Options3 - HOST Flashcards
Compare the price of an option on a stock if the stock price follows mean reversion versus if the stock price does not.
- 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
When can hedging an option position make you take on more risk?
- 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
How do you hedge a written put if I can neither short any stock nor use options?
- 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.
You order a pizza for six people. Diameter is 12 inches. What diameter would you need to feed 8 people?
- 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
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?
- 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
We have 30 day of “representative” stock price data. How do you calculate historical vol to use in BS formula?
- Use Log returns
- Ln(Pt/Pt-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
Why do we use the riskless rate instead of the required return on the stock to derive the BS formula?
- 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
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?
- 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
Why are Theta and Gamma of opposite signs? Are they always the opposite signs?
- 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)
- Only positive:
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?
- 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
Are Asian options cheaper or more expensive than plain Vanilla Options?
- 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
When can plain vanilla American-style Put be treated as a European Put?
- If the risk-less rate is zero then there is no incentive for early exercise of an American-style Put
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 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,…2N
- Total = 2N - 1
Call option is price at c today. What is the expected price tomorrow? (qualitatively)
- 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.
