🎯 Summary of key points included in this course
TOC
Basis RiskOptimal Hedge RatioChange Beta ExposureInterest Rate Quotations & CompoundingForward PriceForward Contract ValuationForward Price v.s. Future PriceFuture Arbitrage StrategiesOption Payoff & ProfitIntrinsic Value and Time Value of OptionFactors Affecting Options’ PriceBounds of Options’ PricePut-Call ParityNever Exercise American Call Option EarlierBinomial Option PricingGirsanov’s TheoremReview of Stochastic ProcessesIto’s LemmaBlack-Scholes ModelGreeks and HedgingImplied VolatilityCVA & DVAIndex Option & Portfolio Insurance1987 Crash & Cascade Theory
Basis Risk
- Long Hedge
- Cash inflow at time between and :
- Short Hedge
- Cash inflow at time between and :
Optimal Hedge Ratio
Take Long Hedge for example:
For one stock long position, short future contracts (equivalent unit)
By shorting , we can eliminate part of stock price’s uncertainty which is linearly correlated with futures’ price (projection). While other parts of stock price’s uncertainty (Orthogonal to Futures’ price) cannot be eliminated by hedging.
When , all uncertainty can be eliminated by hedging, which is called “perfect hedging”
Change Beta Exposure
Take Long Hedge for example:
Hedge using index futures, target beta is ⇒ short futures:
- Optimal Hedge:
- Increase Beta Exposure (higher risk, higer return): , short negative futureslong futures
- Decrease Beta Exposure (lower risk, lower return): , short futures
Interest Rate Quotations & Compounding
Intuition: different interest frequency yield the same return
Forward Price
Institution: No aribitrage ⇒ consider what you gain (decrease the price) and what you sacrifice (increase the price).
- Fair forward price:
- Investment assets:
- No income:
- discrete income:
- yield income:
- Consumption assets:
- No storage costs & no conv. yield:
- discrete storage costs:
- discrete storage costs & convenience yield:
- % storage costs:
- % storage costs & convenience yield:
Forward Contract Valuation
The value of a forward contract at intermediate dates (present value of time , which means “discounted backward”):
- Long T-maturity forward contract at time ,
- Short T-maturity forward contract at time ,
Future valuation is calculated in the same method.
Forward Price v.s. Future Price
One method to compare forward price and future price is based on the correlation between stock (underlying asset) price and the interest rate:
- Interest rate is constant (): forward price = future price
- long futures long forwards
- futures’ price > forwards’ price
- long futures long forwards
- futures’ price < forwards’ price
Future Arbitrage Strategies
When the theoretical Future price () is not identitical to market price , there exist arbitrage opportunities.
If , we should long futures and sell spots. We can gain the profit from Time T or Time 0:
- Gain from Time T
- Cash flow of Time 0: 0
- Cash flow of Time T:
- Gain from Time 0
- Cash flow of Time 0:
- Cash flow of Time T: 0
If , we should short futures and buy spots. We can gain the profit from Time T or Time 0:
- Gain from Time T
- Cash flow of Time 0: 0
- Cash flow of Time T:
- Gain from Time 0
- Cash flow of Time 0:
- Cash flow of Time T: 0
Option Payoff & Profit
Buy Call option
- The call is exercised when
Sell Call option
- The buyers’ gain is the sellers’ loss
Buy put option
- The put is exercised when
Sell put option
- The buyers’ gain is the sellers’ loss
Summary: the payoff functions
- Long call:
- Short call:
- Long put:
- Short put:
Summary: the profit/loss functions
Profit/Loss =
- Long call:
- Short call:
- Long put:
- Short put:
Intrinsic Value and Time Value of Option
Intrinsic Value: Max(Payoff, 0)
- Payoff > 0: In the money (ITM)
- Payoff = 0: At the money (ATM)
- Payoff < 0: Out of the money (OTM)
Time Value: defined as option price minus intrinsic value
- It is a measure of the possibility that the option could move further ITM as time passes
- Time value is non-negative and is highest for ATM options
Factors Affecting Options’ Price
Bounds of Options’ Price
ㅤ | Upper Bound | Arbitrage Strategy
when UB is violated | Lower Bound | Arbitrage Strategy
when LB is violated |
European Call Option | Buy stock and sell call option | Buy Portfolio A and Sell Portfolio B | ||
European Put Option | Sell put option and invest at | Buy Portfolio C and Sell Portfolio D | ||
American Call Option | Buy stock and sell call option | — | — | |
American Put Option | Sell put option and invest at | — | — | |
American Call - American Put (C-P) | — | — |
Note
- above options are all based on non-dividend paying stock
- Portfolio A: One European call and a zero-coupon bond providing a payoff of at
- Portfolio B: One share of stock
- Payoff of Porfolio A at Payoff of Portfolio B at . Thus
- Portfolio C: One European put and one share of stock
- Portfolio D: A zero-coupon bond providing a payoff of at
- Payoff of Porfolio C at Payoff of Portfolio D at . Thus
Put-Call Parity
Note: Put call parity is only established for European options. Since American options can be exercised earlier, there’s no put-call parity.
Never Exercise American Call Option Earlier
It’s always better off to sell American call option rather than exercise it earlier.
Binomial Option Pricing
Replication Portfolio
Buy stocks and short one call option, where
Risk-neutral Probability
General definition for :
is defined as follow:
- for a non-dividend paying stock
- for a stock index where is the index dividend yield
- for a currency where is the foreign risk-free rate
- 1 for a futures contract
Risk-neutral Valuation
Multiple-step Binomial Tree
Value the nodes in the last step and then move back.
