mathematical framework for estimating the price of options, based on variables like the current stock price, strike price, time to expiration, risk-free rate, and implied volatility.

The primary equation for pricing a European call option is given by:
  • \[C = FN(d_1) - Ke^{-rT}N(d_2)\]
  • For a European put option, the formula is:\[P = Ke^{-rT}N(-d_2) - FN(-d_1)\]
    Variables in the Black-Scholes equation include:
    • \(F\): Current futures price.
    • \(K\): Exercise (strike) price.
    • \(r\): Risk-free interest rate.
    • \(T\): Time to expiration.
    • \(N(d)\): Cumulative distribution function for a standard normal distribution.

      One of the challenges in using the Black-Scholes model is estimating the implied volatility, which is now widely considered a measure of market sentiment or expected volatility over the life of the option.
futures pricing
Futures pricing involves the determination of the market value of a future contract, which is an agreement to buy or sell an asset at a future date for a predetermined price. In the context of the given exercise, we are dealing with soybean futures and how these prices impact European put options.

Important components of futures pricing include:
  • Current futures price (spot price): The price at which the market currently values the future contract.
  • Expected future spot price: Anticipated market price at the contract's expiration.
  • Risk-free rate: Used to discount future payments to present value.

    In practice, the futures pricing formulas consider:
    • The cost of carry, which includes storage and financing costs.
    • The arbitrage opportunities, which keep futures prices aligned with expectations.
    • Market expectations and news, influencing immediate and long-term pricing.

      Futures contracts are a cornerstone in the calculation of options pricing since they impact values like the underlying asset price within the Black-Scholes model calculations.
option pricing theory
Option pricing theory is a framework used to determine the value of options based on various influencing factors. At its core, it tries to address the inherent uncertainty in financial markets by calculating the theoretical price of an option.

Here are the key elements of option pricing theory:
  • Volatility: A crucial component that reflects the price fluctuations of the underlying asset over the option's lifespan. Implied volatility, which the exercise seeks to calculate, gauges the market's forecast of a likely movement.
  • Time value: This accounts for the option's value over time before its expiration.
  • Intrinsic value: The actual value an option would have if it were exercised immediately.
  • Risk-free rate: Used to adjust future cash flows to present value.

    Option pricing theory also makes assumptions about market efficiency, constant risk, and log-normal distribution of prices, all of which help models like Black-Scholes deliver practical and actionable insights. Understanding these theories and methods provides a foundational insight into the sophisticated world of financial derivatives.

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Most popular questions from this chapter

A futures price is currently \(60 .\) It is known that over each of the next two 3 -month periods it will either rise by \(10 \%\) or fall by \(10 \%\). The risk- free interest rate is \(8 \%\) per annum. What is the value of a 6 -month European call option on the futures with a strike price of \(60 ?\) If the call were American, would it ever be worth exercising it early?

A portfolio is currently worth \(\$ 10\) million and has a beta of \(1.0 .\) The \(\operatorname{SeP} 100\) is currently standing at \(500 .\) Explain how a put option on the S\&P 100 with a strike of 480 can be used to provide portfolio insurance.

Consider a stock index currently standing at \(250 .\) The dividend yield on the index is \(4 \%\) per annum and the risk-free rate is \(6 \%\) per annum. A 3 -month European call option on the index with a strike price of 245 is currently worth \(\$ 10 .\) What is the value of a 3-month European put option on the index with a strike price of \(245 ?\)

Calculate the value of a 5 -month European put futures option when the futures price is \(\$ 19,\) the strike price is \(\$ 20,\) the risk-free interest rate is \(12 \%\) per annum, and the volatility of the futures price is \(20 \%\) per annum.

Suppose that the spot price of the Canadian dollar is US \(\$ 0.75\) and that the Canadian dollar/US dollar exchange rate has a volatility of \(4 \%\) per annum. The risk-free rates of interest in Canada and the United States are \(9 \%\) and \(7 \%\) per annum, respectively. Calculate the value of a European call option to buy one Canadian dollar for US \(\$ 0.75\) in 9 months. Use put-call parity to calculate the price of a European put option to sell one Canadian dollar for US \(\$ 0.75\) in 9 months. What is the price of a call option to buy US \(\$ 0.75\) with one Canadian dollar in 9 months?
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