Derivatives

Question 1

Describe how interest rate forwards and futures are priced, and calculate and interpret their no-arbitrage value.

Answer 1

The “price” of an FRA is the implied forward rate for the period beginning when the FRA expires to the maturity of the underlying “loan.”

The value of an FRA at maturity is the interest savings to be realized at maturity of the underlying “loan” discounted back to the date of the expiration of the FRA at the current LIBOR. The value of an FRA before maturity is the interest savings estimated by the implied forward rate discounted back to the valuation date at the current LIBOR.

Question 2

Describe how fixed-income forwards and futures are priced, and calculate and interpret their no-arbitrage value.

Answer 2

forwards on coupon-paying bonds, the price is calculated as the spot price minus the present value of the coupons times the quantity one plus the risk-free rate:

FP (on a fixed income security) = (S0 – PVC) × (1 + Rf)T = S0 × (1 + Rf)T – FVC

The value of a forward on a coupon-paying bond t years after inception is the spot bond price minus the present value of the forward price minus the present value of any coupon payments expected over the term of the contract:

If given the current forward price (FPt) on the same underlying and with the same maturity:

In a futures contract, the short may have delivery options (to decide which bond to deliver). In such a case, the quoted futures price is adjusted using the conversion factor for the cheapest-to-deliver bond:

Question 3

Describe how equity swaps are priced, and calculate and interpret their no-arbitrage value.

Answer 3

The fixed-rate side of an equity swap is priced and valued just like an interest rate swap. The equity side can be valued by multiplying the notional amount of the contract by 1 + the percentage equity appreciation since the last payment date. Use the difference in values to value the swap.

Question 4

Describe how equity forwards and futures are priced, and calculate and interpret their no-arbitrage value.

Answer 4

The calculation of the forward price for an equity forward contract is different because the periodic dividend payments affect the no-arbitrage price calculation. The forward price is reduced by the future value of the expected dividend payments; alternatively, the spot price is reduced by the present value of the dividends.

The value of an equity forward contract to the long is the spot equity price minus the present value of the forward price minus the present value of any dividends expected over the term of the contract:

If given the current forward price (FPt) on the same underlying and with the same maturity:

We typically use the continuous time versions to calculate the price and value of a forward contract on an equity index using a continuously compounded dividend yield.

Question 5

Describe how interest rate swaps are priced, and calculate and interpret their no-arbitrage value.

Answer 5

The fixed periodic-rate on an n-period swap at initiation (as a percentage of the principal value) can be calculated as:

The value of a swap on a payment date has a simple relationship to the difference between the new swap fixed rate and the original swap fixed rate

Question 6

Describe how currency swaps are priced, and calculate and interpret their no-arbitrage value.

Answer 6

The fixed rates in a fixed-for-fixed currency swap are determined using the yield curves for the relevant currencies. The notional principal amounts in the two currencies are of equal value, based on exchange rate at inception of the swap. Use the difference in values to value the currency swap. The conversion of value from one of the two currencies into the common currency is based on the exchange rate on the valuation date.

Question 7

Calculate and interpret the value of an interest rate option using a two-period binomial model.

Answer 7

The value of an interest rate option is computed similarly to the value of options on stocks: as the present value of the expected future payoff. Unlike binomial stock price trees, binomial interest rate trees have equal (risk-neutral) probabilities of the up and down states occurring.

Question 8

Identify an arbitrage opportunity involving options and describe the related arbitrage.

Answer 8

Synthetic call and put options can be created using a replicating portfolio. A replication portfolio for a call option consists of a leveraged position in h shares where h is the hedge ratio or delta of the option. A replication portfolio for a put option consists of a long position in a risk-free bond and a short position in h shares. If the value of the option exceeds the value of the replicating portfolio, an arbitrage profit can be earned by writing the option and purchasing the replicating portfolio.

Question 9

Delta

Answer 9

Input: Asset price (S)

Calls: Positively related Delta > 0

Puts: Negatively related Delta < 0

Delta is the change in the price of an option for a one-unit change in the price of the underlying security. e–δTN(d1) from the BSM model is the delta of a call option, while –e–δTN(–d1) is the put option delta.

As stock price increases, delta for a call option increases from 0 to e–δT, while delta for a put option increases from –e–δT to 0.

