Derivatives Test 2

Option Contract

Confers the right, but not the obligation, to buy/sell a specific asset and quantity on (or before) a specific future expiration date

Buyer of an Option Contract

Always short- acquires the right to buy/sell

Seller of an Option Contract

Always long- issues the right to buy/sell

Call Option

The right to buy shares of stock for a strike price

Put Option

The right to sell shares of stock for a strike price

Strike Price

The contract price for the option (remains fixed at all times during the contract)

American Option

Can be done on or before the expiration date

European Option

Can only be exercised at expiration

Valuing American vs. European

VAM > VEUR

AM contains EUR with additional privileges regarding selling before expiration

Premium (V0)

The price a buyer pays to the seller at initiation of the contract

Payoff Bounds on Option Prices

VcT = max(ST - k, 0)

-Max payoff earned is the call - the strike (does not factor in premium paid)

-Lowest payoff is 0, if the option finishes OTM we will not exercise it

VpT = max(k -ST, 0)

-Max payoff earned is the put - the strike (does not factor in premium paid)

-Lowest payoff is 0, if the option finishes OTM we will not exercise it

Put-Call Parity

For European options with the same underlying asset, strike k, and time T to expiration:

Vpt + St = Vct + ke^(-r)(T-t)

Put Option + Underlying Stock = Call Option + Risk-Free Bond (pays k @ T)

-Proves no arbitrage- must be equal (does not apply to American)

Put Option Valuation

The higher the strike price of a put, the more valuable it is (easier to exercise) to a buyer as they can sell the underlying for a higher price

In a falling market, the difference between the stock price and the strike is increasing, making the payoff more valuable

Higher Strike = Insurance

Call Option Valuation

The higher the strike price of a put, the less valuable it is (harder to exercise)

Lower likelihood of exercise makes the option cheaper

Call Profit/Loss

Buyer:

-Breaks-even @ K + Premium

-Profit Begins @ St > K + Premium

Seller:

-Breaks-even @ K + Premium

-Profit Begins @ Premium (as long as stock finishes at or below K. Seller profit is capped at K as the buyer will not exercise if St < K)

Put Profit/Loss

Buyer:

-Breaks-even @ K - Premium

-Profit Begins @ St < K - Premium

Seller:

-Breaks-even @ K - Premium

-Profit Begins @ Premium (as long as stock finishes at or aboe K. Seller profit is capped at K as the buyer will not exercise if St > K)

Option Value

Option Value = Time-Value + Intrinsic Value

Time-Value

Time remaining on a contract has potential for good things to happen, more likely to finish ATM or ITM

An option with t = 0.5 is more valuable than an option with t = 0.75 (T = 1 for both) as there is more time for become profitable

Vlong > Vshort

Intrinsic Value

Considers worth at expiration

You use it (exercise if ITM) or you lose it!

Moneyness Valuation

VITM > VATM > VOTM

-ITM already has intrinsic value and less uncertainty (∆ close to 1)

-ATM has 0 intrinsic value, but has the highest time value (however, maximum uncertainty)

-OTM has 0 intrinsic value, some time value but less than ITM and overall the lowest total value

Option Greeks

Let V be a function of two variables plus parameters: V(St, t; k, sigma, r, ∂)

Take "partial derivatives" with respect to inputs to define the function value, v.

Delta ∆

Measures how much the value of an option changes for a $1 change in the underlying asset

1. How many shares of the stock does the option replication

(think ∆ = 0.6 -> owning 0.6 shares, ∆ = -0.4 -> shorting 0.4 shares)

2. Slope?

(∆ is the tangent line slope on an option's payoff graph. Fastest (0 or +/- 1 deep OTM or ITM, steepest ∆ ATM)

3. Approximate probability of exercising the option?

(roughly approximates chances of finishing ITM)

Call ∆

0 - 1 range, option gains value as the stock rises

Put ∆

-1 - 0 range, option gains value as the stock falls

Gamma

Measures how much ∆ changes when the stock moves by $1. It is an acceleration of the option's sensitivity. Gamma is always positive

1. How quickly is ∆ changing?

(Think if ∆ = 0.5 and Gamma = 0.1, a $1 increase in the stock makes ∆ = 0.6)

2. Where is the option most sensitive to price moves?

(Tells us where ∆ is most reactive, how risky is it to sell this option?)

