Financial Engineering 3

Brownian Motion

B(t) - stochastic process, which is 
continuous, with independent and normally distributed increments
Important properties:
	E(dW)=0
	((E(dW))^2) = dt
	dW∼N(0, (Sqrt[dt]))
	W(t+dt)-W(t) and W(t)-W(t-dt) are independent

First order rules:
	dt*dt = 0
	dt*dW = 0
	dW*dW=dt

Diffusion process

General form	dS = μ(S,t)dt + σ(S,t)dW, here W is the Standard Brownian motion = Random Walk = Wiener Process

Stock price dynamic (diffusion process)	dS = Sμdt + SσdW , also called Geopmetrical BM
Here
	dS - change in the stock price,
	μ - the drift (will be replaced later by r - risk free interest rate)
	σ - volatility of stock's returns (not price)

Lemma Ito

if x is a stochastic variable following a diffusion process dx = μdt + σdW, then
					df(x) = \(TraditionalForm
      \`\(∂f\/∂x\) dx + 
        \(1\/2\) \(∂\^2 f\/∂\^2 x\) \((dx)\)\^2\)

Bond Pricing

The value of a bond is a function time only and follows dynamic dP=rPdt, where r is the riskless interest rate.
The solution of this equation is P(t)=Exp[rt] (up to a multiplication by a constant).

Contingent Claims Pricing

The value of a contingent claim is a function of the time to maturity and stock price F(S,t)
Construct a riskless portfolio based on the contingent claim and the stock
F+δS - the portfolio that contains the contingent claim F ans δ units of the stock.
The change in value of the portfolio during a short time period is
					d(F+δS)=\(TraditionalForm
      \`\(-\(∂F\/∂t\)\) dt + 
        \(∂F\/∂S\) dS\)\(TraditionalForm
      \`\(+\(1\/2\)\) \(∂\^2 F\/∂S\^2\) \((dS)\)\^2 + 
        δdS\)

					d(F+δS)=\(TraditionalForm
      \`\(-\(∂F\/∂t\)\) dt + 
        \(∂F\/∂S\) \((μSdt + σSdW)\)\)\(TraditionalForm
      \`\(+\(1\/2\)\) \(∂\^2 F\/∂S\^2\) 
          \((μSdt + σSdW)\)\^2 + δdS\)
					
					d(F+δS)=\(TraditionalForm
      \`\((\(-\(∂F\/∂t\)\) + 
          μS ∂F\/∂S + 
          \(\(\(σ\^2\) S\^2\)\/2\) ∂\^2 F\/∂S\^2 + 
          δμS)\)\)\(TraditionalForm
      \`dt + \((∂F\/∂S + δ)\) σSdW\)
in order to make this portfolio riskless we choose δ=-∂(F/∂)S, so that the uncertain term disappears.  Then the portfolio F-∂(F/∂)SS is riskless and by no arbitrage assumption should earn the riskless rate of return.
A riskless portfolio dynamic must be d(F-∂(F/∂)SS)=r(F-∂(F/∂)SS)dt
Thus the pricing equation is 
					r(F-∂(F/∂)SS)dt=\(TraditionalForm
      \`\((\(-\(∂F\/∂t\)\) + 
            μS ∂F\/∂S + 
            \(\(\(σ\^2\) S\^2\)\/2\) 
              ∂\^2 F\/∂S\^2 - 
            \(∂F\/∂S\) μS)\) dt\)
which can be written as
					\(TraditionalForm
      \`\(\(\(σ\^2\) S\^2\)\/2\) ∂\^2 F\/∂S\^2 - 
          ∂F\/∂t + \(∂F\/∂S\) rS - 
          rF = 0\)		Black-Scholes-Merton equation

Bounday conditions
	F(0,t)=0
Terminal conditions
	F(S,0)=payoffAtMaturity(S)

The Black-Scholes-Merton Formula

The solutioon of the BSM equation for European Call and Put options with strike K and time to maturity t is
C = SN((Subscript[d, 1])) - (Ke^(-rt)) N((Subscript[d, 2])), where \(TraditionalForm
      \`d\_1 = \(ln(S\/K) + \((r + σ\^2\/2)\) t\)\/\(σ \@ t\), 
      \ d\_2 = d\_1 - σ \@ t\)
here N is the cumulative normal distribution.
the value of European Put option can be found from the put-call parity		P = C - S + (Ke^(-rt))