![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]](https://image.slidesharecdn.com/modelinganassetprice-140521094513-phpapp01/85/Modeling-an-Asset-Price-21-320.jpg)
Modeling an Asset Price
- 1. 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
- 2. The absolute change in the asset price is not by itself useful
- 3. 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 $
- 4. 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
- 5. 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
- 6. in a small interval of time dt, asset price changed to S+ds
- 7. 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
- 8. 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
- 9. 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
- 10. 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.
- 11. Wiener Process dX , the randomness, is called Wiener Process.
- 12. 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
- 13. 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 −1 2 𝜑2
- 14. 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
- 15. 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)
- 16. 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
- 17. Ito’s Lemma suppose f(S) is a smooth function of S using Taylor’s series: 𝑑𝑓 = 𝑑𝑓 𝑑𝑆 𝑑𝑆 + 1 2 𝑑2 𝑓 𝑑𝑆2 𝑑𝑆2 + … But dS = µ S dt + σ S dX Then (𝑑𝑆)2 = (µ S dt + σ S dX)2 (𝑑𝑆)2 = µ2 𝑆2 𝑑𝑡2 + 2σµ𝑆2 dtdX + 𝜎2 𝑆2 𝑑𝑋2 Since 𝑑𝑋 = 𝑂 𝑑𝑡 Then the last term is largest for small dt and dominate the other two terms. So we have: 𝑑𝑆2 = 𝜎2 𝑆2 𝑑𝑋2 Using the result 𝑑𝑋2 → 𝑑𝑡 then: 𝑑𝑆2 → 𝜎2 𝑆2 𝑑𝑡
- 18. Ito’s Lemma Substituting in Taylors expansion: 𝑑𝑓 = 𝑑𝑓 𝑑𝑆 𝑑𝑆 + 1 2 𝑑2 𝑓 𝑑𝑆2 𝑑𝑆2 𝑑𝑓 = 𝑑𝑓 𝑑𝑆 µ S dt + σ S dX + 1 2 𝑑2 𝑓 𝑑𝑆2 (𝜎2 𝑆2 𝑑𝑡) Or 𝑑𝑓 = σ S 𝑑𝑓 𝑑𝑆 dX + (µS 𝑑𝑓 𝑑𝑆 + 1 2 𝜎2 𝑆2 𝑑2 𝑓 𝑑𝑆2 )𝑑𝑡 relating the small change in a function of random variable to the small change in the variable itself
- 19. Ito’s Lemma this can be generalized to f(S,t) 𝑑𝑓 = 𝑑𝑓 𝑑𝑆 𝑑𝑆 + 𝑑𝑓 𝑑𝑡 𝑑𝑡 + 1 2 𝑑2 𝑓 𝑑𝑆2 𝑑𝑆2 And doing the same, we will get: 𝒅𝒇 = σ S 𝒅𝒇 𝒅𝑺 dX + (µS 𝒅𝒇 𝒅𝑺 + 𝟏 𝟐 𝝈 𝟐 𝑺 𝟐 𝒅 𝟐 𝒇 𝒅𝑺 𝟐 + 𝒅𝒇 𝒅𝒕 )𝒅𝒕 Which is Ito’s Lemma
- 20. Black-Scholes equation suppose we have an option V(S,t) depends on S and t using Ito’s Lemma: 𝑑𝑉 = σ S 𝑑𝑉 𝑑𝑆 dX + (µS 𝑑𝑉 𝑑𝑆 + 1 2 𝜎2 𝑆2 𝑑2 𝑉 𝑑𝑆2 + 𝑑𝑉 𝑑𝑡 )𝑑𝑡 assuming V has at least one t derivative and two for S
- 21. 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]
- 22. 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 𝑑𝑆 dX + µS 𝑑𝑉 𝑑𝑆 + 1 2 𝜎2 𝑆2 𝑑2 𝑉 𝑑𝑆2 + 𝑑𝑉 𝑑𝑡 𝑑𝑡 − ∆ (σ S dX + µ S dt ) = 𝜎𝑆 𝑑𝑉 𝑑𝑆 − ∆ dX +(µS ( 𝑑𝑉 𝑑𝑆 − ∆) + 1 2 𝜎2 𝑆2 𝑑2 𝑉 𝑑𝑆2 + 𝑑𝑉 𝑑𝑡 )𝑑𝑡
- 23. 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 2 𝜎2 𝑆2 𝑑2 𝑉 𝑑𝑆2 + 𝑑𝑉 𝑑𝑡 )𝑑𝑡 But: ∏= ∏0ert then d∏= r∏0ertdt d∏= r∏dt So ( 1 2 𝜎2 𝑆2 𝑑2 𝑉 𝑑𝑆2 + 𝑑𝑉 𝑑𝑡 )𝑑𝑡= r∏dt ( 1 2 𝜎2 𝑆2 𝑑2 𝑉 𝑑𝑆2 + 𝑑𝑉 𝑑𝑡 )𝑑𝑡= r(V-∆S)dt
- 24. Black-Scholes equation ( 1 2 𝜎2 𝑆2 𝑑2 𝑉 𝑑𝑆2 + 𝑑𝑉 𝑑𝑡 )𝑑𝑡= r(V-∆S)dt But 𝑑𝑉 𝑑𝑆 = ∆ ( 1 2 𝜎2 𝑆2 𝑑2 𝑉 𝑑𝑆2 + 𝑑𝑉 𝑑𝑡 )𝑑𝑡= r(V- 𝑑𝑉 𝑑𝑆 S)dt Dividing by dt 1 2 𝜎2 𝑆2 𝑑2 𝑉 𝑑𝑆2 + 𝑑𝑉 𝑑𝑡 = rV- 𝑑𝑉 𝑑𝑆 𝑟 S Or 𝒅𝑽 𝒅𝒕 + 𝟏 𝟐 𝝈 𝟐 𝑺 𝟐 𝒅 𝟐 𝑽 𝒅𝑺 𝟐 + 𝒅𝑽 𝒅𝑺 𝒓𝑺 − 𝒓𝑽 = 𝟎 Which is the Black-Scholes equation