Equivalent Probability Measures
- Two meastures, such as \(\mathbf{P},\mathbf{Q}\) are equivalent if
\[ P(A) > 0 \leftrightarrow Q(A)>0 \text{ for any outcome A}\]
- The two measures agree on the possible outcomes, however may disagree on the individual probabilities
Radon-Nikodym Derivative
- If we consider the probability of following a specific path/ending at a set node, noting these are probabilities related to maturity, i.e, \(t=T\):
Path Probability Examples
- We can encode the differences between two measures as the ratio:
\[ \frac{\overbrace{\pi_i'}^{\text{q prob}}}{\underbrace{\pi_i}_{\text{p prob}}}\]
- This ratio can be defined as the Radon Nikodym Derivative:
\[ \frac{\pi_i'}{\pi_i} = Y = \frac{dQ}{dP}\]
Expection and Radon-Nikodym
- This derivative acts naturally when considering expectations, allowing manipulations such as these:
\[ \mathbb{E}_P[X] = \sum_i\pi_ix_i\] \[ \mathbb{E}_Q[X] = \sum_i\pi_i'x_i = \sum_i\left(\frac{\pi_i'}{\pi_i}\right)\pi_ix_i = \sum_i\pi_i\left(\frac{dQ}{dP}x_i\right)\]
- Therefore we can achieve an important result:
\[ \mathbb{E}_Q[X] = \mathbb{E}_P\left[\frac{dQ}{dP}X\right] \]
Radon-Nikodym as a Process
- We can define \(\xi(t)\) to be the Radon-Nikodym derivation up to time \(t\):
\[ \xi(t) = \mathbb{E}_P\left[\frac{dQ}{dP}|\mathcal{F}(t)\right]\]
- We now have an expression for \(0\leq t\leq T\):
\[ \mathbb{E}_Q[X|\mathcal{F}(t)] = \xi^{-1}(t)\mathbb{E}_P[\xi(T)X|\mathcal{F}(t)]\]
Brownian Motion
We can consider a special discrete binomial process with \(n\rightarrow\infty\):
Arithmetic random walk \(W_n(t)\)
- Binomial process
- \(W_n(0)=0\)
- Layer spacing of \(\frac{1}{n}\)
- Up and down jumps of equal size \(\frac{1}{n}\)
- With the measure \(\mathbf{P}\), there is a probability of 0.5 of up and down movement
Considering the random variable for whether the process moves up or down:
\[ X_i = \begin{cases}1,\quad p=0.5\\-1,\quad p=0.5\end{cases}\] * Then for \(i=1,2,...,nt\)
\[ W_n\left(\frac{1}{n}\right)=W_n\left(\frac{i-1}{n}\right)+\frac{X_i}{\sqrt{n}}\]
- Therefore we can consider the whole process up to time t:
\[ W_n(t) = 0 + \sum_{i=1}^{nt}\frac{X_i}{\sqrt{n}} = \sqrt{t}\left(\sum^{nt}_{i=1}\frac{X_i}{\sqrt{nt}}\right)\]
- Under the CLT:
\[ \left(\sum^{nt}_{i=1}\frac{X_i}{\sqrt{nt}}\right)\rightarrow N(0,1),\quad\text{as }n\rightarrow\infty \]
- Therefore:
\[ W_n(t)\rightarrow N(0,t),\quad\text{as }n\rightarrow\infty\] + Formally this is convergent to Brownian motion + This type of relationship is true for all: - Martingale distributions - Condition distributions, such as those conditional on \(\mathcal{F}(t)\)
Formal Brownian Motion Definition
- A stochastic process \(\{W(t),t\geq 0\}\) is said to be a \(\mathbf{P}\) Brownian motion iff:
- \(W(t)\) is continuous and \(W(0)=0\)
- Value of \(W(t)\) is distributed, under \(\mathbf{P}\) as \(N(0,t)\)
- Increment \(W(s+t)-W(s)\) is distributed under \(\mathbf{P}\) as \(N(0,t)\) and is independent of \(\mathcal{F}_s\)
- Other notes on Brownian Motion:
- \(W\) is continuous but differentiable nowhere
- It will hit every real value no matter how large or negative
- Once \(W\) hits a particular value it immediately hits it again infinitely often and then again from time to time in the future
- Scaling doesn’t matter as it is fractal.
