Continuous time financial models

Suppose a sample space $\Omega$ probability measure $P$ and filtration $(\mathcal{F}_t)$ .

Have $d+1$ assets, with prices $(S_i(t))_{0 \le t \le T}$ , $i = 0, ..., d$ , and $S_i(t) \in \mathcal{F}_t$ . These prices are Ito diffusions: $dS_i (t) = U_i (t) dt + V_i (t) dW_i (t)$ . The numeraire $S_0 (t)$ is strictly positive, and usually $= e^rt$ .

Main example

Geometric brownian model (or Black-Scholes model)

Or equivalently:

An arbitrage opportunity is a strategy $\phi$ with $V_\phi(0) = 0$ , $V_\phi (T) \gt 0$ , $P( V_\phi (T) \gt 0) \gt 0$ . Should not exist in model with consistent prices.

Martingale measures

A martingale measure is a probability measure under which all discounted asset prices, $\tilde{S}_i$ , are martingales. (sometimes called strong marginale measure).

Proposition: Under a martingale measure, the discounted value $\tilde{V}_\phi$ of any self-financing strategy is a martingale. Proof: $d\tilde{V}_\phi (t) = \sum \phi_i (t) d\tilde{S}_i (t)$ $ = \sum \phi_i V_i (t) dW_i (t)$ , so $\tilde{V}_\phi$ is also a martingale.

Theorem: If there is a martingale measure $Q$ equivalent to the orginal $P$ , then the model contains no arbitrage opportunities.

Proof: For any self-financing strategy $\phi$ , $\tilde{V}_\phi$ is a martingale under Q. So:

For an arbitrage opportunity, $V_\phi(0) = 0$ , but $V_\phi (T) \ge 0$ and $P(V_\phi (T) \gt 0) \gt 0$ so $P(V_\phi (T) \gt 0) \gt 0$ so $E_Q{[\frac{V_\phi (T)}{S_0 (T)}]} \gt 0$ .

Note: Reverse is not true – there are arbitrage-free models in continuous time with not EMM.

Pricing contingent claims

Definition: A contingent claim is a random variable $X \in \mathcal{F}$ , thought of as amount of money to paid to the holder at time $T$ . The claim is attainable if there is a self-financing strategy $\phi$ with $V_\phi (T) = X$ . The model is complete if all contingent claims are attainable.

Risk-neutral pricing formula

Theorem: If $X$ is an attainable contingent claim and $Q$ an EMM, then any strategy that replicates $X$ will have price process $\pi_X (t) = V_\phi(t) = S_0 (t) E_Q {[ \frac{X}{S_0 (T)} | \mathcal{F} ]} $

Doesn’t depend on $\phi$ so can find $\pi_X$ without knowing $\phi$ (but $X$ must be attainable). If there is more than one EMM any can be used – they will all give the same results.

Proposition: If there is a unique EMM $Q$ then any contingent claim $X$ with $E {[ {| \frac{X}{S_0 (T)} |} ]} \lt \infinity$ is attainable. Note: no converse (like discrete case). Not very useful.

Proposition: For the Black-Scholes model any contingent claim $X$ with $E {[ {| \frac{X}{S_0 (T)} |} ]} \lt \infinity$ is attainable.

Risk-neutral pricing in the Black-Scholes model

Under $Q$ :

European options ($X = f(S_1 (T))$ )

Then $\pi_X (0) = e^{-rt} E_Q [ f(S_1 (T) ] = ... = e^{-rt} \int_{-\infinity}^\infinity f(s_0 exp((r-\sigma^2/2)T + \sigma \sqrt{T}x) \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}} dx$ .

European call ($X = (S_1 (T) - K)_{+}$ )

$\pi_X(0) = S_1(0) \Phi(z_0 + \sigma \sqrt{T}) - K e^{-rT} \Phi(z_0)$ , where $z_0 = \frac{(r-\sigma^2/2)T + log(S_1(0) / K)}{\sigma \sqrt{T}}$ – the Black-Scholes model.

European put ($Y = (K - S_1 (T) )_{+}$ )

Can value using integral as above, or using put-call parity: $X - Y = S_1(T) - K$ , so $Y = X - S_1(T) + K$ .

\[ \pi_Y (0) = E_Q [Y e^{-rT} ]

Replicating strategies

Theorem: If the arbitrage value of claim $X$ satisfies $\tilde{\pi}_X (t) = G(t, S_1(t))$ then a replicating strategy for $X$ is $\phi_1 (t) = e^{rt} G_s(t, S_1(T))$ and $\phi_0(t) = G(t, S_1(t)) - S_1(t)G_s(t, S_1(t))$ .

Proof: $\tilde{\pi} = G(t , S_1(t)) = G(t, s_1(0) exp{( (r - \sigma^2 /2)t + \sigma \tilde{W}_t)}$ . By Ito’s lemma, $d\tilde{\pi}_X (t) = (F_t + \frac{1}{2} F_ww)dt + F_w d\tilde{W}_t$ Drift = 0, so $ = G_s(t, S_1(t)) S_1(t) \sigma d\tilde{W}_t = h(t) d\tilde{W}_t$ , say.