Question 10

Gamma

Answer 10

Input: Delta

Calls: Positively related Gamma > 0

Puts: Positively related Gamma > 0

Question 11

Vega

Answer 11

Input: Volatility (σ)

Calls: Positively related Vega > 0

Puts: Positively related Vega > 0

Question 12

Rho

Answer 12

Input: Risk-free rate (r)

Calls: Positively related Rho > 0

Puts: Positively related Rho < 0

Question 13

Theta

Answer 13

Input: Time to expiration (T)

Calls: Time value → $0 as call → maturity Theta < 0

Puts: Time value → $0 as put → maturity Theta < 0

Question 14

Describe the role of gamma risk in options trading.

Answer 14

When the price of the underlying stock abruptly jumps, a violation of BSM, the delta of the option would change (captured by the option gamma), leaving a previously delta hedged portfolio unhedged. This is the gamma risk of a delta hedged portfolio.

Question 15

Describe how the Black–Scholes–Merton model is used to value European options on equities and currencies.

Answer 15

European options on dividend-paying stock can be valued by adjusting the model to incorporate the yield on the stock: the current stock price is adjusted by subtracting the present value of dividends expected up until option expiration. Options on currencies incorporate a yield on the foreign currency based on the interest rate in that currency.

Question 16

Describe how a delta hedge is executed.

Answer 16

The goal of a delta-neutral portfolio (or delta-neutral hedge) is to combine a long position in a stock with a short position in call options (or a long position in put options) so that the portfolio value does not change when the stock value changes. Given that delta changes when stock price changes, a delta hedged portfolio needs to be continuously rebalanced. Gamma measures how much delta changes as the asset price changes and, thus, offers a measure of how poorly a fixed hedge will perform as the price of the underlying asset changes.

Question 17

Identify assumptions of the Black–Scholes–Merton option valuation model.

Answer 17

The assumptions underlying the BSM model are:

1. The price of the underlying asset changes smoothly (i.e., does not jump) and has a normally distributed continuously compounded return.
2. The (continuous) risk-free rate is constant and known.
3. The volatility of the underlying asset is constant and known.
4. Markets are “frictionless.”
5. The underlying asset generates no cash flows.
6. The options are European.

Question 18

Define implied volatility and explain how it is used in options trading

Answer 18

Implied volatility is the volatility that, when used in the Black-Scholes formula, produces the current market price of the option. If an option is overvalued, implied volatility is too high.

Question 19

Describe how the value of a European option can be analyzed as the present value of the option’s expected payoff at expiration.

Answer 19

Option values can be calculated as present value of expected payoffs on the option, discounted at the risk-free rate. The probabilities used to calculate the expected value are risk-neutral probabilities.

Question 20

Describe how the Black model is used to value European interest rate options and European swaptions.

Answer 20

The Black model can be used to value interest rate options by substituting the current forward rate in place of the stock price and the exercise rate in place of exercise price. The value is adjusted for accrual period (i.e., the period covered by the underlying rate).

A swaption is an option that gives the holder the right to enter into an interest rate swap. A payer (receiver) swaption is the right to enter into a swap as the fixed-rate payer (receiver). A payer (receiver) swaption gains value when interest rates increase (decrease).

Question 21

Interpret the components of the Black–Scholes–Merton model as applied to call options in terms of a leveraged position in the underlying.

Answer 21

Calls can be thought of as leveraged stock investment where N(d1) units of stock is purchased using (e^–rT x XN(d2)) of borrowed funds. A portfolio that replicates a put option can be constructed by combining a long position in N(–d2) bonds and a short position in N(–d1) stocks.

Question 22

Calculate the no-arbitrage values of European and American options using a two-period binomial model.

Answer 22

The value of a European call option using the binomial option valuation model is the present value of the expected value of the option in the 2 states.

The value of an American-style call option on a non-dividend paying stock is the same as the value of an equivalent European-style call option. American-style put options may be more valuable than equivalent European-style put options due to the ability to exercise early and earn interest on the intrinsic value.

Question 23

Describe and interpret the binomial option valuation model and its component terms.

Answer 23

To value an option using a two-period binomial model:

Calculate the stock values at the end of two periods (there are three possible outcomes because an up-down move gets you to the same place as a down-up move).
Calculate option payoffs at the end of two periods.
Calculate expected values at the end of two periods using the up- and down-move probabilities. Discount these back one period at the risk-free rate to find the option values at the end of the first period.
Calculate expected value at the end of period one using the up- and down-move probabilities. Discount this back one period to find the option value today.
To price an option on a bond using a binomial tree, (1) price the bond at each node using projected interest rates, (2) calculate the intrinsic value of the option at each node at maturity of the option, and (3) calculate the value of the option today.