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                                 OPTION PRICING THEORY

                                          - REVISION –

A/ Arbitrage proofs of basic relations for European options

* Assumption

      There are no transactions costs.
      All trading profits (net of trading losses) are subject to the same tax rate.
      Borrowing and lending are possible at the risk-free interest rate.

S0 = current stock price

K = Strike price of option

T = Time to expiration of option

ST = Stock price on the expiration date

r = continuously compounded risk-free rate of interest for an investment maturing in time T

c/ C = Value of European/ American call option to buy one share

p/ P = value of European/ American put option to sell one share

1. The upper and lower bounds for European call options

a. Upper bounds

      Call option gives the holder the right to buy one share of a stock for a certain price. No
       matter what happens, the option can never be worth more than the stock. Hence, the stock
       price is an upper bound to the option price: ���� ≤ ����0
      At maturity the option cannot be worth more than K. It follows that it cannot be worth
       more than the present value of K today: ���� ≤ �������� −��������

b. Lower bounds

      A lower bound for the price of a European call option on a non-dividend-paying stock S0
       – Ke-rT

c. General argument

      Consider the following two portfolios
       Portfolio A: one European call option plus a zero-coupon bond that provides a payoff of
       K at time T.
       Portfolio B: one share of stock
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      In portfolio A, the zero-coupon bond will be worth K at time T.
      If ST > K, the call option is exercised at maturity and portfolio A is worth ST.
      If ST < K, the call option expires worthless and the portfolio is worth K.
      Hence, at time T, portfolio A is worth max(ST,K)
      Portfolio B is worth ST at time T. Hence, portfolio A is always worth as much as, and can
       be worth more than, portfolio B at the option’s maturity. The zero-coupon bond is worth
       Ke-rT today. Hence,
                                  ���� + �������� −�������� ≥ ����0 �������� ���� ≥ ����0 − �������� −��������
      Because the worst that can happen to a call option is that it expires worthless, its value
       cannot be negative. This means that c≥0 and therefore ���� ≥ max⁡ 0 − �������� −�������� , 0)
                                                                                   (����

2. The lower bounds for European put options

a. Lower bounds

      For a European put option on a non-dividend-paying stock, a lower bound for the price is
       Ke-rT – S0

b. General argument

      Consider the following two portfolios
       Portfolio C: one European put option plus one share
       Portfolio D: an amount of cash equal to Ke-rT (or equivalently a zero-coupon bond paying
       off K at time T)
      If ST < K, the option in portfolio C is exercised at option maturity and the portfolio
       becomes worth K.
      If ST > K, the put option expires worthless and the portfolio is worth ST at this time.
       Hence, portfolio is worth max(ST,K) in time T.
      Portfolio D is worth K in time T. Hence, portfolio C is always worth as much as, and can
       sometimes be worth more than, portfolio D in time T.
                                ���� + ����0 ≥ �������� −�������� �������� ���� ≥ �������� −�������� − ����0
      Because the worst that can happen to a put option is that it expires worthless, its value
       cannot be negative. This mean that ���� ≥ max⁡ −rT − S0 , 0)
                                                         (Ke

3. The put-call parity relationship for European options

      Consider the following two portfolios
       Portfolio A: one European call option plus a zero-coupon bond that provides a payoff of
       K at time T.
       Portfolio C: one European put option plus one share
      Assume that the stock pays no dividends. The call and put options have the same strike
       price K and the same time to maturity T

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      The zero-coupon bond in portfolio A will be worth K at time T. If the stock price ST at
       time T proves to be above K, then the call option in portfolio A will be exercised. This
       means portfolio A is worth (ST-K)+K=ST at time T. If ST proves to be less than K, then
       the call option in portfolio A will expire worthless and the portfolio will be worth K at
       time T

       Value of portfolio at time T                            ST > K             ST < K
       Portfolio A         Call option                         ST – K                0
                           Zero-coupon bond                      K                  K
                           Total                                 ST                 K
       Portfolio C         Put option                             0               K - ST
                           Share                                 ST                 ST
                           Total                                 ST                  K
      In portfolio C, the share will be worth ST at time T. If ST proves to be below K, then the
       put option in portfolio C will be exercised. This means that portfolio C is worth (K- ST)+
       ST=K at time T. If ST proves to be greater than K, then the put option in portfolio C will
       expire worthless and the portfolio will be worth ST at time T.
      Both portfolio are worth ST at time T if ST > K and worth K if ST < K. In other words,
       both are worth max(ST,K) when the options expire at time T.
      Because of European, the options cannot be exercised prior to time T. Value today have
       to be identical. The components of portfolio A are worth c and Ke-rT today, the
       components of portfolio C are worth c and S today. Hence,
                                          ���� + �������� −�������� = ���� + ����0

