MIDTERM EXAM

PART I:  

Q1. Suppose that Stock X is currently selling for $60, but it can go up to $65 or down to $50 in 2 months. There is a European stock call option with an exercise price of $54. The risk-free rate of interest today is 2%.

a. How would you replicate a long position in a call?

b. How would you formulate a riskless portfolio with stocks for each call sold short?

c. Compute the option delta.

d. Compute the no-arbitrage call price.

e. Compute the price for the call, but now by using the risk neutral valuation method.

Q2. Consider a 6-month European put with a strike price of $42 on a stock whose current price is $40. Assume that there are two timesteps, and in each timestep the stock price either moves by 10% or moves down by 10%.  We also suppose that the risk-free rate of interest is 2%.  Compute the value of the put using the recombining binomial tree.

Q3. Repeat the same but now by matching volatility equal to 30% a year.

Q4. Repeat the same as in (2) but assume that the put option is American.

Q5. Discuss the delta hedging strategy in the question (2) above.

PART II - (Modeling the Stock Price Behavior)

Q1. Suppose that the daily stock price "drifts" at the rate of μ over time in a year.  How would you write the change in the stock price over a short time interval Δt?

Q2. Suppose that the amount of return uncertainty for a dollar of investment in a year is constant and is measured by the return standard deviation σ. How would you write the change in the stock price movement over a short time interval Δt?

Q3. Taking the both of the above into account, write down the equation for the stock price changes.

Q4. However, the stock price evolves through time, even though we are able to describe the stock price changes as above. If the stock price undergoes a stochastic Wiener process, how would you modify the equation in (3)?

Q5. Given the equation in (4), can you forecast the stock price in 10 days?  Assume that the stock is currently selling for $48.27 and pays no dividend. The stock's volatility is 60% per annum and a continuously compounded return of 21% a year. REMARKS: Pick any arbitrary value of epsilons with ε ∼ N(0, 1). The best way is to use the Excel macro function.

Q6. We know that a little bit of mathematics of Ito's lemma results that the log price is normally distributed, in which case we say that the stock price lognormally distributed.  Suppose that an initial price is $40.  If the annual expected return on this stock is 20% and a volatility is 20%, how would the stock price behave, given a 95% probability?

ln S_T ∼ ∅[3.759, 0.02]

PART III - (Black-Scholes-Merton Model)

Q1. Suppose that at a particular point in time, the relationship between the call option price and the stock price is given by: Δc = 0.4 ΔS

Let's assume that you happen to be short 100 shares of the stock now and the price is expected to rise. Would you be interested in buying calls or selling calls to hedge? If so, how many calls? Suppose instead that you happen to own 100 shares of the stock and the price is expected to rise.

What would you do with respect to calls if you are interested in forming a riskless portfolio?

Q2. The return on a riskless portfolio of going short one derivative and long "delta" shares of stocks is the risk-free rate of interest.  This results in the stochastic differential equation (SDE) in the Black-Scholes-Merton Formula:

c = S₀N(d₁) - Ke^-rTN(d₂)

p = Ke^-rTN(-d₂) - S₀N(-d₁)

d1 = (ln(S₀/K)+(r+(σ²/2))T/σ√T); d₂ = d₁ - σ√T

Compute the call and put premium under the following conditions.

The current stock price: $42.

The strike price: $40

The risk-free rate of interest: 10%

Annual volatility: 20%

The options maturity: 6 months

PART IV - (Value at Risk, VaR)

Q1. Suppose that we are interested in computing the 10-day VaR of our $10 million investment in Microsoft with a 99% confidence about the maximum loss.  Assume that the daily volatility is 2%.

Q2. Consider a portfolio which consists of a $10 million investment in Microsoft and $5 million investment in AT&T.  These stocks are correlated in returns at 0.3.  AT&T stock has a daily volatility of 1%.  Compute the 10-day 99% VaR of this portfolio.