exam2

Question 1

General Motors stock price is $59, the strike price is $60, the expiration is in 2 months, the implied volatility of the underlying stock σ is 30 percent per year, and the continuously compounded risk-free rate is 3.3 percent per year

Answer 1

Question 2

– Each share of Berkshire Hathaway (BRK-B) is currently trading at $150.

If the risk-free rate is 5% and the expected volatility of BRK-B is 20%, what is the Black-Scholes-Merton price of an at-the-money European call option on BRK-B with two years until expiration?

Answer 2

Question 3

Each share of Tesla Motors, Inc. (TSLA) is currently trading at $300.

If the riskfree rate is 10% and the expected volatility of TSLA is 30%, what is the BlackScholes-Merton price of a European put option on TSLA with three months until expiration and strike price of $250?

Answer 3

Question 4

Each share of Alphabet Inc (GOOG) is currently trading at $500.

If the risk-free rate is 10% and the expected volatility of GOOG is 30%, what is the BlackScholes-Merton price of an at-the-money European put option on GOOG with one year until expiration?

Answer 4

Question 5

Each share of Ford Motor Co. (F) is currently trading at $200.

If the risk-free rate is 5% and the expected volatility of F is 20%, what is the Black-Scholes Merton price of a European put option on F with 6 months until expiration and strike price of $250?

Answer 5

Question 6

Please use the following:

What is the American Put Price?

Answer 6

Question 7

Please use the following to calculate the American call price:

Answer 7

Question 8

Calculate the portfolio delta of a bull call spread buying 100 calls with K1 = 40 and selling 100 calls with K2 = 60:

How can we become delta neutral on this strategy?

Answer 8

Delta Neutral: Short sell 47 shares of the underlying

Question 9

– Calculate the portfolio delta of a long straddle using 100 puts and 100 calls with:

Question 10

How can we become delta neutrtal on this strategy?

Answer 10

SHort sell 16 shares of the underlying.

Question 11

– What will our estimated P&L be if the underlying increases by $1 for a portfolio consisting of selling 100 puts with a strike price of 50 and one year expiration and buying 100 calls with a strike price of 60 and one year to expiration.

The current spot price is 70 and the underlying has an annual volatility of 20% and the risk free rate is 5%?

How can we delta hedge this position?

Answer 11

In order to delta hedge, we’d need to short sell 89 shares of the underlying.

Question 12

– What will our estimated P&L be if the underlying increases by $1 for a portfolio consisting of selling 100 at-the-money straddles with a strike price of $50 and six months to expiration if the underlying has an annual volatility of 30% and the risk free rate is 10%? How can we delta hedge this position?

Answer 12

In order to delta-hedge these straddles, we’d need to buy 26 shares of the underlying.

Question 13

Φ(d₁)

Answer 13

represents the proportion of that total stock value that is attributable to states in which the stock “is in the money”, or the risk-neutral probabilities in these states weighted by the terminal stock price in these states.

Question 14

Φ(d₂)

Answer 14

is simply the risk-neutral probability that the stock will be “in the money” at expiration.

Question 15

Implied Volatility

Answer 15

We can easily look up option prices.

Instead of using volatility to compute option prices, we can use option prices to compute volatility.

We can let the market do the work for us and extract the level of volatility they are pricing into option premiums

Although we can’t solve for volatility algebraically, we can solve for numerically.

The actual methods used are quite complex, and are constantly being improved. Conceptually, however, these methods are essentially guided/intelligent trial-and-error procedures.

You will “solve” for vol in the group project using Excel’s Goal Seek (or Solver) function.

Volatilities computed in this way, from observed option prices, are called implied volatilities.

