SOA Exam MFE Flash Cards

Spring 2007

Forwards

Put Call Parity

Spreads

A Bull Spread consists of either buying a call with strike price and selling

one with strike price , or buying a put with strike price and selling a put

with strike price .

A Bear Spread is the opposite of a bull spread; either for calls or for

puts .

Exchange Rate Terminology

Spot Exchange Rate

Risk Free Rate for Domestic Currency

Risk Free Rate for Foreign Currency

Pre-paid forward exchange rate

Forward exchange rate

Currency Options

Asset = Call – Put + PV(Strike Price)

Call – Put +

A call to purchase a foreign with domestic is equivalent to a put to sell domestic for foreign.

Binomial Trees

Option Price = ∆ S + B =

∆ = = B =

p* = =

u = d =

d < < u σ = Volatility

Alternative Binomial Trees

Cox-Ross-Rubinstein

Lognormal

Arbitraging a Mispriced Option

The Option is OverpricedIf we know that the option is overpriced we would like to sell it for guaranteed profit (Arbitrage). This is achieved by selling the option and simultaneously buying a synthetic option for the correct value.

The Option is Under-pricedIf we know that the option is under-priced we would like to buy it for guaranteed profit (Arbitrage). This is achieved by buying the option and simultaneously selling a synthetic option for the correct value.

Real Probability Pricing

In reality, the expected return on a stock is α > r. Let p denote the true probability of an up move:

p =

γ is the discounting rate for the option.

Option Cost: C =

Using γ and α yields the same option price as risk-neutral probability. To obtain value for γ, compute the option price, C, using risk-neutral method.

Then plug in parameters into equation below.

Black-Scholes Formula

General Black-Scholes Formula

where

and are the pre-paid forward prices for the underlying and strike assets, respectively.Assumptions of the Black-Scholes Formula

Continuously compounded returns on the stock are normally distributed and independent over time.

o Lognormality of .

Continuously compounded returns on the strike asset (the risk-free rate) are known and constant.

Volatility is known and constant.

Dividends are known and constant. There are no transaction costs or taxes. It is possible to short-sell any amount of stock and to borrow any amount of

money at the risk-free rate.

Black-Scholes Formula for Common Stock Options

where

Black-Scholes Formula for Currency Options

where

This is known as the Garman-Kohlhagen model.

Currency-Denomination

An option is currency-denominated if the strike price and premium are denominated in that currency.

dollar-denominated option on eurosdollars to euros

Risk Free Rate for dollars

Risk Free Rate for euros

Black-Scholes Formula for Options on Futures

where

The pre-paid forward price for a future, is the future discounted at the risk-free rate.

Option Greeks

DeltaDelta measures the option price change when the stock price

increases by $1.

GammaGamma measures the change in Delta when the stock price

increases by $1. Gamma is the convexity of the option.

where V is the derivative of S.

VegaVega measures the change in the option price when there is an

increase in volatility, , of one percentage point.

Vega = where V is the derivative of S.

ThetaTheta measures the change in the option price when there is a

decrease in the time to maturity of one day.

= where V is the derivative of S.

Rho

Rho measures the change in the option price when there is an increase in the interest rate of one percentage point (100 basis points).

= where V is the derivative of S.

PsiPsi measures the change in the option price when there is an

increase in the continuous dividend yield of on percentage point (100 basis points).

= where V is the derivative of S.

Option Elasticity

An option is an alternative to investing in the stock. Delta tells us the dollar risk of the option relative to the stock: If the stock price changes by $1, by how much does the option price change? The option elasticity, by comparison, tells us the risk of the option relative to the stock in percentage terms: If the stock price changes by 1%, what is the percentage change in the value of the option?

Percentage Risk of the OptionThe Option Elasticity, , computes the percentage change in the option price relative to the percentage change in the stock price.

Option Volatility

Option Risk Premiumwhere

α = Expected Stock Returnγ = Expected Option Return

Sharp RatioThe Sharp Ratio for any asset is the ratio of the risk premium to volatility.

Sharp Ratio for Call =

Sharp Ratio for Put =

Delta-Hedging

Overnight Profit on a Delta-Hedged Portfolio

A delta-hedged portfolio consists of selling or buying an option, buying ∆ shares of stock, and borrowing or loaning the cash-flow.

The overnight profit has 3 components:

1. The change in the value of the option.2. ∆ times the change in the stock price.3. Interest on the cash-flow.

Profit =

Profit = {Change in Option Value}+{∆ * Change in stock price}-{Interest}Overnight Profit Formulas

Black-Scholes Equation

Delta-Gamma-Theta Approximation

where

Option Cost at time t+h

Option Cost at time t

= Change in stock price from time t to time t+h

Market Making

Market maker profitability is dependent primarily on the stock price movement.

We define a one-standard deviation move as:

If the stock price moves less than one standard deviation then the market maker profits, if the price moves one standard deviation then we break-even, finally if the price moves more than one standard deviation then we suffer a loss.

Daily h = ; Monthly h = ; Weekly h =

Rehedging

Let x be the number of standard deviation movements:

is the period i return after a stock moves and rehedging is done every h.

=

Var( ) = this is the 1/h period variance,

For annual variance:

AnnVar( ) =

Perpetual Options

Exotic OptionsAsian Option

An Asian option pays based on the average price. 3 Parameters

o Call or Puto Geometric or Arithmetic Average

o Average Price, or Average Strike,

For an Average Price option, the payoff is based on the difference between the strike price and the average price.

o Call Payoff = max(0, - K)

o Put Payoff = max(0, K - ) For an Average Strike option, there is no given strike price. The payoff is based

on the difference between the final price and the average price.o Call Payoff = max(0, - )

o Put Payoff = max(0, - )

Barrier Option

A Barrier Option has a payoff which depends on whether over the lifetime of the option the underlying asset hits the barrier. 3 Types.

1. Knock-Out Options Up and Out Down and Out

2. Knock-In Options Up and In Down and In

3. Rebate Options Pays a fixed amount if barrier is hit. Up Rebates Down Rebates

Parity Relationship for Barrier OptionsKnock-In + Knock-Out = Ordinary

Compound Option

A Compound Option is an option to buy an underlying option.

is the current time

is the time of exercise for the Compound Option

T is the time of exercise for the Underlying Option x is the strike price of the Compound Option;

o x is the price we pay for the underlying option

The price of the underlying option at time is x = , we will

exercise if this option price is less than the price we would pay for the regular option anyone could purchase.

The price at time for a regular option that anyone could purchase is

Compound Options continued… So then the value of the compound option at time is

Let be the Critical Stock Price above which the compound option is exercised. Then

o

o The compound option is exercised for

Compound Option Parity

Compound Option Diagram