(R49) Basics of Derivative Pricing and Valuation

Question 1

The price of the underlying assumes:

Answer 1

Risk Aversion

Question 2

Principle of Arbitrage

Answer 2

A transaction used when two assets produce identical results but sell for different prices. Pressures for prices to converge. (This means everything is correctly priced and you should buy low, sell high). Returns a risk-free rate

Question 3

Law of one price

Answer 3

Two identical assets can only have one true market price. Equivalent to the principle that no arbitrage opportunities are possible.

Question 4

Short/long Asset + long/short derivative =

Answer 4

Risk-free bond (perfectly hedged portfolio)

Question 5

Asset - risk free bond =

Answer 5

Negative derivative (which is a short position)

Question 6

Negative Asset + risk free bond =

Answer 6

Positive derivative (which is a long position)

Question 7

What is replication in derivatives?

Answer 7

The creation of an asset/portfolio from another asset/portfolio/derivative. In the absence of arbitrage, replication would not produce excess return. Replication can reduce transaction costs

Question 8

What is risk neutrality in derivatives?

Answer 8

This means that investor’s risk aversion is not a factor in determining the price of a derivative but it is important in determining price of assets. Prices of derivatives assume risk neutrality

Question 9

Distinguish between value and price of futures/forwared contracts?

Answer 9

• Forward/future price represents represents a fixed price at which the underlying asset will be purchased at a later date
• Value of forward/future represents the change in the contract price from inception to the valuation date
• At inception both value and price equal zero

Question 10

Calculate the value of a forward/futures contract at expiration?

Answer 10

V₀ = S_T - F₀(T)

S_T = spot price at future date

F₀(T) = forward price agreed upon today

Question 11

Calculate the price of a forward/futures contract at inception with and without costs and benefits?

Answer 11

Without costs and benefits: F₀(T) = S₀ (1+r)^T

With costs and benefits: F₀(T) = (S₀ - PV of benefits + PV of costs) (1+r)^T

S₀ = spot price today

Question 12

Calculate the value of a forward/futures contract during the life of the contract?

Answer 12

V_T(T) = S_T - ( Y - gamma)(1+r)^t - F₀(1+r)^-(T-t)

Y = PV of benefits

gamma = PV of costs

Question 13

What is an off-market forward?

Answer 13

A forward transaction that starts with a nonzero value (each rate is the same)

Question 14

Explain how swap contracts are similar to but different from a series of forward contracts

Answer 14

A swap involves the exchange of cash flows (exchange floating rate for fixed rate with another party). A swap contract is equivalent to a series of forward contracts, each created at the swap price.

For a swap contract, the rate is fixed at each period. For forward contacts, the rate/price is different at each period.

Value of swap is zero at inception

Question 15

Why do forward and futures prices differ?

Answer 15

• Futures are marked-to market daily, while forwards are not
• Differences in the cash flows can also lead to pricing differences.
• If futures prices are positively correlated with interest rates, futures contracts are more desirable if in long position

Question 16

Define a forward rate agreement and its uses

Answer 16

• A forward contract calling for one party to make a fixed interest payment and the other to make an interest payment at a rate to be determined at the contract expiration
• A contract where the underlying is an interest rate
• FRAs are based on libor and represent forward rates

Question 17

What is moneyness of an option?

Answer 17

• This is comparing the underlying at expiration (Spot price at future date) to the strike price.
• In the money: underlying at expiration > strike price
• Out of the money underlying at expiration < strike price
• At the money: underlying at expiration = strike price

Question 18

Identify factors that determine the value of an option and explain how each factor effects the value of an option

Answer 18

1. Value of the underlying: value of call is directly related to S_T (put inversely related)
2. Exercise price: call option values are inversely related to X (put directly related)
3. Risk-free rate: when rates fall, call option prices fall (Put prices rise)
4. Time: value of a call option is directly related to time (same for put)
5. Volatility: greater volatililty in the underlying increases both call and put prices

Question 19

Formula for put-call parity

Answer 19

S₀ + P₀ = C₀ + (x/(1+r)^T)

S = asset

P = put

C = call

Formula with X = bond

A positive value in this equation is in a long position, negative value is in short position. Both the put and call have the same payoff at expiration and also have the same cost

Question 20

Formula for put-call forward parity

Answer 20

F₀/(1+r)^T + P0 = C0 + (x/(1+r)^T)

Formula with F₀ = forward contract

P = put

C = call

Formula with X = bond

A positive value in this equation is in a long position, negative value is in short position. Both the put and call have the same payoff at expiration and also have the same cost

Question 21

How is the value of an option determined using a one-period binomial model?

Answer 21

A model for pricing options in which the underlying price can move to only one of two possible new prices.

The expected payoff/value based on risk-neutral probabilities is discounted at the risk-free rate