Put-Call Parity Equation
$C-P = F_{0,T}^P (S)- F_{0,T}^P(K)$
$max[0,S-K] – max[0,K-S] = S-K$
Binomial Tree Model
$u = e^{(r-\delta)h+\sigma\sqrt h}$ | $d = e^{(r-\delta)h-\sigma\sqrt h}$
$p^* = \frac{e^{(r-\delta)h}-d}{u-d}$ [btw] = $ \frac{1}{1+e^{\sigma \sqrt h}} $[/btw]
$C_u = e^{-rh} [p^* .C_{uu} + (1-p^*).C_{ud}]$ | $C_d = e^{-rh} [p^* .C_{ud} + (1-p^*).C_{dd}]$
$ \Delta = e^{-\delta h}\frac{C_u – C_d}{S_0 (u-d)} $ | $ B = e^{-rh}\frac{uC_d -d C_u}{ u-d}$
$\text{Option Price} = e^{-r(2h)}[\binom{2}{2}p^{*^2}C_{uu} + \binom{2}{1}p^{*}(1-p^*)C_{ud} + \binom{2}{0}(1-p^*)^2C_{dd}]$
No dividends => American call option Price = European call option price. This is because, early exercise is never optimal.
Exchange Options
C(A,B) -> C(Receive A, Give up B)
P(A,B) -> P(Give up A, Receive B)
$C(A,B) – P(A,B) = F_{t,T}^P(A) – F_{t,T}^P(B)$
$C(A,B) = P(B,A)$
European Exchange option pricing
$ F^P(S)= Se^{-\delta_{s} (T-t)} $ | $ F^P(Q) = Qe^{-\delta_{q} (T-t)} $ | $ v^2 = \sigma ^2 (T-t) $ | $ \sigma ^2 = \sigma_s^2 + \sigma_q^2 – 2\rho\sigma_s\sigma_q$
$ d_{1} = \frac{ln(\frac{F^P(S)}{F^P(Q)}) + [0.5*v^2]}{v}$
$ d_{2} = d_{1} – v$
[icon name=”bell” class=””] Price of a call to exchange stock S for stock Q. Payout at expiration = $(S_T – Q_T)_+$ :
Call Price $ = [ F^P(S) * N( d_{1})] – [ F^P(Q) * N( d_{2})]$
Put Price $ = – [ F^P(S) * N(-d_{1})] + [ F^P(Q) * N(-d_{2})]$
[icon name=”bell” class=””] Determining Price of a European call option on the mixed Geometric Average $ M_t = (S_tQ_t)^{\frac{1}{2}} $ with strike K. [Payout at Expiration = $(M_t-K)_+$]
$ M_t = (S_tQ_t)^{\frac{1}{2}} $
=> $ M_t^2 = S_tQ_t = [S_0e^{(r-\delta_s-\frac{1}{2}\sigma_s^2)t+\sigma_sZ_s \sqrt t}][Q_0e^{(r-\delta_q-\frac{1}{2}\sigma_q^2)t+\sigma_qZ_q \sqrt t}]$
=> $ M_t = [S_0Q_0e^{(2r-\delta_s-\delta_q-\frac{1}{2}\sigma_s^2-\frac{1}{2}\sigma_q^2)t+(\sigma_sZ_s + \sigma_qZ_q )\sqrt t}]^\frac{1}{2}$
$ln M_t = ln[S_0Q_0]^\frac{1}{2} + [r-\frac{\delta_s+\delta_q+\frac{1}{2}\sigma_s^2+\frac{1}{2}\sigma_q^2}{2}]t + [\frac{\sigma_sZ_s + \sigma_qZ_q}{2}]\sqrt t$
$E[ln M_t] = ln[S_0Q_0]^\frac{1}{2} + [r-\frac{\delta_s+\delta_q+\frac{1}{2}\sigma_s^2+\frac{1}{2}\sigma_q^2}{2}]t$
$Var[ln M_t] = \sigma_m^2 = Var[[\frac{\sigma_sZ_s + \sigma_qZ_q}{2}]\sqrt t] = \frac{1}{4}[\sigma_s^2 + \sigma_q^2 +2\rho\sigma_s\sigma_q]t$
Compare $ M_t$ to the generic expression $M_0e^{(r-\delta_m-\frac{1}{2}\sigma_m^2)t+\sigma_mZ_m \sqrt t}$:
$\delta_m + \frac{1}{2}\sigma_m^2 = \frac{[\delta_s+\delta_q+\frac{1}{2}\sigma_s^2+\frac{1}{2}\sigma_q^2]}{2} $
Having identified $\sigma_m$ and $\delta_m$
Price of call = $c_M = [M_0e^{-\delta_m} * N(d_1) – Ke^{-\delta_q} * N(d_2)]$
Bounds for Option Prices
European Call: $F^P(S) \geq C_{Eur}(S,K,T) \geq max(0, F^P(S) – F^P(K))$
European Put: $F^P(K) \geq P_{Eur}(S,K,T) \geq max(0, F^P(K) – F^P(S))$
American Call: $S \geq C_{Amer}(S,K,T) \geq max(0, S-K)$
American Put: $K \geq P_{Amer}(S,K,T) \geq max(0, K-S)$
Call: $S \geq C_{Amer}(S,K,T) \geq C_{Eur}(S,K,T) \geq max(0, F^P(S) – F^P(K))$
Put: $K \geq P_{Amer}(S,K,T) \geq P_{Eur}(S,K,T) \geq max(0, F^P(K) – F^P(S))$
Early Exercise of American Options
Check for American CALL
Exercising a call option entails purchasing the stock at the pre-determined ‘Strike price’. For an American call, an early exercise entails the following:
- You no longer are able to leverage the interest on the Strike Price [K] which you otherwise would have gained. [Thus an effective indirect loss]
- Forgoing the benefit of no-exercise at expiry, thus losing an implicit put option!