**Important: Gamma does not change sign if the stock moves down as it solely measures for $1 increases in the stock

Gamma and Moneyness

-Deep ITM: 0 (∆ is stable, option already ITM)

-ATM: Highest (∆ is changing, small-movements change moneyness)

-Deep OTM: 0 (∆ is stable, option is unlikely to go ITM)

Gamma for Sellers

Negative: short position becomes more sensitive to market moves

Gamma for Buyers

Positive: moves option towards moneyless, long position moves towards the strike

Theta ø

Measures how much the value of an option decreases per day as it approaches expiration, can be seen as time decay

If a call is worth $5 today with a ø = 0.5, the option will be worth $4.5 tomorrow

Long-term ø is negative, the rate of decay hurts the option holder as it increases towards expiration

Short-term ø is more explosive and impactful

ø for Buyers

Negative: losing value and potential for the option to go ITM as time goes on

ø for Sellers

Positive: collect the premium, time decay means the option may not be exercised (do not have to payout the buyer)

ø and Moneyness

ATM: Most negative (high decay)

OTM/ITM: Less negative (slow decay)

Vega V

Measures the value of the option changes with a 1% change in implied volatility

Positive for both puts and calls

Vega and Moneyness

Deep ITM/OTM: Lower

ATM: Highest (may change moneyless, each movement matters)

Rho P

Measures how much the value of an option changes when the risk-free rate r changes by 1%

Rho for Calls

Positive: higher interest rates lead to a lower present value of the strike

If P = 0.25, if interest rates rise by 1%, the call option value increases by $0.25

Rho for Puts

Negative: the strike is now worth less than if you received it today

If P= -0.3, if interest rates rise, the put option value decreases by $0.30

Epsilon E

Measures how much the value of an option changes when the dividend yield ∂ changes by 1%

Epsilon for Calls

Negative: dividends make us want to hold the underlying stock itself rather than an option

Epsilon for Puts

Positive: stock prices fall around dividend dates

Omega Ω

(Elasticity) measures the percentage change in the option's value relative to a 1% change in the underlying stock price

If the stock moves up by 1%, what % will my option price move?

(return on option vs. the return on the stock, can reflect volatility)

Ω is positive for calls, negative for puts

Ω and Call Moneyness

-Deep OTM: Very high (>1, to infinity: option is cheap, small stock move results in big change in value)

-ATM: Moderate (option value is higher -> leverage decreases, but still responds to market moves)

-Deep ITM: Low < 1 (option behaves like the stock -> high price means less % change from movement)

Ω and Put Moneyness

-Deep OTM: Very negative (to -infinity: option is cheap, small stock move results in big change in value)

-ATM: Moderate (option value is higher -> leverage decreases, but still responds to market moves)

-Deep ITM: Low negative (option behaves like the stock -> high price means less % change from movement)

Portfolio Greeks

Additive, portfolio greek = sum of all individual greeks

Black-Scholes

A formula for pricing European options

At every single point in time t and at every single stock price St, the option should be exactly replicated by the mimicking portfolio

Foundational Assumptions of BSM

-Frictionless markets (no taxes, transaction costs)

-No short-sale restrictions, no price impacts from large trades

-Continuous ∆-hedging from the representative market maker (most unrealistic!)

-Option's value V(St, t) is a twice-differentiable function of St, once differentiable in time

Parametric Assumptions of BSM

-Constant risk-free rate, r

-Limitless borrowing and spending at r

-Stock price follows a stochastic differential equation (SDE)

Market Maker's Portfolio

-Seller's portfolio replicates the payoff of a call option to sell it at a fair price

-Sells 1/∆ call options with price V

-Buys one share with price S

Goal = no risk = no arbitrage

Partial Different Equation (PDE)

PDE

r[∆S - V] + (1/2)(gamma)(sigma ^2)(s^2) + ø = 0

Gamma is positive and shows us how the option price moves with the stock price as it moves in our favor

ø is negative as time decay is always negative for buyers

Call Prices in BSM

Think:

Call = Expected Stock Gain - Expected Cost of exercise

Put Prices in BSM

Think:

Put = Expected Benefit of Selling K - Cost of Owning the Stock

N(D1)

