Forwards and Futures (and a lil option payoff and trading strategies) Flashcards
What is the distinction between investment and consumption assets?
Investment assets: assets that do not provide utility beyond monetary payoffs, i.e. stocks, bonds, and derivatives.
Consumption assets: assets that provide utility beyond monetary payoffs, i.e. oil, meat, and corn.
What is a forward and how is its payoff specified?
The forward contract on an asset S, entered into at time t, is a commitment made by the buyer/seller to buy/sell the asset S at maturity T, for the predetermined delivery price K.
The payoff on the buyer side is S(T)-K and the payoff on the sell side is K-S(T).
It is custom to set up the contract such its initial value is zero. It is custom to use the notation K=F(t) for the delivery price that makes the initial value of the contract equal to zero.
Explain the replicating portfolio technique (1-step binomial model)
The replicating portfolio technique builds on one simple assumption: that financial markets are complete and there are no arbitrage opportunities. The replicating portfolio technique creates a synthetic portfolio that exactly copies the payoff of a given derivative at maturity and therefore, by no arbitrage, the portfolio and the derivative must have the same value.
Step 1: calculate the payoff of the derivative in the up- and down-states of the world. The payoff of a put option is K-S(T) and the payoff of a call option is S(T)-K - REMEMBER that if it is a forward the payoff in the down-state of the world isn’t 0 - it can very well be negative.
Step 2: calculate the position in the money market and the stock market that has to be taken to replicate the payoff of the derivative. This is done using below formulas.
Step 3: as the last thing the value of the derivative can be calculated using the formula V^h(0) = n(B) + S x n(S)
Explain the arbitrage strategy the can be conducted under the following circumstances
Answer is below
Explain how a forward is valued when the underlying pays a discrete dividend
Step 1: calculate the present value of all future dividend payments.
Step 2: subtract the PV of future dividend payments from the current stock price.
Step 3: the forward price can be found by finding the FV of the stock price (less future dividends): F(t) = S^ex x e^r(T-t).
Step 4: the value of the forward contract is initially always 0.
OBS: if you’re given like a “two-step” assignment where you have to calculate the price and value of the forward today and “after 3 months” where some dividends have been paid out in the meantime than the forward price (step 3) above will be the exercise of the forward going forwards.
If it’s now “3 months later” and one dividend payment has been made but there’s one more payment in 2 months you could be asked to calculate the price and value of the forward again. You are most likely going to be given a new stock price here “3 months later”.
Step 1: calculate the present value of all future dividend payments.
Step 2: the forward price is one again calculated using the same formula: F(t) = S^ex x e^r(T-t).
Step 3: to calculate the value of the forward you have to use the forward price (step 3 above) as this is the strike price. The forward value is calculated using below formula
Explain the risk neutral valuation technique
The risk neutral valuation techniques involves the following steps:
Step 1: calculate the payoff of the derivate in the up- and down-states of the world.
Step 2: calculate u and d - u = stock price in the up-state of the world divided by current stock price and d = stock price in the down-state of the world divided by the current stock price.
Step 3: calculate the risk neutral probabilities q(u) and (q)d.
Step 4: calculate the value of the derivative.
The valuation formula and formulas for calculating q(u) and q(d) are seen below:
Explain forwards on currencies
See below
Explain what minimum variance hedge ratio is and how it’s calculated
Minimum variance hedge ratio is relevant when talking about cross hedging. Consider, for example, an airline that is concerned about the future price of jet fuel. Because jet fuel futures are not actively traded, it might choose to use heating oil futures contracts to hedge its exposure. The hedge ratio is the ratio of the size of the position taken in futures contracts to the size of the exposure. When the asset underlying the futures contract is the same as the asset being hedged, it is natural to use a hedge ratio of 1.0. When cross hedging is used, setting the hedge ratio equal to 1.0 is not always optimal. The hedger should choose a value for the hedge ratio that minimizes the variance of the value of the hedged position.