4. An arbitrage argument why the value of American and European calls on non-dividend paying
stock have the same value

      Share of the stock is worth S0 and the zero-coupon bond is worth Ke-rT today
                                            ���� ≥ ����0 − �������� −��������
      Because the owner of an American call has all the exercise opportunities open to the
       owner of corresponding European call, we have���� ≥ ����. Hence,
                                            ���� ≥ ����0 − �������� −��������
      Given r>0, it follows that C>S0-K. If it were optimal to exercise early, then C would
       equal S0-K. We reduce that it can never be optimal to exercise early.
      An American call on a non-dividend-paying stock should not be exercised early because
       of 2 reasons:
           o Once the option has been exercised and the strike price has been changed for the
               stock price, insurance vanishes.
           o Time value of money, the later the strike price is paid out the better.
      Because American call options are never exercised early when there are no dividends,
       they are equivalent to European call options.


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* Arbitrage (3)

      Show c+Ke-rT < P + S0 false
          o At time t=0 (today)

                              Position                   Cash flow
                  + Long 1 call                             -c
                  + Long cash                             -Ke-rT
                  + Short 1 put                             +P
                  + Short 1 share                           +S0
                  + Long in the extra cash                  -X
                  -> Total                                   0


           o After T years, close position

                            Condition                     Cash flow
                + ST < K                                       0
                                                              K
                                                            -K+ST
                                                             -ST
                                                            +XerT
                                                         Total: +XerT
                + ST = K                                       0
                                                             +K
                                                               0
                                                             -ST
                                                            +XerT
                                                         Total: +XerT
                + ST > K                                    -K+ST
                                                             +K
                                                               0
                                                             -ST
                                                            +XerT
                                                         Total: +XerT
          o Whatever happens, cash flow is positive. Value of call option increases ->
             everyone put money in the bank -> rate decreases -> value of c+Ke-rT increases.
          o Share price increases and value of put option decreases -> P+S0 decreases.
      Show c+Ke-rT > P + S0 false
          o At beginning, t=0

                              Position                   Cash flow
                  + Long 1 put                              -P
                  + Long share                              -S0
                  + Short 1 call                            +c
                  + Short cash                            +Ke-rT
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                + Long in the extra cash                               -X
                -> Total                                                0
           o After T-years, t=T
                                  ST < K                      ST = K                           ST > K
             Cash flow             K-ST                          0                                0
                                   +ST                         +ST                              +ST
                                     0                           0                              K-ST
                                    -K                          -K                               -K
                                  +XerT                       +XerT                            +XerT
             Total                +XerT                       +XerT                            +XerT


B/ Binomial Tree Technique for Pricing Options

1. Arbitrage technique

      A derivative lasts for time T and is dependent on a stock

                                                         Su

                          S                              fu

                          f
                                                         Sd

                                                         f
      Consider the portfolio that is long ∆ shares and d
                                                        short 1 derivative
                                                       Su∆ - fu



                                                       Sd∆ - fd
                                                                             �������� − ���� ����
      The portfolio is riskless when �������� ∆ − ���� = �������� ∆ − �������� or ∆ =
                                                ����                          ����0 ����− ����0 ����
      Value of the portfolio at time T is �������� ∆ − ��������
      Value of the portfolio today is �������� ∆ − �������� ���� −��������
      Another expression for the portfolio value today is �������� ∆ − ��������
      Hence, ���� = ����∆ − (�������� ∆ − ���� )���� −��������
                                    ����
                                                                                            ���� �������� −����
      Substituting for ∆ we obtain ���� = [�������� + 1 − ���� �������� ]���� −�������� where ���� =
                                             ����                                              ����−����
      Explain




