Lecture 46
Implied Volatility
CMT LEVEL - I

What is Implied Volatility – IV?
• Implied volatility is a metric that captures the
market's view of the likelihood of changes in a
given security's price.
•Investors can use it to project future moves and
supply and demand, and often employ it to price
options contracts.
•Implied volatility is not the same as historical
volatility, also known as realized volatility or
statistical volatility.

What is Implied Volatility – IV?
•Implied volatility is the market's forecast of a likely
movement in a security's price.
•It is a metric used by investors to estimate future
fluctuations (volatility) of a security's price based on
certain predictive factors.
•Implied volatility, denoted by the symbol σ (sigma),
can often be thought to be a proxy of market risk.
•It is commonly expressed using percentages and
standard deviations over a specified time horizon.

What is Implied Volatility – IV?
•When applied to the stock market, implied volatility
generally increases in bearish markets, when investors
believe equity prices will decline over time.
•IV decreases when the market is bullish, and investors
believe that prices will rise over time.
•Bearish markets are considered to be undesirable,
hence riskier, to the majority of equity investors.
•Implied volatility does not predict the direction in
which the price change will proceed.

Summarize
•Implied volatility is the market's forecast of a likely
movement in a security's price.
•Implied volatility is often used to price options
contracts: High implied volatility results in options with
higher premiums and vice versa.
•Supply/demand and time value are major determining
factors for calculating implied volatility.
•Implied volatility increases in bearish markets and
decreases when the market is bullish.

Historical volatility (HV)
•Historical volatility (HV) is the volatility experienced by
the underlying stock, stated in terms of annualized
standard deviation as a percentage of the stock price.
•Historical volatility is helpful in comparing the volatility
of one stock with that of another stock or to the stock
itself over a period of time.

Historical volatility (HV)
•For example, a stock that has a 15 historical volatility is
less volatile than a stock with a 25 historical volatility.
•Additionally, a stock with a historical volatility of 35 is
now more volatile than it was when its historical
volatility was, say, 20.

Historical V/s Implied Volatility
• Implied or projected volatility is a forward-looking metric
used by options traders to calculate probability.
• Implied volatility, as its name suggests, uses supply and
demand, and represents the expected fluctuations of an
underlying stock or index over a specific time frame.
• With historical volatility, traders use past trading ranges of
underlying securities and indexes to calculate price changes.
•Calculations for historical volatility are generally based on
the change from one closing price to the next.

Put Call Parity
• Put-call parity is a principle that defines the relationship
between the price of European put options and European
call options of the same class, that is, with the same
underlying asset, strike price, and expiration date.
•If the prices of the put and call options diverge so that this
relationship does not hold, an arbitrage opportunity exists,
meaning that sophisticated traders can theoretically earn a
risk-free profit.
• Such opportunities are uncommon and short-lived in liquid
markets.

Put Call Parity Theorum
• The equation expressing put-call parity is:
•C + PV(x) = P + S
• where:
• C = price of the European call option
• PV(x) = the present value of the strike price (x),
discounted from the value on the expiration date at the risk-
free rate
• P = price of the European put
• S = spot price or the current market value of the underlying
asset

Conclusion
• Put/call parity shows the relationship that has to exist
between European put and call options that have the
same underlying asset, expiration, and strike prices.
•Put/call parity says the price of a call option implies a
certain fair price for the corresponding put option with
the same strike price and expiration (and vice versa).
•When the prices of put and call options diverge, an
opportunity for arbitrage exists, enabling some traders
to earn a risk-free profit

Put-Call Parity And Arbitrage
• Let's say that the risk-free rate is 4% and that TCKR stock is currently trading at $10.
Let's continue to ignore transaction fees and assume that TCKR does not pay a
dividend. For TCKR options expiring in one year with a strike price of $15 we have:
• C + (15 ÷ 1.04) = P + 10
• 4.42 = P - C
• In this hypothetical market, TCKR puts should be trading at a $4.42 premium to their
corresponding calls. This makes intuitive sense: with TCKR trading at just 67% of the
strike price, the bullish call seems to have the longer odds. Let's say this is not the
case, though, for whatever reason, the puts are trading at $12, the calls at $7.
• 7 + 14.42 < 12 + 10
• 21.42 fiduciary call < 22 protected put

Put-Call Parity And Arbitrage
• When one side of the put-call parity equation is greater than the
other, this represents an arbitrage opportunity.
• You can "sell" the more expensive side of the equation and buy the
cheaper side to make, for all intents and purposes, a risk-free profit.
• In practice, this means selling a put, shorting the stock, buying a call
and buying the risk-free asset (TIPS, for example).
• In reality, opportunities for arbitrage are short-lived and difficult to
find.
• In addition, the margins they offer may be so thin that an enormous
amount of capital is required to take advantage of them.

Option Pricing
• Option pricing models calculate a variety of pieces of useful
information, such as the impact of changes in pricing factors. These
outputs are known as the option Greeks.
• Option Pricing Calculator—Option Value Output

Calculating Single-day Implied Volatility
• Assuming there are 252 trading days in a year, the denominator of this
formula turns out to be the square root of the number of trading days for the
year.
• 1 Day Movement = Implied Volatility / Square Root of 252
• Volatility is the only estimated factor in this model. The volatility that is used is
forward volatility. Forward volatility is the measure of implied volatility over a
period in the future.
• Implied volatility shows the "implied" movement in a stock's future volatility.
It tells you how traders think the stock will move. Implied volatility is always
expressed as a percentage, non-directional and on an annual basis.
• The higher the implied volatility, the more people think the stock's price will
move.