Options3 - HOST

Question 1

Compare the price of an option on a stock if the stock price follows mean reversion versus if the stock price does not.

Answer 1

• Mean-reversion means that asset price tends to move back towards its long term average.
• This may reduce the volatility of the option and make the option cheaper
  
  • Mean reversion makes sense for interest rates
  • I am not sure if it make sense for the stock prices
• Although Fama reported that stock returns can be negatively correlated at long term horizon

Question 2

When can hedging an option position make you take on more risk?

Answer 2

• Hedging can increase the risk
  
  • if we are forced to buy “short-dated” options and hedge them
  • In this case, we are short the stock
  • If the stock rises upto the strike (not below/above), the option might expire worthless
    
    • I would also lose in the short stock position
  • By hedging, we end up worse than if we had not hedged

Question 3

How do you hedge a written put if I can neither short any stock nor use options?

Answer 3

• Use an asset whose returns are correlated with returns on the underlying stock
• If the shorting is not possible, then short some index futures (this would be an imperfect hedge)
• We need to know either Beta or Correlation of the stock relative to the index to apportion the hedge correctly.

Question 4

You order a pizza for six people. Diameter is 12 inches. What diameter would you need to feed 8 people?

Answer 4

• Area = pi*r^2 = 36 pi
• This is similar to the option pricing formula:
  
  • Six month ATM call is worth $12
  • What is 8 month ATM Call?
• Sqrt(8) / sqrt(6)
• 2.8/ 2.5 = 1.15
  
  • 1.15*12 = 13.8

Question 5

Land in Arizon, tiny piece of beach in Florida. The filed in Arizon is idle, no plans to develop it in a land. Tiny beach in florida - popular, you can charge a small entrance fee for beachgoers.

Both land has offer for $1 million, which piece of land has the lower forward value?

Answer 5

• Fair price for future delivery depends upon the spot price and the cost of carry
• Cost of carry includes the cost of money (interest rates), dividend income, storage costs and the convenience yield
• Entrance fee for the Florida land
  
  • Florida property has lower forward value

Question 6

We have 30 day of “representative” stock price data. How do you calculate historical vol to use in BS formula?

Answer 6

• Use Log returns
• Ln(P_t/P_t-1) [we have 29 returns]
• Make sure to use T - 1 = 28 in the variance estimator to get the unbiased sample estimator of historical vol

Question 7

Why do we use the riskless rate instead of the required return on the stock to derive the BS formula?

Answer 7

• Option must cost the same as the replicating portfolio - else there is money to be made
• The result is driven by no-arbitrage and hence it is independent of risk-preference

Question 8

According to BS: which is more valuable?

(1) European Call option that is 10% out-of-the money?
(2) European Put option that is 10% out of the money?

Answer 8

• Assume: Stock = $100
• Call Strike = $110, Put strike = $90
• Distribution of the future stock price is Lognormal
  
  • Thus it is skewed with its mean higher than the median (which in turn higher than the mode - the peak)
• Median of the distribution is [Se^{(r-.5*sig^2)}]
• Both Call and Put (at 10% out-of-the money) are roughly equidistant from the median
  
  • Half of the distance is above the median (and half below)
• Probablity that call finishes in the money:
  
  • .5 - P ( 100 < S < 110)
• Probablity that put finishes in the money:
  
  • .5 - P (90 < S < 100)
• However the distribution is downward sloping, the call is more likely to finish in the money than the put
• Call is more valuable

Question 9

Why are Theta and Gamma of opposite signs? Are they always the opposite signs?

Answer 9

• Gamma is bounded by 0 to infinity
• Theta can be positive or negative.
  
  • Only positive:
    
    • If the Put option is deep-in-the money, the life does not get better
    • For the deep in-the-money European call (if the dividend yield is high enough)

Question 10

Riskless rate = 0, Stock = $100, One year from now; stock will be either at $130 or $70 with proability .80 and .20 respectively? No dividends. What is the value of one-year European Call with strike $110?

Answer 10

• Trick: Discount rate is not zero (just the riskless rate is zero)
• Discount rate is some leveraged version of the discount rate on the stock and we do not have that information
• We must use the risk-neutral valuation:
  
  • Risk neutral proabability q:
  • S = e^-r*T [q * Su + (1-q)Sd]
  • 100 = [q * 130 + (1-q) *70]
  • Solve for q =.5
• call = e(-r*T) [q*max(0,130-110) + (1-q)*max(0,70-110)] = .5*(20) = $10

Question 11

Are Asian options cheaper or more expensive than plain Vanilla Options?

Answer 11

• An Asian option is an average rate option
• Underlying is time series of average of prices
• Changes in average prices are less volatile
• Lower vol = lower option value
• Asian options are less expensive

Question 12

When can plain vanilla American-style Put be treated as a European Put?

Answer 12

• If the risk-less rate is zero then there is no incentive for early exercise of an American-style Put

Question 13

How many nodes are there in a recombining binomial tree with N time steps? How many nodes are there in non-recombining binomial tree with N time steps?

Answer 13

• Answer for recombining Binomial tree: for t = 0, 1, 2,3,…N
  
  • N = 1, 2,3 ,4, ..N, N+1
  • Total = (N+1)(N+2)/2
• Answer for non-recombining Binomial Tree:
  
  • N = 2⁰, 2¹, 2², 2³,…2^N
  • Total = 2^N - 1

Question 14

Call option is price at c today. What is the expected price tomorrow? (qualitatively)

Answer 14

• Most stocks have positive Betas
• Since Call is a leveraged instrument - it has very large positive Beta and high Expected Return
• Ex: $50 Stock, Beta = 1.10, r = 0.05, Vol = .30
  
  • Beta of Call = 6.7
  • Beta of Call = [N(d1) * S] / c
    
    • This is the elasticity of call price with respect to Stock
• This is instantaneous Beta.
• So the option’s expected return is positive and tomorrow’s expected price of option is higher.