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           o Suppose that the option lasts for time T and that during the life of the option the
             stock price can either move up from S0 to a new level S0u (u>1) or move down
             S0d (d<1)
           o If the stock price moves up, the proportional is u-1 and the payoff is fu. If the
             stock price moves down, the proportional is 1-d and the payoff is fd.
           o Imagine a portfolio consisting of a long position in ∆ shares and a short position
             in one option.
           o If there is a up movement, the value will be ����0 ����∆ − ���� . If there is a down
                                                                        ����
                                                                                             �������� − ���� ����
               movement, the value will be ����0 ����∆ − �������� or ∆ =                            ����0 ����− ����0 ����
           o The portfolio is riskless and for there to be no arbitrage opportunities it must earn
             the risk-free interest rate. If we denote the risk-free interest rate by r, the present
             value of the portfolio is (����0 ����∆ − �������� )���� −��������
           o The cost of setting up the portfolio is ����0 ∆ − ����
           o It follows that ����0 ∆ − ���� = ����0 ����∆ − �������� ���� −�������� or ���� = ����0 ∆ 1 − �������� −�������� + ���� ���� −��������
                                                                                                  ����
                                              �������� −���� ����
           o We obtain ���� = ����0                                 1 − �������� −�������� + ���� ���� −��������
                                                                                  ����
                                           ����0 ����− ����0 ����
                         �������� 1−�������� −�������� +���� ���� (�������� −�������� −1)
               Or ���� =                                              or ���� = ���� −�������� [������������ + 1 − ���� �������� ]
                                           ����−����
                               ���� �������� −����
               Where ���� =       ����−����

2. Risk-neutral technique

      ���� = �������� + 1 − ���� �������� ���� −��������
               ����
      The variables p and (1-p) can be interpreted as the risk-neutral probabilities of up and
       down movements. The value of a derivative is its expected payoff in a risk-neutral world
       discounted at the risk-free rate.

                                                            p                          Su

                                 S                                                     fu

                                 f
                                                                                       Sd
                                                            1-p
      Since p is a risk-neutral probability ����0 ���� �������� = �������� ∗f���� + �������� ∗ (1 − ����)
                                                                   d
                                                                        ���� �������� − ����
      Alternatively, we can use the formula ���� =                         ���� −����




C/ Black-Scholes formula

1. Compute option value

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                                    ���� = ����0 ���� ����1 − �������� −�������� ����(����2 )
                                  ���� = �������� −�������� ���� −����2 − ����0 ���� −����1
Where:
                                          ln ����0 ���� + ���� + ���� 2 2 ����
                                  ����1 =
                                                       ���� ����
                                  ln ����0 ���� + ���� − ���� 2 2 ����
                          ����2 =                                   = ����1 − ���� ����
                                              ���� ����
2. Assumptions underlying the formula

        Stock price behaviour corresponds to the lognormal model with ���� and ���� constant.
        There are no transactions costs or taxes. All securities are perfectly divisible.
        There are no dividends on the stock during the life of the option.
        There are no riskless arbitrage opportunities.
        Security trading is continuous.
        Investors can borrow or lend at the same risk-free rate of interest.
        The short-term risk-free rate of interest, r, is constant.

D/ Applying derivative strategies (Chapter 11)

1. Bull spreads

        Be created by buying a call option on a stock with a certain strike price and selling a call
         option on the same stock with a higher strike price. Both option have the same expiration
         date.
        Strategy




        The profit from two options: dashed line. The profit from the whole strategy: solid line
         because the call price always decreases as the strike price increases, the value of option
         sold is always less than the value of the option bought.
        An investor who enters into a bear spread is hoping that the stock price will decline.
        Payoff from a bull spread created using calls

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       Stock price range      Payoff from long Payoff from short Total payoff
                              call option      call option
              �������� ≤ ����1                0                0                 0
          ����1 < �������� < ����2           ST – K1             0              ST – K1
              �������� ≥ ����2             ST – K1          K2 – ST           K2 – K1

       Where: K1 = strike price of the call option bought
                K2 = strike price of the call option sold
                ST = stock price on the expiration date of the options
      Three types of bull spread can be distinguished
          o Both calls are initially out of the money.
          o One call is initially in the money, the other call is initially out of the money.
          o Both calls are initially in the money.

