#### Asian Options

An option with payoff that depends on the average strike price over a period of time. Hence these options are path dependent

$G(S)$- Geometric Average  |  $A(S)$ – Arithmetic Average  |  $G(S) \leq A(S)$

$V_{Asian} \leq V_{Ordinary}$ This is because of the averaging feature leading to lesser volatility and hence lower payoff

Average PRICE options *Replace Stock price with average price

• $\text{Payoff}_{Call} = [\overline{S} – K]_+$
• $\text{Payoff}_{Put} = [K – \overline{S}]_+$
• As N Increases V decreases

Calculating the payoffs

1. $x = \frac{N-1}{N}$  |  $y = \frac{N+1}{N}$  |  $z = \frac{(N+1)(2N+1)}{6N^2}$ | $s = \delta + 0.5 \sigma ^2$

2. $\delta ^* = 0.5 [rx + sy – \sigma ^2 z]$ | $\sigma ^{*2} = \sigma^2z$

3. $F^P(S) = Se^{-\delta^*T}$ | $F^P(K) = Ke^{-rT}$ | $v^2 = \sigma^{*2}T$

Average STRIKE options *Replace strike with average price

• $\text{Payoff}_{Call} = [S – \overline{S} ]_+$
• $\text{Payoff}_{Put} = [ \overline{S} – S]_+$
• As N Increases V Increases

Calculating the payoffs

1. $x = \frac{N-1}{N}$  |  $y = \frac{N+1}{N}$  |  $z = \frac{(N+1)(2N+1)}{6N^2}$ | $s = \delta + 0.5 \sigma ^2$

2. $r ^* = 0.5 [rx + sy – \sigma ^2 z]$ | $\sigma ^{*2} = \sigma^2 [z+1-y]$ | $K = S$  | $\rho = 0.5\frac{y}{\sqrt{z}}$

3. $F^P(S) = Se^{- \delta T}$ | $F^P(K) = Ke^{-r^*T}$ | $v^2 = \sigma^{*2}T$

#### Gap Options

$K_1$ – Strike price Determines amount of option payoff

$K_2$ – Trigger price Determines whether or not the option will have a payoff

$\text{Payoff}_{ Call} = \begin{cases} 0 & \text{ if } S_t \leq K_2 \\ S_T – K_1 & \text{ if } S_t > K_2 \end{cases}$

$\text{Payoff}_{ Put} = \begin{cases} K_1 – S_T & \text{ if } S_t < K_2 \\ 0 & \text{ if } S_t \geq K_2 \end{cases}$

A Gap option MUST be exercised when $K_2$ is hit. *Thus leading to possible NEGATIVE payoffs and premiums

Pricing Gap Options

1. $F^P(S) = Se^{- \delta T}$ | $F^P(K_1) = K_1e^{-rT}$ | $F^P(K_2) = K_2e^{-rT}$ | $v^2 = \sigma ^2 T$
2. $d_1 = \frac{ln \frac{F^P(S) }{F^P(K_2) } + 0.5v^2}{v}$ | $d_2 = d_1 – v$ * Use $K_2$ only to calculate $d_1$ and $d_2$
3. Call = $+F^P(S) N(+d_1) – F^P(K_1)N(+d_2)$
4. Put = $-F^P(S) N(-d_1) + F^P(K_1)N(-d_2)$

Parity Equation

$GapCall – GapPut = S_0e^{- \delta T} – K_1 e^{-rT}$

Delta for Gap Calls

$\frac{\partial C_{GapCall} }{\partial S} = N(d_1)+ S\frac{\partial N(d_1)}{\partial S} – K\frac{\partial N(d_2)}{\partial S}$

$= N(d_1)+ S\frac{\partial N(d_1)}{\partial d_1}\frac{\partial d_1}{\partial S} – K\frac{\partial N(d_2)}{\partial d_2}\frac{\partial d_2}{\partial S}$

$= N(d_1)+ SN'(d_1)\frac{\partial d_1}{\partial S} – KN'(d_2)\frac{\partial d_2}{\partial S}$

