5 Applications in Finance Flashcards
assuming set up
Assume we have probability space
brownian motion
(W(t))_{t in [0,T]}
take filtration to be the sigma algebra generated by this brownian motion
F_t=σ(W(s); s <= t)
WLOG assume F=F_T
for financial markets we need
processes we need two models
riskless asset
one models cash
by differential equation
dB(t)=rB(t).dt
B(0)=1
B(t) is the value of £1
thinking of it as a bond, the value in the future? invested in account with interest rate r considered a riskless asset
solution of this
B(t)= B_0exp(rt)
r will be the interest rate of the market
reminder solution of the diff
risky asset model
stock model
dS(t) = µS(t) dt + σS(t) dW(t),
S(0) = s_0,
where µ > 0 is known as the trend of the stock
we know the solution of this!!
linear stochastic differential equation
Black scholes model
the two models given
recap solution of linear stochastic differential eq
if we didint have the stochastic part solution would be exponential
dS(t) = µS(t) dt + σS(t) dW(t),
coefficients
µ
σ
coefficients
µ risk coefficient
how fast to infinity as behaviour contributes to exponential
σ volatility
larger sigma larger fluctuations about” exponential trend”
example:
you own a stock
and we agree today that in one year you will sell the stock, I have the right to buy from you the stock for fixed value K
if value of stock at terminal time
X(T)<K i will not buy/not exercise
X(T)≥ K I will exercise the right
Gain/PAYOFF is X(T)-K is positive
What is the fair price?
dep of value of the stock?
European call option
The most classical derivatives in the market are the European options (call and put). In the
European call option the owner has the right (but no obligation, in contrast to a forward contract)
to buy the stock at time T at a predetermined price K, which is called the strike price. If the value
of a stock is above K, that is, if S_T > K then the owner of the contract will buy the stock for K,
which leads to a profit (S_T − K). If the price of the stock is below K, that is, if S_T ≤ K then the
owner will not exercise his right to buy the option because that would lead to a loss. In this case
the profit is just 0
Therefore, the pay-off is (S_T − K)+, that is, g(x) = (x − K)+
european put option
The European
put option, is a derivative that gives to its owner the right to sell the stock at the maturity T for a
fixed price K. Similarly to the above, it follows that the pay-off is (K − ST )
+, that is, the pay-off
function in this case is given by g(x) = (K − x)+
fair price?
In order to find the fair price of a contract with pay-off g(ST ), the idea is to “replicate” the
derivative by trading the riskless and the risky asset. This means, we want to find a portfolio (a
strategy for trading those two assets) which at the terminal time has value exactly g(ST ). It is
natural (fair) then to say that the price of the derivative at time zero is equal to the value of the
portfolio replicating the derivative (provided that this value is unique). Let us make this more
precise mathematically.
also as it turns out the coefficient for exponential growth doesnt play any role in this price of the contract: if we had two stocks one which grows faster than the other but both fluctuate the same we might expect different prices but this is not the case
thm 5.1.1.
consequence of girsanovs thm
There exists a unique probability measure P∗ on FT such that
* P and P∗ are equivalent, that is, for all A ∈ FT , P(A) = 0 if and only if P∗(A) = 0
* The process S˜ is a martingale under P∗
The above theorem is a consequence of Girsanov’s theorem. Indeed, if we define
P∗(A) := E[1_A exp (− θW(T) −1/2θ^2T]
, θ =(µ − r)/σ
then under P∗the process W∗(t) = θt + W(t) is an F-Wiener process
summary
we have the thm previously
There exists a unique probability measure P* on (omega F) with F_T=F
s.t the original measure P and P* are equivalent when one 0 iff other 0 for same event
Process S*(t)= exp(-rt)S(t))
account for discounted interest rate
This adjustment accounts for the time value of money, reflecting the fact that future cash flows are typically worth less in present terms due to the opportunity cost of capital
which is a martingale wrt F generated by the BM
this is a stochastic process on the probability space (Ω,F,P*)
and martingale on this space
dep on Girsanov thm
also
E[characteristic set of a * exp(-θ_p(t_ - θ^2-0.5θ^2t)
where θ is coeffient in u -inrerest rate over C
By girasanovs thm this is a new probability measure
BM W* on this probability space
from now on we work in this probability space
filtration stays the same
W star and P star used
not visible camera:notes summary
For the stock price i have an equation in W
I want to express wrt W*
W∗(t) = θt + W(t) is an F-Wiener process
with , θ =(µ − r)/σ
dS(t) = µS(t) dt + σS(t) dW(t)
= rS(t) dt + σS(t) (θ dt + dW(t)
(or use trick +θ(t)-θ(t)
dS(t) = rS(t) dt + σS(t) dW∗(t), S(0) = s0.
we see that in this equation no µ: the rate of drift doesnt affect/play a role in the solution
The only thing we see here is volatility an intersect
to summarise:
If we want with this brownian motion in this probability space
then equation for the stock price becomes
dS(t) = µ σ(t)dt + σ(t)S(t) dW*(t)
From now we always work in this probability space
reminder solution of this differential eq
____
solution equals
S(t)=S_0 exp((r-0.5σ²)t) +σW*(t))
so this is the value of the stock
let’s justify this is a solution….!!
We can check that this is indeed the solution to the Black-Scholes equation, using Ito’s lemma:
dS(t) = rS(t) dt + σS(t) dW∗(t),
S(0) = s0.
I’m going to check (hinting?) this stochastic process satisfies this stochastic differential
dS(t) = rS(t) dt + σS(t) dW∗(t),
S(0) = s0.
S(t)=S_0 exp((r-0.5σ²)t) σW*(t))
so this is the value of the stock
this is an ito process
deterministic part and stochastic part
as usual set
Y(t)=(r-0.5σ²)t+σdW*(t)
u(x)=S_0exp(x)
u’(x)=u”(x)=u(x)
also
dY(t) = (r-0.5σ²)dt +σdW*(t)
so differential of S(t)= differential of u(Y(t)
by itos formula
dS(t) = du(Y(t))
=u’(Y(t)** (r-0.5σ²)dt + u’(Y(t)) (σ)dW*(t) +0.5u’‘(Y(t)) (( |σ|)**^2) dt
=u(Y(t)** (r-0.5σ²)dt + u(Y(t)) (σ)dW*(t) +0.5u(Y(t)) (( |σ|)**^2) dt
=s_0(exp(Y(t)) (r-0.5σ²)dt +s_0(exp(Y(t)) ** (σ)dW*(t) +0.5s_0(exp(Y(t))σ²dt
using the original S(t)=S(0)=s_0 exp(Y(t))
=S(t)r dt +S(t)(-0.5σ²)dt + S(t)σdW(t) +0.5S(t)σ²
so these two cancel
=rS(t)dt +σdW(t)
showing S solves the equation
S(0)=s_0 exp((r-0.5σ²)0+σW*(0))
=s_0exp(0)=s_0
showing satisfies IC
by the way since we found the solution:unique by thm as coefficients are linear functions of the sol
stochastic eq with lipshitz we have unique sol which we have found
