Describe binary calls and puts
Payoff = 1 when the option is inthemoney
Define gamma, theta, speed, vega, rho
 Gamma
 Second derivative of the option price V w.r.t. the underlying S
 Represents:
 Sensitivity of delta w.r.t. the underlying
 How much or how often a position must be rehedged to maintain a deltaneutral position
 Theta
 Derivative of the option price V w.r.t. time t
 Quantifies how much t contributes in a completely certain way
 Option value decreases when T increases
 Speed
 Third derivative of the option price V w.r.t. underlying S
 Quantifies the rate at which gamma changes w.r.t. the underlying S
 Vega
 Derivative of the option price V w.r.t. volatility parameter σ
 Can add significant model risk because it relies on whether volatility is modeled correctly
 Downfalls:
 Only meaningful for options with singlesigned gamma everywhere
 Not as useful for analyzing binary options
 Rho
 Sensitivity of an option value V w.r.t. parameter for interest rate r
 Same model risk concern as vega, but less of it because r is easier to estimate than σ
 Typically separated into buckets and term structure interest rates so r(t) can vary over time
 Second derivative of the option price V w.r.t. the underlying S
 Represents:
 Sensitivity of delta w.r.t. the underlying
 How much or how often a position must be rehedged to maintain a deltaneutral position
 Derivative of the option price V w.r.t. time t
 Quantifies how much t contributes in a completely certain way
 Option value decreases when T increases
 Third derivative of the option price V w.r.t. underlying S
 Quantifies the rate at which gamma changes w.r.t. the underlying S
 Derivative of the option price V w.r.t. volatility parameter σ
 Can add significant model risk because it relies on whether volatility is modeled correctly
 Downfalls:
 Only meaningful for options with singlesigned gamma everywhere
 Not as useful for analyzing binary options
 Sensitivity of an option value V w.r.t. parameter for interest rate r
 Same model risk concern as vega, but less of it because r is easier to estimate than σ
 Typically separated into buckets and term structure interest rates so r(t) can vary over time
Define portfolio insurance
 Strategy when you:
 Reduce stock holdings when prices fall,
 Increase stock holdings when prices rise
 Overall, option values due to portfolio insurance are balanced because of mean reversion
 Reduce stock holdings when prices fall,
 Increase stock holdings when prices rise
What are two types of hedging w.r.t. models?
 Model independent
 Few and far in between
 e.g., violations in PutCall parity
 Model dependent
 Most hedging strategies
 Requires some kind of volatility model
 Few and far in between
 e.g., violations in PutCall parity
 Most hedging strategies
 Requires some kind of volatility model
What are some types of hedging w.r.t. greeks?
 Delta
 Exploits perfect correlation between option and underlying
 Gamma
 Reduces transaction costs, rebalancing needs
 More accurate than delta hedging
 Vega
 Trading strategy that results in zero vega
 Exploits perfect correlation between option and underlying
 Reduces transaction costs, rebalancing needs
 More accurate than delta hedging
 Trading strategy that results in zero vega
Define static, margin and crash (platinum) hedging
 Static hedging
 Buying/selling a set of more liquid contracts to reduce the CFs of the original contract
 Positions are left to expiry
 Margin hedging
 Portfolio set up such that margin calls are covered by refunds of hedging contracts
 Crash (platinum) hedging
 Minimizes worst possible outcome for the portfolio
 Buying/selling a set of more liquid contracts to reduce the CFs of the original contract
 Positions are left to expiry
 Portfolio set up such that margin calls are covered by refunds of hedging contracts
 Minimizes worst possible outcome for the portfolio
Define implied volatility
 Volatility of the underlying which, when used in BS formula, results in market prices
 Market consensus or estimate of volatility
Define actual volatility
Amount of randomness of a financial quantity that actually transpires at any given point
 Amount of noise int he stock price
