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Intermediate Finance (Derivatives markets (Options (Different types of…

- - - - It is always reasonable to think that a investor will prefer more over less, and hence prefere more reward than less
        
        The utility function is increasing on reward
      - Investors dislike risk
        
        Investors always prefer less risk over more risk
        
        Utility function is decreasing on risk
      - What is risk and what is reward?
        
        Reward = Mean (expected return)
        
        Risk = Variance
        
        Concepts of risk tolerence:
        
        Risk averse - wil always pick the least risky option when confronted with two investments with the same E(r)
        
        For a risk averse investor - A is positive
        
        Risk neutral- the mean/E(r) is the only thing that matters
        
        For risk neutral investor - A is zero
        
        Risk loving -the riskier the better
        
        For a risk loving investor - A is negative
  - - - We use the arithmetic average of past obervations
    - - Var(x) = E((x-E(x))^2)
        
        Often denoted as Var(x) = sigma^2(x)
        
        The variance is the expectation of the squared deviation of a random variable from its mean, and it measures how far a set of (random) numbers are spread of from its mean . The variance itself is diffucult to interpretate
      - In reality we dont know probabilities, hence we replace the expected value with the arithmetic average of past observations
    - - Sigma(x) = (var(x))^0,5
        
        Is a measure that is used to quantify the amount of variation or dispertion of a set of data values. A low std indicates that the data points tend to be close to its mean value (expected value). A high std indicated that there is a big spread over a wider range of values. (easy to interpretate)
    - - To find the risk of a portfolio, one must know the degree to wich stocks move together
      - Cov(x,y) = E(xy)-E(x)E(y)
      - The correlation coefficent: p(xy) = Cov(xy)/std.dev(x)std.dev(y)
      - The correlation always between -1 and 1
  - - - The utility of a given risky return is just that (risk-free return) - the higher the risk-aversion, the lower certainty equivalent
  - - - C is a given level of utility
- - - - U(rc) = E(rc) - 1/2Avariance(rc) = rf + y(E(rp)-rf) - 1/2Ay^2variance(rp)
      - We wish to maximize the utility of the investor with respect to the proportion y
        
        Max. for y: rf + y(E(rp)-rf) - 1/2Ay^2*variance(rp)
        
        The firs derivative is set to zero - which gives: y=E(rp)-rf/Avariance(rp)
        
        Interpretation:
        -Higher level of risk aversion gives smaller amount invested in risky portfolio.
        -Higher level of volatility gives smaller amount invested in rp.
        
        Larger risk premium gives larger amount invested in rp.
  - - - For y < 1, the lending rate applies, which implies -> E(rc)=rfl + std.dev(rc)*(E(rp)-rfl)/std.dev(rp)
      - For y>1, the borrowing rate applies ->E(rc)=rfb + std.dev(rc)*(E(rp)-rfb)/std.dev(rp)
- - - - Independent risk sources across firms. Its possible to reduce to negligible levels. Its non systematic and diversifable.
    - - Risk that affect all firms, typical macro events. Related to marketwide risk. It is systematic and non diversifiable.
  - - - Minimizing -> var((1-as)rb+asrs)
  - - - Max. as: E(rp)-rf/std.dev(rp)
      - Se notes for this formula (long and tidious)
      - Optimal risky portfolio p: point of tangency with CAL with opportunity set of risky assets
        
        To solve:
        
        Obtain its weights
        
        Obtain its variance and expected return
  - - - To solve:
        
        Obtain fraction y
        
        Obtain the final weights
- - - - Tyipical a element of a matrix X is usually denoted Xij, where i denotes the row number and j denotes the column
      - A matrix with N rows and 1 column is known as a (column) vector - it has a dimension of N*1, also called a N-dimensional (column) vector
    - - Calculated entrywise - (array x + array y)
      - Both matrices must have the same dimension (N*K)
    - - Scalar product: let c and A be a scalar and a matrix, respectively
      - The matrix product: X23*Y32
      - The matrix product: In every matric product we have -> XnkYkm = Znm
        
        Where n and m denotes dimension, and k denotes conformability
      - Transposition
        
        The transpose of a matrix A, denoted by A´, is the original matrix A with its rows (rader) and columns (kolonner) interchanged (ombyttet)
        
        Transposition wil allow us to write summation like: SumXiYi -> as x´y (important thing to understand)
        
