Question 1:
You have just sold an at-the-money European call option contract to one of your clients. The option will expire in 12 months and it is trading at a 50% implied volatility. The stock will not pay any dividends during the contract's life. The risk-free rate is equal to 2% per annum, continuously compounded, and the term structure is flat. This is the only position in your trading book.
Based on the following simulated end-of-day price and implied volatility sequence for the next 5 trading days, delta-neutralize this option position and calculate your trading account's balance at the end of each day, including today (day 0). Assume that the money required to implement this trade is borrowed at the risk-free rate. Report each component of your trading account balance in a neat table.
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Day
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0
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1
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2
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3
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4
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5
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Stock price
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60
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62
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59
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57
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58
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61
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Implied volatility
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0.50
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0.52
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0.48
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0.50
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0.52
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0.50
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Question #2:
You have a long position in 1,500 units of option contract #1, whose characteristics are provided in the table presented below. The underlying stock is trading for $75 and it pays dividends at a continuously compounded rate of 1% per year. The risk-free rate is equal to 3% per annum, across all tenors. You wish to gamma-, vega-, and delta-neutralize your option position. For this purpose, you have identified counterparties who are willing to trade with you two additional option contracts written on the same stock, as described below.
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Option contracts
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#1
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#2
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#3
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Style
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European
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European
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European
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Type
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Call
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Call
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Call
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Strike price:
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$75
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$70
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$80
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Tenor (in years):
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1
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.5
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.75
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Implied volatility (%):
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55
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60
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58
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A) Using Professor John Hull's DerivaGem MS Excel Add-On, calculate each option contract's delta, gamma, and vega.
B) Determine the number of units of option contract #2 required to gamma-neutralize your option position. What position will you need initiate to delta-neutralize this gamma-neutral portfolio?
C) Determine the number of units of option contract #3 required to vega-neutralize your option position. What position will you need initiate to delta-neutralize this vega-neutral portfolio?
D) Determine the number of units of option contracts #2 and #3 required to jointly gamma-and vega-neutralize your option position. What will you need to do to delta-neutralize this gamma- and vega-neutral portfolio?
Question 3:
Today is February 21, 2018, and you are issuing a floating rate note with face value of $500 million. The note will expire on February 21, 2019, and will pay interest on the dates specified in the table below. You are concerned about the risk of a steep increase in LIBOR during that period, so you are considering the possibility of limiting your exposure with an interest rate collar. The current term structures of interest rates and forward volatilities are as follows:
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Term
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Rate
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End date
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Forward volatility
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0x3
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0.0100
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2018-05-22
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0.15
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0x6
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0.0135
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2018-08-20
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0.19
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0x9
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0.0167
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2018-11-21
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0.23
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0x12
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0.0195
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2019-02-21
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0.20
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0x15
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0.0210
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2019-05-23
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0.19
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0x18
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0.0235
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2019-08-22
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0.22
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0x21
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0.0255
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2019-11-23
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0.23
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0x24
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0.0284
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2020-02-21
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0.22
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A) Calculate the market price of the three interest rate caplets spanning the term of the note. The cap rate is set equal to 3%.
B) Calculate the flat volatility that should be quoted for the above interest rate cap to preclude arbitrage opportunities between cap and caplet prices.
C) Calculate the market price of the three floorlets spanning the term of the note. The interest rate floor is set equal to 1.5%.
D) Establish the interest rate floor, which, in combination with the above interest rate cap, would enable you to create a zero-cost collar.
E) On February 21, 2019, you anticipate issuing a new floating rate note with notional amount of $500 million that will mature on February 21, 2020. On the note's issuance date, you anticipate swapping into fixed to reduce your interest rate risk exposure. The swap will entail quarterly payments coinciding with the end dates provided in the above table. How much are you be willing to pay today for an at-the-money swaption giving you the right to enter into a one-year pay-fixed-receive-floating swap on the new note's issuance date? The volatility of the forward swap rate is 15%.
Question 4:
The proprietary trading desk in your shop is implementing a very simple strategy, which traders refer to as Long Tech/Short Energy. This strategy consists of a long position in the stocks of Amazon, Apple, and Google and it is funded by a short position in the stocks of BP, Chevron, and Schlumberger. The notional amount of each leg is equal to USD 60 million, equally divided into each stock. Using the price series provided in the MS Excel workbook accompanying this assignment, address the following questions:
A) In one single chart, plot the price series for each stock using a base value of 100 at the start of the sample period to make the series comparable. Based on the trend that you observe, does this trading strategy seem to make any sense to you?
B) Provide descriptive statistics (mean, standard deviation, min, p10 p25, p50, p75, p90, min) for each stock's return series for the entire sample period. Express returns as Ln(St/St_i). From this perspective, does the strategy make any sense to you?
C) Obviously, this trade will require economic and regulatory capital. Using the historical simulation procedure, determine the value-at-risk of this strategy using the usual 99% confidence level. For this purpose, retain the 500 most recent complete historical scenario sets, i.e. sets for which all six individual historical scenarios are available. Provide the one-day and the ten-day VaR 99% estimate and report the date associated with the VaR 99% scenario.
D) One of the traders responsible for implementing this trading strategy informs you that, according to his analysis, the date associated with the VaR 99% scenario for the overall portfolio is different from the component stocks' individual VaR 99% scenarios. Is this true? Please, verify this assertion. Does this make any sense to you?
Question 5:
The Monte-Carlo simulation procedure is a very powerful tool, which financial engineers use not only to value complex derivatives, for which we have no closed-form solution, but also to assess the maximum potential counterparty credit risk exposure associated with derivatives.
Consider a one year forward contract on a generic asset that pays income at a continuous rate of 1% per year. The asset is currently trading for $600 and at-the-money options on this asset are trading at an implied volatility of 50%. The term structure of risk-free interest rates is flat at 3% per year, continuously compounded, and will remain at this level until the contract expires. Assume that the asset's price is distributed log-normally and that its weekly price sequence over the next 52 weeks can be simulated according to the following equation:
St+At = Ste(r-q--2 )At+o->olaxNORMSINV(RAND( ))
where St is the asset price at time t, At is length of a time step (1/52 year), r is the risk-free rate, q is the income yield, a is the implied volatility, and NORMSINV(RAND()) is the combination of MS Excel functions that produces random draws from the standardized normal distribution. To facilitate my grading process, when possible, present your results in a neat tabular format. Please, do not include your simulated price paths in your paper.
A) Using the above equation, simulate 5,000 random price paths for the underlying asset, each path consisting of 52 weekly random draws, and calculate the replacement cost of the forward contract at the end of each week for each simulated price path. Pick any one of the simulated price paths and plot the simulated underlying asset's price as well as the contract's replacement cost over the 52-week period.
B) Based on the underlying price paths generated in A), estimate the maximum potential counterparty credit risk exposure associated with the forward contract, at the 95% confidence level, based on the first 1,000, 2,000, 3,000, 4,000, and all 5,000 price paths, respectively.
C) In a simple figure, plot the 95% counterparty credit risk exposure profile associated with the forward contract, based on the first 1,000, 2,000, 3,000, 4,000, and all 5,000 price paths, respectively.
D) What can you conclude from C), regarding the forward contract's credit risk exposure profile, as the number of random price paths unerlying the simulation goes from 1,000 to 5,000?
Attachment:- Data.rar