 Wiener process coefficient in the stock returns model
Define historical, forward and hedging volatility

Historical volatility
 Backwardlooking statistical measure
 Forward volatility
 Actual or implied, for some time in the future
 Hedging volatility
 What is plugged into the detla calculation
 Backwardlooking statistical measure
 Actual or implied, for some time in the future
 What is plugged into the detla calculation
List types of models used for volatility
 Econometric
 Timeseries analysis to estimate current and future expected actual volatility, e.g., GARCH
 Deterministic
 Deterministic volatility surface
 Set σ(S,t) in the BS model
 Does not capture dynamics of volatility very well
 Stochastic
 Better captures the dynamics of traded option prices compared to deterministic
 Poisson
 Volatility jumps
 Uncertain
 Define a range of σ ⇒ range of prices
 Timeseries analysis to estimate current and future expected actual volatility, e.g., GARCH
 Deterministic volatility surface
 Set σ(S,t) in the BS model
 Does not capture dynamics of volatility very well
 Better captures the dynamics of traded option prices compared to deterministic
 Volatility jumps
 Define a range of σ ⇒ range of prices
Compare and contrast pros/cons of hedging with actual and implied volatility
 Actual volatility
 PROS:
 Known profit at expiration, assuming continuous hedging
 Most reasonable if marktomodel strategy is followed
 More leeway than implied volatility, as long as forecast is "good enough"
 CONS:
 Daily P&L volatility can be substantial ⇒ risks
 Need to estimate actual volatility forecast for ∆
 Implied volatility
 PROS:
 Minimal local fluctuations in P&L (i.e., continual profit)
 No need exact actual volatility estimation, just the right side of the trade
 Easy to calculate because implied vol is observable
 More reasonable if market value approach is used
 CONS:
 Final profit is unknown, just know that it will be positive
 PROS:
 Known profit at expiration, assuming continuous hedging
 Most reasonable if marktomodel strategy is followed
 More leeway than implied volatility, as long as forecast is "good enough"
 CONS:
 Daily P&L volatility can be substantial ⇒ risks
 Need to estimate actual volatility forecast for ∆
 PROS:
 Minimal local fluctuations in P&L (i.e., continual profit)
 No need exact actual volatility estimation, just the right side of the trade
 Easy to calculate because implied vol is observable
 More reasonable if market value approach is used
 CONS:
 Final profit is unknown, just know that it will be positive
What are two trains of thought for the rationale behind option pricing movements?
What are some considerations when pricing options?
 Valuation (theory)
 Prices are driven by BS (theoretica, parameters, assumptions)
 Option values are consistent with the price of the underlying
 Pricing (practice)
 Prices are driven by supply and demand
 Prices are driven by BS (theoretica, parameters, assumptions)
 Option values are consistent with the price of the underlying
 Prices are driven by supply and demand
Considerations:
 OTM options sell at a premium
 American options are difficult to price becasue early exercise is seldom done optimally
 Embedded options are priced high because Σ parts > whole security
What is the power law survival function?
S(x) = K / x^{α}
Compare/contrast normal and fractal distributions
 Normal/Gaussian
 Nonscalable
 Typical member is mediocre
 Winner takes a piece of the pie
 Ancestral environment
 Not determined by a single instance
 Tyranny of collective
 Easy to predict from the past
 Fractal
 Scalable
 No typical member
 Winnter takes all
 Modern environment
 Determined by a few events
 Tyranny of accidental
 Hard to predict from the past, need large window of observation
 Nonscalable
 Typical member is mediocre
 Winner takes a piece of the pie
 Ancestral environment
 Not determined by a single instance
 Tyranny of collective
 Easy to predict from the past
 Scalable
 No typical member
 Winnter takes all
 Modern environment
 Determined by a few events
 Tyranny of accidental
 Hard to predict from the past, need large window of observation
What is volatility smile?
How can it be built into pricing?
