Ito's Lemma

$V(S,t) = \text{value of an asset}$

$dS  = h[S,t] dt + k[S,t] dZ(t)$

$dV = V_SdS  +  0.5V_{SS}[dS]^2  +  V_tdt$

  • $dt * dt = 0$
  • $dt * dZ = 0$
  • $dZ * dZ = dt$

For a process {U(t)}, the quadratic variation over [0,T], T>0 can be calculated as $\int_{0}^{T}[dU(t)]^2$

Sharpe Ratio

$\phi = \frac{\text{Risk Premium}}{Volatility} = \frac{\alpha – r}{\sigma}$

Two Ito processes with the same $dZ(t)$ will have the same $\phi$

  • $\frac{dX(t)}{X(t)} = adt + bdZ(t)$
  • $\frac{dY(t)}{Y(t)} = cdt + ddZ(t)$
  • Then, $\phi_X = \phi_Y$

Risk Free Portfolio

$\frac{dS_1(t)}{S_1(t)} = (\alpha _1 – \delta _1)dt + \sigma _1dZ(t)$

$\frac{dS_2(t)}{S_2(t)} = (\alpha_2 – \delta_2)dt + \sigma_2dZ(t)$

  • Risk free => No dZ term => $N_1[\sigma _1S _1] + N_2[\sigma _2S _2] = 0$
  • Return on Portfolio = 1 + 2
    1. $=dS_1 + \delta_1 S_1dt$
    2. $=dS_2 + \delta_2 S_2dt$
  • + = long => Buy, – = short => sell

 

Risk Neutral Pricing

True/Realistic:

$\frac{dS(t)}{S(t)} = (\alpha – \delta)dt + \sigma dZ(t)$

$d [ lnS(t) ] = (\alpha – \delta – 0.5 \sigma ^2)dt + \sigma dZ(t)$

  • Risk Neutral : $\frac{dS(t)}{S(t)} = (r – \delta)dt + \sigma d\widetilde{Z}(t)$
  • $ d\widetilde{Z}(t) = dZ(t) + \phi dt$
  • $\widetilde{Z}(t) = Z(t) + \phi t$
  • Since $Z(t) \sim N(0,t)$ => $\widetilde{Z}(t) \sim N(\phi t, t)$

If we consider the risk-neutral probability measure as our base measure

  • Since $\widetilde{Z}(t) \sim N(0, t)$
  • $Z(t) \sim N(- \phi t, t)$


Proportional Portfolio

Assume  portfolio W has $x%$ of Asset A and $(1-x)%$ of Asset B. Let $w(t)$ denote the value of the portfolio at time t and $\delta _A$, $\delta _B$, $\delta _W$ be the corresponding deltas of the Asset A, B and the portfolio respectively. Then

Instantaneous % return on Portfolio W = Instantaneous % return on Asset A + Instantaneous % return on Asset B

=> $\frac{dW(t)}{W(t)} + \delta_W dt  = x [\frac{dA(t)}{A(t)} + \delta_A dt] + (1-x)[\frac{dB(t)}{B(t)} + \delta_B dt]$

Note: If Asset A is a risk free asset,

=> $\frac{dW(t)}{W(t)} + \delta_W dt  = x [r dt] + (1-x)[\frac{dB(t)}{B(t)} + \delta_B dt]$

Black-Scholes Equation

  • $V$: Value of derivative at time t
  • $r$: risk-free rate
  • $\alpha$: Continuously compounded expected return of the stock
  • $\delta$: Dividend yield on the stock
  • $\delta ^*$: Dividend yield on the derivative

$(r-\delta)V_SS + 0.5 \sigma ^2 V_{SS} S^2  + V_t = (r-\delta ^*)V$

+ $ V_SdS  +  0.5V_{SS}[dS]^2  +  V_tdt = dV$

$(r-\delta) \Delta S + 0.5 \sigma ^2\Gamma S^2   + \Theta = (r-\delta ^*)V$

 

$S^a$

 

  • $\delta$: Dividend yield on the stock
  • $\delta ^*$: Dividend yield on the derivative $S^a$
  • $\nu$:  Continuously compounded return on the derivative $S^a$
  • $\lambda _{true}= \alpha – \delta + 0.5(a-1)\sigma ^2$ + * Note the + and the (a-1)
  • $\lambda _{rn}= r – \delta + 0.5(a-1)\sigma ^2$

$\frac{dS^a}{S^a} = a[\lambda _{true}dt + \sigma dZ(t)]$ + * When a = 1, the volatility factor is reduced to $\sigma$ and for the drift the $\sigma ^2$ factor escapes to thin air just leaving behind $\alpha-\delta$  . $\frac{dS(t)}{S(t)} = (\alpha – \delta)dt + \sigma dZ(t)$

  • $E[S(T)^a] = [S(t)e^{\lambda_{true} (T-t)}]^a$
  • $F_{t,T}[S(T)^a] = E^*[S(T)^a] = [S(t)e^{\lambda _{rn} (T-t)}]^a$
  • $\delta ^* = r – a\lambda _{rn}$
  • $\nu  = a\alpha – r(a-1)$

 

 


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