Question 16

Delta

Answer 16

Delta measures the change in the option’s value when there is a

small change in the underlying stock price

Question 17

Gamma

Answer 17

Gamma measures the sensitivity of the delta to changes in the

underlying stock price

Question 18

Vega

Answer 18

Vega measures the sensitivity of the option value to changes in the

volatility

Question 19

Theta

Answer 19

Theta measures the sensitivity of the option value to changes in

the time to expiration

Question 20

Theta Decay

Answer 20

Time (Theta) Decay of Options

Option time values often decay in a non-linear way, at the money options have the most time-value and the most non-linear time decay. Also a function of volatility (more vol, more time value/non-linear decay)

Question 21

Rho

Answer 21

Rho measures the sensitivity of the option value to changes in the interest rate

For a put, rho is negative because an increase in rates decreases the PV of the strike the put owner recieves if the option is exercised

Question 22

Delta Hedging

Answer 22

A delta hedge will fully immunize us against market risk (become

“delta neutral”), but only for small price changes in the underlying.

Option dealers will often delta hedge options that they sell in order

to manage risk.

Conceptually, if your option trade is bullish, you need to short sell ∆ shares of the underlying, and if your option trade is bearish, you need to buy ∆ shares of the underlying to hedge.

Question 23

Delta-Gamma Hedging

Answer 23

But what about delta???

i.e. If I want to gamma hedge a 100 long calls, sell 100 puts with the same strike and expiration. ( S 0 = 59 ; K = 60 ; T = 1 ; = 0 : 3 ; r = 0 : 033 ) ∆=( 100 0 : 58 + 100 0 : 42 )= 100! Since the delta of the underlying is always 1, then its Gamma is zero, we can preserve our Gamma Hedge and obtain a Delta-Gamma hedge by simply shorting 100 shares of the underlying.

Question 24

What are the principal sources of uncertainty facing Aqua Bounty?

Answer 24

The uncertainty for regulatory approval for their products by government & commercialization levels.

Futher delay of FDA process will decrease the value of the products. This could be because a myriad of reasons. Delay in the FDA process may result in unfavorable public opinion and a decrease in shareholder interest. Also, the company will not be able to generate a steady cash flow and thus will have SG&A expenses pile up as well as R&D loan payments. This will also increase the premium about of the call on the option which results in a larger IPO cost.

Question 25

How can we interpret Φ(d₁) and Φ(d₂)?

Answer 25

To get the call option value, we multiply the stock price, S₀ by the fraction of its value attributable to states in which the option will expire “in the money” Φ(d₁) and then subtract the present value of the strike price multiplied by the risk-neutral probability that the strike price will be paid Φ(d₂)

Question 26

Does it make sense for the Aqua Bounty to launch an IPO now?

Answer 26

Absolutely not. If they chose to conduct IPO now, it might suffer from the high possibility of being refused by FDA, as well as the risk of commercialization; also, if they launch IPO and FDA refused their products later, the company would not generate revenues and leading to the risk of bankrupt. While if they chose to conduct IPO after FDA approved their products, they only need to worry about the commercialization uncertainties, and keep away the risk of being refused. They should launch IPO only if they get approval from FDA as well as a positive NPV brought by commercialization. Thus launching an IPO before getting approval would incur a lower NPV and higher risk thus does not make sense.

Question 28

volatility surface”

Answer 28

ompute implied volatilities for puts and calls with the

same underlying asset, but different strike prices and different time

to expirations?

Question 29

Volatility Smile

Answer 29

The relationship between implied volatility and strikes usually presents a smile-like shape, indicating that those contracts with extreme strike levels (either deep

out -of-the-money or deep in-the-money) are assigned larger volatility inputs by traders.

The volatility smile is a relatively recent phenomenon. Barely twenty years old. October 19th, 1987 is conventionally credited as its birth date. It is not too difficult to understand why. That day, when equity markets worldwide plummeted (with the Dow Jones tumbling by almost a quarter, its worst single-day percentage decline ever), market players experienced the first serious crash since the introduction of the Black-Scholes option pricing model in 1973.

The first unavoidable wake-up call reminding traders of the unworldly unrealistic assumptions of Black-Scholes when it came to the probability distribution underlying financial markets.

The model assumes that asset returns follow a Normal distribution (thus ruling out the possibility of extreme moves), but clearly there was nothing “normal” about daily market declines of 23% (New York), 15% (Tokyo) or 10% (London). Today ́s equivalents of such plunges would be the Dow Jones, the Nikkei, and the FTSE falling by 3200, 2700, and 650 points respectively. Truly monstruous.