Question
1. Calculate the value of a $40-strike at-the-money European call option expiring in 18 months. The risk-free rate is equal to 4% per annum, with
1. Calculate the value of a $40-strike at-the-money European call option expiring in 18 months. The risk-free rate is equal to 4% per annum, with continuous compounding, The stocks volatility is equal to 50%. Construct a three-period binomial lattice to value this option.
2. Calculate the price of a $55-strike European put option expiring in 9 months when the
stock is trading for $50. The risk-free rate is equal to 5% per annum, with continuous
compounding, and the stocks volatility is equal to 35%. Use a four-date binomial lattice
to value this option.
3. Calculate the price of a three-year $40-strike in-the-money American call option on a
stock that is trading for $50. The stocks volatility, its dividend yield, and the risk-free
rate are equal to 40%, 3%, and 5%, respectively. Rates are expressed with continuous
compounding. Use a four-date binomial lattice to value this option and spell-out the
value of its early-exercise privilege.
4. Calculate the price of a $35-strike American put option expiring in 6 months when the
stock is trading for $30. The stock will not pay dividends during this period. The
risk-free rate is equal to 5% per annum, with continuous compounding, and the stocks
volatility is equal to 40%. Use a three-period binomial lattice to value this option. Will
the option ever be exercised prematurely given that the stock pays no dividends? If so,
determine the value of its early exercise privilege.
5. Calculate the price of a $78-strike out-of-the-money American call option on a stock that
is trading for $75. The option will expire in 18 months. The stocks volatility is equal to
30%, the risk-free rate is equal to 5% per annum, and the stocks dividend yield is equal
to 2%. Both rates are expressed with continuous compounding. Using a three-period
binomial lattice, determine the value of this options early-exercise privilege. Explain
your result.
6. A stock is trading for $75 on the NYSE and at-the-money options on this stock expiring
in 18 months are currently trading at an implied volatility of 50% on the CBOE. The
risk-free rate is equal to 2% per annum, continuously compounded. Using a three-period
binomial lattice:
(a) Estimate the value of a European call option on this stock, along with the value of its
American counterparts early exercise privilege, should the stock pay no dividends
during the options life.
(a) $20.3172, $0.0000
(b) Estimate the value of a European call option on this stock, along with the value
of its American counterparts early exercise privilege, should the stock pay a single
dividend of $2.25 in 9 months.
(b) $18.8769, $0.0000
(c) Estimate the value of a European call option on this stock, along with the value
of its American counterparts early exercise privilege, should the stock pay a single
dividend of $2.25 in 15 months.
(c) $18.8912, $0.2660
(d) What does this analysis teach us about what drives the value American call options
early exercise privilege?
(d)
Discuss
7. A stock is trading for $75 on the NYSE and at-the-money options on this stock expiring
in 18 months are currently trading at an implied volatility of 50% on the CBOE. The
risk-free rate is equal to 2% per annum, continuously compounded. Using a four-date
binomial lattice:
(a) Estimate the value of a European put option on this stock, along with the value of its
American counterparts early exercise privilege, should the stock pay no dividends
during the options life.
(b) Estimate the value of a European put option on this stock, along with the value
of its American counterparts early exercise privilege, should the stock pay a single
dividend of $2.25 in 9 months.
(c) Estimate the value of a European put option on this stock, along with the value
of its American counterparts early exercise privilege, should the stock pay a single
dividend of $2.25 in 15 months.
(d) What does this analysis teach us about what drives the value American put options
early exercise privilege?
8. An at-the-money call option on the Euro will expire in three months. The spot exchange
rate is $2 per Euro. The domestic interest rate is equal to 5% and the interest rate in
the Eurozone is equal to 6%. The exchange rate volatility is equal to 10%. Calculate
the value of this option without and with early exercise privilege using a binomial lattice
consisting of three periods of equal length.
9. Construct a four-date binomial tree to value a European and an American call option on
the U.S. dollar using Rendelman and Bartters equal probability method. The relevant
inputs are provided in the table below.
Parameter input
Value
Spot exchange rate (USD/CAD)
1.34
Strike
1.30
Tenor (in years)
0.75
Risk-free rate (CAD)
0.0450
Risk-free rate (USD)
0.0472
Volatility
0.15
10. Using the Black-Scholes-Merton (BSM) formula, calculate the price of a 50-strike, at
the-money European call option on a stock trading at an implied volatility of 50%. The
option will expire in one year and the risk-free rate is equal 3% per year. Using put-call
parity, calculate the price of the corresponding put option
11. Using the BSM formula, calculate the price of an $80-strike European put option on
a stock that is trading for $70. The option will expire in 9 months and it is trading
at a 60% implied volatility. The risk-free rate is equal to 2% per year, continuously
compounded. Using put-call parity, calculate the price of the corresponding call option
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