- Splitting Brownian Motion results in Brownian motion
Brownian Motion for Stock Prices
- Brownian motion by itself is not good enough for modelling stock prices as it needs extra properties that stock prices tend to have:
- Stock prices tend to increase in the real world under \(\mathbf{P}\)
- Stock prices do not become negative.
- Therefore, in general functions of Brownian motion are considered.
Stochastic Processes
- In the Brownian motion framework a stochastic process \(X(t)\) is a continuous process (on \(t\)) such that:
- Where \(\sigma(s)\) and \(\mu(s)\) are (possibly random) \(\mathcal{F}_s\) processes
\[ X(t) = X(0)+\int^t_{0}\sigma(s)dW(s)+\int^t_0\mu(s)ds\]
- The differential form can be written as (with \(X(0)\) as a constant):
\[ \underbrace{dX(t)}_{\text{Change}} = \underbrace{\sigma(t,X(t))dW(t)}_{\text{Diffusion/Uncertain}} + \underbrace{\mu(t,X(t))dt}_{\text{Drift/Certain}}\]
- Note that \((dW(t))^2\approx dt\)
Ito’s Lemma
- Consider a stochastic process \(X(t)\) with:
\[ dX(t) = \sigma(X(t))dW(t) + \mu(X(t))dt\]
- If we consider a function of this stochastic process:
\[ Y(t) = f(X(t)) \]
- We can apply Ito’s lemma to get the dynamics of this process:
\[ dY(t) = \frac{\partial f}{\partial x}dX(t) + \frac{1}{2}\frac{\partial^2f}{\partial x^2}\underbrace{(dX(t))^2}_{\sigma^2(X(t))dt}\] * We can also consider t as a variable, however this typically would cancel out:
\[ df(t,X_t) = \left(\frac{\partial f}{\partial t}+\mu_t\frac{\partial f}{\partial x}+\frac{1}{2}\sigma^2\frac{\partial^2f}{\partial x^2}\right)dt + \sigma_t\frac{\partial f}{\partial x}dW(t)\]
- Written more generally:
\[ df(t,X_t) = \frac{\partial f}{\partial t}dt + \frac{\partial f}{\partial x}dX_t + \frac{\partial^2 f}{\partial x^2}(dX_t)^2\]
- Ito’s Product Rule can be used for the product of two stochastic processes:
\[ F(X_t,Y_t) = X_tY_t\]
- The simplified equation is, noting the third term cancels out if either stochastic process has no diffusion:
\[ dF(X_t,Y_t) = X_tdY_t + Y_tdX_t + dX_tdY_t\]
- The more general form can be used for simple functions of 2 processes, such as \(Z(t) = \frac{S(t)}{B(t)}\):
\[ dF(X_t,Y_t) = \frac{\partial F}{\partial x}dX_t + \frac{\partial F}{\partial y}dY_t + \frac{\partial^2F}{\partial x\partial y}dX_tY_t\]
Martingales in Continuous Time
- Stochastic process M(.) is a martingale with respect to a measure \(Q\) and filtration \(\mathcal{F}\) if:
- Aka there is no drift
\[ \mathbb{E}_Q[M(t)|\mathcal{F}_s] = M(s),\quad\forall s\leq t\]
- Examples of brownian motion:
- Constant process
