## Accounts-AW658 Online Services

**Question 1.1 **

On April 10, ABC inc. Enters in a swap contract for 10 years with a chartered bank to turn a fixed rate on liability of $150 million to floating rate. ABC wants to receive interest payments at a fixed rate in exchange for interest payments at a floating rate. The floating reference to the Canadian market is the bankers’ acceptance rate 6 months. Given the average rate that you will find in the worksheet, calculate

a) Explain briefly why such swap transforms a fixed rate liabilities of ABC inc. in floating rate liabilities.

b) The value of cash flows that ABC pays to the financial institution (Table 7.2 in your textbook should help).

c) The value of the cash flows that ABC receives from financial institution.

d) The net cash flows received (paid) by ABC.

**Question 1.2 M**

The comparative advantage argument is often used to explain the benefits of swaps. In the Excel file, you will find the cost of fixed rate and floating rate for two companies that do not have the same credit quality. AAA company wants to borrow at floating rate, while the BBB company wants to borrow at fixed rates.

a) Calculate the total gain (in terms of interest rates) that the two companies can achieve by entering into a swap between them.

b) Assuming that the gain is divided equally between the two companies, provide new financing costs for AAA (the example in your lecture notes on swaps should help)

c) Assuming that the gain is divided equally between the two companies, provide new funding costs for BBB (for help, use the example from your lecture notes on swaps).

d) Suppose the fixed rate for the BBB has been 7.4% instead of 8% (all other rates are unchanged). Would it have been possible to gain by entering into a swap between AAA and BBB? Why?

e) Explain briefly why the comparative advantage argument is misleading.

**Question 1.3 **

To understand the dynamics of the swap, you will have to assess the value of a swap at various times of its life, using the approach of a bond portfolio. The date is January 1, the swap horizon is 1 year, the notional amount is $100 and the fixed rate associated with swap is indicated in the worksheet.

a) Convert rates from semi-annual to continuously compounded.

b) Calculate the value of the floating rate bond on January 1, April 1, July 2, 28 September and 31 December.

c) Calculate the value of fixed-rate bond on January 1.

d) Calculate the value of the swap for fixed payment for January 1st.

e) Calculate the value of fixed-rate bond on April 1.

f) Calculate the value of the swap for fixed payment for April 1st.

g) Calculate the value of the fixed rate bond at 31 December.

h) Calculate the value of the swap for fixed payment for December 31.

Question 2.1

Mr. Lambert currently holds a share of stock of the company ABC. He wants to keep the stock because he believes that the prospect of higher long-term are very good. On the other hand, in short term, it is anticipated that the share price will drop significantly for the next month. He would protect against such a decline in value in case it needs liquidity and would be forced to sell the security in the coming months. So he decides to protect himself by buying a put.

a) What are the advantages and disadvantages of such a strategy? Make sure to include in your reply the maximum possible loss and state the value that the underlying stock must reach at the time of maturity of the option in order for Mr. Lambert to take advantage of your strategy.

b) Using the data in the Excel spreadsheet, calculate: • the profit associated with the derivative;

• the profit associated with holding the underlying;

• the total profit.

c) Graph the total profit associated with the suggested strategy. What do you see?

**Question 2.2 **

A speculator believes that the share price of the company DEF will not change by much in the coming months.

a) List four strategies that this trader can use to gain profit if his prediction comes true.

b) Now assume that speculator decides to opt for a butterfly spread. Using the data in the Excel spreadsheet, calculate: • the profit associated with each of the three options;

• the total profit associated with this strategy.

c) Graph the total profit associated with the butterfly spread.

**Question 2.3 **

An investor is considering the purchase of a strip strategy by buying four call option contracts and eight put option contracts.

a) Explain the likely incentives of the investor.

b) Using the data in the Excel spreadsheet, calculate: • the profit associated with each option of the strip;

• the total profit associated with this strategy.

c) Make a graph of profit as a function of the price at maturity.