$= N(d_1)+ S[\frac{1}{\sqrt {2 \pi}}e^{-0.5d_1^2}]\frac{1}{S} – K[\frac{1}{\sqrt {2 \pi}}e^{-0.5d_2^2}]\frac{1}{S}$

#### Barrier Options

An option that goes into / out of existence if the price of the underlying asset reaches the barrier during the life of an option. And hence these options are path dependent

Knock-in Barrier Options

• Goes into existence if barrier is reached during the life of the option

Knock-out Barrier Options

• Goes out of existence if barrier is reached during the life of the option

Rebate Barrier Options

• Makes a fixed payment if barrier is reached during the life of the option

Up/Down option

• Stock price has to go UP to meet the barrier. [$S_0 < Barrier$]
• Stock price has to go DOWN to meet the barrier. [$S_0 > Barrier$]

Parity Relationship

• Knock-in + Knock-out option = Ordinary Option
• Barrier Option $\leq$ Ordinary Option
• If Barrier $\leq$ Strike ,
• Up-and-in call = ordinary call
• Up-and-out call = 0 * Since Knock-in + Knock-out option = Ordinary Option
• If Barrier $\geq$ Strike ,
• Down-and-in put = ordinary put
• Down-and-out put = 0 * Since Knock-in + Knock-out option = Ordinary Option

#### Compound Options

An option that allows you to buy/sell another option at a specified price.

Compound Call

• Call on Call
• Call on Put

Compound Put

• Put on Call
• Put on Put

How Compound options work

$t_1$ = Time when compound call or compound put expires. You need to decide at this point in time if you are/are not going to exercise the compound option at strike $x$

$T$ = Time when the underlying asset with strike $K$ expires. *$t_1 \leq T$

• If you buy a compound call at t=0 and you decide to exercise it at time $t_1$ then you receive the underlying asset(With strike K and time to expiry – $T-t_1$) and give-up the strike x. If you do not decide to exercise the compound call, the option would expire worthless. * C(Receive A, Give up B) . Value of compound call = $max[0, V(S_{t_1}, K, T-t_1)-x]$
• If you buy a compound put at t=0 and you decide to exercise it at time $t_1$ then you give-up the  underlying asset(With strike K and time to expiry – $T-t_1$) and receive x. If you do not decide to exercise the compound put, the option would expire worthless. * P( Give up A, Receive B) . Value of compound call = $max[0, x – V(S_{t_1}, K, T-t_1)]$

Compound option Parity

• $\text{Call on Stock} – \text{Put on Stock} = F^P(S) – Ke^{-rT}$
• $\text{Call on Call} – \text{Put on Call} = C_{Eur} – xe^{-rt_1}$
• $\text{Call on Put} – \text{Put on Put} = P_{Eur} – xe^{-rt_1}$

#### All or Nothing Options

Asset or Nothing Options

Asset-Call

$\text{Payoff}_{ AssetCall(K)} = \begin{cases} 0 & \text{ if } S_t \leq K \\ S_T & \text{ if } S_t > K \end{cases}$

$\text{AssetCallPrice}_t = S_te^{-\delta (T-t)} N(+d_1)$

Asset-Put

$\text{Payoff}_{ AssetPut(K)} = \begin{cases} S_T & \text{ if } S_t < K \\ 0 & \text{ if } S_t \geq K \end{cases}$

$\text{AssetPutPrice}_t = S_te^{-\delta (T-t)} N(- d_1)$

Cash or Nothing Options

Cash Call

$\text{Payoff}_{ CashCall(K)} = \begin{cases} 0 & \text{ if } S_t \leq K \\ 1\$ & \text{ if } S_t > K
\end{cases}\text{CashCallPrice}_t = e^{-r(T-t)} N(+d_2)$Cash Put$\text{Payoff}_{ AssetPut(K)}  = \begin{cases}
1\$& \text{ if } S_t < K \\ 0 & \text{ if } S_t \geq K \end{cases}$

