Pricing a Forward Contract: Pricing a Forward Contract let d(0,T) be discount rate for T periods.
S = d(0,T) F --- Why?
Buy one unit a S today and Forward contract to deliver it at T at price F. Cash flow is (-S,F) which is fully determined at t=0. Same as lending over (0,T).
Suppose F > S/d(0,T) ? Then what?
Slide2: Let d(0,T) = 1/(1+r) = 1/1.1. S = $1. Then F should be $1.10. If F > $1.10, then
borrow S today, buy a unit on spot market (long) and take a one unit short position in the forward market. At T, deliver the asset at price F receiving:
F -S/d(0,T).
Slide3: If F < S/d(0,T), then short the asset, sell it, take the proceeds and invest at 1/d(0,T). At time T, buy it with the proceeds, deliver it back and keep the profit S/d(0,T) - F.
Single Period Option Valuation: Single Period Option Valuation At period’s close price will be (uS, dS) with probability (p, 1-p) and u>d>0.
It is possible to borrow at the risk-free rate r such that R = 1+r.
To avoid arbitrage opportunities, u>R>d.
Let the exercise price be K.
Pricing a Call Option-Single period case: Pricing a Call Option-Single period case Think of a stock whose spot price S can go up (uS) or down (dS) at the end of the period and a risk free loan whose return is R.
S uS dS 1 R R
Pricing a Call: Pricing a Call A Call on a share will be worth:
C max(uS-K,0) max(dS-K,0)
Slide7: S uS dS R R C max(uS-K,0) max(dS-K,0) 1 We can use the first two to mimic the payoff for the third
Combining the first two allows us to construct any outcome:: Combining the first two allows us to construct any outcome: Let Cu = max(uS-K,0)
Let Cd = max(dS-K,0)
Purchase $x of the stock and $b of the risk-free asset and construct the Replicating portfolio:
ux + Rb = Cu
dx + Rb = Cd
Solve for x and b: Solve for x and b x = (Cu-Cd)/(u-d) ; b = (uCd-dCu)/R(u-d).
x+b = 1/R[(R-d)/(u-d) Cu + (u-R)/(u-d) Cd]
Simplifying, we get
x+b = 1/R[qCu + (1-q) Cd]
But x+b must equal C, the value of the call due to the no-arbitrage principle.
C = 1/R[ q Cu + (1-q) Cd]
For example: For example R = 1.10, u = 1.2, d= .9 , S = 100, k = 50.
C Cu=max(uS-K,0) = 120-50=70 Cd=max(dS-K,0)=90-50=40 C = 1/1.1[2/3 Cu + 1/3 Cd] = $54.54
For two periods...: For two periods... Expand the lattice as follows:
C Cu Cd Cuu Cud Cdu Cdd
Where do we get u,d ?: Where do we get u,d ? E(lnS) = p ln(u) + (1-p) ln(d)
V(lnS) = p(ln u)^2 + (1-p)(ln d)^2 - [E(lnS)]^2.
Let U = ln u , D = ln d, E(lnS)=vdt, V(ln S) = s^2 dt. Note that D = -U.
Square (1), add it to (2), solve for U to get U = ln u = s^2 dt + (v dt)^2. Take anti-log.
Continued: Continued Plug this back into either (1) or (2) and solve for p (using quadratic formula).
Taking limits as dt goes to zero…
p = -1/2 + 1/2 (v/s) dt^.5
u = exp(s dt^.5)
d = 1/u.
Example from Excel and Portfolio: Example from Excel and Portfolio
Put Valuation: Put Valuation To value a put, we used the same information and logic except that we choose:
Pu = max(K-uS,0); Pd = max( K-dS,0)
See spreadsheet for an example.
Early Exercise: Early Exercise American Options can be exercised at any time up to expiration.
In general, it does not pay to exercise a call option early. Why not?
In general, it does pay to exercise a put option early. Under what condition? Why?
See spreadsheet again for put options.
Simplico Gold Mine: Simplico Gold Mine Consider leasing a gold mine (with unknown remaining deposits) for a period of ten years.
the cost of mining gold is $200/ounce and 10,000 ounces can be mined per year.
the market price of gold is $400/ounce.
the value of the lease is...
Simplico Gold Mine cont.: Simplico Gold Mine cont. profit = 10,000x($400-$200) = $2 million per year.
PV = sum $2M/(1+r)^k , k = 1,…,10.
PV = $2M[1-(1/1.1)^10]x10 = $12.29 M.
Expand this thinking now: Expand this thinking now We are assuming the price of gold is constant at $400.
We are assuming interest rates are constant.
When the price of gold varies over time... It isn’t.
Interest rates can be highly volatile.
Then the lease is a derivative instrument whose underlying security is gold.
Simplico Revisited: Simplico Revisited Assume the price of gold is random from year to year.
Each year, the price either increases by a factor of 1.2 or decreases by a factor of .9.
The lease values are determined first for the final nodes and are…
= 10,000($value-$200)/(1+r).
Valuing the Lease: Valuing the Lease Lease values for the penultimate node(s) are determined as weighted averages of the current value plus the future values.
the weights are determined as before:
q = (R-d)/(u-d) and (1-q).
We price this derivative similar to a call option. (see spreadsheet presentation).
Valuation: Valuation At any node, the value of the lease is equal to the sum of the profit that can be made that year plus the risk-neutral expected profit of the lease in the next year, both discounted back one year.
Some thoughts: Some thoughts Simplico gold mine is worth about twice as much as we would have thought using our simpler analysis.
We assumed then that the price of gold was constant. We should realize now that by doing so we have an inconsistency -- if gold prices were constant, then gold is a risk-free asset with a zero rate of return.
More Examples: More Examples You have a portfolio consisting of two assets A & B which have negatively correlated returns. You expect R(A) to rise. How can a combination of puts and calls minimize the risk on your portfolio?
Buy a call on A and a put on B.
Slide25: Suppose the returns on A and B are positively correlated. Then what?
Buy a call on A (or B) and sell (write) puts on the other.
Slide26: Why would an investor simultaneously write a put and buy a call on one asset?
If the price rises, you exercise the call but the put will not be exercised.
If the price falls, you do not exercise the call but the put will be exercised.
You must think that the price is going to rise.
Butterfly Spread: Butterfly Spread Consider buying two calls with strike prices equal to K1 and K3 and simultaneously…
Writing two calls at a strike price of K2.
Characterize the payoff.
Slide28: S K2 chosen to be near current S 0 $ K2 K3 K1
Slide29: A manufacturer has total costs T = F + Vx where x is output. Profit is px - F-Vx. Clearly, if p > V will operate at x equal to optimal capacity. What’s the option here? The strike price?
The firm has the option to operate as long as price exceed the strike price of V.