        Remember that in order to perform matrix addition the dimensions need to be the same. That is why we transpose matrices?? (sjekk dette opp)
  - - - Where rf1 denotes the risk-free rate multiplied by a vector of 1 in order to get into matrices form
      - a´is the transposed version of matrix a
    - - U = rf + a´(Er-rf1)-1/2AaSum*a
- - - - Allocation: The separation property - all investors choose the same portfolio of risky assets p (the tangency portfolio)
      - Pricing:: Mean variance return, therefor the expected beta-return representation holds
        
        Er-rf = Beta(r,A)(E(rA)-rf)
        Where beta(r,A)=Cov(r,A)/var(A)
        
        The expected value of any return is given by:
        Er -rf = Beta(r,rm)(E(rm)-rf)
        Where Beta(r,rm)=cov(r,rm)/var(rm)
  - - - The relevant measure of risk is the beta from the market portfolio, due to the fact that investors dont get any reward for bearing firm-specific risk
- - - - The systematic components are grouped into one single random variable with different sensitivities across different stocks: mi=Beta(i)(rm)
    - - Cov(Rhp,Rs&p)=BetaHP*variance(s&p)
      - BetaHP=cov(Rhp,Rs&p)/variance(s&p)
      - Then the relationship of any return is: E(Rhp)=alfa(hp)+beta(hp)*E(Rs&p)
    - - The idea is to find an optimal active portfolio with two concerns in mind:
        
        Attention can only be devoted to a limited numer of securities
        
        Balance between agressive strategies and diversification
      - A model for security selection
      - Passive strategy = rm (marked/index)
      - Active strategy: a portfolio A consisting of limited set of securities with return rA
      - The return of the portfolio: rp = (1-aA)rm + aArA
        
        The weight on A is the one that provides the highest slope of the CAL of P (see notes)
        
        Optimal A is the one with highest information ratio
- - - - Issuers are borrowers
        
        Can be governments (state or local) or companys
      - Holders are creditors
  - - - Indexed bonds
      - Catastrophe bonds
      - Asset-backed bonds
      - Inverse floaters
  - - - If the cupon rate is higher than the yield to maturity, then the price of the bond is lower 1 year forward (IMPORTANT)
      - If the yield to maturity is higher than the coupon rate, then 1 year forward the price wil be higher (IMPORTANT)
  - - - Bond issuers may not satisfy their obligations to repay their debt: bond default risk = credit risk
      - Yield to maturity has to be interpreted as the maximum possible yield to maturity (IMPORTANT)
    - - Is the yield spread between the bond and a riskless asset (US og Norwegian bonds for instance)
- - - - A forward contract is a agreement between two counterparties - a buyer and a seller. The buyer agrees to buy an underlying asset from the seller
      - The delivery of the asset occurs at a later time, but the price is determined at the time of purchase
    - - Futures are similar to forward, but feature formalized and standarized characteristics
      - An agreement to buy an underlying asset at some specified time in the future at a predetermined price
      - Key terms:
        
        Futures price: agreed-upon price at maturity
        
        Long position: agree to purchase
        
        Profit = spot price minus original futures price
        
        Short position: agree to sell
        
        Profit = original futures price minus spot
      - Types of contracts:
        
        Agricultural commodities, metals and minerals (including energy contracts), forreign currencies, fiancial futures (interest rates or stock indexes) etc..
      - Spot - futures parity: There are two ways of acquire an asset for some specified date in the future.
        
        Purchase it now and store it
        
        Take a long position in futures
        
        The two strategies must have the same financial cost - therefor it is a parity
        
        With a perfect hedge the futures payoff is certain - there is no risk!
        
        A perfect hedge should return the riskless rate of return (no arbitrage opportunity)
        
        This relationship can be used to develop futures pricing relationship
        
        If the spot-futures parity is violated, arbitrage is possible (not realistic): If the futures price is too high, short the futures and acquire the stock by borrowing at rf. If the futures price is too low, go long futures, short the stock and invest the rest in rf
    - - Forwards
        
        Trade OTC (known counterparty)
        
        Customized
        
        Counterparty risk
      - Futures
        
        Exchange traded (unknown counterparty)
        