The minimum variance hedge ratio depends on the relationship between changes in the spot price and changes in the futures price.
The formulas for calculating the minimum variance hedge ratio (H) are shown below
Show and understand the payoff diagrams of long and short calls and long and short puts - and also the profit functions
See below
Explain how a covered call strategy is set up
If you own a stock and believe that the price will not move much (up or down), you can earn some extra money by issuing a call.
The call is not “naked” (unprotected). The call is “covered” (insured) by the stock. This is because you already own the stock so you don’t have to go out and buy it in case it’s exercised against you – you already own the stock. The call is said to be covered because it’s already insured by the stock. Note that this is equivalent to selling a (naked) put.
We know the stock’s profit is very similar to having a forward contract on the stock because we could use the forward, which costs 0 to set up, to move it around the K. We have the profit diagram for a short position in the call, which is what we are issuing. We’ll be earning money if S_t
Explain how a bull spread trading strategy is set up
If you believe that a stock will increase in value you can buy a call option with K(1) on the stock. This call has a price and if you’d like to decrease the cost you can short a call with K(2) on the same stock as well. The strike price of the shorted call has to be higher than the strike price of the long call, K(2) > K(1), but the options have to have the same expiration date.
You will now gain the stock price increases beyond K(1), but the profit is capped at K(2) - K(1) so both the upside and downside are capped. We see that in the beginning we have reduced the cost of our long position in the call option equivalently to what we are earning on the short position. The profit is increasing after K_1 because here we will begin to exercise our long position but since the stock price is below K_2 our short position will not be exercised against us. The profit increases linearly until we hit K_2 where the short position is exercised against us.
Bull spreads can also be constructed by buying a European put with a low strike price and selling a European put with a high strike price, as illustrated in Figure 12.3. Unlike bull spreads created from calls, those created from puts involve a positive up-front cash flow to the investor (ignoring margin requirements) and a payoff that is either negative or zero.
Explain how a bear spread strategy is created
An investor who enters into a bear spread is hoping that the stock price will decline. Bear spreads can be created by buying a European put with one strike price K(1) and selling a European put with another strike price K(2). The strike price of the option purchased is greater than the strike price of the option sold, K(1) > K(2) - so opposite of a bull spread.
A bear spread created from puts involves an initial cash outflow because the price of the put sold is less than the price of the put purchased. In essence, the investor has bought a put with a certain strike price and chosen to give up some of the profit potential by selling a put with a lower strike price. In return for the profit given up, the investor gets the price of the option sold.
Like bull spreads, bear spreads limit both the upside profit potential and the downside risk. Bear spreads can be created using calls instead of puts. The investor buys a call with a high strike price and sells a call with a low strike price, as illustrated in Figure 12.5. Bear spreads created with calls involve an initial cash inflow (ignoring margin requirements).
Explain how a straddle trading strategy is created
If you believe that that the stock price will move a lot, but you do not know whether it will rise or fall, you can then create a straddle:
o Buy a call with strike K.
o Buy a put with strike K.
This has an initial cost of setting up which is the price of the call and the price of the put. Hence, you will lose money if the stock price does not change. However, you stand to profit if the change in the price is sufficiently large. You are buying volatility! We will be earning money when either the stock price increases or decreases.
A short straddle is where you sell volatility.
A long straddle is where you buy volatility.
Explain how a butterfly spread trading strategy is created
Butterfly spreads can be created using put options. The investor buys two European puts, one with a low strike K(1) price and one with a high strike price K(3), and sells two European puts with an intermediate strike price K(2). A butterfly spread leads to a profit if the stock price stays close to K(2), but gives rise to a small loss if there is a significant stock price move in either direction. It is therefore an appropriate strategy for an investor who feels that large stock price moves are unlikely. The strategy requires a small investment initially.
A butterfly spread can also be created using call options (to see the profit from this see figure 12.6 on p. 262)
Explain how a forward is valued when the underlying pays continuous dividends
See below
Explain how a forward on currencies is valued
See below
