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Financial engineering

Andrey Dung

I. FORWARDS, FUTURES AND SWAPS

A. Forwards

1) Motivation: A farmer devotes his whole land to plant wheat this year. One of his biggest

concerns would be the future price of wheat. If the price goes down, then he won’t be able to

repay bonds.

The same thing happens to a fast food restaurant. They usually fix the price of a burger, and

therefore need to make sure that the price of the bread won’t change over time.

So how is that done? In finance there is forward contract, which allows people to fix the

delivery price at K at a future date T. There are two parties of a contract:

• Long party: agree to buy the asset at the price K at time T

• Short party: agree to sell the asset at the price K at time T

Several notes:

• It costs nothing to enter forward contract, but after the contract, the agreement is a must.

• It can be either through a financial market, or over-the counter contract.

• No win situation: as for example if the real future price goes up, one of the two parties

will get the lost.

2) Arbitrage Argument: Due to the arbitrage argument: no existence of risk-free profit

• Underlying provides no income

F (t, T ) = er(T−t)St

• Known cash income

F (t, T ) = er(T−t)(St−It)

• Known yield q

F (t, T ) = e(r−q)(T−t)St

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3) Different types of forward: Stock indices

Foreign currencies

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Commodities Interest rate

B. Futures

C. Swaps

II. HEDGING

III. OPTIONS

IV. BLACK-SCHOLE-MERTON MODEL

1) Model for the stock price: In Black- Scholes model, the stock price is modeled as an

exponential of a Brownian motion

St = S0e(m−σ2/2)t+σBt

dST = mSTdt+ σStdBt

where m is the interest rate, and Bt is normal distribution with mean 0 and variance t used

to create the randomness. At a fixed time t, St follows log normal distributionE[St] = emtS0

var[St] = S20e

2mt(eσ2t − 1)

Below is a typical realization of St

2) Model for options: Setup a portfolio

Π = ∆S − f

For no arbitrage, the hedge ratio must be:

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./lognormal.png

∆ = ∂f∂S

Then we have the no arbitrage equation dΠ = rΠdt and this leads to the formula to find f(s,t)

is

12σ2S2 ∂2f

∂S2 + rS fS

+ ∂S∂t

= rf

Important note: this equation does not contain m, but only utilizes the risk-free rate r

The solution to the above equation is the risk-neutral Pricing formula

f(S, t) = e−rtEt,S[F (ST )]

and the boundary condition: f(s, T ) = F (S)

Now let’s apply this formula to different options

1) Forward Contracts

Forward Contracts F (S) = S−K then we have f(S, t) = e−rt(Et,S[ST ]−K) = S−e−rtK

2) Call option

European call option F (S, T ) = (S −K)+ then we have

C(S, t) = SN(d+)− e−rtKN(d−)

d− = ln(S/K)+µτσ√τ

, d+ = d− + σ√τ

τ = T − t, µ = r − σ2/2

3) Put option

Put option F (S, T ) = (K − S)+ then we have

P (S, t) = e−rτKN(−d−)− SN(−d+)

Call and put pricing formulas satisfy the put call parity C − P = S − e−rτK

3) In case with dividends:

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