- Driftless P Brownian motion
- Driftless Q Brownian motion
- P Conditional Expectation
- Q Conditional Expectation
- To prove a process is a martingale, can either:
- Use Ito’s lemma or related to find SDE and show there is no drift term
- Prove \(X(s) = \mathbb{E}_P[X(t)|\mathcal{F}_s]\)
- When doing this, it is often beneficial to split the Brownian motion
- \(W_t - W_s\) is independent of \(\mathcal{F}_s\) and \(\mathbb{E}[W_s|\mathcal{F}_s]=W_s\)
Change of Measure
- In continuous time, the radon-nikodym derivative is as follows:
\[ \frac{dQ}{dP} = \exp(-\gamma W_T-\frac{1}{2}\gamma T) \]
- Note the following relation:
\[ \mathbb{E}_P[X_T] = \mathbb{E}_Q\left[\frac{dQ}{dP}X_T\right]\]
- Also note that the MGF of a normal random variable is often useful in calculations:
- Brownian motion and increments of Brownian motion are normally distributed
\[ \mathbb{E}_P[e^{\theta X}]=e^{\mu\theta+\frac{1}{2}\sigma^2\theta^2} \]
Cameron-Martin-Girsanov Theorem
If \(W(.)\) is a Brownian motion and we have a preversible process \(\gamma()\), then there exist a measure \(Q\) such that:
- \(Q\) is equivalent to \(P\)
- The continuous time radon-nikodym:
\[ \frac{dQ}{dP} = \exp\left(-\int^T_0\gamma(t)dW(t)-\frac{1}{2}\int^T_0\gamma^2(t)dt\right) = \exp(-\gamma W_T-\frac{1}{2}\gamma T)\]
- A \(Q\) Brownian motion is a \(P\) Brownian motion with altered drift:
\[ W_Q(t) = W(t)+\int^t_0\gamma(s) ds \leftrightarrow dW_Q(t) = dW(t) + \gamma(t)dt\]
To change the measure, the following is often subbed into the SDE:
\[ dW_P(t) = dW_Q(t)-\gamma(t)dt \]
Martingale Representation Theorem
- Assuming \(M(.)\) and \(N(.)\) are two martingales and \(\phi(.)\) is a preversible process:
\[ N(t) = N(0) + \int^t_0\phi(s)dM(s)\]
\[ dN(t) = \phi(t)dM(t)\]
Replicating Portfolio
- A replicating portfolio strategy has associated portfolio value:
- Note that preversibility in continuous time means \(\phi(t)\) is known at time \(t\), therefore the notation differs from the discrete case.
\[ V(t) = \phi(t)S(t) + \psi(t)B(t)\]
The self financing definition for discrete and continuous cases:
Discrete
\[ V_i-V_{i-1}=\phi_i(S_i-S_{i-1})+\psi_i(B_i-B_{i-1})\]
- Continuous
- The change in portfolio value between points only depends on the change of the underlying securities
\[ dV(t) = \phi(t)dS(t) + \psi(t)dB(t)\]
For a continuous portfolio to be self financing, the dynamics of \(V(t) = \phi(t)S(t)+\psi(t)B(t)\), calculated by Ito, should match the above which is known by plugging in values. If these are not equivalent the portfolio is not self financing.