2. Bear spreads

      An investor who enters into a bull spread is hoping that the stock price will increase.
      Be created by buying a put with one strike price and selling a put with another strike
       price.
      The strike price of the option purchased is greater than the strike price of the option sold.
      Strategy




      Payoff from a bear spread created with put options
       Stock price range     Payoff from long Payoff from short Total payoff
                             call option            call option
              �������� ≤ ����1            K2 – ST                ST – K1     K2 – K1
          ����1 < �������� < ����2          K2 – ST                   0        K2 – ST
              �������� ≥ ����2               0                      0           0


3. Box spreads
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      A combination of a bull call spread with strike price K1 and K2, and a bear put spread
       with the same two strike price.
      Strategy




      Payoff from box spread
       Stock price range    Payoff from long Payoff from short Total payoff
                            call option      call option
              �������� ≤ ����1              0             K2 – K1           K2 – K1
          ����1 < �������� < ����2         ST – K1          K2 – ST           K2 – K1
              �������� ≥ ����2           K2 – K1             0              K2 – K1


4. Butterfly spreads

      Involves positions in options with three different strike prices.
      Be created by buying a call option with a relatively low strike price K1, buying a call
       option with a relatively high strike K3, and selling two call options with a strike price K2,
       half way between K1 and K3.
      Strategy




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      Payoff
        Stock price range     Payoff from first   Payoff from second     Payoff from       Total payoff
                                 long call             long call          short call
              �������� ≤ ����1              0                    0                  0                 0
          ����1 < �������� ≤ ����2        ST – K1                  0                  0              ST – K1
          ����2 < �������� < ����3        ST – K1                  0             -2(ST – K2)         K3 - ST
              �������� ≥ ����3          ST – K1               ST – K3          -2(ST – K2)            0


5. Calendar spreads

      The options have the same strike price and different expiration date.
      Can be created by selling a call option with a certain strike price and buy a longer-
       maturity call option with the same strike price. The longer the maturity of an option, the
       more expensive it usually is.
      Strategy




6. Diagonal spreads

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      Both the expiration date and the strike price of the calls are different.
      This increases the range of profit patterns that are possible.

7. Straddle

      Involves buying a call and put with the same strike price and expiration date.
      The strike price is denoted by K. If the stock price is close to this strike price at
       expiration of the options, the straddle leads to a loss. However, if there is a sufficiently
       large move in either direction, a significant profit will result.
      Strategy




      Payoff from a straddle
       Stock price range      Payoff from call         Payoff from put             Total payoff
             �������� ≤ ����                  0                     K – ST                      K – ST
             �������� > ����               ST – K                      0                        ST – K


8. Strips and straps

      A strip consists of a long position in one call and two puts with the same strike price and
       expiration date.
      A strap consists of a long position in two calls and one put with the same strike price and
       expiration date.
      Strategy




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9. Strangles

      Sometimes called bottom vertical combination, an investor buys a put and a call with the
       same expiration date and different strike prices.
      Strategy




      Payoff
       Stock price range     Payoff from call       Payoff from put       Total payoff
              �������� ≤ ����1               0                  K1 – ST                K1 – ST
          ����1 < �������� < ����2             0                      0                     0
              �������� ≥ ����2           ST – K2                    0                  ST – K2


E/ The Greeks

1. Delta



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      Delta of an option, ∆, is defined as the rate of change of the option price with respect to
       the price of the underlying asset.
      Delta of European stock options ∆ ���������������� = ����(����1 ) where d1 is in BS formula and N(x) is
       the cumulative distribution function for a standard normal distribution. ∆ ������������ =
       ���� ����1 − 1.
                                                                                                             ∆Π
      Delta of a portfolio: depends on a single asset whose price is S is given by Δ���� where Δ���� is
       a small change in the price of the asset and ΔΠ is the resultant change in the value of the
       portfolio. The delta of the portfolio can be calculated from the deltas of the individual
       options in the portfolio. If the portfolio consists of a quantity wi of option i 1 ≤ ���� ≤ ����, the
       delta of the portfolio is given by ∆= ���� �������� ∆���� where ∆���� is the delta of the ith option.
                                                 ����=1

2. Gamma

      The gamma, Γ, of a portfolio of options on an underlying asset is the rate of change of the
       portfolio’s delta with respect to the price of the underlying asset.
      If gamma is small, delta changes slowly, and adjustments to keep a portfolio delta neutral
       need to be make only relative infrequently. However, if gamma is highly negative or
       positive, delta is highly sensitive to the price of the underlying asset.
      When the stock price moves from S to S’, delta hedging assumes that the option price
       moves from C to C’, when in fact it moves from C to C’’. The different C’ and C’’ leads
       to a hedging error.
      Suppose that ΔS is the change in the price of an underlying asset in a small interval of
       time Δt and ΔΠ is the corresponding change in the price of the portfolio. For a delta-
       neutral portfolio, it is approximately true that
                                                            1
                                            ΔΠ = Θ Δ���� + ΓΔ���� 2
                                                            2
       Where: Θ is the theta of the portfolio
      For European call or put option on a non-dividend-paying stock, the gamma is given by
            ���� ′ (���� 1 )
       Γ=                  where d1 and N’(x) are defined in BS formula.
            ����0 ���� ����

