Options in Continuous Time Flashcards
What are the three assumptions of a Brownian motion?
- The process {B: t >=0} has continuous paths; is continuous in time
- For all time 0 = t0 < t1 < … < tn = T the random variables Btn - Btn-1, … , Bt2 - Bt1 are independent
- The distribution of the increment Bt+h - Bt does not depend on t, independent on everything before
- is normally distributed with zero mean and variance h (length)
What is the mean and variance of a Brownian Motion?
Mean = zero; Variance = delta t
What is the Quadratic Variation of a Brownian motion
One can show that for a Brownian motion, the quadratic variation over the interval [0, T ] is T
This implies that the total variation is infinite, where total variation is defined as the sum of the absolute changes
How is the plot of a Brownian motion
Less smooth than that of standard function, jiggles rapidly moving up and down in an erratic (random) way, captured by concept of quadratic variation
What is an Ito process?
As the number of periods in the Binomial tree model goes to infinity, a price process there converges to a simple transformation of a Brownian motion. This transformation is called Ito Process (brownian motion with two adjusted features).
A GBM is always an Ito process
Ito process evolves continuously over time:
dX_t=μ_t dt+σ_t dB_t where B is a Brownian motion, and μ and σ can also be random processes. In particular, μ and σ can depend on X.
This equation implies that X_(t+Δ)-X_t≅μ_t Δ+σ_t (B_(t+Δ)-B_t ). An Ito process evolves continuously over time. μ_t is the drift and σ_t is the diffusion coefficient of X at t. μ_t dt is the expected change and σ_t dB_t is the unexpected change in X in an instant dt.
Change in BM^2?
Given by change in time intervall.
Assumptions of the BSM model:
Log-returns are normally distributed with constant volatility, σ
Asset prices change continuously and cannot “jump”
Interest rate r is constant, no taxes, no transactions costs, no short-sale constraints
Trading strategies can change continuously, i.e., at each instant of time
What is a digital call option?
Pays 1 if the stock price is above K and else 0
Pros and Cons of using continious time models
Disadvantages
→ The mathematics is initially more difficult
→ Data is often discrete, so you will need to reconcile discretely-sampled data with a model set in continuous time
Advantages
→ The mathematics is easier once you get used to it
→ There are a large number of results available for continuous-time stochastic processes that you will be able to use
→ For many markets, data is now being collected at a high-frequency level, so continuous time models may be more appropriate
What is a quadratic variation of the brownian motion?
We calculate the sum of the squared changes of the brownian motion over time intervals. The limit of this sum, as we get the partitions get finer and finer, is called the quadratic variation.
For a Brownian motion, the quadratic variation over time interval [0,T] is T. This implies that the total variation is infinite where total variation is defined as the sum of the absolut changes: lim n → ∞ sum |delta B_it| = ∞
What is the quadratic variation of a continuously differentiable function?
lim n → ∞ a^2 (T^2 / N) = 0
Essentially, the same argument shows that the quadratic variation of any continuously differentiable function is zero, because such a function is approximately linear at each point.
To sum up: for continuously differentiable funtions, the quadratic variation is zero, and the total variation is finite.
What do we have to check to prove the Black scholes formula?
- Boundary condition max [St - K, 0] is satisfied
- Black Scholes PDE is satisfied
For 1:
If S > K then ln(S/K) > 0 and d1, d2 → ∞ and N(d1) = N(d2) = 1
If S < K then ln(S/K) < 0 and d1, d2 → -∞ and N(d1) = N(d2) = 0
This means that at expiration the BS formula for a call equals S-K if S>K and 0 if S<K which is exactly the boundary condition.
For 2: We calculate the derivatives theta, delta, gamma and plug these into the black scholes PDE:
C_t + 0.5*σ^2S^2C_ss +rSC_s −rC = 0
How can we calculate the price of a put using Black Scholes Call formula?
We can use Put Call parity c + Ke^(-rT)= p + S_0 and the fact that N(x) + N(-x) = 1 for any x
1-N(d2) = N(-d2)
put = Ke^(-r(T-t))N(-d_2 ) - sN(-d_1)
Extension of BS: underlyings with payouts
Known dividend amounts: replace S with S − PV (D)
Example: Assume we are pricing a 1-year option and it is known that there is a dividend of $0.5 in three months and $0.5 in nine months
Then: PV (D) = 0.5e−r ×3/12 + 0.5e−r ×9/12
Continuous dividend yield δ: replace S with Se−δ(T−t) Don’t forget that S also appears in d1 and d2 terms Apply these formulae to stocks, indices, currencies etc
BS and the binomial approach
C = ∆S + B
Comparing the BS formula to this expression, we see (intuitively, and this can be shown to be correct) that
→ ∆=N(d1)
→ B=−e−r(T−t)K×N(d2)
Thus, the Black-Scholes formula provides us with a precise composition of the replicating portfolio for the European call option
Replicating PF for put
→ ∆=-N(d1)
→ B=e−r(T−t)K×N(-d2)
