Optimal Hedge Ratio

(Page 57)

Basis Risk

•Basis is usually defined as the spot price minus the futures price

•Basis risk arises because of the uncertainty about the basis when the hedge is closed out

Measuring Interest Rates

•When we compound m times per year at rate R an amount A grows to A(1+R/m)^{m} in one year

Continuous Compounding formula etc.

•In the limit as we compound more and more frequently we obtain continuously compounded interest rates

•$100 grows to $100eRT when invested at a continuously compounded rate R for time T

•$100 received at time T discounts to $100e-RT at time zero when the continuously compounded discount rate is R

Contiuous Compounding Conversion formulas

If the spot price of an investment asset is S0 and the futures price for a contract deliverable in *T* years is *F*_{0}, then....

*F*_{0}* *=* S*_{0}*e ^{rT}*

where *r* is the *T*-year risk-free rate of interest.

In our examples, *S*_{0} =40, *T*=0.25, and *r*=0.05 so that

F_{0 } = 40*e*^{0.05×0.25} = 40.50

Forward price when an Investment Asset Provides a known Yield/Return

F_{0} = S_{0} *e*^{(r–q )T }

where *q* is the average yield during the life of the contract (expressed with continuous compounding)

The value of a Forward Contract

•A forward contract is worth zero (except for bid-offer spread effects) when it is first negotiated

•Later it may have a positive or negative value

•Suppose that *K* is the delivery price and *F*_{0} is the forward price for a contract that would be negotiated today

•By considering the difference between a contract with delivery price K and a contract with forward price F0 we can deduce that:

–the value of a long forward contract, ƒ, is

*(F _{0} – K )e^{–rT} *

–the value of a short forward contract is

(*K* – *F*_{0} )*e ^{–rT}*

Forward vs Futures Prices

•When the maturity and asset price are the same, forward and futures prices are usually assumed to be equal.

•

•When interest rates are uncertain they are, in theory, slightly different:

–A strong positive correlation between interest rates and the asset price implies the futures price is slightly higher than the forward price

–A strong negative correlation implies the reverse

Forward Stock Index Pricing

•Can be viewed as an investment asset paying a dividend yield

•The futures price and spot price relationship is therefore

*F*_{0} = *S*_{0} *e*^{(r–q )T}

where *q* is the average dividend yield on the portfolio represented by the index during life of contract

Index Arbitrage

•When *F*_{0} > *S*_{0}^{e(r-q)T} an arbitrageur buys the stocks underlying the index and sells futures

•When F_{0} < S_{0}*e ^{(r-q)T}*

^{ }an arbitrageur buys futures and shorts or sells the stocks underlying the index

•Index arbitrage involves simultaneous trades in futures and many different stocks

•Very often a computer is used to generate the trades

•Occasionally simultaneous trades are not possible and the theoretical no-arbitrage relationship between F0 and S0 does not hold (19 October 1987)

Futures and Forwards on Currencies

•A foreign currency is analogous to a security providing a yield

•The yield is the foreign risk-free interest rate

•It follows that if rf is the foreign risk-free interest rate

F_{0}=S_{0}e^{(r-rf)T}

Consumption Assets Formula

Storage is Negative Income

The Cost of Carry

Valuation of an Interest Rate Swap

•Initially interest rate swaps are worth close to zero

•At later times they can be valued as the difference between the value of a fixed-rate bond and the value of a floating-rate bond

•Alternatively, they can be valued as a portfolio of forward rate agreements (FRAs)

The Put-Call Parity

•Both are worth max(ST , K ) at the maturity of the options

•They must therefore be worth the same today. This means that

*c* + *Ke ^{ -rT}* =

*p*+

*S*

_{0}Reasons For Not Exercising a Call Early (No Dividends)

• No income is sacrificed

• You delay paying the strike price

• Holding the call provides insurance against stock price falling below strike price

Bull Spread Using Calls

Bull Spread Using Puts

Bear Spread Using Puts

Figure 11.4, page 240

Bear Spread Using Calls

Figure 11.5, page 241

What is a Box Spread?

