Derivatives and Risk Management FINAL EXAM

Please answer the following questions-

Q1. Using TOS, find an option on and S&P500 security that is currently at the money.

a. Create a replicating portfolio based on four (4) call options and the appropriate number of shares of the underlying. Take a picture of this in the Analyze tab in TOS and post it to your Word document.

b. The replicating portfolio should be approximately delta neutral (i.e., the delta should be close to zero), but the value of the position changes as the price of the underlying moves away from the strike price. Explain why this is so, and discuss whether this has implications for options pricing models.

c. Using the same underlying, create a synthetic position for one (1) long put on the stock. Compare the pricing for this vs. the actual put option (i.e., use this position to describe put-call parity). Is the Law of One Price and the role of synthetic securities evident in this case? Provide a TOS screenshot to support your answer.

Q2. The equation for the Black-Scholes option pricing model is shown below.

C = S₀N(d₁) - Xe^-rtN(d₂)

Discuss the structure and logic of this equation and each of its components. Consider the following when creating your answer.  

a. What is meant by the variable σ, and is why the variable σ√T important in the B-S model?

b. Discuss what is represented by the first and second terms. This should include the individual components of each term, what each term represents, how it relates to the other term, and how the two terms jointly reflect the equilibrium value of a call option.

c. In addition to your discussion in part b, describe how the N(d₂) variable relates to the "Probability OTM" values quoted in TOS for a call and put.

Q3. The range of values for the delta of an option is -100 to +100. Address the following short answer questions relative to delta, gamma, theta, and vega.

a. When is delta positive and when is it negative? Explain why.

b. For a positive delta option, when will delta have a value of 50, when will delta approach 100, and when will delta approach 0? Explain why.

c. Explain how the delta can be used to estimate a hedge ratio and create a hedge position as part of a replicating portfolio. (Note that this refers to the B-S option pricing model.)

d. We discussed that gamma and vega must be proportional, and opposite in sign, to theta. Explain why this is so and how this relates to the intrinsic and extrinsic values for a call option.

Q4. Pull up an at-the-money long call option with a July 21^st maturity for a DJIA stock. Estimate the implied volatility of the stock using the B-S model in Excel (include dividend yield as quoted in TOS). Paste a picture of the Risk Profile page from the Analyze Tab into your Word file (note: this is so I can verify that you estimated implied vol correctly). If there is a difference between your estimate of the implied volatility and the implied volatility reported by TOS, explain why this might be so.

Q5. Provide the following for the stock you selected in problem #4.

a. What is the expected range of values for this stock on June 16^th and also on July 21^st as priced into the at-the-money call option? (Please provide a 95% probability range, which is +/- 2 standard deviations. Show your work on how you calculated the range - don't just show a screen capture of the Probability Analysis page from TOS).

b. Create a graph that shows the probability distribution of prices for the underlying on these dates. Describe what these two graphs represent as they evolve through time.

• By "evolve through time" I meant how does the bell curve change as the maturity date of the option gets farther out? How do the graphs for the two dates relate to each other?

Q6. Assume that you have a portfolio consisting of 10 long call options for a stock that is in the S&P500. (Please limit the price for your stock to $100 max to make this work well.) You wish to create a simultaneous delta, gamma, vega hedge for this portfolio.

a. Estimate the position needed to neutralize delta, gamma, and vega. (HINT: use options on the SPY, options on a different stock within the same industry, and a spot position on the underlying for your original position as your hedging instruments.)

b. Estimate the net delta, gamma, and vega positions for the hedged portfolio (I recommend creating a table in Excel). Execute the required trades for this hedge, and include relevant pictures from the Monitor tab in Think or swim to verify your results.

c. Discuss any basis risk that may exist in this position. 

d. Has the position you created here addressed any of the issues raised in Question #1b? If so, then discuss.

Q7. After the great success of your last principal protected note product, you are now the rising star in the Structured Products Division at RBC Capital. You have been given a new challenge, which is to design a synthetic security with the features described below. The product will be marketed as a Buffered TwinWin Security (or BTWS for short).

a. Design the model by determining the appropriate bond, call, put, and binary option package to create the desired payoff structure for the BTWS.

b. Use Excel to price the product, and demonstrate the profit to the bank per $100 par value unit of the BTWS.

c. Describe the structure of your product and how you were able to create a security that generated the promised payoffs to the customer while also making money for the bank.

TERMS OF THE BWTS:

The BWTS will have a term of approximately 5.5 years. It will be issued on February 17, 2017 and mature on August 18, 2022. The total days to maturity will be 2,018 days on the date of issue.

The cost of the BWTS to the client is $100 per note. The payout on maturity is linked to the return of the S&P500.The underlying is indexed to the S&P500, with a base value of 100. 

The BTWS will yield 100% of the appreciation of the S&P500 index if the return is positive. There is no cap on the upside potential.

The BTWS will return the absolute value of the percent decline in the S&P500 if the index return is negative. This absolute return value is capped at 20%.

The BTWS will have 20% downside protection. This means that if the S&P500 goes down by more than 20%, then the BTWS will lose value along with the S&P500, but it will not participate in the first 20% loss in the index.

• For example, if the S&P500 loses 20%, then the BTWS will gain 20%, but if the S&P500 loses 21%, then the BTWS will lose 1%.
• A payout graph from the Prospectus is shown below.

The yield to maturity on the zero-coupon bond is 212BP.

The dividend yield on the S&P500 is 200BP.

The risk-free rate for options is 50BP.

The implied volatility on SPX options is currently 12%.

Attachment:- BWTS Template.rar