Liam Ellison

OV&P notes

The following notes are based on my readings from Option Volatility & Pricing by Sheldon Natenberg.

Chapter 1 – Financial Contracts

option contract = gives the buyer the right (but not the obligation) to buy or sell an underlying asset at a predetermined strike price, on or before a certain date (maturity/expiration date). premium = price of the contract strike price = price at which the underlying asset can be bought/sold call option = buyer gets the right to purchase the underlying asset at a later date put option = buyer gets the right to sell the underlying asset at a later date

Typically, in order to sell something, you must first own it. With options, you can agree to sell something before you've bought it. Example:

  1. You first buy a put option for 10 shares of AAPL:
    • Current market value of AAPL: $100
    • Premium: $10 per share or $100
    • Strike Price: $90 per share or $900.
  2. On or before the expiration date, AAPL's market value drops to $80.
  3. You decide to purchase 10 shares of AAPL at $80 for a total of $800.
  4. You then turn around and exercise your put option at the strike price of $900, effectively selling the 10 shares you just purchased for $800.
    • PnL: +$100

The first trade to take place (buying or selling) is an opening trade, resulting in an open position. A subsequent trade that reverses the opening trade is called a closing trade.

open interest = the # of contracts traded on an exchange that have not yet been closed out.

If a trader's opening trade is to buy a contract, they are now long the contract. If a trader's opening trade is to sell a contract, they are now short the contract.

Selling first results in an open short position, because you are now short the amount of shares you've agreed to buy (assuming no prior holdings). In order to close a short position, you will have to buy the shares later before you have to fulfill the contract (assuming the buyer exercises the option to buy the shares from you at the strike price).

Buying first results in an open long position, because you are now long the amount of shares that you will eventually need to sell in order to cash out (hopefully with a profit).

Notional or Nominal Value of a forward contract notional value = units x price per unit

If you buy 10 shares at $50, and the market price rises to $60, you have an unrealized profit of $100. Until you sell your 10 shares, the profit remains unrealized since you have not converted the profit to cash (because of this, the profit can sometimes be referred to as paper profit, since on you've profited on paper but have yet to actually claim any cash profits).

If you then sell the 10 shares for $60/share, you now have a realized profit of $100.

When two traders enter a contract on an exchange, the exchange collects a margin deposit from both the buy and seller incase either party defaults on their end of the agreement.

clearinghouse = an exchange division or independent entity made up of member clearing firms responsible for processing and guaranteeing all trades made on the exchange. Assumes the ultimate responsibility for ensuring the integrity for all exchange-traded contracts. clearing firm = a member of the clearinghouse that processes trades made by individual traders and agrees to fulfill any financial obligation(s) arising from those trades.

If an individual trader defaults, the trader's clearing firm guarantees fulfillment of that trader's responsibilities. No individual trader may trade on an exchange without first becoming associated with a clearing firm.

The clearing firm is also in charge of collecting any margin deposits from its traders and depositing these funds with the clearinghouse. Dependent on market conditions or other factors, the clearing firm may require the trader to deposit more than is required of the firm by the clearinghouse for a given transaction. This is typically negotiated between the trader and their clearing firm.

Chapter 2 – Forward Pricing

a fair forward price f for a contract can be calculated by: fp = current cash price + cost of buying underlying now - benefit of buying underlying now

examples:

  • stock options: fp=c∗(r∗t)−d\displaystyle fp = c * (r * t) - d c = current stock price, r = 1 + annual cash interest rate, d = dividend payouts (aggregate over contract period), t = length of contract (time in years)

    The benefit of buying a stock now are the dividend payouts to be received before maturity, and the cost of buying a stock now is losing out on the interest rate you would've received from holding your cash over the contract period.

  • foreign currency (FX): fp=cr∗(t∗d)(t∗f) \displaystyle fp = cr * \frac{(t * d)}{(t * f)} cr = conversion/spot exchange rate, d = domestic currency interest rate, f = foreign currency interest rate, t = length of contract (time in years)

arbitrage = buying and selling of the same or very closely related instruments across different markets to profit from an apparent mispricing

dividends

declared date = date that the company will announce the amount of the upcoming dividend payment and which date it will be paid record date = date at which the stock must be owned in order for the holder to receive the dividend payment ex-dividend date/ex-date = the first day that a stock is traded without rights to a dividend. typically in the US, this is two days prior to the record date. payable date = date for when the dividend payouts are paid out to qualifying shareholders (those holding before the record date).

