Trident Advisory Group manages assets for high-net-worth individuals and family trusts. Alice Lee, chief investment officer, is meeting with a client, Noah Solomon, to discuss risk management strategies for his portfolio. Solomon is concerned about recent volatility and has asked Lee to explain options valuation and the use of options in risk management.

Options on Stock

Lee begins: We use the BlackScholesMerton (BSM) model for option valuation. To fully understand the BSM model valuation, one needs to understand the assumptions of the model. These assumptions include normally distributed stock returns, constant volatility of return on the underlying, constant interest rates, and continuous prices. Lee uses the BSM model to price TCB, which is one of Solomons holdings. Exhibit 1 provides the current stock price (S), exercise price (X), risk-free interest rate (r), volatility (), and time to expiration (T) in years as well as selected outputs from the BSM model. TCB does not pay a dividend.

Exhibit 1.BSM Model for European Options on TCB

BSM Inputs
S	X	r				T
$57.03	55	0.22%		32%		0.25
BSM Outputs
d1	N(d1)		d2		N(d2)		BSM Call Price	BSM Put Price
0.3100	0.6217		0.1500		0.5596		$4.695	$2.634

Options on Futures

The Black model valuation and selected outputs for options on another of Solomons holdings, the GPX 500 Index (GPX), are shown in Exhibit 2. The spot index level for the GPX is 187.95, and the index is assumed to pay a continuous dividend at a rate of 2.2% () over the life of the options being valued, which expire in 0.36 years. A futures contract on the GPX also expiring in 0.36 years is currently priced at 186.73.

Exhibit 2.Black Model for European Options on the GPX Index

Black Model Inputs
GPX Index	X	r			T	Yield
187.95	180	0.39%		24%	0.36	2.2%
Black Model Call Value			Black Model Put Value			Market Call Price	Market Put Price
$14.2089			$7.4890			$14.26	$7.20

Option Greeks
Delta (call)	Delta (put)	Gamma (call or put)	Theta (call) daily	Rho (call) per %	Vega per % (call or put)
0.6232	0.3689	0.0139	0.0327	0.3705	0.4231

After reviewing Exhibit 2, Solomon asks Lee which option Greek letter best describes the changes in an options value as time to expiration declines.

Solomon observes that the market price of the put option in Exhibit 2 is $7.20. Lee responds that she used the historical volatility of the GPX of 24% as an input to the BSM model, and she explains the implications for the implied volatility for the GPX.

Options on Interest Rates

Solomon forecasts the three-month Libor will exceed 0.85% in six months and is considering using options to reduce the risk of rising rates. He asks Lee to value an interest rate call with a strike price of 0.85%. The current three-month Libor is 0.60%, and an FRA for a three-month Libor loan beginning in six months is currently 0.75%.

Hedging Strategy for the Equity Index

Solomons portfolio currently holds 10,000 shares of an exchange-traded fund (ETF) that tracks the GPX. He is worried the index will decline. He remarks to Lee, You have told me how the BSM model can provide useful information for reducing the risk of my GPX position. Lee suggests a delta hedge as a strategy to protect against small moves in the GPX Index.

Lee also indicates that a long position in puts could be used to hedge larger moves in the GPX. She notes that although hedging with either puts or calls can result in a delta-neutral position, they would need to consider the resulting gamma.

1. Lees statement about the assumptions of the BSM model is accurate with regard to:
   1. interest rates but not continuous prices.
   2. continuous prices but not the return distribution.
   3. the stock return distribution but not the volatility.

2. Based on Exhibit 1 and the BSM valuation approach, the initial portfolio required to replicate the long call option payoff is:
   1. long 0.3100 shares of TCB stock and short 0.5596 shares of a zero-coupon bond.
   2. long 0.6217 shares of TCB stock and short 0.1500 shares of a zero-coupon bond.
   3. long 0.6217 shares of TCB stock and short 0.5596 shares of a zero-coupon bond.

3. To determine the long put option value on TCB stock in Exhibit 1, the correct BSM valuation approach is to compute:
   1. 0.4404 times the present value of the exercise price minus 0.6217 times the price of TCB stock.
   2. 0.4404 times the present value of the exercise price minus 0.3783 times the price of TCB stock.
   3. 0.5596 times the present value of the exercise price minus 0.6217 times the price of TCB stock.

4. What are the correct spot value (S) and the risk-free rate (r) that Lee should use as inputs for the Black model?
   1. 186.73 and 0.39%, respectively
   2. 186.73 and 2.20%, respectively
   3. 187.95 and 2.20%, respectively