# Volatility in Stocks

1. Jul 28, 2009

### John Creighto

Options prices are often based upon the volatility of a stock. I'm left to wonder how we might estimate volatility. Let:

$$r$$ is the yearly expected rate of return
$$\sigma_r$$ is the uncertainty in the yearly expected rate of return.
$$\sigma_y$$ the daily volatility.

Then we might model the current price as follows:

$$y(n)=\sum_{i=1}^{\infty} w_i (r+\sigma_{r,i})^{(i-n)/365}(y(n-i)+\sigma_{y,i})$$

where:

$$w_i$$ is how much weight we use each past value to determine the future value

and

$$1=\sum_{i=1}^{\infty} w_i$$

Once the $$w_i$$'s are chosen then $$r$$, $$\sigma_r$$ and $$\sigma_y$$ are chosen so that they minimize:

$$E \left[\left( y(n)-\hat{y}(n) \right)^2\right]$$

where:

$$\hat{y}(n)=\sum_{i=1}^{\infty} w_i E\left[(r+\sigma_{r,i})^{(i-n)/365}\right]y(n-i)$$

Last edited: Jul 28, 2009
2. Aug 1, 2009

3. Aug 3, 2009

### John Creighto

The expected rate of return has no bearing on the value of an option under Black-Scholes because of assumption 2.

On average most stocks do go up in value over time so this assumption isn't completly accurate. However, since the expiry time for an option is about one month this assumption may be okay, because a stock that is expected to increase in value by 8% of the year, will only increase by 0.64% on average in one month.

I just now decided to google long term options. I found a product called:
http://www.investopedia.com/terms/l/leaps.asp

They have a higher premium and in this case the expected rate of return would have a bearing on there value.

4. Aug 3, 2009

### John Creighto

On the topic of Leaps, I belive you can find leaps on exchanges by deriving the symbol as follows:

http://www.cboe.com/Products/EquityLEAPS.aspx

5. Aug 3, 2009

### BWV

No, the expected return has no bearing on option prices because of arbitrage opportunities. If the expectation of return within the option price varies from the risk-free rate there is an arbitrage using some combination of cash, a going short or long the underlying and going short or long the option. The basic formula is put-call parity - Stock + Put = Call + Cash

if you price options based upon an 8% discount rate based on the expected return on the underlying instead of the risk free rate then the call will have a higher price than in BS and the put will have a lower. Per put call parity you could buy the stock, put on a costless collar (short call and long put) and achieve the 8% return risk-free - this of course cannot exist in a competitive financial market therefore the options have to be priced to the risk free rate

The geometric brownian motion with a constant drift could be a positie rate of return - the 8% in your example is the drift

6. Aug 3, 2009

### John Creighto

-Sell a call (European Style) which will give the buyer the risk free rate of return
-Buy a put that represents how much loss you are willing to take. (Strike price equal to stock value or some percentage less (say 20%).

-If the stock tanks your losses are limited to the stock price you paid for the stock minus the strike price of the put.
-If the stock price performance is marginal, your losses are limited to the risk free rate of return.
-If the stock does awesome the upside potential is infinite.

Last edited: Aug 3, 2009
7. Aug 3, 2009

### BWV

That strategy is called a collar

you have capped your upside by selling the call - say the stock price is $10 and you buy a put and sell a call both with a$10 strike price. Then the position should give you the risk free rate (excluding any transaction costs). By selling the call, you have to pay the difference between the stock price and the strike price if S>X so without being long the underlying your potential loss in selling a call is infinite

phrased another way, at the same strike price:
Long put + short call = synthetic short position
Long call + short put = synthetic long position