3. Theta

      The theta of a portfolio of option, Θ, is the rate of change of the value of the portfolio
       with respect to the passage of time with all else remaining the same.
                                 ΔΠ
      Specifically, Θ =               where ΔΠ is change in the value of the portfolio when an amount of
                                 ����
       time ���� passes with all else remaining the same.
      For European option on a non-dividend-paying stock
                                       ����0 ���� ′ (���� 1 )����
           o Θ ���������������� = −                                 − ������������ −�������� ����(����2 )
                                            2 ����
                                                                                      1              2
                 Where d1 and d2 are defined as ���� ′ ���� =                                   ���� −����       2
                                                                                      2����

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                              ����0 ���� ′ (���� 1 )����
            o Θ ������������ = −                         + ������������ −�������� ����(−����2 )
                                   2 ����
        Theta is usually negative for an option. This is because as time passes, with all else
         remaining the same, the option tends to become less value.
        When the stock price is very low, theta is close to zero.
        For an at-the-money call option, theta is large and negative.
        As the stock price becomes larger, theta tends to –rKe-rT
        There is uncertainty about the future stock price, but there is no uncertainty about the
         passage of time. Many traders regard theta as a useful descriptive statistic for a portfolio.

4. Rho

        The rho of a portfolio of options is the rate of change of the value of the portfolio with
         respect to the interest rate. It measures the sensitivity of the value of a portfolio to a
         change in the interest rate when all else remains the same.
        For European option
             o rho(call) = KTe-rTN(d2)
             o rho(put) = -KTe-rTN(-d2)

5. Vega

        The vega of a portfolio of derivative, ����, is the rate of change of the value of the portfolio
         with the respect to the volatility of the underlying asset.
        If vega is highly positive or negative, the portfolio’s value is very sensitive to a small
         changes in volatility. If it is close to zero, volatility changes have relatively little impact
         on the value of the portfolio. A position in the underlying asset has zero vega.
        For European call or put option on a non-dividend-paying stock, vega is given by
         ���� = ����0 �������� ′ (����1 )

6. Positions and strategies

a. A naked position

        Open to the financial institution is to do nothing.
        Works well if stock price below $50 at the end of the 20 weeks.
        The option then costs the financial institution nothing and it makes a profits.
        A naked position works less well if the call is exercised. The cost to the financial
         institution is 100,000 times the amount by which the stock price exceeds the strike price.

b. A covered position

        Involves buying 100,000 shares as soon as the option has been sold.


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      If the option is exercised, this strategy works well, but in other circumstances it could
       lead to a significant loss.

c. A stop-loss strategy

      Consider an institution that has written a call option with strike price K to buy one unit of
       a stock.
      The hedging procedure involves buying one unit of the stock as soon as its price rises
       above K and selling it as soon as its price falls below K. The objective is to hold a naked
       position whenever the stock price is less than K and a covered position whenever the
       stock price is greater than K.
      The procedure is designed to ensure that at time T the institution owns the stock if the
       option closes in the money and does not own it if the option closes out of the money.
      Strategy




      A stop-loss strategy, although superficially attractive, does not work particularly well as a
       hedging procedure. If the stock price never reaches the strike price of K, the hedging
       procedure costs nothing. If the path of the stock price crosses the strike price level many
       times, the procedure is quite expensive.

d. Delta hedging

      The delta hedging procedure just described is an example of dynamic hedging. It can be
       contrasted with static hedging, where the hedge is set up initially and never adjusted.
       Delta is closely related to the BS analysis.
      Expressed in terms of ∆, the portfolio is
          o – 1: option
          o + ∆: shares of the stock
      Black, Scholes and Merton valued options by setting up a delta-neutral position and
       arguing that the return on the position should be the risk-free interest rate.


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      Delta is negative means a long position in a put option should be hedged with a long
       position in the underlying stock, and a short position in a put option should be hedged
       with a short position in the underlying stock.

