Estimating Stock Volatility with Returns & Options Prices

[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:

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

 

[BWV]

The standard deviation of the log price change is the standard method

the expected return of a stock has no bearing on the value of an option

http://en.wikipedia.org/wiki/Black–Scholes#Black.E2.80.93Scholes_PDE

the only problem is that price changes, particularly over shorter time horizons, are not normally distributed

 

[John Creighto]

The standard deviation of the log price change is the standard method

the expected return of a stock has no bearing on the value of an option

http://en.wikipedia.org/wiki/Black–Scholes#Black.E2.80.93Scholes_PDE

the only problem is that price changes, particularly over shorter time horizons, are not normally distributed

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

* It is possible to borrow and lend cash at a known constant risk-free interest rate.
* The price follows a geometric Brownian motion with constant drift and volatility.
* There are no transaction costs.
* The stock does not pay a dividend (see below for extensions to handle dividend payments).
* All securities are perfectly divisible (i.e. it is possible to buy any fraction of a share).
* There are no restrictions on short selling.

On average most stocks do go up in value over time so this assumption isn't completely 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.

 

[John Creighto]

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

Symbol:
Normally, symbols for all equity Long-term Equity AnticiPation Securities (LEAPS) expiring in any calendar year are based on the underlying stock symbol, but are modified by either the letter L, V, W, or Z. For example, LEAPS on XYZ might use the symbol "LXY" and LEAPS on PQR might use "LQR"; similarly, LEAPS on XYZ expiring in the next year might use the symbol "WXY", while those expiring in the year following that might use "VYZ". Due to conflicts with pre-existing security symbols, it is not possible to consistently alter the underlying stock symbol in the same manner for every LEAP expiring in a certain year, nor for each LEAP expiration on a specific underlying stock.

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

 

[BWV]

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 completely 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.

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

 

[John Creighto]

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

Interesting. So how about this strategy.

-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%).
-Buy the underlying stock.

-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.

 

[BWV]

Interesting. So how about this strategy.

-Sell a call 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%).
-Buy the underlying stock.

-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.

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