Forwards and Futures: Forward prices and the no-arbitrage channel

Question 1

How can we replicate the forward purchase of a non-dividend paying stock (cash and carry)?

Answer 1

At time 0:
1. Buy a share at S₀
2. Borrow S₀ at the risk-free rate r for maturity T
This replicates the purchase of a forward, arbitrage can be done with step 3
3. Sell a forward contract on the share at F₀
At time 0 net cashflow is equal S₀ - S₀ = 0
At maturity the payoff is equal to -S₀ * e^rt + F₀
No arbitrage condition implies F₀ = S₀ * e^rt

Question 2

What is the equilibrium forward price with no dividends?

Answer 2

F^*=S₀ * e^rt

Question 3

What is the reverse cash and carry?

Answer 3

A reverse cash and carry strategy is just the opposite of a cash and carry strategy, and implies:
* selling the asset spot;
* investing the proceeds at the risk-free rate;
* buying the stock forward

Question 4

How does the need to borrow the stock for a reverse cash and carry affect the arbitrage profit?

Answer 4

To borrow a stock costs fees to the lender from the borrower, as such the profit you can make by borrowing the stock decreases, contracting the arbitrage opportunity, to the point that no arbitrage may be possible with borrowing.

Question 5

What are some of the transaction costs to consider for cash and carry/reverse cash carry, and what effect do they have?

Answer 5

1. A bid-ask spread on the spot transaction
2. A bid-ask spread on the forward transaction
3. A bid-ask spread on borrowing/investment
   These costs result in the creationg of a no-arbitrage channel. As long as the market forward price does not differ enough from the fair price F^* (does not fall outside of the channel), arbitrage is no feasible.

Question 6

What situations are cash & carry and reverse cash & carry used for?

Answer 6

1. If F_actual > F^* the forward is overvalued and a cash and carry arbitrage can be made
2. If F_actual < F^* the forward is undervalued and a reverse cash and carry arbitrage can be made

Question 7

Is there a way to benefit from mispricing of a forward even if there is no arbitrage possible?

Answer 7

Yes! In case one simply wants t obuy a forward, replicating it instead could be a bit cheaper.

Question 8

What is the formula for the implied repo rate?

Answer 8

F_mkt = S * e ^{r imp * t}

Hence, if markte forward price is equal to the equilibrium forward price, the implied repor ate equals the risk-free rate

Question 9

What is a repo?

Answer 9

Repo is a repurchase agreement. A deal in which a counterparty either:
1. Buys a bond/stock spot and sells it forward at a specified price (investment repo) or
2. Sells spot and buys fowrad (financing repo)

Question 10

What happens to the optimal forward price if the underlying for a forward pays a dividend?

Answer 10

In the case of one dividend the cost of carry is reduced and the fair forward price is:
F₀ = (S₀ - I) * e^rt
where I = present value of the dividend

Question 11

What is the value of a forward contract of the buyer and the seller?

Answer 11

Buyer
f = e^-rt(F-K)
Seller
f = e^-rt(K-F)

Where t is time from today (valuation date) to the forward deal’s maturity, r the risk-free interest rate on maturity t, K the originally contracted forward price, F the new forward price.