"sell stock short" = to sell a stock that one does not own short-stock rebate = the interest rate received on the short sale of stock

Chapter 3 – Contract Specifications & Options Terminology

underlying = underlying asset for a contract serial options = options w/ expiration dates that don't correspond w the underlying futures month. In this case the nearest futures contract with an exp date after the exp date of the option will be the underlying futures contract. flex options = buyer & seller get to negotiate the contract specifications (exp date, underlying quantity, strike price & settlement style) round lot = 100 shares odd lot = more or less than 100 shares midcurve options = short-term options on long-term futures

exercise styles

-European-style = option must be exercised on the last business day prior to expiration -American-style = option can be exercised at any time until the contract expires. options on futures or individual stocks tend to be American, while options on indexes tend to be European

option pricing

option pricing is determined by supply and demand any premium/price paid for an option can be broken down into two parts: an option's intrinsic value and an option's time value*. intrinsic value = the difference between the current market price and the strike price time value/premium or extrinsic value = the added premium a trader is willing to pay for the protection afforded from buying an option rather than buying or selling the underlying outright.

in-the-money call = strike price lower than current market price in-the-money put = strike price higher than current market price

automatic exercise policy = any in-the-money option is automatically exercised by the exchange on expiration date

Chapter 4 – Expiration P&L

buyers of options have limited risk (they can only lose the price of the option) and unlimited profit potential. sellers of options have limited profit potential (they can only profit from the premium paid for the option) and unlimited risk.

parity graph = graph representing the intrinsic value of a position as a function of the price of the underlying contract. it's slope can be represented by: change in position value / change in underlying price

A long call & put on the same underlying, both with the same exercise price, results in an increase in position value as the underlying price strays from the exercise price, regardless of which direction the underlying price falls.

expiration P&L graph = the parity graph for a given position, shifted downward by the amount of any debit or upward by the amount of any credit.

Chapter 5 – Theoretical Pricing Models

A trader in the underlying market who goes long on a position, expecting the market to move up, is guaranteed a profit, whenever the market moves upward. An options trader however, is not guaranteed a profit, but is only able to profit if the underlying moves enough to offset/breakeven with the contract's premium before the expiration date.

Let P = the probability of an underlying value U, and S = the strike price. The expected value of a call option can be represented as: ∑i=1nPi∗max(Ui−S,0)\sum_{i=1}^{n} P_i * max(U_i-S, 0) Let P = the probability of an underlying value U, and S = the strike price. The expected value of a put option can be represented as: ∑i=1nPi∗max(S−Ui,0)\sum_{i=1}^{n} P_i * max(S-U_i, 0) The theoretical value of an option is just the same as the expected value, except it incorporates value loss from interest, or the concept of change in the expected value from now until some later date. In the context of options, this just means that while the expected value of a 6-month call option may be equal to around $95, the theoretical value incorporates the loss incurred from interest over the 6 month period where you are short the cash and lose out on interest. Typically, this is an adjustment that will move the theoretical value of an option slightly lower than the expected value.

Now if we want to apply these concepts to a model, we need to:

  1. Propose a series of possible prices at expiration for the underlying contract.
  2. Assign a probability to each possible price (with the assumption of an underlying market that is arbitrage-free, i.e. fair forward pricing where the forward price = expected value of the contract)
  3. Using the series of possible expiration/strike prices in step 1 and the probabilities for each from step 2, calculate the expected value of the option.
  4. Dependent on the settlement procedure, calculate the theoretical/present value given the expected value.

Black-Scholes Model Developed by Fischer Black of UChicago and Myron Scholes of MIT in 1973 in concurrence with the opening of the Chicago Board Options Exchange (CBOE), the Black-Scholes model relies on 5 characteristics of an option and its underlying contract to determine the theoretical value of an option:

  1. The exercise/strike price of the option
  2. The time left until expiration
  3. The current price of the underlying
  4. The applicable interest rate over the remaining contract period
  5. The volatility of the underlying

hedge ratio = the correct proportion of underlying contracts needed to establish a risk-less hedge.

| Option Position | Corresponding Market Position | Appropriate Hedge | | ---------------------------- | ----------------------------- | ----------------- | | Buy call (option to buy) | Long | Sell underlying | | Buy put (option to sell) | Short | Buy underlying | | Sell call (agreeing to sell) | Short | Buy underlying | | Sell put (agreeing to buy) | Long | Sell underlying |

Chapter 6 – Volatility

Volatility = the speed at which a market or an instrument changes in value, typically represented in the form of standard deviation (σ\sigma).

Options on higher volatility markets are more valuable than ones on low-vol markets b/c they have a higher probability of moving into the money on a given option contract.

