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Flashcards in Financial Engineering Deck (58):

Optimal Hedge Ratio 

(Page 57)

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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

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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

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If the spot price of an investment asset is S0 and the futures price for a contract deliverable in T years is F0, then....

F0 = S0erT


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

  In our examples, S0 =40, T=0.25, and r=0.05 so that

F = 40e0.05×0.25 = 40.50


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



F0 = S0 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 F0 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  

        (F0 – K )e–rT        
–the value of a short forward contract is

  (KF0 )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  

                F0 = S0 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 F0 > S0e(r-q)T an arbitrageur buys the stocks underlying the index and sells futures
•When F0 < S0e(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 



Consumption Assets Formula

Storage is Negative Income

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The Cost of Carry

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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 + S0        


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 


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Bull Spread Using Calls


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Bull Spread Using Puts


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Bear Spread Using Puts
Figure 11.4, page 240


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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)


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Butterfly Spread Using Calls
Figure 11.6, page 242


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Butterfly Spread Using Puts
Figure 11.7, page 243


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Calendar Spread Using Calls
Figure 11.8, page 245


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Calendar Spread Using Puts
Figure 11.9, page 246


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A Straddle Combination
Figure 11.10, page 246


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Strip & Strap
Figure  11.11, page 248


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A Strangle Combination
Figure 11.12, page 249





Binomial Tree



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Binomial Tree 

probability of an up move formula

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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

S0 N (d1) - K e-rTN(d2)


Black Scholes Put Option Formula

K e-rT N (-d2) - SN(-d1)


Black Scholes d1 and d2

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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 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 (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 (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 (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 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

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

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Put-Call parity with dividend

c + Ke-rT+D = p + S0


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 ST  is normal, ST  is lognormally distributed

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N'(d1) (N Prime) formula

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Hedge BEta

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


where VA is the value of the portfolio, B is its beta, and VF is the value of one futures contract.

(Gamla B - Nya B) om nya aer mindre. 

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Put-Call Parity - Market to BS

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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

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Standard Deviation of Portfolio

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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

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Quadratic Model

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