To be replicating:
\[ V(T) = \phi(T)S(T)+\psi(T)B(T) = X\]
Black Scholes Model
- Black Scholes assumes:
\[ dB(t) = rB(t)dt \]
\[ dS(t) = \mu S(t)dt + \sigma S(t)dW(t)\]
- These SDE have the solutions:
\[ B(t) = e^{rt}\]
\[ S(t) = S(0)e^{\sigma W_Q(t)+(r-\frac{1}{2}\sigma^2)t}\]
- Note that stock price dynamics is that of geometric Brownian motion
Pricing Steps
- Find the Q measure and dynamics
- Under \(P\) the discounted stock price dynamics is as follows:
\[ dZ(t) = \sigma Z(t)\left(dW(t)+\frac{\mu - r}{\sigma}dt\right)\]
- This is converted to \(Q\) measure:
\[ dZ(t) = \sigma Z(t)\left(dW_Q(t)+\left(\frac{\mu - r}{\sigma}-\gamma(t)\right)dt\right)\]
- To ensure that the process is a martingale it has to be driftless, therefore we consider \(\gamma(t) = \frac{\mu - r}{\sigma}\):
\[ dZ(t) = \sigma Z(t)dW_Q(t)\]
- Martingale Representation
- Creation of a \(Q\) martingale as the expectation of the discounted payoff of the derivative
\[ Y(t) = \mathbb{E}_Q\left[\frac{1}{B(t)}X|\mathcal{F}_t\right]\]
- Therefore, as we have two martingale processes \(Y(t)\) and \(Z(t)\) we can invoke the martingale representation theorem:
\[ dY(t) = \phi(t)dZ(t)\]
- Self Financing and Self Replicating Portfolio
- We use the portfolio strategy:
\[ V(t) = \phi(t)S(t)+\psi(t)B(t)\]
- \(\phi(t)\) of stock \(S(t)\)
- \(\psi(t) = Y(t)-\phi(t)Z(t)\) of bond
Self Replicating Proof
\[ \begin{split}V(t) &= \phi(t)S(t)+\psi(t)B(t)\\ &= \phi(t)S(t) + \left(Y(t) - \phi(t)\frac{S(t)}{B(t)}\right)B(t) \\ &= Y(t)B(t) \end{split}\]
Self Financing Proof
- Note that by Ito:
\[ dV(t) = d(Y(t)B(t)) = B(t)dY(t)+Y(t)dB(t)\]
- Also note that by the martingale representation theory:
\[ dY(t)=\phi(t)dZ(t)\]
- Also note the rearrangement of \(V(t)\):
\[ Y(t) = \phi(t)Z(t)+\psi(t)\]
- Hence:
\[ \begin{split}dV(t)&= B(t)dY(t)+Y(t)dB(t)\\ &= B(t)(\phi(t)dZ(t))+(\phi(t)Z(t)+\psi(t))dB(t)\\ &= \phi(t)dS(t)+\psi(t)dB(t)\end{split}\]
- Therefore for no arbitrage:
\[ \begin{split} V(t) &= Y(t)B(t) \\ &= \mathbb{E}\left[\frac{1}{B(T)}X|\mathcal{F}_t\right]B(t)\\ &= \mathbb{E}_Q[e^{-r(T-t)}X|\mathcal{F}_t]\end{split}\]
Derivative Valuation under Black Scholes
- We can value a derivative at time 0:
\[ V(0) = e^{-rT}\mathbb{E}_Q[f(S_T)]\]
- For a call, the Black Scholes formula is:
\[ V(S(t), t) = S(t)\mathcal{N}(d_1) - Ke^{-r(T-t)}\mathcal{N}(d_2)\]
- For a put, the Black Scholes formula is:
\[ p(t,S_t) = Ke^{-r(T-t)}\mathcal{N}(-d_2) - S_t\mathcal{N}(-d_1)\]
\[ d_1 = \frac{\ln\left(\frac{S_0}{K}\right)+(r+\frac{1}{2}\sigma^2)T}{\sigma\sqrt{T}} \quad d_2 = \frac{\ln\left(\frac{S_0}{K}\right)+(r-\frac{1}{2}\sigma^2)T}{\sigma\sqrt{T}} = d_1-\sigma\sqrt{T} \]
Notes on Calculating Option Prices through Black Scholes
- Consider the example of the European call option:
\[ f(S_T) = (S_T-K)^+\]
- Consider the distribution of the stock price process, GBM for example:
\[ S_T = S_0e^{(r-\frac{1}{2}\sigma^2)t+\sigma W_Q(t)} \sim LN(S_0e^{rT}, S^2_0e^{2rT}(e^{\sigma^2T}-1))\]
\[ x = \ln\left(\frac{K}{S_0}\right) \sim N((r-0.5\sigma^2)T,\sigma^2T)\]
- Consider the possible payouts and when they pay out:
\[ f(S_T) = \begin{cases}S_T-K,\quad S_T>K\\0, \quad S_T < K\end{cases}\] * The price at time \(0\) (can be generalised to time \(t\)) is, noting this is the same price as the replicating portfolio at this time:
\[ V(0) = e^{-rT}\mathbb{E}_Q[f(S_T)]\]
- We can write this expectation in terms of the payout:
\[ e^{-rT}\underbrace{(\mathbb{E}[(S_T-K)\mathbb{1}_{x>\ln(\frac{K}{S_0})}]}_{(1)} + \underbrace{\mathbb{E}[0 * 1_{x<\ln(\frac{K}{S_0})}]}_{(2)})\] * Equation 1 can be decomposed into two components in this case, which shows an expectation and pure probability: + We can often get the equations into probabilities or expected values so similar values can be used, with \(d_1,d_2\) slightly modified.