∆ of the call, risk-adjusted probability of exercise (think: share replicated)

N(D2)

Risk-neutral probability of finishing ITM

Put-Call Parity for BSM

Put-call parity ensures that the value of a European call plus the present value of the strike (invested at the risk-free rate) equals the value of a European put plus the value of holding the underlying stock (until exercise, including any dividend gains)

∆ Put-Call Parity

∆c - ∆p = 1

The call option replicated part of a share of stock (0 < ∆ < 1)

The put option replicates a short (negative share) as it increases as the stock falls

So ∆c - ∆p gives us 1 total unit of stock exposure

Empirical Anomalies in Option Prices

1. The BSM model tells us that the option trading volume is predicted to be 0

(why would we trade options and pay premiums if the replication portfolio of stocks and options can give us the same return without paying a premium?)

2. Hetergeneous Implied Volatilities

(BSM assumes volatility is constant across all options, but in reality it is not constant but heterogeneous)

Implied Volatility

Option price is monotonic (1-to-1) in implied volatility

Implied volatilities (IV) are estimates of sigma "recovered" or "backed out: from trade option prices

Since BSM is derived on constant volatility, it fails to capture real-world implications from market volatility

Two IV Anomalies

1. IV Smirk/Smile/Skew

2. Time-Series Variation in IV

IV Smirk/Smile/Skew

OTM options are "too expensive"- investors demand protection from high volatilities, ATM are underpriced and can be used as arbitrage

An OTM put should have the same IV as an ITM call

Options expiring soon have higher IV not accounted for under the constant sigma of BSM. BSM struggles to price short-dated options

Time-Series Variation in IV

"Stochastic Volatility"- volatility changes over time ("vol of vol")

-VIX Index

Binomial Trees

Numerical approximation technique, converges in the limit to the true option value. These are used on American options since BSM cannot be used.

The stock can go up (u) or down (d) in each period (h)

How to Use a Binomial Tree

1. Stock prices can only take on Ust or Dst

2. Assume the returns are constant for any interval h

3. Apply the forward stock price Ft, t+h, to the potential upwards or downwards movement of the stock

4. Plug-in and solve

5. Find profit u or loss d

P*

The risk-neutral probability of an up move in t -> t+h

1- P*

The risk-neutral probability of a down move in t -> t+h

Solving Via Backwards Induction

1. Start at time T; solve for option value at each terminal node

2. Work backwards using the formula at each step to find Ct-1

3. Repeat until Ct

Callable Bond

Gives the issuer the option to redeem the bond before maturity

YieldCallable > YieldStraight (makes less profitable to the bondholder as upside is capped. Yield is also adjusted higher as compensation of the bond potentially being called early)

Putable Bond

The issuer has the right to sell the bond back before expiration

YieldPutable < YieldStraight (acts as "insurance", reduces the risk to the bondholder and therefore the yield)

Single-Stock and Option Strategies

Covered Calls and Protective Puts

-∆-hedged options positions involving a single option and a position in the stock that offsets the option's ∆ (if the option has a positive ∆, then the tock option is short and vice versa)

This allows us to see the put-call parity relationship. If a long protective put (long option, short stock), produces a payoff diagram for the combined strategy that looks like a payoff diagram for a call option with the premium adjusted by an amount equal to Ke^-rt per share

Covered Calls

-Sell the call option (collect premium)

-Long the stock (protects us as we own the stock and can sell at strike if the option is exercised

-Limited upside potential (if the stock goes up and the option is exercised, only keep the premium. If the stock falls and the option is not exercised, we keep the stock and premium)

Protective Puts

-Own the underlying stock

-Put a put option on the stock

-Pay a premium as "insurance"- earn upside either way

Spreads/"Verticals"

Formed by combining multiple options of the same type (calls/puts). They involve buying or writing at multiple strikes, usually arranged vertically. These strategies allow the trader to place levered meters on the direction (up or down) and/or the volatility (small or large movement) at the stock price over the option's life 0 -> T

Bull Spread

(usually done with calls)

-Bets on the stock price increasing

-Upside capped outside of the strike

-Trader has three options:

1. So < K: both calls OTM- high risk, high expected return

2. K1 < S0 < K2: one call is ITM, one is OTM. Medium risk, medium expected return

3. K2 < So: both calls are ITM, low risk expected return

Bear Spread

(usually done with puts)

-Bets the stock price will decrease

-Max Profit = K1 - K2 - Net Premium

-Max Loss = premium paid

-Break-even = K1 - net premium

-Buy a put at a higher strike k1

-Sell a put at a lower strike k2

(Intuition Check) How would the risk and expected return on one of these strategies change if the trader picked strikes k1 and k2 that were closer together or further apart?

