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Financial Engineering
Discerning complexity of Financial Market
S. ARUNAGIRI
Presentation
Financial Engineering – p. 1
Overview
A quick perspective is presented on
complex nature of financial products [assets(stocks), derivatives (options)]
approach to quantify (measure) thecomplexity
Financial Engineering – p. 2
Financial Institutions
Trade financial products
Manage portfolio minimising risk
Financial Engineering – p. 3
Financial Engineering
Pricing financial products correctly
Hedging risks effectively
Financial Engineering – p. 4
Financial Engineering
Mathematically replicates
growth of assets and options
risk free portfolio
Growth of assets and options in time
Unpredictable although expected to beearning
Unpredictability making the growth complex
Financial Engineering – p. 5
Financial Engineering
Understanding and incorporating thecomplexity
in mathematical framework
ensuring positive payoff
How complex is asset and option evolution?
How to represent and quantify thecomplexity?
Financial Engineering – p. 6
Brownian Motion
Diffusion of ink in water
The diffusion pattern during a finite time cannot be reproduced.
This is said to be Stochastic Process.
Financial Engineering – p. 7
Asset growth
Let S be the stock price. Growth of the asset
dS = St − S0 {0 ≤ t < T}
Expected risk free return @ µ during dt is
dS = µSdt deterministic
Volatility @ σ during dt is
dS = σSdW random
Financial Engineering – p. 8
Asset growth
dS = µSdt+ σSdW
Asset growth consists of two interdependentparts, as above,
deterministic
random.
What is dW?
dW is the stochastic process known asGeometric Brownian Motion
Financial Engineering – p. 9
Wiener Process (GBM)
dW = W (ti+1)−W (ti) {dt = ti+1 − ti}
where i = 0, 1, 2, 3, · · ·
Financial Engineering – p. 10
Wiener Process (GBM)
for 0 < t1 < t2 <· · · < ti < ti+1,W (ti+1)−W (ti)are
stationeryincrements
dependent onlyon ti+1 − ti
mutuallyindependent
Financial Engineering – p. 11
Wiener Process (GBM)
W (0) = 0
W (t) normally distributed for every t > 0
Financial Engineering – p. 12
Asset Price
The Wiener process (Gemetric Brownian Motion)is
a mathematical model that replicates theevolution of asset over time.
According to this, the asset evolved in time to be
S(t) = S0e(µ− 1
2σ2)t+σW
This shows that the growth rate is interdependenton µ and σ.
Financial Engineering – p. 13
Option Pricing: BS formula
Option, V (S, t, · · · ),
is written on an asset, S(t)
has expiry time, t, etc.
Black-Scholes-Merton (1973):
Emergence of modern financial market in 90’s
Birth of new discipline in 2000’s called
Financial Engineering
Quantitative Finance
Mathematical Finance
Financial Engineering – p. 14
That’s it!
THANKS!
Financial Engineering – p. 15