        No counterparty risk
        
        Clearing houses and margin deposits
        
        Standardized
        
        Secondary trading - liquidity
  - - - Most trades are reversed and do not involve actual delivery
  - - - Short - positioned for fall in price
      - Long - positioned for increased price
    - - Long hedge - protecting against a rise in price
      - Short hedge - protecting against a fall in price
  - - - A call option gives the holder the right to purchase an asset for a specified price, called exercise or strike price, on or before some specified expiration date
      - A put option gives the holder the right to sell an asset for a specified price, called the exercise or strike price, on or before som specified expiration date
      - The purchase price of the option is called the premium
    - - Buy - long
      - Sell - short
      - Call
      - Put
    - - In the money: exercise of the option is profitable if
        
        Call: market price > exercise price
        
        Put: exercise price > market price
      - Out of the money: Exercise not profitable if
        
        Call: market price < exercise price
        
        Put: exercise price < market price
      - At the money: exercise price and asset price are equal
    - - The option can be exercised at any time before the specified expiration date
    - - The option can only be exercised on the specified expiration date
      - Put-call parity:(only valid for European options)
        
        Since the payoff on a combination of a long call and a short put are equivalent to leveraged equity, the prices must be equal. If prices are not equal there wil be arbitrage possibilities
        
        C + X/(1+rf)^T = S0 + P
    - - Stock options
      - Index options
      - Futures options
      - Forreign currency (FX) options
      - Interest rate options
      - Optionlike securities
        
        Callable bonds
        
        Convertible securities
        
        Warrants
        
        Collateralized loans
      - Exotic options
        
        Asian options
        
        Barrier options
        
        Lookback options
        
        Currency translated options
        
        Binary options
    - - An unlimited variety of payoff patterns can be achieved by combining puts and calls with different maturities (its like like Lego)
      - Protective put
        
        Limit loss - position: Long the stock and long the put
      - Covered call
        
        Some downside protection at the expense of giving up gain potential. Position: Own the stock and write a call
      - Other strategies
        
        Straddle: Long call and long put
        
        Spreads: A combination of two or more call options or put options on the same asset with different exercise prices or time to expiration
        
        Vertical or money spread: Same maturity, different exercise price
        
        Horizontal or time spread: Different maturity dates
    - - Intrisic value:: Profit that could be made if the option was exercised immediately
      - Time value: The difference between the option price and the intrisic value
      - Factors influencing option values: CALL
        
        Stock price (increases)
        
        Exercise price (decreases)
        
        Volatility of stock price (increases)
        
        Time to expiration (increases)
        
        Interest rate (decreases)
        
        Dividend rate (decreases)
      - Restrictions on option value: CALL
        
        Value cannot be negative
        
        Value cannot exceed the stock value
        
        Value of the call must be greater than the value of levered equity
      - Early exercise
        
        It never pays to exercise a Call option before expiration
        
        Always worth more than its intrisic value, but for American options early exercise is sometimes optimal.
    - - The ability to create a perfect hedge is key to replication arguments
      - Since we have locked in the year-end we can discount using the risk-free rate
      - Hence, we can express the opions value in terms of the current stock price, the stock price in year end is redudant
      - Dont need to know the options beta or expected return
    - - Set up perfectly hedged portfolio on each node, and work backwards through the tree. The portfolios are perfectly hedged over a tiny period. By continously revising the hedge ratio, the portfolio can remain hedged and earn the risk-free rate over each tiny interval
      - As the sub period get infinetessmall we approach what is called continously time, and with that assumption we can derive closed form formula for option valuation
        
        Black-Scholes Merton model
        
        N(d) can (loosely) be viewed as risk-adjusted probability that the call option will expire in-the-money
        
        If both N(d) terms is close to 1, then c=0 (the value of the call is about zero)
        
        For midrange values of N(d), c can be viewed as the present value of the calls potensial payoff, adjusted for the probability of expire in-the-money
        
        Assumptions for BSM:
        
        The stock will pay no dividends during the lifetime of the option
        
        Both r and std.dev are constant through the lifetime of the option
        
        Stock prices are continous (no jumps)
        
        Using the BSM: Hedge ratio - the number of stocks required to hedge against the price risk of holding one option
      - Portfolio insurance:
        
        Buy put: results in downside protection with unlimited upside potential
        
        Limitations:
        
        Tracking errors (if indexes are used)
        
        Maturity of puts may be too short
        
        Hedge ratios/delta change as stock values change