•A combination of a bull call spread and a bear put spread

•If all options are European a box spread is worth the present value of the difference between the strike prices

•If they are American this is not necessarily so (see Business Snapshot 11.1)

Butterfly Spread Using Calls

Figure 11.6, page 242

Butterfly Spread Using Puts

Figure 11.7, page 243

Calendar Spread Using Calls

Figure 11.8, page 245

Calendar Spread Using Puts

Figure 11.9, page 246

A Straddle Combination

Figure 11.10, page 246

Strip & Strap

Figure 11.11, page 248

A Strangle Combination

Figure 11.12, page 249

FIX FORMULAS FOR BINOMIAL TREE

LECTURE 5

Binomial Tree

u

d

Binomial Tree

probability of an up move formula

Stochastic Processes

•Describes the way in which a variable such as a stock price, exchange rate or interest rate changes through time

•Incorporates uncertainties

Markov Processes (See pages 280-81)

•In a Markov process future movements in a variable depend only on where we are, not the history of how we got to where we are

•Is the process followed by the temperature at a certain place Markov?

•We assume that stock prices follow Markov processes

•In Markov processes changes in successive periods of time are independent

•This means that variances are additive

•Standard deviations are not additive

Weak-Form Market Efficiency

•This asserts that it is impossible to produce consistently superior returns with a trading rule based on the past history of stock prices. In other words technical analysis does not work.

•A Markov process for stock prices is consistent with weak-form market efficiency

Black Scholes Call Option Formula

*C *= *S _{0} N *(

*d*

_{1}) -

*K e*(

^{-rT}N*d*)

_{2}Black Scholes Put Option Formula

*p *= *K e ^{-rT} N *(

*-d*) - S

_{2}_{0 }N(-d

_{1})

Black Scholes d_{1} and d_{2}

What is implied volatility?

The implied volatility is the volatility that makes the Black–Scholes-Merton price of an option equal to its market price. It is calculated using an iterative procedure.

Theta

Theta of a derivative (or portfolio of derivatives) is the rate of change of the value with respect to the passage of time

The theta of a call or put is usually negative. This means that, if time passes with the price of the underlying asset and its volatility remaining the same, the value of a long call or put option declines

Gamma

•Gamma (G) is the rate of change of delta (D) with respect to the price of the underlying asset

•Gamma is greatest for options that are close to the money

Delta

•Delta (D) is the rate of change of the option price with respect to the underlying

Delta Hedging

•This involves maintaining a delta neutral portfolio

•The delta of a European call on a non-dividend paying stock is N (d 1)

•The delta of a European put on the stock is

Vega

•Vega (v) is the rate of change of the value of a derivatives portfolio with respect to volatility

Managing Delta, Gamma, & Vega

•Delta can be changed by taking a position in the underlying asset

•To adjust gamma and vega it is necessary to take a position in an option or other derivative

Rho

Rho is the rate of change of the value of a derivative with respect to the interest rate

All formulas for

Delta, Gamma, Theta, Vega, Rho

Lower Bound for European Call Option Prices; No Dividends (Equation 10.4, page 220)

Put-Call parity with dividend

c + Ke^{-rT}+D = p + S_{0}

What is implied volatility? How can it be calculated?

The implied volatility is the volatility that makes the Black–Scholes-Merton price of an option equal to its market price. It is calculated using an iterative procedure.

The Lognormal Property

•Since the logarithm of S_{T} is normal, S_{T} is lognormally distributed

N'(d1) (N Prime) formula

Hedge BEta

To hedge the risk in a portfolio the number of contracts that should be shorted is

[image]

where V_{A} is the value of the portfolio, B is its beta, and V_{F} is the value of one futures contract.

(Gamla B - Nya B) om nya aer mindre.

Put-Call Parity - Market to BS

Daily Volatilities

•In option pricing we measure volatility “per year”

•In VaR calculations we measure volatility “per day”

•Theoretically, sday is the standard deviation of the continuously compounded return in one day

•In practice we assume that it is the standard deviation of the percentage change in one day

Standard Deviation of Portfolio

The Linear Model and Options

Consider a portfolio of options dependent on a single stock price, S. If d is the delta of the option, then it is approximately true that

Quadratic Model