- Since American call options will never be exercised earlier, there’s no difference between the valuation of American call and European call.
- European put is valued in the same method.
- American put can be exercised earlier, thus the value of each node is MAX(intrinsic value,risk-neutral value)
Girsanov’s Theorem
Choice of and
Define as the standard deviation of the underlying asset return over
Choose
to match the volatility of the stock returns. In this way, when we move from the risk-neutral world to the real world:
- The expected rate of return on the stock changes
- The volatility remains the same
Review of Stochastic Processes
Markov Process
Markov processes depend only on the current level of the random variable, histroical values are of no importance.
Example - random walk:
Wiener Process
One kind of Markov process, also known as Brownian motion
A random variable follows a Wiener process if:
- , where
- for any two non-overlapping time periods are independent
Uncertainty (sd) is proportional to the square root of time.
Generalized Wiener Process
A random variable follows a GWP if:
- (drift rate): Mean change per unit of time for a stochastic process
- (variance rate): Variance per unit of time for a stochastic process
- can be constant or functions
Process
That is, drift rate and variance rate are both deterministic functions of and . It is also a kind of Markov Process
Geometric Brownian Motion (GBM)
- is the stock’s expected rate of return
- is the volatility (standard deviation) of the stock price
Ito’s Lemma
Assume variable follows an process:
lemma shows that a function follows the p.d.e
If stock price follows GBM () and assume that , thus
This means that
Prices are lognormally distributed and continuously compounded returns are normally distributed
Black-Scholes Model
Black-Scholes p.d.e
Assume option price is and stock price follows GBM
Risk-neutral valuation
- Applied to European call, European put and American call
- Assume is known and constant
where
Properties:
- As becomes very large, tends to be , tends to be zero
- As becomes large, the values of calls and puts increase
- As increases, the call value converges to the stock value and the put value converges to zero
Dividend & BS Model
Dividend payouts can be modeled in two ways:
- Continuous payouts occuring at a rate of
- Replace in above formula by
- Discrete payouts occuring at specific points in time
- Replace in above formula by
Probability of ending up ITM
Assume no dividends, stock prices follow GBM, then
Risk-neutral probability of European put ending up ITM
Risk-neutral probability of European call ending up ITM
Greeks and Hedging
Delta ()
Calls: , Puts:
For European options in BS world:
- Long call:
- Long put:
Delta Hedging:
where is the quantity (not proportion), is the delta of the i-th asset
Gamma ()
Calls: , Puts:
For European options in BS world:
- Long call or Long put:
Gamma Hedging:
Underlying asset has a Gamma of zero, thus it cannot be used to hedge Gamma risk. We need to take a position in a derivative to hedge gamma risk.
Vega ()
Calls: , Puts:
For European options in BS world:
- Long call or Long put:
where
Vega Hedging:
To hedge vega risk, we need to take a position in a derivative
Theta ()
Calls: , Puts:
Theta is also called time-decay
For European options in BS world:
- Calls:
- Puts:
Implied Volatility
In BS model, we assume that is known and constant. While we cannot observe in real world. We can observe the market price of option and the volatility that satisfy the equation
is called implied volatility
Equation can be solved using numerical search because is positively proportional to option price.
Volatility Smile
Several hypotheses explain the existence of volatility smiles. The simplest and most obvious explanation is that demand is greater for options that are in-the-money or out-of-the-money as opposed to at-the-money options.
Volatility Smirk
One straight explanation to this is that people (especially in stock market) prefer to hold long position and they do not hope the price goes down, thus the put option whose strike price is smaller than its spot price is much demanded. The demand increases the price and since we know IV is positively proportional to the option price, the IV is also higher. Similarly, for some assets people do not want their price go up, the option’s volatility smile will change to the “reverse smirk”.
CVA & DVA
CVA is an adjustment for loss due to counterparty default, while DVA is an adjustment for gain due to own default (for outstanding transactions)
where:
- : the maturity of the longest cash flow
- : counterparty default probability
- : PV of counterparty’s loss given default
- : own default probability
- : PV of own loss given default
Value after credit adjustment: No-default value - CVA + DVA
Index Option & Portfolio Insurance
Entering into put option position to ensure that the value of a portfolio will not fall below a certain level. The strike price is chosen to give the appropriate insurance level.
Methods include
- Using Index put option
- Using Synthetic put options which satisfy the cash flow:
- Sell stocks and deposit
- Since , sell futures and deposit
Protective Put using “put option” is a static hedge, when stock price changes:
- if we use index put option, we should change options we hold which is costly
- it’s better to use synthetic put options then we only need to change or
When decrease, increase, thus increase, we should sell more stocks or futures.
When increase, decrease, thus decrease, we should buy back stocks or futures.
When we try to use “Protective Put” to insure portfolio, things are different when of portfolio is 1 or not.
That’s because CAPM is a total returns relationship including capital gain and dividend yield while hedging using option protects only against capital losses exclude dividend
1987 Crash & Cascade Theory
The theory attribute the crash to two behaviors in the market:
- Dynamic portfolio insurance strategy using synthetic put with futures
- Index futures arbitrage strategies
Cascade theory
- When cash price starts to fall, portfolio insurance scheme triggers massive shorting of index futures
- The sell side pressure on index futures creates an opportunity for index futures arbitrage which leads to a massive selling of stocks in the cash market
- The secondary downward pressure on the stock cash market feeds back to dynamic portfolio insurance which triggers more index futures sales
- This goes on and on till both markets collapsed
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