1. Closer Together

-Narrow spread

-Profit decreases, loss is smaller, higher probability of exercise

2. Further Spread

-Farther, larger spread

-Profit increases, loss is larger, more risky

(Intuition Check) Can you calculate the probability that the stock exceeds the break-even point on a strategy, as well as the rate of return the trader would earn if it did?

Break-even Probability- use BSM N(d2)

- Rate of Return = Profit / Initial Cost

Box Spread

-Arbitrage strategy combining bull and bear spreads (something must be mispriced at different strikes)

-Write a pull-call parity for 2 strikes

If C1 is relatively mispriced to C2, for example, a spread trader can exploit the difference as arbitrage by shorting the overvalued asset and longing the undervalued

Profit = (k2 - k1)e^(-r*T) dividend by the cost of the option positions

Butterfly Spread

-Bet on option volatility rather than direction

-Usually all calls/puts with three strikes:

1. Max profit = k2 - k1 - net premium

2. Max loss = net premium paid

3. Break-even = k1 + net premium and k3 - net premium

-Low risk/low reward

Condor Spread

-Also bet on volatility

-4 legs, 4 strike prices (more forgiving than butterfly)

-Wider profitable range

-Below k, or above, k4 = max loss = net premium

-Between k2 and k3 = max profit

-Between k1-k2 and k3-k4: partial gains/losses

(Intuition Check) The long butterfly strategy involves going long one call each at the high and the low strikes and two short calls at the middle strike. What is the slope of the payoff diagram for the "2x short call" and why?

Slope is -2 as each short call ITM is a ∆ of -1

(Intuition Check) What is the difference between the butterfly and the condor strategies in terms of risk and return?

Butterfly has a more narrow spread (3 strikes) compared to the condor (4 strikes), so it holds higher risk, but higher potential reward

(Intuition Check) Could you recreate these strategies with puts instead of calls? How?

Same logic, mirrored/inverse image to a call

Calendar Spread

-Combines different options with different maturities

-Similar to butterfly

Combination Strategies

Involve using combined positions in options of both types and allow the trader to create a wider variety of payoff diagrams according to their preferences of risk and return

Collar

-Used to hedge a stock position, create a floor and ceiling

-Typically done once a stock already increase in value in order to "lock in" gains for a risk averse investor (Bernie Madoff!)

-Own a stock (long)

-Buy a protected put (downside protection)

-Sell a covered call

Iron Butterfly

-Uses three strikes, both puts and calls

-Use the same middle strike

-Limited risk, limited reward

Iron Condor

-Similar to iron butterfly, use 4 strikes

-More forgiving, more protection

(Intuition Check) When would a trader use an "iron" strategy?

-Synthetically neutral- no need to own the underlying stock

-Fully hedged- requires less capital

-Bet the stock will not move much

Straddle

-Bet on volatility (not specified which direction)

-Long call and long put

-Stock goes up = call earns

-Stock goes down = put earns

-Stock doesn't move = both options OTM (max loss = premiums paid)

Strangle

-Cheaper alternative to a straddle

-Still bet on volatility

-Use different strikes to reduce costs

-Buy 1 put @ k1 (OTM)

-Buy 1 call @ k2 (OTM)

Wider range = cheaper premium

(Intuition Check) Why might. a trader use a strangle as opposed to a straddle?

Using a wider range of multiple strikes is cheaper

(Intuition Check) What is the difference between these payoff diagrams and the butterfly and condor?

-Straddles and Strangles bet on volatility

-Butterfly and Condor bet on stability

Strip

-Bets on downward volatility

-"Bearish Straddle"

(same strike and expiration)

-Buy 1 call at strike K

-But 2 puts at strike K

Strap

-Bets on upwards volatility

-"Bullish Straddle"

(same strike and expiration)

-Buy 2 calls at strike K

-Buy 1 put at strike K

Short Gamma

Always negative