F/ Brownian motion

1. What we mean by a random walk with upwards drift in continuous time

      In the limit, as n goes to infinitely the n step process to standard Brownian motion in
       continuous time. This also called the standard additive random walk in continuous time.
      There are two things wrong with Brownian motion as a model for share movements
           o Share prices should follow a multiplicative random walk process, not an additive
               one. An additive process assumes that the share price changes by p pence
               whatever the level of the share price. It is more realistic to assume that share
               prices rise or fall by a percentage amount each period. Also, share prices can
               never become negative. A multiplicative random walk can never become negative,
               but an additive random walk can.
           o Share prices should tend to drift upwards over time in a rational market. The
               random walk considered so far does not. The mean of B(t) is zero. The expected
               value of the random walk is zero at all times t.
      Brownian motion with drift model C(t) is given by ���� ���� = �������� + ��������(����)
       Here ���� is the drift constant and B(w,t) is standard Brownian motion. ���� is a positive
       constant. The process has upwards drift if ���� is positive.

2. B(0) takes on the value 0 with probability 1

This is obvious. The discrete random walk Vn(t) starts at zero with probability 1, so B(t) does as
well

3. B(1) has the standard normal distribution with expected value, or mean, equal to zero, and
with variance equal to 1

This fact follows from the central limit theorem. In fact this is just a statement of one version of
the central limit theorem. As n goes to infinity the probability table for random variable Vn(1)
gets closer and closer to the density function of the standard normal distribution:
(2����)−1/2 exp⁡ 2 2)
              (−����

4. The probability table for Vn(t) gets closer and closer the density function of the normal
distribution with mean zero and variance t as the number of steps n in the discrete random walk
goes to infinity. This means that the SD of B(t), the square root of the variance is ����

      The normal density with             mean     zero   and    variance    ���� 2   is   given   by
       (2�������� 2 )−1/2 exp⁡ 2 2���� 2 )
                         (−����

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      The normal density with mean                   ����   and   variance   ���� 2   is   given   by
                         (−(���� − ����)2 2���� 2 )
       (2�������� 2 )−1/2 exp⁡

5. For real numbers s<t, B(t)-B(s) has a normal distribution with mean zero and variance t-s

      B(t) is equal to B(t)-B(0), the change in wealth from time 0 to time t.
      Start the game at time s with wealth B(s), then the change in wealth over the time period
       t-s is just B(t) – B(s).
      Start the game at time 0 or at time s. the tree of all possible future wins and losses looks
       the same at time zero as it does at time s or at any other time.
      Tree of all future changes in wealth looks the same at time s as it does at time 0 or at any
       other time. So the distribution of B(t)-B(s) must the same as that of B(t)-B(0).

6. If s<t≤u<v are times, then the random variables B(t)-B(s) and B(v)-B(u) are independent

Knowing the past tells us nothing about the future.

7. B(w,t) is a continuous function of t for fixed w

Any realisation of the random walk follows a path without jumps. The path traced out over time
is a curve with no breaks or gaps. This is reasonable property to expect. In the discrete games the
paths through time do jump, but the jumps get smaller as reducing the time interval between
tosses of the coin.

8. The paths of Brownian motion have no derivative at any point

This is also intuitively obvious. Suppose a curve has a tangent line at point p. The slope of the
tangent line gives us some idea of where the curve is going to next. But a random walk should be
completely unpredictable. So the paths of a random process have no derivative at any point of
the curve. The paths of Brownian motion are “infinitely jagged”.

G/ Volatility smiles

1. Smile in foreign currency options

a. Empirical results

      First step in the production of the table is to calculate the SD of daily percentage change
       in each exchange rate.
      The next stage is to note how often the actual percentage change exceeded one SD, 2
       SDs…
      The final stage is to calculate how often this would have happened if the percentage
       changes has been normally distributed.

b. Reason for the Smile in foreign currency options
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      Two of the conditions for an asset price to have a lognormal distribution are
           o The volatility of the asset is constant.
           o The price of the asset changes smoothly with no jumps.
      The impact of jumps and nonconstant volatility depends on the option maturity. As the
       maturity of the option is increased, the percentage impact of a nonconstant volatility on
       prices becomes more pronounced, but its percentage impact on implied volatility
       becomes less pronounced.

