David's ProTip: The only calculus you really need for the FRM is comfort with the first partial derivative. And option delta is maybe the most common. Delta = dc/dS; i.e., the change in call price with respect to a small change in the stock price. Why small? Because delta is just the first term in the Taylor Series approximation, so like duration, it's a linear approximation that doesn't capture the gamma (convexity). As the numerator (change in the Y-axis, $) and denominator (change in the X-axis, $) are both in dollars (or currency), delta itself is unitless.
GARP likes to test the delta of deeply ITM/OTM calls/puts because we don't need to calculate to know that delta converges to 0, -1.0 or 1.0. So you just want to get comfortable with these asymptotes; e.g., you should know that the delta of deeply ITM call converges to +1.0. Can you intuit each of these asymptotes? Best is to develop the intuition that can easily answer my question 6.6 below.
Finally, you probably already realize that option delta, N(d1) is "embedded" in the Black-Scholes pricing formula. As N(d2) is the probability the option expires ITM in the risk-neutral world (i.e., probability option will be exercised in the risk neutral world), we can re-phrase the non-dividend Black-Scholes in this way:
c = Stock * delta - discounted Strike * Prob[option will expire ITM in risk-neutral world]
AIM: Define delta hedging for an option, forward, and futures contracts. Define and compute delta for an option.
Questions:
6.1. What is, respectively, the delta of an at-the-money (ATM) six-month European call and put option on a non-dividend-paying stock when the riskless rate is 4.0% per annum and the stock price volatility is 28%?
a. 0.20 (ATM call) and -0.20 (ATM put)
b. 0.20 (ATM call) and -0.80 (ATM put)
c. 0.58 (ATM call) and -0.58 (ATM put)
d. 0.58 (ATM call) and -0.42 (ATM put)
6.2. A trader has a short position in 1,000 at-the-money (ATM) one-year put options when the underlying stock price has a volatility of 20% per annum and the riskless rate is 4.0% per annum. Which trade will make the position delta neutral?
a. Long 382 shares
b. Short 382 shares
c. Long 618 shares
d. Short 618 shares
6.3. The spot EUR/USD exchange rate is $1.30 (i.e., USD 1.30 per 1 EUR) with a volatility of 30% per annum. The USD riskless rate is 4% per annum and the EUR riskless rate is 3% per annum. What is the delta of a one-year call option on the Euro with a strike price of EUR/USD $1.36?
a. 0.4980
b. 0.5131
c. 0.5529
d. 0.6078
6.4. The spot price of oil is $80.00 per barrel with a volatility of 26% per annum. The riskfree rate is 5.0% per annum. What is the delta of a one-year futures contract when the one-year futures price is $90.00 per barrel?
a. 0.951
b. 1.000
c. 1.051
d. 1.118
6.5. The current price of the S&P 500 Index is 1200. The one-year futures price is 1262; i.e., +5% continuously compounded. The volatility of the index is 18% per annum and the dividend yield is 2.0% per annum. If the riskfree rate is 4.0% per annum, what is the detla of the the one-year futures contract on the S&P 500 Index?
a. 0.9802
b. 1.0000
c. 1.0202
d. 1.0408
6.6. In sequence FROM LOWEST to highest value of option delta, what is the correct order of the following four options: in-the-money (ITM) call option, out-of-the-money (OTM) call option, in-the-money (ITM) put option, and out-of-the-money (OTM) put option?
a. OTM put, OTM call, ITM call, ITM put
b. OTM call, ITM call, ITM put, OTM put
c. ITM call, ITM put, OTM put, OTM call
d. ITM put, OTM put, OTM call, ITM call
Answers:
GARP likes to test the delta of deeply ITM/OTM calls/puts because we don't need to calculate to know that delta converges to 0, -1.0 or 1.0. So you just want to get comfortable with these asymptotes; e.g., you should know that the delta of deeply ITM call converges to +1.0. Can you intuit each of these asymptotes? Best is to develop the intuition that can easily answer my question 6.6 below.
Finally, you probably already realize that option delta, N(d1) is "embedded" in the Black-Scholes pricing formula. As N(d2) is the probability the option expires ITM in the risk-neutral world (i.e., probability option will be exercised in the risk neutral world), we can re-phrase the non-dividend Black-Scholes in this way:
c = Stock * delta - discounted Strike * Prob[option will expire ITM in risk-neutral world]
AIM: Define delta hedging for an option, forward, and futures contracts. Define and compute delta for an option.
Questions:
6.1. What is, respectively, the delta of an at-the-money (ATM) six-month European call and put option on a non-dividend-paying stock when the riskless rate is 4.0% per annum and the stock price volatility is 28%?
a. 0.20 (ATM call) and -0.20 (ATM put)
b. 0.20 (ATM call) and -0.80 (ATM put)
c. 0.58 (ATM call) and -0.58 (ATM put)
d. 0.58 (ATM call) and -0.42 (ATM put)
6.2. A trader has a short position in 1,000 at-the-money (ATM) one-year put options when the underlying stock price has a volatility of 20% per annum and the riskless rate is 4.0% per annum. Which trade will make the position delta neutral?
a. Long 382 shares
b. Short 382 shares
c. Long 618 shares
d. Short 618 shares
6.3. The spot EUR/USD exchange rate is $1.30 (i.e., USD 1.30 per 1 EUR) with a volatility of 30% per annum. The USD riskless rate is 4% per annum and the EUR riskless rate is 3% per annum. What is the delta of a one-year call option on the Euro with a strike price of EUR/USD $1.36?
a. 0.4980
b. 0.5131
c. 0.5529
d. 0.6078
6.4. The spot price of oil is $80.00 per barrel with a volatility of 26% per annum. The riskfree rate is 5.0% per annum. What is the delta of a one-year futures contract when the one-year futures price is $90.00 per barrel?
a. 0.951
b. 1.000
c. 1.051
d. 1.118
6.5. The current price of the S&P 500 Index is 1200. The one-year futures price is 1262; i.e., +5% continuously compounded. The volatility of the index is 18% per annum and the dividend yield is 2.0% per annum. If the riskfree rate is 4.0% per annum, what is the detla of the the one-year futures contract on the S&P 500 Index?
a. 0.9802
b. 1.0000
c. 1.0202
d. 1.0408
6.6. In sequence FROM LOWEST to highest value of option delta, what is the correct order of the following four options: in-the-money (ITM) call option, out-of-the-money (OTM) call option, in-the-money (ITM) put option, and out-of-the-money (OTM) put option?
a. OTM put, OTM call, ITM call, ITM put
b. OTM call, ITM call, ITM put, OTM put
c. ITM call, ITM put, OTM put, OTM call
d. ITM put, OTM put, OTM call, ITM call
Answers:
