MODELING AN ASSET PRICE

By Adnan Kuhait MSc Business Analytics

Course: Applied Analysis : Financial Mathematics Vrije Universiteit (VU) Amsterdam

Reference book: The mathematics of Financial derivatives, A student introduction by Paul Wilmott, Sam Howison and Jeff Dewynne Cambridge University Press

The absolute change in the asset price is not by itself useful

The absolute change in the asset price is not by itself useful

for example a change of 1$ is more significant in an asset price of 20 $ than if it is 200 $

Return

so we define a “return” to be the change in the price divided by the original price

this relative measurement is better than the absolute measurement

Return

so we define a “return” to be the change in the price divided by the original price

this relative measurement is better than the absolute measurement

in a small interval of time dt, asset price changed to S+ds

the return will be ds/S this has two parts:

First part predictable, deterministic and anticipated return just like investing in risk-free bank which equal to: ds/S = µ dt where µ is the average rate of growth of the asset price (called the drift) and it is considered to be constant. (in exchange rated it could be a function in S and t.

ds / S = µ dt

the return will be ds/S this has two parts:

Second part is the random change in the asset price in response to external effects. it is represented by a random sample drawn from a normal distribution with mean zero. (σ dX) where σ sigma is the volatility, measure the standard deviation of the returns. dX is the sample from the normal distribution.

ds / S = µ dt + σ dX

the return will be ds/S this has two parts:

Second part is the random change in the asset price in response to external effects. it is represented by a random sample drawn from a normal distribution with mean zero. (σ dX) where σ sigma is the volatility, measure the standard deviation of the returns. dX is the sample from the normal distribution.

Or ds = µ S dt + σ S dX

by taking sigma = 0 we will be left with ds= µ dt or ds/dt= µS since µ is constant, then this can be solved: S= S0e(t-t0) where S0 is the asset price at t0

the asset price is deterministic and can predict the future in certainty.

Wiener Process

dX , the randomness, is called Wiener Process.

Wiener Process

dX , the randomness, is called Wiener Process.

Wiener Process properties: • dX is a random variable, drawn from a normal

distribution. • the mean = 0 • the variance is = dt

Wiener Process

dX , the randomness, is called Wiener Process.

Wiener Process properties: • dX is a random variable, drawn from a normal

distribution. • the mean = 0 • the variance is = dt It can be written as:

𝑑𝑋 = 𝜑 𝑑𝑡 Where φ is a random variable drawn from standardised normal distribution which has zero mean and unit variance and pdf:

1

2𝜋 𝑒

−12 𝜑2

Wiener Process

dX , the randomness, is called Wiener Process.

Wiener Process properties: dX is scaled by dt because any other will lead to either meaningless Or trivial when dt0. which we are in particularly interested and it fits the real time data well

suppose todays date is t0and asset price is S0. if in the future time t1and asset price S1, then S1will be distributed around S0 in a bell shaped graph. the future price will be close to S0 the further t1 is from t0the more spread out this distribution is. if S represents the random-walk given by ds= µ S dt + σ S dX then the probability density function represented by this skewed curve is the lognormal distribution, and therefore ds= µ S dt + σ S dX is the lognormal random-walk. properties of the model: does not refer to the past history of the asset price, next asset price depends only on today’s price.(Markov properties)

in real life prices quoted in discrete time intervals, but for efficient solution we use continuous time limit dt 0 Result :

𝑑𝑋2 → 𝑑𝑡 𝑎𝑠 𝑑𝑡 → 0 𝑤𝑖𝑡ℎ 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 = 1

Ito’s Lemma

suppose f(S) is a smooth function of S using Taylor’s series:

𝑑𝑓 = 𝑑𝑓

𝑑𝑆 𝑑𝑆 +

1

2 𝑑2𝑓

𝑑𝑆2 𝑑𝑆2 + …

But dS = µ S dt + σ S dX Then (𝑑𝑆)2 = (µ S dt + σ S dX)2

(𝑑𝑆)2 = µ2 𝑆2𝑑𝑡2 + 2σµ𝑆2dtdX + 𝜎2𝑆2𝑑𝑋2

Since 𝑑𝑋 = 𝑂 𝑑𝑡

Then the last term is largest for small dt and dominate the other two terms. So we have: 𝑑𝑆2 = 𝜎2𝑆2𝑑𝑋2 Using the result 𝑑𝑋2 → 𝑑𝑡 then:

𝑑𝑆2 → 𝜎2𝑆2 𝑑𝑡

Ito’s Lemma Substituting in Taylors expansion:

