# 06 Ito’s lemma and Risk-neutral pricing

#### 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

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

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

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

#### $S^a$

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

$\frac{dS^a}{S^a} = a[\lambda _{true}dt + \sigma dZ(t)]$ [btw] * 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 $\alpha-\delta$  . $\frac{dS(t)}{S(t)} = (\alpha – \delta)dt + \sigma dZ(t)$[/btw]

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

#### Exam FM/2

Financial MathematicsDerivative Markets

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