Replication and risk-neutrality on the binomial tree
8.4 General no-arbitrage condition
Consider a general version of the binomial tree shown in Figure 8.3, where at each node the stock can move from price Sn–1either to price Sn= Sn–1(1 + u) or Sn= Sn–1(1 + d), where d< u. We assume the probability of an up move and a down move are both non-zero. Let the constant interest rate be r, and the size of each time step be T = 1. (In Chapter 10 we will let T become small.)
Theorem The binomial tree is arbitrage-free⇐⇒ d < r < u.
Proof To prove the right implication, we suppose r≥ u and demonstrate that we can con-struct an arbitrage portfolio. In particular, we sell one stock at time T = 0 for S0and invest proceeds at r. At time T = 1 we have S0(1 + r) of cash, and the short stock pos-ition is worth either S0(1 + u)≤ S0(1 + r) or S0(1 + d)< S0(1 + r). Therefore, this is an arbitrage portfolio since the probability of the stock price equalling S0(1 + d) is non-zero.
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A similar argument applies if r≤ d. We construct an arbitrage portfolio by borrow-ing S0and buying one stock.
To prove the left implication, consider all portfolios consisting ofλ stocks and μ bonds, withλ, μ ∈ R, which have zero value at T = 0. We need to show that all such portfolios are arbitrage-free.
The value of the portfolio at T = 0 isλS0+1+rμ , and so portfolios with zero value must be of the form
(λ, μ) =
ν
S0, –ν(1 + r)
, ν ∈ R.
Write V(1) for the value of the portfolio at time T = 1.
(a) Ifν = 0, then trivially V(1) = 0.
(b) Ifν > 0, then if the stock goes down
V(1) =νS0(1 + d) S0
–ν(1 + r) < 0.
(c) Ifν < 0, then if the stock goes up
V(1) =νS0(1 + u)
S0 –ν(1 + r) < 0.
Therefore, there can be no arbitrage portfolios since there is always a non-zero prob-ability of the portfolio having negative value at T = 1. Note that it is important for this argument that both states have non-zero probabilities. 䊏 Suppose the probability of an up move on each node is p> 0, then
E(S1| S0) = S0(p(1 + u) + (1 – p)(1 + d)) = S0(1 + pu + (1 – p)d)
= S0(1 + r)⇐⇒ r = pu + (1 – p)d.
Therefore, the risk-neutral probability p∗is given by
p∗= r – d u – d.
We have thus established the following equivalent conditions.
Result No arbitrage portfolios⇐⇒ d < r < u ⇐⇒ 0 < p∗< 1.
That is, for the binomial tree, the absence of arbitrage portfolios is equivalent to the exist-ence of a unique risk-neutral probability p∗under which prices are discounted expected values of the payout at maturity.
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We can easily show that the replicating portfolio for a general derivative contract, with payoutγ in the down state and γ + β in the up state, γ , β ∈ R, is λ stocks and μ ZCBs, where
λ = β
S0(u – d)andμ = γ –β(1 + d) (u – d) . Therefore, the replication price is given by
β
the discounted risk-neutral expectation. Thus we have shown the equivalence of the replication and risk-neutral price in the general case.
When we have n identical time steps, the risk-neutral probability distribution for Snis binomial with
For a derivative with payout g(Sn) at time n, its price at time T = 0 is, therefore, given by 1 to obtain the Cox Ross Rubinstein formula (see Question 4). However, it is generally easier to take limits and work with continuous distributions. We do this in Chapter 10.
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8.5
exercises
1. Binomial tree: European and American puts
Consider a two-step binomial tree, where a stock that pays no dividends has current price 100, and at each time step can increase by 20% or decrease by 10%. The possible values at time T = 2 are thus 144, 108 and 81. The annually compounded interest rate is 10%.
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(a) Calculate the price of a two-year 106-strike European put using (i) a replication argument and (ii) risk-neutral expectation.
(b) Calculate the price of a two-year 106-strike American put using replication, and hence verify that the American put has price strictly greater than the European.
(c) Calculate the prices of a two-year 86-strike European put and American put. What is different to (b)?
2. Arbitrage on the tree
A stock that pays no dividends has price today of 100. In one year’s time the stock is worth 110 with probability 0.75, and 85 with probability 0.25. The one-year annually compounded interest rate is 5%.
(a) Calculate the forward price of the stock for a forward contract with maturity one year.
(b) Calculate the price of a one-year European put option with strike 100.
(c) Suppose you observe that the put option in part (b) has a market price of 4.
Determine an arbitrage portfolio and calculate how much profit is generated at time T = 1 by this portfolio.
3. General two-state world
A non-dividend paying stock has current price S0. In one year’s time there are two possible states of the world. The stock may be worth SU in state A, or SDin state B, with SU> SD. A derivative contract paysγ + β in state A and γ in state B, for some γ , β > 0. The annually compounded interest rate is a constant r.
(a) What is the forward price for the stock?
(b) Prove that the portfolio consisting of short one derivative and long stocks is hedged—that is, has the same payout in state A and state B—if
= β
SU– SD
.
Hence prove by replication that the price of the derivative is equal to 1
(c) Find the risk-neutral probability p∗ for state A. Hence calculate the discounted expected payout of the derivative under risk-neutral probabilities, and verify that your answer equals the replication price in (b).
(d) What relationships must hold between SU, SD, S0and r for the risk-neutral probab-ility to exist? What have we assumed about the actual probabilities of state A and state B?
(e) Ifβ = γ (SU/SD– 1), what is the derivative contract? Verify the price (b) makes sense.
(f) Ifβ = 0, what is the derivative contract? Again, verify the price (b) makes sense.
80 | replication and risk-neutrality on the binomial tree
4. Cox Ross Rubinstein formula
In the n-step binomial tree, the discounted risk-neutral expectation of the option payout is given by
1 (1 + r)n
n j=0
n j
p∗j(1 – p∗)n–jg
S0(1 + u)j(1 + d)n–j .
Consider a European call option with maturity n where g (Sn) = (Sn– K)+. By choosing m to be the least number such that
S0(1 + u)m(1 + d)n–m> K, show that the call option price is given by
S0B(n, a, m) – K
(1 + r)nB(n, p∗, m)
where B(n, p, k) is defined to be equal to P(X≥ k) with X ∼ binomial(n, p), and a = p∗(1 + u)
(1 + r) .
Hint Use the fact that
(1 – a) = (1 – p∗) (1 + d) (1 + r) .