𝑑𝑓 = 𝑑𝑓

𝑑𝑆 𝑑𝑆 +

1

2 𝑑2𝑓

𝑑𝑆2 𝑑𝑆2

𝑑𝑓 = 𝑑𝑓

𝑑𝑆 µ S dt + σ S dX +

1

2 𝑑2𝑓

𝑑𝑆2 (𝜎2𝑆2 𝑑𝑡)

Or

𝑑𝑓 = σ S 𝑑𝑓

𝑑𝑆dX + (µS

𝑑𝑓

𝑑𝑆+

1

2𝜎2𝑆2

𝑑2𝑓

𝑑𝑆2)𝑑𝑡

relating the small change in a function of random variable to the small change in the variable itself

Ito’s Lemma

this can be generalized to f(S,t)

𝑑𝑓 = 𝑑𝑓

𝑑𝑆 𝑑𝑆 +

𝑑𝑓

𝑑𝑡 𝑑𝑡 +

1

2 𝑑2𝑓

𝑑𝑆2 𝑑𝑆2

And doing the same, we will get:

𝒅𝒇 = σ S 𝒅𝒇

𝒅𝑺dX + (µS

𝒅𝒇

𝒅𝑺+

𝟏

𝟐𝝈𝟐𝑺𝟐

𝒅𝟐𝒇

𝒅𝑺𝟐+

𝒅𝒇

𝒅𝒕)𝒅𝒕

Which is Ito’s Lemma

Black-Scholes equation

suppose we have an option V(S,t) depends on S and t using Ito’s Lemma:

𝑑𝑉 = σ S 𝑑𝑉

𝑑𝑆dX + (µS

𝑑𝑉

𝑑𝑆+

1

2𝜎2𝑆2

𝑑2𝑉

𝑑𝑆2+

𝑑𝑉

𝑑𝑡)𝑑𝑡

assuming V has at least one t derivative and two for S

Black-Scholes equation

consider the portfolio: ∏ = V - ∆S then one jump in the value in one time-step is: d∏ = dV - ∆dS Where ∆ fixed in the interval [t,t+dt]

Black-Scholes equation

using Ito’s Lemma for the value of dV and the first model for dS: d∏ = dV - ∆dS

𝑑∏ = σ S 𝑑𝑉

𝑑𝑆dX + µS

𝑑𝑉

𝑑𝑆+

1

2𝜎2𝑆2

𝑑2𝑉

𝑑𝑆2+

𝑑𝑉

𝑑𝑡𝑑𝑡

− ∆ (σ S dX + µ S dt )

= 𝜎𝑆𝑑𝑉

𝑑𝑆 − ∆ dX +(µS (

𝑑𝑉

𝑑𝑆− ∆) +

1

2𝜎2𝑆2

𝑑2𝑉

𝑑𝑆2+

𝑑𝑉

𝑑𝑡)𝑑𝑡

Black-Scholes equation we want ∏ to be a bond thus it should be deterministic, that's mean the random term dX should be dropped. so take

𝑑𝑉

𝑑𝑆 = ∆

Then:

𝑑∏ = (1

2𝜎2𝑆2

𝑑2𝑉

𝑑𝑆2+

𝑑𝑉

𝑑𝑡)𝑑𝑡

But:

∏= ∏0ert

then d∏= r∏0ertdt d∏= r∏dt So

(1

2𝜎2𝑆2 𝑑2𝑉

𝑑𝑆2 +𝑑𝑉

𝑑𝑡)𝑑𝑡= r∏dt

(1

2𝜎2𝑆2 𝑑2𝑉

𝑑𝑆2 +𝑑𝑉

𝑑𝑡)𝑑𝑡= r(V-∆S)dt

Black-Scholes equation (

1

2𝜎2𝑆2 𝑑2𝑉

𝑑𝑆2 +𝑑𝑉

𝑑𝑡)𝑑𝑡= r(V-∆S)dt

But 𝑑𝑉

𝑑𝑆 = ∆

(1

2𝜎2𝑆2 𝑑2𝑉

𝑑𝑆2 +𝑑𝑉

𝑑𝑡)𝑑𝑡= r(V-

𝑑𝑉

𝑑𝑆 S)dt

Dividing by dt 1

2𝜎2𝑆2 𝑑2𝑉

𝑑𝑆2 +𝑑𝑉

𝑑𝑡= rV-

𝑑𝑉

𝑑𝑆𝑟 S

Or

𝒅𝑽

𝒅𝒕+

𝟏

𝟐𝝈𝟐𝑺𝟐 𝒅𝟐𝑽

𝒅𝑺𝟐 + 𝒅𝑽

𝒅𝑺 𝒓𝑺 − 𝒓𝑽 = 𝟎

Which is the Black-Scholes equation