**Question 2.4 **

Consider the following options strategy: buy a call option and two puts with a strike price of $ 35, and sell a call option with a strike price of $40. All options have the same maturity. Prices of calls and puts are provided in the Excel spreadsheet.

a) Calculate the benefits associated with this strategy and graph the gains and losses.

b) Explain the advantages and disadvantages of this strategy. Give an example of an investor who might be interested in such a strategy.

c) Analyze the differences of this strategy relative to straddle and strip.

**Question 3.1 **

Using a one-period binomial tree, estimate the value of a call option with a strike price of $52 and a maturity of two years. The risk free rate is 5%, the value of the underlying is $50, u = 1.2 and d = 0.8.

a) Calculate the value of the option using the Δ method.

b) Calculate the value of the option using the risk-neutral probabilities method.

c) Explain briefly what does Δ represent.

**Question 3.2 **

The risk free rate is 5%, the value of the underlying asset is $50, u = 1.2 and d = 0.8. Using a two-period binomial tree, estimate the value of European and American style options, with an exercise price of $52 and a maturity of two years using the following steps:

a) Calculate the values of the underlying asset at the various nodes of the tree.

b) Estimate the value of a European-style call option at different nodes of the tree using the Δ method. To do this, you must

i. Calculate option values at t = 2

ii. Calculate the Δ values at t = 1

iii. Calculate the values of the option at t = 1 using the Δ values found in the previous step

iv. Calculate the Δ value at t = 0

v. Calculate the value of the option at t = 0, using the calculated Δ value in the previous step

c) Estimate the value of the European-style call option at different nodes of the tree using the risk-neutral method.

d) Assess the value of the European-style put option at different nodes of the tree using the risk-neutral method.

e) Assess the value of the American-style put option at different nodes of the tree using the risk-neutral method.

**Question 3.3 **

The seller of a call option must be able to pay the holder of the option in the event that the value of the underlying security is higher than the strike price at the maturity date of the contract. We will see that it is possible to use the binomial model to achieve a hedging strategy that ensures that the seller of the option has the necessary amount to pay to the buyer of the option at maturity. Here we examine this strategy.

As explained in your textbook, combining Δ underlying securities and -1 option, we can obtain a riskless portfolio. This can be written as

BCS=−Δ0

where is the value of the portfolio without risk. In the next period, the portfolio is: B

or dt rBeCS=−Δ1111 dt rBeCS=−Δ1212

with representing the value of the underlying asset at date 1 following an upward movement and where is the value of the underlying asset at date 1 following a downward movement. It is possible to use this principle to develop a hedging strategy. In fact, since it is possible to write 11S12SBCS=−Δ0

CBS=−Δ0

which takes the values

or 1111CBeSdt r=−Δ 1212CBeSdt r=−Δ

In other words, it is possible to form a portfolio (buying Δ underlying securities financed by a loan at the risk-free rate) that replicates the cash flows of an option! The person who sold the option has to form the portfolio at the beginning of each period to ensure a value exactly equal to the value of the option at the end of each period.

Apply this principle to the case of a seller of options wishing to hedge. Worksheet Question 3 includes data relating to the problem. Assume that realized prices are the prices in the yellow cells of the tree and fill the columns in the space response.

**Question 3.4 **

In practice, the parameters of the binomial tree are selected as described in section 11.7 of your textbook. For this question you need to evaluate the price of a European-style call option with the underlying index S&P500 with a strike price of $1,470, maturity of 30 days and a value of the S&P500 1468.36. The risk free rate is 5%. To do this, use a 4-steps binomial tree.

