Derivatives (Terms)

to price a plain vanilla interest rate swap at initiation, the value of the fixed-rate bond must equal

the value of the floating-rate bond.

is 1 (par) because the interest payment is set to match the discount rate. Recall that when the YTM (discount rate) of a bond is equal to the coupon rate, the bond sells at par.

Hedge Ratio

the value of the call if the underlying goes up minus the value of the call if the underlying goes down divided by the value of the underlying if it goes up minus the value of the underlying if it goes down

Delta is

defined as the change in a given portfolio for a given small change in the value of

the underlying instrument, holding everything else constant.

Gamma

is defined as the change in a given portfolio delta for a given small change in the value of the underlying instrument, holding everything else constant

Theta

is defined as the change in the value of an option given a small change in calendar

time, holding everything else constant.

Vega

is defined as the change in a given portfolio for a given small change in volatility,

holding everything else constant

Rho

is defined as the change in a given portfolio for a given small change in the risk-free

interest rate, holding everything else constant.

d1

A standardized distance measure representing the expected log-return of the underlying stock relative to the option's exercise price, adjusted for volatility and time to expiration.

d2

The standardized distance measure adjusted specifically for the uncertainty/volatility over the option's life

N(d1)

Option Delta: Measures the sensitivity of the call option price to a $\$1$ change in the underlying stock price. For this call option, hedging requires holding $0.687$ shares of stock per option.

Leveraged Exposure Factor: Represents the weight applied to the current stock price in the call valuation formula.

N(d2)

Probability of Call Expiring In-the-Money: Under risk-neutral probabilities, this is the precise probability that the stock price at expiration will exceed the exercise price ($S_T > X$), meaning the call option will be exercised.

Present Value Factor Weight: Represents the probability-adjusted weight applied to the present value of the exercise price payment.

N(-d1)

Put Option Delta (Magnitude): Equals $1 - N(d_1)$ and represents the absolute magnitude of a European put option's delta (actual put delta is $-0.313$).

N(-d2)

Probability of Put Expiring In-the-Money

An arbitrageur dynamically manages a portfolio of options on a single stock. Using the Black–Scholes–Merton model, when the dividend yield on the stock increases, the arbitrageur should:

lower the number of stocks to buy for calls and raise the number of stocks to short sell for puts.