\[ S_0\mathbb{E}\underbrace{[e^x\mathbb{1}_{x>\ln(\frac{K}{S_0})}]}_{e^{rT}\Phi(d_1)} - K\underbrace{\mathbb{E}[\mathbb{1}_{x>\ln(\frac{K}{S_0})}]}_{\Phi(d_2)}\] * In the original equation \((2) = 0\). However, if considering a put, this part would result to:
\[ K\underbrace{\mathbb{E}[\mathbb{1}_{x<\ln(\frac{K}{S_0})}]}_{\Phi(-d_2)} - S_0\mathbb{E}\underbrace{[e^x\mathbb{1}_{x<\ln(\frac{K}{S_0})}]}_{e^{rT}\Phi(-d_1)}\]
- Note that the put call parity can be used with black scholes as they relate to call/put prices.
- We have the full proof for call, can use put call parity to prove put from here
\[ p_t + S_t = c_t + Ke^{-r(T-t)} \]
- Useful Identity to turn everything into distributions:
\[ e^xf(x;\theta,\gamma^2) = e^{\theta+\frac{1}{2}\gamma^2}f(x;\theta+\gamma^2,\gamma^2)\] * Another useful identity:
\[ sn(d_1) = Ke^{-r\tau}n(d_2)\]
Terminal Pricing
- We consider the value of a derivative at time \(t\):
\[ V(s,t) = \exp(-r(T-t))\mathbb{E}[f(S_T)|S_t=s]\]
- The trading strategy we employ is:
\[ \phi(t) = \frac{\partial V(s,t)}{\partial s}\] * For \(\psi\) we note $ V(t) = Y(t)B(t)$ by replicating portfolio arguments: + \(V(t)\) is the value of the portfolio, which by no arbitrage arguments is the value of the option at time \(t\)
\[ \psi(t) = Y(t)-\phi(t)Z(t) = \frac{V(t)-\phi(t)S(t)}{B(t)} = \frac{V(t)-\frac{\partial V}{\partial s}S(t)}{B(t)}\]
For a European call, we derive the Black Scholes formula with respect to the stock price to get:
\[ \phi(t)=\mathcal{N}\left(\frac{\ln\left(\frac{S_t}{K}\right)+(r+\frac{1}{2}\sigma^2)(T-t)}{\sigma\sqrt{T-t}}\right)\]
Therefore the value of the bond holding at any time is:
\[ B_t\psi_t=-ke^{-r(T-t)}\mathcal{N}\left(\frac{\ln\left(\frac{S_t}{K}\right)+(r-\frac{1}{2}\sigma^2)(T-t)}{\sigma\sqrt{T-t}}\right) \]
The Black Scholes Partial Differential Equation:
\[ \frac{\partial v}{\partial t} + \frac{\partial v}{\partial S_t}rS_t + \frac{1}{2}\frac{\partial^2V}{\partial S_t^2}\sigma^2S_t^2 - V(t)r = 0\]