a. To evaluate the option, you must first obtain a volatility parameter. To do this, you will use the time series of the S&P500. To estimate the parameter, you must first estimate the daily continuously compounded rate of return associated with S&P500. To do this, you must use the formula where is the daily return on day t and is the value of the S&P500 at time t. )ln()ln(1−−=tttPPr trtP

b. Once the historical returns are calculated, you must estimate the standard deviation associated with the series of daily returns. This standard deviation is the standard deviation of daily returns. What you need to enter into binomial tree, however, is the standard deviation of annual returns. You must annualize the standard deviation of daily multiplying by. 365

c. You must then obtain values for, and . For the value of, you must calculate the time period associated with one step in your tree. Thus, for a maturity of 30 days and a 4-step tree, we will have T=0.0821918 years, which means that one step will correspond to years. And values and can be found by using the equations from section 11.7. t δu dt δ4/Tt =δ u d

d. Calculate the values in the trees.

**Question 4.1 **

A key assumption of the Black-Scholes option-pricing model is the process of evolution of the price of the underlying security. In fact, Black-Scholes adopt as an assumption a process that involves annual expected return and annual volatility of returns. μσ

To get a good intuition about these parameters and how they influence the evolution of the share price over time, it is useful to use simulation experiments in which we simulate prices under the assumptions of Black-Scholes. In the sheet “Question 1” you will find simulated series of prices for a stock based on the assumptions of Black-Scholes. This simulation represents a plausible future series for the share price. This is not the only plausible series, there are an Project 2. MOS 3312A (570): Derivative Securities Markets Prepared by Jacques Raynauld and Jean-Guy Simonato, HEC Montréal. Translated and adopted by Grigori Erenburg, King’s University College, UWO. and change.

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**Question 4.2**

In your textbook it states that the logarithm of the share price atthe date Tin the Black-Scholesmodel is assumed to follow a normal distribution i.e.

2lnS~NlnS+μ−σT,σT.

T0

2

We will check this(and few other) property using simulated data under the assumptions of Black-Scholes.In the previous question, we simulated a single series. For this question, we will check the properties of the underlying price with 500 series.

a)Calculate the logarithm of the stock price for each series at each point in time.

b)Check the property mentioned above using simulated series i.e. check if the average log price of the security at the timeT=1/2(across the series)corresponds to the theoretical value mentioned above.As the series were simulated using known valuesμandσ, it is

possible to know the theoretical value of this average.Do the same forthestandard deviation of the logarithm ofthe share price.

c)Calculate the continuously compounded rate of return associated with price series i.e.calculate, for each series, the return

S

RT=lnTfor T= 1/12, 2/12, …, 6/12.

S

0

d)In your textbook, Equation (12.4) states that the continuously compounded return is also distributed according toa normal distribution, i.e.

STσ2

ln~Nμ−T,σT

S02

Check this property in the same way as question b)Prepared by Jacques Raynauld and Jean-Guy Simonato, HECMontréal. Translatedand adopted byGrigori Erenburg, King’s University College, UWO.Project 2. MOS 3312A (570): Derivative Securities Markets Prepared by Jacques Raynauld and Jean-Guy Simonato, HEC Montréal. Translated and adopted by Grigori Erenburg, King’s University College, UWO.

e) In your textbook, it states that in the world of Black-Scholes model, the price of the underlying security is distributed lognormally while the logarithm of the price of the underlying security is distributed normally. Check these properties by plotting histograms for the price and the logarithm of the price for the period of 6 months.

f) Using the series # 1, estimate the parameter that represents the volatility of stock returns. σ

**Question 4.3 **

Using the Black-Scholes model, evaluate call and put options described in the example on page 281 of your textbook.

a) Start by calculating the values and . 1d 2d

b) Calculate the values , , and the value of the call. To calculate and use the NORMSDIST Excel function. )(1dN)(2dN)(1dN )(2dN

c) Calculate the value of the put.

d) Since the parameter is unobservable, analysts using the Black-Scholes model calculate, what is frequently called, implied volatility. Implied volatility is the value that equates the theoretical price of Black-Scholes with the observed price for the option. Given an observed market price of $ 5.00 for a call with the same parameters (except for) as in previous questions, calculate implied volatility using Excel Solver. σσ σ