- Benefiting from the stocks dividends since you now own the stock.
When is exercise optimal for an American Call ?
- Stock does not pay dividends => Exercise is never optimal
- Stock pays dividends
- Early exercise may be optimal if PV(Dividends)[btw] $= S_0 – F_{t,T}^P(S)$ [/btw] > PV(Interest on the Strike) [btw]$ = K(1-e^{-r(T-t)})$ [/btw]
- Early exercise is optimal if PV(Dividends) > PV(Interest on the Strike) + Implicit Put
Check for American PUT
- Irrespective of stock paying dividend
- Early exercise may be optimal if PV(Dividends) < PV(Interest on the Strike)
- Early exercise is optimal if PV(Dividends) < PV(Interest on the Strike) – Implicit call
Market Makers' Profit
Delta Hedging
Since traders mostly purchase options, Market-makers are often exposed to short positions and hence losses are theoretically limitless.
In a Delta-Hedged portfolio, for a change from $S_t$ to $S_{t+h}$
Delta Hedging -> Short a call, Long delta shares of stock and borrow.
Market Makers’ Profit = Change in value of $\Delta$ shares + Change in option value – Interest Expense
Market Makers’ Profit =$ [\Delta _t (S_{t+h} – S_t) ] + [ – (C_{t+h} – C_t)] – [rh( \Delta _t S_t – C t)]$
From $\Delta \Gamma \Theta $ approximation,
$C_{t+h} – C_t = \epsilon \Delta_t + 0.5 \epsilon ^2 \Gamma + h \Theta$ [btw] $\epsilon = S_{t+h} – S_t $ [/btw]
[icon name=”bullseye” class=””] $V_{ud}-V = (S_{ud}-S) \Delta (S,0) + 0.5 (S_{ud}-S)^2 \Gamma (S,0) + 2h \Theta$
- Since $\Gamma $ is positive for both puts and calls, Market makers profit is decreasing with $\epsilon ^2$ . He thus looses for huge changes in the stock price in either direction. [btw] For large changes in $ | \epsilon | $ market-makers profit is less [/btw]
- He breaks-even when $\epsilon = \sigma \sqrt h S_t $.
- $\epsilon < \sigma\sqrt h S_t$ – Makes Profit
- $\epsilon > \sigma\sqrt h S_t$ – Goes into loss
Delta Gamma Hedging [Gamma neutral portfolio]
- Use another option on the same stock to neutralize $\Gamma $ of his existing position. [btw] Greeks in a portfolio are additive! [/btw]
- Calculate the $\Delta$ of the new portfolio
- Now Delta hedge using the underlying stock. [btw] $\Gamma$ will remain neutralized because $\Gamma _{Stock} = 0$ [/btw]
State Prices /Utility
$Q_i$ = Price of a security that pays 1\$ iff the stock moves to state $i$ = Partial expected value of PV(1)
$U_i$ = PV(1), given the stock is in state $i$
$Q_u = pU_u = \frac{p^*}{1+r}$ | $Q_d = (1-p)U_d = \frac{1-p^*}{1+r}$ | $Q_u + Q_d = e^{-r}$
$C_0 = Q_uC_u + Q_dC_d$ | $S_0 = Q_uS_u + Q_dS_d$
$p* = \frac{Q_u}{Q_u + Q_d}$