# Valuation of a Portfolio: Bonds, Forward Contract, and Call Options and Using a One-Step Binomial Tree to Value Options

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## Valuation of a Portfolio: Bonds, Forward Contract, and Call Options and Using a One-Step Binomial Tree to Value Options

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### ASSIGNMENT INSTRUCTIONS:

QUESTION 1
The question involves the valuation of a portfolio consisting of bonds, a forward contract, and European call options. The bonds have a face value of £200 and pay an annual coupon of £5 semi-annually, with an original term of 4 years and a remaining maturity of 40 months. The forward contract is to buy an investment vehicle worth £550, two years from now, which provides an income equivalent to a continuous dividend yield at the nominal rate of 2.6% p.a. The portfolio also includes 6 European call options on a stock with a strike price of £420 and an expiry of 36 months, with a spot price of £400 and a market volatility of 25%. Zero rates for various maturities are provided. The question requires the calculation of the value of the portfolio, the yield to maturity of the bond, the new value of the portfolio in case of a shift in interest rates, the \$rho\$, \$Gamma\$, and \$Delta\$ for the option, and an explanation of how to hedge the portfolio against a large loss.
QUESTION 2
The question involves the use of a one-step binomial tree to value a call and a put option for an ounce of gold with a current price of £1,146, an expected price in a year of £1,200 or £1,050, and an interest rate of 4.2%. The strike price for both options is £1,150. The first step involves constructing the binomial tree, and then the options’ values can be calculated using backward induction. The call option’s value is the maximum of either the expected value at the end node minus the strike price or zero, while the put option’s value is the maximum of the strike price minus the expected value at the end node or zero.

### HOW TO WORK ON THIS ASSIGNMENT (EXAMPLE ESSAY / DRAFT)

The value of a portfolio made up of bonds, a forward contract, and European call options is the subject of this article. Calculating the value of the portfolio, the yield to maturity of the bond, the new value of the portfolio in the event of a change in interest rates, the rho, Gamma, and Delta for the option, and a description of how to protect the portfolio from a significant loss are all necessary for its valuation.

Body: The portfolio consists of bonds having a face value of £200 that bear a 4-year original term and a 40-month remaining maturity and pay an annual coupon of £5 semi-annually. In two years, the forward contract will pay for an investment vehicle worth £550 that will provide an income equal to a continuous dividend yield at a nominal rate of 2.6% annually. Six European call options on a company with a strike price of £420 and a 36-month expiration, a spot price of £400, and a market volatility of 25% are also included in the portfolio. There are zero rates for a range of maturities.

We must apply the bond valuation formula, the Black-Scholes formula, and the method for calculating the value of a forward contract to determine the portfolio’s worth. Using the bond pricing formula, we can determine the bond’s yield to maturity. If interest rates change, we must recalculate the bond and forward contract values using the new zero rates. We must apply the Black-Scholes algorithm to determine the rho, Gamma, and Delta for the option. Finally, we can employ the delta hedging method to protect the portfolio against a big loss.

In conclusion, a thorough understanding of bond pricing, option pricing, and hedging strategies is necessary for the valuation of a portfolio made up of bonds, a forward contract, and European call options. Investors can make wise investment decisions by estimating the value of the portfolio and the many aspects related to it.

Essay Subject 2:

Title: One-Step Binomial Tree for Call and Put Option Valuation

In this article, the call and put options for an ounce of gold with a current price of £1,146, an anticipated price in a year of either £1,200 or £1,050, and an interest rate of 4.2% are valued using a one-step binomial tree. Both of the options’ strike prices are £1,150.

Body: The construction of the binomial tree is the first phase, which entails figuring out the up and down factors depending on the projected price and the present price of gold. After constructing the binomial tree, we can use backward induction to determine the call and put option values at each node. The maximum value of a call option is equal to the expected value at the end node minus the strike price, whereas the maximum value of a put option is equal to the strike price less the expected value at the end node.

In conclusion, call-and-put option pricing utilizing a one-step binomial tree offers investors a handy tool for making educated investing decisions. Investors can choose based on the expected value of the options and the corresponding risk by estimating the value of the options at each node.

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