Derivatives Stuff

### Put-Call Parity

From Wikipedia: put-call parity defines a relationship between the price of a call option and a put option—both with the identical strike price and expiry. To derive the put-call parity relationship, the assumption is that the options are not exercised before expiration day, which necessarily applies to European options.

C(t) + K * B(t,T) = P(t) + S(t)

C(t)     = value of call

P(t)     = value of put
S(t)     = value of underlying share
K          = string price
B(t,T) = value of a bond maturing at T.

Let's break this down from the right side:
P(t) + S(t) is a position of the put and a stock. This position exposes us to all the upside of the stock and none of the downside.

Now let's break this down from the left side

C(t) + K * B(t,T) is a position consisting of a call and a bond, such that the value of bond at maturity is enough to cover the strike price. This position also exposes the investor to all of the upside and none of the downside of the stock.

### Fixed Rate on a Vanilla Swap

A swap is equivalent to two bonds: the fixed payer can issue a fixed-coupon bond and invest the proceeds in a floating rate bond. In a swap, fixed rate must be set such that value of float and value of fixed bonds are the same.

Cash flows for the fixed note:

1000 = (FR /[1+R1]) + (FR /[1+R2]) + (FR /[1+R3]) + (FR /[1+R4]) + (1000 /[1+R4])
1000 = FR*{(1 /[1+R1]) + (1/[1+R2]) + (1/[1+R3]) + (1/[1+R4])} + (1000 /[1+R4])

This works out to:

1 - 1/(1 + R4)
FR =  ----------------------------------------------------- * 1000
(1/(1+R1)) + (1/(1+R2)) + (1/(1+R3)) + (1/(1+R4))

Rn is the 'discount factor' rather than periodic rates. The computation is horrible but there's a possible simplification. 1/(1+Rn) can be viewed as the present value of a zero-coupon bond with par of one dollar if the bond matures in n periods. So, if we have the spot rates: Z1, Z2, Z3, Z4:

1 - Z4

FR = --------------------
Z1 + Z2 + Z3 + Z4

### put/call parity for an option on a futures contract

```   Call + (X - Ft)/(1+Rf)^t = Put
Ft is the price of that futures contract today

Call = Put - (X - Ft)/(1+Rf)^t
Which makes sense because:
The call gets cheaper as the strike price goes up
The call gets more valuable as the Ft (current price) goes up.
```