 It is the graph of strike (K) vs. implied volatility, which may result in higher implied vol for OTM calls/puts
 Can be built into pricing by:
 Deterministic volatility surface
 May not describe actual dynamics very well
 Stochastic volatility models
 Sources of randomness are stock returns and volatility
 Greater potential to capture dynamics
 Jump diffusion model
 Accommodate for excess kurtosis
 Deterministic volatility surface
 May not describe actual dynamics very well
 Stochastic volatility models
 Sources of randomness are stock returns and volatility
 Greater potential to capture dynamics
 Jump diffusion model
 Accommodate for excess kurtosis
What is vonma?
 The second derivative of V w.r.t. σ
 It is negative close to ATM, and >> 0 for ITM/OTM
 Results in higher price and implied volatility for OTM options
 If vonma > 0 ⇒ vega is positively related to volatility changes
 If vonma
 Results in higher price and implied volatility for OTM options
What causes volatility smiles?
Due to:
 Supply and demand
 ↑ OTM puts demand for insurance protection ⇒ ↑ price and σ of OTM puts
 ↑ OTM calls supply to earn premium ⇒ ↓ price and σ of OTM calls
 Kurtosis / fat tails
 Correlation between stock prices and volatility
 Dramatic ↓ in price ⇒ ↑ implied vol for puts with lower K
 Volatility gamma (vonma)
 OTM puts have higher vonma ⇒ ↑ implied vol
List 10 assumptions in BS and how to take advantage of them
 Volatility is known
 If σ ↑, buy a straddle/strangle
 No jumps
 If expecting symmetric jumps, buy OTM options
 Constant rfr
 If r ↑, buy calls/stocks and sell puts
 Borrowing = lending rates; infinite borrowing
 If r > lending rates + borrowing limits ⇒ buy calls
 If r < lending (no borrowing limits) ⇒ borrow instead of buy calls
 If implied r ↑, buy options instead of stock
 Short sales can be invested
 Instead of short stock, hold put or naked short call
 No transaction costs
 Use arbitrage bands
 No taxes
 No dividends
 European options
 No early exercise or takeover events
 May affect shortterm OTM options dramatically
 If σ ↑, buy a straddle/strangle
 If expecting symmetric jumps, buy OTM options
 If r ↑, buy calls/stocks and sell puts
 If r > lending rates + borrowing limits ⇒ buy calls
 If r < lending (no borrowing limits) ⇒ borrow instead of buy calls
 If implied r ↑, buy options instead of stock
 Instead of short stock, hold put or naked short call
 Use arbitrage bands
 May affect shortterm OTM options dramatically
List 7 assumptions in BS and how to relax them
 Discrete hedging
 Expected value is the same as continuous
 Transaction costs
 Use volatility range to represent bidask spreads
 Timedependent volatility
 Use rootmeansquare average variance over the remaining lifetime (Tt)
 Arbitrage opportunities
 Use BS to deltahedge and determine how much profit you would like
 Nonlognormal underlying
 Nothing to do
 Borrowing costs
 Adjust drift, similar to dividend adjustment
 Nonnormal returns
 Nothing to do only need finite variance of returns due to CLT
 Expected value is the same as continuous
 Use volatility range to represent bidask spreads
 Use rootmeansquare average variance over the remaining lifetime (Tt)
 Use BS to deltahedge and determine how much profit you would like
 Nothing to do
 Adjust drift, similar to dividend adjustment
 Nothing to do only need finite variance of returns due to CLT
What is the total PV of profit when hedging using actual volatility? And from time t to t + dt?
V^{a}  V^{i}
e^{r(tt0)} d(V^{a}  V^{i})
What is the total PV of profit when hedging using implied volatility? And from time t to t + dt?
dV^{i} = 1/2 (σ^{2}  σ_{imp}^{2}) S^{2} Γ^{i} dt
dV^{i} = 1/2 (σ^{2}  σ_{imp}^{2}) ∫_{[t0, T] }e^{r(tt0)}S^{2} Γ^{i} dt
What is the general total PV of profit formula?
V(S,t;σ_{h})  V(S,t;σ_{i}) + 1/2(σ^{2}  σ_{h}^{2}) ∫_{[t0, T]}e^{r(tt0)}S^{2}Γ^{h} dt