Module 69: Arbitrage, Replication, and Carrying Costs Flashcards
(26 cards)
What it Arbitrage?
Possibility to make a riskless profit
It arise when the laws of one price does not price
What are examples of arbitrage?
2 different assets with identifical future cash flows (Bond A & Bond B), same issuer, same maturity and same coupon.
Investor who spot this, take the overpriced Bond, Buy Brond A
What is replication?
Replication is mirrowing cash flows from a derivative using long and short posisitions in the underlying, and lending/borrowing cash
What is the cost of carry?
The Cost of Ownership - Benefits of owning the underlying asset
What is the net costs?
- Risk free rater): If R is higher, than the difference between the forward price and spot price widens
- Other costs of ownership (C,c): e.g. storage costs, transportation costs, insurance costs, so the owner of the asset must be compensated for these costs
Owner of te asset must be compensated for these costs - Benefits of ownership (I, i): Dividends, coupons
Owner enjoys these benefits
If R+C > i then Forward > Spot
If R+C = i Forward = spot
If R+C < i then Foward < spot
What compounding do we apply to derivative pricing
Discrete: If there’s an individual asset
Continuous: Used for a portfolio of assets and FX
How does Discrete Compounding work?
Forward Price = [Spot Price - PV(I) + PV (C)] * (1+r)^t
How does continuous compounding work?
Forward Price = Spot Price * E^(r+c-i)t
What is the convenience yield?
Any non-cash benefit of holding a physical commodity, it works in to reduce the difference between the future price and spot price.
Forward Rate = Spot Rate * e ^(rf-rd) * d
rf = interest rate in price curreecny
Rd = interest rate in base currency
d = months / years
Forward Rate = Spot Rate * e ^(rf-rd) * d
rf = interest rate in price curreecny
Rd = interest rate in base currency
d = months / years
What’s the one rule
The currency with a higher IR trades at a discount in the forward market, and then currency with a lower IR trades at a premium in the forward market
What do we use a non-arbitrage conditions to do?
Can be used to determine the current value of a derivative based on the known value of a portfolio of assets that have the same future payoff as the derivative, regardless of future events
Why will there be a small difference in price?
There are transaction costs of exploiting an arbitrage opportunity, so differences in price may persist when the arbitrage gain is less than the transaction costs of exploting it
What costs should be the same
1) Buy the asset today and hold it for x amount of months
2) Locking into a forward contract to buy the asset in x amount of months
What is replication?
Replicating a derivative by creating a portfolio that has future payoffs identical to the derivative
Give an example of replication
- Long forward contract to buy a share of Acme at 31.50, when its currently trading at 30
- Can borrow 30 at 5% (30)(1.05) = 31.50 to buy a share of Acme, and holding it for one year
- So (1+Rfr)^T = forward price
Initial cost = 0.
Forward: payoff = Stock - 31.50
Borrow and Buy: Stock - 31.50
31.50 is the 0 arbitrage 1-year forward price
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How do you workout No Arbitrage Forward Price
F0(T) = S0 (1+Rf)^t
F0(T) = Sell asset forward for F0(T) at time = 0
S0 (1+Rf)^t = Buy Asset Now, hold until time T
What do you do if if the Forward Price is more than the risk free arbitrage value
(Cash and Carry Arbitrage)
Market Observation: The forward price ($F$) is higher than the spot price ($S$) plus the cost of carrying the asset to the delivery date (interest on a loan to buy the asset).
Asset Purchase: Buy the underlying asset in the spot market at price $S$.
Financing: Borrow money at the risk-free rate to finance the purchase.
Short Forward: Simultaneously, sell a forward contract to deliver the asset at the future date at price $F$.
Settlement: At the delivery date, deliver the asset, receive $F$, repay the loan (principal + interest).
Profit: The profit is the difference between $F$ and the total cost ( $S$ + borrowing costs).
What do you do if if the Forward Price is lower than the risk free arbitrage value, what arbitrage is it called
(Reverse Cash and Carry)
Market Observation: The forward price ($F$) is lower than the spot price ($S$) minus the return from investing the proceeds of selling the asset.
Asset Sale: Sell the underlying asset in the spot market at price $S$.
Investment: Invest the proceeds at the risk-free rate.
Long Forward: Simultaneously, buy a forward contract to receive the asset at the future date at price $F$.
Settlement: At the delivery date, receive the asset, sell it to cover the initial short sale, and collect the investment returns.
Profit: The profit is the difference between the investment returns and the cost of buying the asset at price $F$.
How do you add benefits and costs to the relationship between forward price and spot prices
There are costs and benefits for buying the underlying asset and holding it
Costs = PV of storage/insurance costs (Monetary Costs)
Benefits = PV of cash flows (monetary beenfits) and convenience yield (non monetary benefits)
What is the non-arbitrage forward price w/costs and benefits
Forward Price = Spot Price - PV (benefits) + PV(Costs) x (1+Risk free rate/opportunity cost)^t
As benefits increase, then forward price decrease, the benefits are reducing the cost of buying the underlying asset in the spot market and holding it.
As costs increase, the forward price increases, the costs increase the price of buying the underlying asset
What othe calculation can you use to calculate forward price
Forward Price = Spot price (1+R)^t - FV (Benefit) + FV (Cost)
How do you work out PV and FV from continous compounding
FV = Spot Price * e^rt (compounding)
PV = Spot Price * e^-rt (discounting)
e.g. with R = 3% and continuous compounding
FV of spot in 2 years = S2 = Spot * e^0.03(2)
PV of Spot = Spot in 2yrs * e^-0.03(2)
How do you use continous compounding with using costs and benefits
Forward Price = Spot Price * e^(Risk free rate + costs - benefits)(n/12)