3. Smile in equity options

      The volatility decreases as the strike price increases. The volatility used to price an option
       with a low strike price is significantly higher than that used to price an option with a high
       strike price.
      Reasons
           o Leverage: as a company’s equity declines in value, the company’s leverage
               increases. This means that the equity becomes more risky and its volatility
               increases.
           o A company’s equity increases in value, leverage decreases. This equity then
               becomes less risky and its volatility decreases.

H/ Derivatives mishaps, failures, and scandals, LTCM and/or other case study (Chapter 25)

1. For all users of derivative

      Define risk limits
          o All companies define in a clear and unambiguous way limits to the financial risks
              that can be taken.
          o Companies monitor risks carefully when derivatives are used. Without close
              monitoring, it is impossible to know whether a derivatives trader has switched
              from being a hedger to a speculator or switched from being an arbitrageur to
              being a speculator.
      Take the risk limits seriously
          o It is tempting to ignore violations of risk limits when profits result. However, this
              is short sighted. It leads to a culture where risk limits are not taken seriously, and
              it paves the way for a disaster.
          o The penalties for exceeding risk limits should be just as great when profits result
              as when losses result. Otherwise, traders who make losses are liable to keep
              increasing their bets in the hope that eventually a profit will result and all will be
              forgiven.
      Do not assume you can outguess the market
          o Some traders are quite possibly better than others. But no trader gets it right all
              the time.

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           o A trader who correctly predicts the direction in which market variables will move
               60% of the time is doing well.
       Do not underestimate the benefits of diversification
           o When a trader appears good at predicting a particular market variable, there is a
               tendency to increase the trader’s limits.
           o Diversification enables the investor to reduce risks by over half. Another way of
               expressing this is that diversification enables an investor to double the expected
               return per unit of risk taken.
       Carry out scenario analyses and stress tests
           o The calculation of risk measures such as Var should always be accompanied by
               scenario analyses and stress testing to obtain an understanding of what can go
               wrong.
           o It is important to be creative in the way scenarios are generated and to use the
               judgement of experienced managers.

2. For financial institutions

       Monitor traders carefully
           o It is important that all traders – particularly those making high profits – be fully
               accountable.
           o It is important for the financial institution to know whether the high profits are
               being made by taking unreasonably high risks.
       Separate the front, middle and back office
           o Front office: traders who are executing trades, taking positions, and so forth.
           o Middle office: risk managers who are monitoring the risks being taken.
           o Back office: where the record keeping and accounting takes place.
       Do not blindly trust models
           o If large profits are reported when relatively simple trading strategies are followed,
               there is a good chance that the models underlying the calculation of the profits are
               wrong.
           o Getting too much business of a certain type can be just as worrisome as getting
               too little business of that type.
       Be conservative in recognizing inception profits
           o When a financial institution sells a highly exotic instrument to a nonfinancial
               corporation, the valuation can be highly dependent on the underlying model.
           o Recognizing inception profits immediately is very dangerous.
       Do not sell clients inappropriate products
           o It is tempting to sell corporate clients inappropriate products, particularly when
               they appear to have an appetite for the underlying risks.
           o This is short sighted.
       Do not ignore liquidity risk
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          o Financial engineers usually base the pricing of exotic instruments and other
              instruments that trade relatively infrequently on the prices of actively traded
              instruments.
          o It is dangerous to assume that less actively traded instruments can always be
              traded at close to their theoretical price.
      Beware when everyone is following the same trading strategy
          o It sometimes happens that many market participants are following essentially the
              same trading strategy.
          o This creates a dangerous environment where there are liable to be big market
              moves, unstable markets, and large losses for the market participants.
      Do not finance long-term assets with short-term liabilities
          o It is important for a financial institution to match the maturities of assets and
              liabilities.
          o If it does not to do this, it is subjecting itself to significant interest rate risk.
      Market transparency is important
          o Investors should have demanded more information about the underlying assets
              and should have more carefully assessed the risks they were taking.
          o It is easy to be wise after the event.
      Manage incentive
          o When loans are securitized, it is important to align the interests of the party
              originating the loan with the party who bears the ultimate risk so that the
              originator does not have an incentive to misrepresent the loan.
          o Require the originator of a loan portfolio to keep a stake in all the tranches and
              other instruments that are created from the portfolio.
      Never ignore risk management
          o When times are good or appear to be good, there is a tendency to assume that
              nothing can go wrong and ignore the output from stress tests and other analyses
              carried out by the risk management group.

