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# P1.T3.732. Exchange option, volatility swap, and static option replication (Hull Ch. 26 continued)

#### Nicole Seaman

##### Director of FRM Operations
Staff member
Subscriber
Identify and describe the characteristics and pay-off structure of the following exotic options: exchange. Describe and contrast volatility and variance swaps. Explain the basic premise of static option replication and how it can be applied to hedging exotic options.

Questions:

732.1. Consider a one-year exchange option to give up 14.0 units of Ethereum (aka, ether or ETH) for one unit of Bitcoin (BTC). The current price of Bitcoin is $4,200.00 BTC and the current price of one unit of ether is$300.00 ETH. The risk-free interest rate is 2.0% per annum with continuous compounding. The per annum volatility of Bitcoin is 50.0% and the volatility of Ethereum is 38.0%. Their correlation, ρ(BTC, ETH) = 0.540. We can price an exchange option with a simple variation on the Black-Scholes-Merton called the Margrabe variation. Using the Margrabe under these assumptions, the price of this BTC-for-ETH exchange option is \$723.11. Further, each of the following statements, ceteris paribus, is true EXCEPT which is false?

a. A higher riskfree rate has no impact on this exchange option's value
b. A higher correlation ρ(BTC, ETH) implies a lower exchange option value
c. An increase in Ethereum's (ETH) spot price will increase the exchange option's value
d. An increase in the Bitcoin (BTC) spot price implies an increase in the exchange option's value

732.2. Assume that V^2(r) signifies some measure of realized asset price variance and V(r) signifies some measure of realized asset price volatility. Also, F(T, V^2) is a prespecified fixed variance and F(T, V) is a prespecified fixed variance rate; aka, forward price. If (N) is the notional amount, then the payoffs to a variance and a volatility swap are given by:
• Payoff to a variance swap: [V^2(r) - F(T, V^2)]*N
• Payoff to a volatility swap: [V(r) - F(T, V)]*N
Against that mathematical context, each of the following is true about variance or volatility swaps EXCEPT which is false?

a. The VIX is an exchange-traded fund (ETF) that tracks the the weighted-average price of a static portfolio of S&P 500 variance future contracts
b. It is easier to price a variance swap than a volatility swap because the variance swap can be replicated with a static hedge (portfolio of puts and calls)
c. Unlike a regular option or futures contract that settles based on a final spot price at maturity, the variance and volatility swap settle based on a series of prices
d. Because the volatility is the square root of the variance, Jensen’s inequality implies that the volatility forward price will be less than the square root of the variance forward price of the variance.

732.3. Patricia wants to hedge her portfolio of exotic options. The portfolio consists mostly of barrier options. She is comparing a classic delta-hedge to a static options replication; the static option replication entails shorting a portfolio that replicates certain boundary conditions. Each of the following is a good argument in favor of a static option replication, for the purpose of hedging her portfolio, EXCEPT which is WEAKEST argument?

a. The delta of barrier options is discontinuous at the barrier and consequently difficult to delta-hedge
b. The delta-hedged portfolio can still experience losses due to large moves in the underlying asset price
c. Static options replication has a key advantage over delta-hedging in that it does not require frequent rebalancing
d. Her underlying position is already options and static option replication is not designed to hedge options, as it would add risk to hedge options with options