Question 1 [Black Scholes and Binomial]

Suppose Bitcoin (BTC) is presently trading at $22,000/coin with an historical average volatility of 20% per year. You believe that if Coinbase collapses that this will be very bad news for BTC and that it will immediately lose 20% of its value. Alternatively, if Coinbase survives that will be a bullish signal for BTC and you expect it to rise 15% immediately.

A crypto expert, if such exists, estimates the likelihood of Coinbase surviving at 75%

You have just been offered an opportunity to buy an American call option on BTC with a strike of $23,000 and with 3 months to expiration. You feel that the fate of Coinbase will be known by the end of those 3 months.

You also know the risk-free rate to be 3.75% EAR.

A. [4 pts] Find the delta number of call options that will hedge a single bitcoin and find the price of those options using the binomial model. Compute the risk-neutral probability of Coinbases collapse and comment/contrast this probability to the one offered by the crypto expert in terms of risk preferences.

B. [3 pts] Compute the value of the BTC call options using the Black-Scholes model and compare the result to the Binomial model you just used in A. State which value is higher and argue why each of the models is more realistic than the other.

[A Normal CDF is on the final page of this exam booklet]

C. [1.5 pts] In the real world, stock prices probably exhibit some momentum, occasionally change dramatically and the level of volatility is also probably changing over time as well. Explain how these would affect the application of the original Black-Scholes model.

D. [1.5 pts] Define the option smile, and state how it is computed. Offer at least one possible and also plausible explanation for the smile.