3. For nonfinancial corporations

      Make sure you fully understand the traders you are doing
         o Corporations should never undertake a trade or a trading strategy that they do not
             fully understand.
         o It is surprising how often a trader working for a nonfinancial corporation will,
             after a big loss, admit to not knowing what was really going on and claim to have
             been misled by investment bankers.
      Make sure a hedger does not become a speculator
         o One of the unfortunate facts of life is that hedging is relatively dull, whereas
             speculation is exciting.
         o Clear limits to the risks that can be taken should be set by senior management.

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      Be cautious about making the treasury department a profit center
          o The treasurer is motivated to reduce financing costs and manage risks as
              profitably as possible.
          o The problem is that the potential for the treasurer to make profits is limited.
          o The goal of a hedging program is to reduce risks, not to increase expected profits.

4. Long term capital management (LTCM)

a. LTCM

      LTCM was doing risk arbitrage
           o In real world all arbitrage trades involve some risk.
           o Many institutional prefer to buy on-the-run bonds and avoid the off-the-run bonds.
      It doesn’t really take a genius to understand what LTCM was doing
           o The Treasury bond arbitrage is something that can be understood by anyone who
               has some knowledge of finance.
           o One of the myths about LTCM was that they were doing something so complex
               that no one else could understand it.
      It had all been done before away
           o Every investment bank has an arbitrage desk.
           o Bond desk trade: A bond arbitrage desk may have hundreds of traders, all doing
               arbitrage trades.
           o Salomon arbitrage trade hedged away risk in the swaps market.
      If there is mis-pricing in derivatives or bond markets it is almost always possible to
       exploit the mis-pricing using arbitrage
           o In studying the 1-step and multi-step binomial models, if call options or put
               options on stocks are mis-priced, then mis-pricing can be exploited through
               arbitrage.
           o Once mis-pricing has been identified we need to identify the arbitrage that we can
               use to exploit it.
      Meriwether set up LTCM, not Merton and Scholes
           o Merton’s and Scholes role was to help raise the finance, trading on their
               reputations as the smartest minds in academic finance.
           o Meriwether was the driving force behind the creation of LTCM.
           o Merton and Scholes were partners, but they did not trade themselves, and their
               active role seems to have been confined to the weekly risk meetings where
               strategies and trades were discussed.
      LTCM’s comparative advantage
           o Got financing on the best possible terms
           o Because LTCM did not pay the haircut, it took positions far larger than any other
               institution,

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          o There is no great mystery as to why LTCM made so much money in its first few
            years. Through taking on more leverage than other arbitrageurs.
          o It is not obvious that LTCM was smarter than the other arbitrageurs. They made
            more money because they got cheaper financing and had greater leverage.

b. LTCM’s mistake

      Lack of risk management
          o Meriwether was not a good risk manager.
          o Meriwether was always arguing for more capital to do larger trades.
      Greed
          o In some years, there are many good opportunities for arbitrage. But in other years
              not. Because the exploitation of arbitrage opportunities by LTCM and other
              arbitrageurs had eliminated opportunities.
          o The smart thing to do would be to cut back on the size of LTCM’s positions.
          o Merger arbitrage: the stock price of the target will be slightly below the offer
              price, since there is some risk that the takeover will be abandoned. Buying target
              company stocks will yield a small profit if the takeover goes ahead, but a huge
              loss if the takeover fails.
      Clever idiocy
          o The BS model assumes that securities returns are normal distribution. The LTCM
              worldview was that the BS assumption is true for real world market.
          o In real world, markets correlations go to 1 or to -1 in a panic. All the riskier and
              less liquid assets go down together and all the really safe assets go up together.
              LTCM of course held the riskier side in every trade.
          o Assuming that the historical correlations were correct for the future, all of
              LTCM’s trades would go wrong at the same time about once every ten lives of the
              universe -> short sale.
      The collapse
          o When LTCM’s model stops working, mis-pricing increase dramatically.
          o In a panic other things go wrong as well.
          o LTCM’s positions were too big to liquidate. Selling them would tell the whole
              market what their positions were. And all the time their competitors were selling
              to the same assets LTCM owned, and buying the assets LTCM had shorted.
      The aftermath
          o Markets had returned to a more normal state, and some of LTCM’s positions
              returned to profit.
          o After LTCM Myron Scholes founded another hedge fund, Platinum Grove Asset
              Management.



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