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    Risk-Neutral Probabilitites

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    Show that the risk-neutral probabilities in the Cox-Ross-Rubinstein
    model are given by equations (6.15).

    -------------------------------------------------------------------------------------
    (See attached pages for relevant information from the textbook)

    188 Chapter 6
    is preferable to taking a position in the futures contract and getting them only in the future.
    For example, there may be benefits to owning the commodity in the case of shortages,
    or to keep a production process running, or because of their consumption value. In such
    situations, the value of the futures contract becomes smaller than in the case in which there
    are no benefits in holding the commodity, so we conclude
    F(t) ? [S(t) + U¯ (t)]er (T?t)
    or
    F(t) ? S(t)e(r+u)(T?t)
    as the case may be. As a measure of how much smaller the futures price becomes, we define
    the convenience yield (which represents the value of owning the commodity) as the value
    y for which
    F(t)ey(T?t) = [S(t) + U¯ (t)]er (T?t)
    or
    F(t)ey(T?t) = S(t)e(r+u)(T?t)
    In general, any type of cost or benefit associated with a futures contract (whether it is a
    storage cost, a dividend, or a convenience yield) is called cost of carry. We define it as the
    value c for which
    F(t) = S(t)e(r+c)(T?t)
    6.3 Risk-Neutral Pricing
    We have hinted before that the absence of arbitrage implies that the contingent claims that
    can be replicated by a trading strategy could be priced by using expectations under a special,
    risk-neutral probability measure. In the present section we explain why this is the case. The
    main results of this section are summarized in figure 6.1.
    6.3.1 Martingale Measures; Cox-Ross-Rubinstein (CRR) Model
    The modern approach to pricing financial contracts, as well as to solving portfoliooptimization
    problems, is intimately related to the notion of martingale probability
    measures. As we shall see, prices are expected values, but not under the "real-world" or
    "true" probability; rather, they are expected values under an "artificial" probability, called
    risk-neutral probability or equivalent martingale measure (EMM).
    Arbitrage and Risk-Neutral Pricing 189
    Complete markets 
    Unique risk-neutral measure 
    One price, the cost of replication
    Financial Markets
    Arbitrage  No risk-neutral measures
    Incomplete markets 
    Many risk-neutral measures 
    Many possible no-arbitrage prices
    No arbitrage  Risk-neutral measures
    Expected value Solution to a PDE
    Figure 6.1
    Risk-neutral pricing: no arbitrage, completeness, and pricing in financial markets.
    We first recall the notion of a martingale: Consider a process X whose values on the
    interval [0, s] provide information up to time s. Denote by Es the conditional expectation
    given that information. We say that a process X is a martingale if
    Es [X(t)] = X(s), s ? t (6.14)
    (We implicitly assume that the expected values are well defined.) This equation can be
    interpreted as saying that the best possible prediction for the future value X(t) of the
    process X is the present value X(s). Or, in profit/loss terminology, a martingale process,
    on average, makes neither profits nor losses. In particular, taking unconditional expected
    values in equation (6.14), we see that E[X(t)] = E[X(s)]. In other words, expected values
    of a martingale process do not change with time.
    Recall our notation A¯ that we use for any value A discounted at the risk-free rate. We
    say that a probability measure is a martingale measure for a financial-market model if the
    discounted stock prices S¯ i are martingales.
    Let us see what happens in the Cox-Ross-Rubinstein model with one stock. Recall that
    in this model the price of the stock at period t +1 can take only one of the two values, S(t)u
    or S(t)d, with u and d constants such that u > 1+r > d, where r is the constant risk-free
    rate, and we usually assume d < 1. At every point in time t, the probability that the stock
    takes the value S(t)u is p, and, therefore, q := 1 ? p is the probability that the stock will
    take the value S(t)d. Consider first a single-period setting. A martingale measure will be
    ? given by probabilities p? and q := 1? p? of up and down moves, such that the discounted
    190 Chapter 6
    stock price is a martingale:
    ¯ ? S(0)d
    S(0) = S(0) = E?[S¯ (1)] = p? S(0)u + (1 ? p )
    1 + r 1 + r
    Here, E? = E0
    ? denotes the (unconditional, time t = 0) expectation under the probabilities
    ? ? p , 1 ? p?. Solving for p we obtain
    ? (1 + r ) ? d ? u ? (1 + r )
    p = , q = (6.15)
    u ?d u? d
    We see that the assumption d <1+r <u guarantees that these numbers are indeed positive
    probabilities. Moreover, these equations define the only martingale measure with posi-
    ? tive probabilities. Furthermore, p? and q are strictly positive, so that events that have zero
    probability under the "real-world" probability measure also have zero probability under the
    martingale measure, and vice versa. We say that the two probability measures are equiv-
    ? alent and that the probabilities p?, q form an equivalent martingale measure or EMM.
    In order to make a comparison between the actual, real probabilities p, 1? p and the risk-
    ? neutral probabilities p?
    , 1? p , introduce the mean return rate ? of the stock as determined
    from
    S(0)(1 + ?) = E[S(1)]
    Then a calculation similar to the preceding implies that we get expressions analogous to
    equations (6.15):
    (1 + ?) ?d u? (1 + ?)
    p = , 1 ? p = (6.16)
    u ?d u? d
    Thus we can say that
    the risk-neutral world is the world in which there is no compensation for holding the risky
    assets, hence in which the expected return rate of the risky assets is equal to the risk-free
    rate r .
    We want to make a connection between the price of a contingent claim and the possibility
    of replicating the claim by trading in other securities, as discussed in chapter 3. Denote now
    by ? the number of shares of stock held in the portfolio and by x the initial level of wealth,
    X(0) = x. The rest of the portfolio, x ? ?S(0), is invested in the bank at the risk-free
    rate r . Therefore, from the budget constraint of the individual, the discounted wealth X¯ (1)
    at time 1 is given by
    ¯X
    (1) = ?S¯ (1) + x ? ?S(0)

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    This solution calculates the risk-neutral probabilities in the Cox-Ross-Rubinstein model. Risk-neutral probability models are provided.

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