Standard deviation approximations: ±\pm 0.5 stddev (σ\sigma) ≈\approx 38.3% (about 3/8) of all occurrences ±\pm 1 stddev (σ\sigma) ≈\approx 68.27% (about 2/3) of all occurrences ±\pm 2 stddev (σ\sigma) ≈\approx 95.45% (about 19/20) of all occurrences ±\pm 3 stddev (σ\sigma) ≈\approx 99.73% (about 369/370) of all occurrences

Volatility is always expressed as an annualized number, same way as interest rates. However, volatility is different from interest rates in that it is proportional to the square root of time, and not 1:1 proportional with time directly. So, in order to determine the volatility for a given time period, you must use the following calculation: Volatilityt=volatilityannual∗t Volatility_t = volatility_{annual} * \sqrt{t} Two things to note:

  1. The input value for volatility should be in decimal format (i.e. 30% = 0.3)
  2. The amount of trading days (or days that a market/asset value can change) in a given year is usually not all 365 days in the year depending on the instrument and exchange. For this reason, most exchanges fall between 250-260 trading days in a given year, and to make our vol calculations easy, we typically choose 256 since it's a perfect square.

Taking this assumption to determine daily volatility, we get the following calculation: Voldaily=Volannual∗1/256=Volannual/16Vol_{daily} = Vol_{annual} * \sqrt{1/256} = Vol_{annual}/16 Given that there typically aren't any full weeks without trading, we can also calculate the weekly volatility given 52 weeks in a year: Volweekly=Volannual∗1/52=Volannual/7.2Vol_{weekly} = Vol_{annual} * \sqrt{1/52} = Vol_{annual}/7.2

For contracts on instruments that produce some sort of yield (e.g. bonds), there are two volatility metrics that we can model:

  • yield volatility = the volatility calculated from the current yield produced by the instrument
  • price volatility = the volatility of the current market price for the instrument.

The Black-Scholes model is a continuous-time model, in that it calculates compounded volatility over time. If the percent-price change distribution is normal, then the continuous compounding of these price changes will result in a lognormal distribution.

Thus, the Black-Scholes model assumes a lognormal distribution for the distribution of expiration price probabilities.

Continuous rates of return can be calculated using the exponential function denoted by either expr(x)expr(x) or exe^x. Given an example interest rate of 12% or 0.12 and a cash value of $1000, the expected rate of returns for gains and losses are: 1000×e0.12=1127.501000 \times e^{0.12} = 1127.50 and 1000×e−0.12=886.921000 \times e^{-0.12} = 886.92

realized volatility = volatility associated w/ the underlying contract implied volatility = volatility associated w/ options

In a sense, the implied volatility is the marketplace's consensus of what the future realized volatility will be on the underlying contract over the life of the associated option.

value of an option = determined by the future realized volatility of the underlying (theoretical, as future realized volatility is never certain, simply just a prediction or forecasting of the future market value of the underlying) price (premium) of an option = reflects a market consensus on the implied volatility of an option

Most market activity around options tend to be trades made on at-the-money or out-the-money options, since they rely most on volatility for their theoretical value.

Chapter 7 – Risk Management I

risk characteristics of options:

| event | call values | put values | | ---------------------- | ----------- | ---------- | | underlying price rises | rise | fall | | underlying price falls | fall | rise | | volatility rise | rise | rise | | volatility falls | fall | fall | | time passes | fall* | fall* | *In some unusual cases, there may be possibility that an option's value rises as time passes, even w/ all other market conditions unchanged. These situations will be discussed in a later chapter.

Greeks

delta (Δ\Delta) = the rate of change of an option's value with respect to movement in the underlying contract. the hedge ratio for an option contract can be calculated by dividing the delta of the underlying contract by the delta of the option contract. the delta of an underlying is always 1 or 100, so the calculation for the hedge ratio of an option contract with a delta of 50 would be 100/50=2100/50 = 2, meaning for every option contract we buy, we need to hedge (in the opposite direction) with 2 underlying contracts to establish a neutral/riskless hedge.

gamma (Γ\Gamma) = the rate of change of an option's delta.

  • given a graph with an x-axis of the underlying contract value and a y-axis of the option's intrinsic value, the gamma is the curve's derivative.
  • given a graph with an x-axis of the underlying contract value and a y-axis of the option's delta, the gamma is the graph's slope or rate of change.

theta (θ\theta) = the time decay, or the rate at which an option's theoretical value decreases as time passes.

  • this metric is a representation of how, as time passes and an option subsequently moves towards expiration, the theoretical value of the option (intrinsic + time value) converges on the value of the option at expiration (intrinsic value).

vega / kappa (KK) = the sensitivity of an option's theoretical value to a change in volatility.

rho (P) = the sensitivity of an option's theoretical value to a change in interest rates