**Question 5.1 **

The delta of a call option is defined as

where CS∂Δ=∂ CS∂=Δ∂

which can be interpreted as the change in the option price due to a small change in the price of the underlying security. Thus, delta gives us a way to approximate the change in the value of our option due to a change in the value of the underlying. Using the price series found in your worksheet, check this interpretation:

a) For each date in the table, calculate the Black-Scholes price associated with a call option whose parameters are given in the worksheet. For this calculation, use BLACKSCHOLES function available in your worksheet. To use this function

• Press the Insert Function

• Go to the User Defined category

• Choose the function BLACKSCHOLES

• The parameters of the function: S is the price of the underlying asset, K is the strike price, r is the interest rate, T is the maturity, SI is the volatility parameter. For the last parameter, OTYPE, you must specify, by entering the letter c, if it is a call or p if it is a put.

• The function DELTA is used in the same way.

Given the price series found in the sheet, calculate the option price and its delta on each date.

b) Check the equation . To do this C S∂=Δ∂

i. calculate changes ini.e. . S 1ttSSS−∂=−

ii. Calculate the following predicted changes by the approximation equation for delta i.e. calculate . SΔ∂

iii. Calculate the real changes in the price of the option i.e. calculate where is the value of the option on the date t calculated using the Black-Scholes model. 1ttcc−− tc

iv. Calculate the errors of the approximation.

c) Why do we observe differences between the price changes predicted by the delta and real changes in the option price? In what situation the errors of the approximations are more critical?

**Question 5.2 **

Reproduce Table 15.2 on page 332 of your textbook as well as some calculations associated with it. To achieve this:

a) Calculate the Black-Scholes price associated with option described in this table.

b) Reproduce the numbers appearing in the table. To make these calculations and have the same rounding errors as in the textbook, you will have to do the calculations according to the precision used in the table. For example, the third column that provides the delta is calculated with three digits after the point. You will need to use the Excel ROUND function to keep three digits after the point.

c) Check if the current value of the costs associated with the hedging strategy is close to the theoretical value of the option.

d) The delta of an option is often used to approximate the probability that an option is in-the-money at maturity date i.e. that . Is this interpretation confirmed in the table that you built? Explain.XST>

Question 5.3

The gamma of an option is defined as

/SΓ=∂Δ∂

which can be interpreted as the change in the delta of the option following a small change in the price of the underlying security. Therefore, the gamma provides us with a way to know if a small change in price can cause a wide variation in the delta. It is useful to know this, because it is an indicator of the quality of our delta hedging strategies. Indeed, if the gamma is small, delta changes slowly and required adjustments (sales and purchases of shares) to keep a portfolio delta neutral (delta hedging) do not have to be performed frequently. By contrary, a large gamma tells us that the delta changes quickly. Adjustments in our strategy of delta hedging should be done more frequently.

Using the series of prices found in your worksheet

a) Check the equation . To do this /SΓ=∂Δ∂

i. calculate changes in i.e. . S 1ttSSS−∂=−

ii. Calculate the following changes in delta i.e. calculate . 1tt−∂Δ=Δ−Δ

iii. Calculate the approximate value of gamma i.e. calculate . /S∂Δ∂

iv. Calculate the true values of the gamma using the Excel function GAMMA.

v. Calculate the approximation errors.

**Question 5.4 **

If we want to guard against the risk associated with rapid changes in a delta using delta hedging strategy, we must obtain a gamma neutral portfolio. As mentioned on page 338 of your textbook, in the context of a delta hedging strategy, it is possible to make a portfolio gamma neutral by buying additional options. Using the prices series provided in your worksheet, build a portfolio gamma and delta neutral:

a) For each week, calculate the gamma of the option # 1 and # 2.

b) Calculate the delta of the option # 2.

c) Calculate the number of options # 2 to hold to make the portfolio gamma neutral.

d) Calculate the number of shares held in the portfolio to make it delta neutral.

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