Recall that in a Black-Scholes model, if a strategy has $d\tilde{\pi}_X (t) = h(t) d\tilde{W}_t$ it is replicated by $\phi$ with $\phi_1 (t) = h(t) / \sigma \tilde{S}_t (t)$ , hence $\phi_1(t) = e^{rt} G_s(t, S_1(t))$ .

Let’s apply it to the Euro call:

\[ \tilde{\pi}_X (t) = e^{rt} \int_{-\infinity}^{\infinity} (S_1(t) exp{( (r-\sigma^2/2)(T -t) + \sigma\sqrt{T - t}x)} - K)_{+} \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}} dx \] \[ \phi_1(t) = \Phi {( \frac{ log(S/K) + (r + \frac{1}{2}\sigma^2)(T-t)}{\sigma \sqrt{T-t}} )} \]

The Black-Scholes PDE

Suppose the arbitrage value $\pi_X (T) = F(t, S(t))$ then $F(t,s)$ satisfies

\[ F_t + rsF_s + \frac{1}{2} \sigma^2 s^2 F_ss = rF$

By Ito’s lemma

\[ d\tilde{\pi}_X (t) = (h_t + \frac{1}{2} h_ss)dt + h_s d\tilde{W}_t \]

Setting $(h_t + \frac{1}{2} h_ss)$ to 0 ($\tilde{\pi}_X$ is a martingale) gives the Black-Scholes PDE.

The Greeks

These are the partial derivatives of arbitrage values with respect to various parameters. If $\pi_X (0) = p = g(S_1(0), T, r, \sigma)$ then:

Process of holding a replicating portfolio and attempting to keep up to date with current $\Delta$ called delta holding. Delta-gamma hedging uses both $\Delta$ and $\gamma$ .

Volatility ($\sigma$ )

$\sigma$ is the only parameter that is hard to estimate. Can estimate using historical or implied volatility.

Historical volatility

Fit a GBM to historical data on stock prices. $S_1(t) = S_1(0) exp {( (b-\frac{1}{2}\sigma^2)t + \sigma W_t )}$

Suppose we have observations $s_0, s_1, ..., s_n$ of stock prices at times $t_i = i \Delta t$ , $i = 0, ..., n$ . By the model $log(S_i / S_{i-1})$ are independent $N((b-\frac{1}{2}\sigma^2)t, \sigma^2 \Delta t)$ , so a sample standard deviation of $log(S_i / S_{i-1})$ is an estimate of $\sigma$ .

\[ \hat{\sigma} = \sqrt{ \frac{ \sum (x_i - \bar{x})^2}{(n-1)\Delta t} } \]

Implied volatility

We know $p, s, T, r$ so solve $p = g(s,T,\sigma,r)$ for $\sigma$ . Note $\nu = \frac{\partial p}{\partial \sigma}$ so $\frac{1}{\nu} = \frac{\partial \sigma}{\partial p}$ . If $\nu$ is large, $\sigma$ not very sensitive to changes in $p$ , so easy to estimate $\sigma$ well and vice versa. When computing $\sigma$ from different options often get a “volatility smile” (because GBM isn’t adequate for modelling tail behaviour).

Path-dependent options

Payoff dependents on the full path, eg. the knockout barrier function. For most exotic options numerical methods (or discretisation + CRR model) are required.

Black-Scholes models with multiple risky assets

Consider a model with $d+1$ assets

Equivalent martingale meausre

Look for equivalent measure $Q$ with:

\[ \frac{dQ}{dP} = exp{(\sum^n {( \int^T_0 \gamma_i (t) dW_j (t) - \frac{1}{2} \int^T_0 \gamma(t)^2 dt )} )} \]

Under Q $\tilde{W}_k (t) = W_j(t) - \int^T_0 (s) ds$ , which are Brownian motion for $j= 1, ..., n$ , so:

\[ dS_i(t) = b_i(t)S_i(t)dt + S_i(t) \sum \sigma_ij(t) )d\tilde{W}_j(t)

The first part must be 0 so there is no drift. Hence we need $\gamma_1, ..., \gamma_n$ s.t. $\sum \sigma_ij(t) \gamma_j(t) = r - b_i(t)$ for $i = 1,...,d$ . A special case is when $\sigma_ij$ and $b_i$ are constant. Then $\gamma_j$ will be constants too, satisfying a system of $d$ linear equations, eg. $\Delta \gamma = r 1_d - b$ , where $\Delta = (\sigma_ij)$ , $b = (b_i)$ .

Pricing contingent claims

Can have claims with payoff based on more than one risky asset, eg. exchange option $X = (S_2(T) - S_1(T))_{+}$ . As usual $\pi_X (t) = S_o (t) E_Q {[ \frac{X}{S_0(T)} | \mathcal{F}_t ]}$ , but now we will have a joint density to integrate out over.