$\text{CashPutPrice}_t = e^{-r (T-t)} N(- d_2)$

#### Replicating an option with all-or-nothing options

Ordinary Call

$C_t = \underbrace{S_te^{-\delta(T-t)}N(+ d_1)} – K \underbrace{e^{-r(T-t)}N(+ d_2)}$

——–[Asset Call]——————-[Cash Call]———

$C_t = AssetCall(K) – K. CashCall(K)$ * Buying one unit of ordinary call is equivalent to buying one unit of asset call with trigger K and shorting K units of cashcall with trigger K

Ordinary Put

$P_t = \underbrace{-S_te^{-\delta(T-t)}N(- d_1)} + K \underbrace{e^{-r(T-t)}N(- d_2)}$

——–[- Asset Put]——————-[Cash Put]———

$P_t = -AssetPut(K) + K. CashPut(K)$ * Buying one unit of ordinary put  is equivalent to selling one unit of asset put with trigger K and buying K units of cashput with trigger K

Gap Call

$GapCall = \underbrace{S_te^{-\delta(T-t)}N(+ d_1)} – K_1 \underbrace{e^{-r(T-t)}N(+ d_2)}$

——–[Asset Call($K_2$)]——————-[Cash Call($K_2$)]———

$GapCall = AssetCall(K_2) – K_1. CashCall(K_2)$

Gap Put

$Gap Put = \underbrace{-S_te^{-\delta(T-t)}N(- d_1)} + K_1 \underbrace{e^{-r(T-t)}N(- d_2)}$

——–[- Asset Put$K_2$]——————-[Cash Put($K_2$]———

$Gap Put = -AssetPut(K_2) + K_1. CashPut(K_2)$

#### Maxima and Minima

$max(A,B) = max(0, B-A) + A$

$max(A,B) = max(A-B, 0) + B$

$max(cA, cB) = c. max(A,B)$ If $c > 0$

$max(cA, cB) = c. min(A,B)$ If $c < 0$

$max(A,B) + min(A,B) = A+B$ => $min(A,B) = A+B – max(A,B)$

#### Forward Start Option

A Prepaid forward on an option that provides the owner an option at a specified time in the future.

Forward start call

t = time when the European call option is delivered

T = time when call expires

$K = XS_t$ Strike price is a % of the Stock price at t

$V_t$  = Value of call at t = $C(S_t, XS_t, T-t)$ = $S_te^{-\delta (T-t)}N(d_1)-XS_te^{-r (T-t)}N(d_2)$ = $S_t[e^{-\delta (T-t)}N(d_1)-Xe^{-r (T-t)}N(d_2)]$ = $S_t$*constant

$V_0$ = Time 0 price of the forward start call option = $F_{0,t}^P(S)[constant]$

Forward start put

t = time when the European put option is delivered

T = time when put expires

$K = XS_t$ Strike price is a % of the Stock price at t

$V_t$  = Value of put at t = $C(S_t, XS_t, T-t)$ = $- S_te^{-\delta (T-t)}N(-d_1) + XS_te^{-r (T-t)}N(-d_2)$ = $S_t[-e^{-\delta (T-t)}N(-d_1)+Xe^{-r (T-t)}N(-d_2)]$ = $S_t$*constant

$V_0$ = Time 0 price of the forward start call option = $F_{0,t}^P(S)[constant]$

#### Chooser Options

An option that allows one at time t to take either a call or put option expiring at T. The call and put have the same strike price

$V_t$ = Value of chooser option = $max[C(S_t, K , T-t),P(S_t, K, T-t)]$

= $max[0,P(S_t, K, T-t)-C(S_t, K , T-t)] + C(S_t, K , T-t)$

= $max[0,Ke^{-r(T-t)} – S_te^{-\delta (T-t)}]+ C(S_t, K , T-t)$

= $e^{-\delta (T-t)}max[0,Ke^{-(r-\delta )(T-t)} – S_t]+ C(S_t, K , T-t)$

$V_0 = e^{-\delta (T-t)}P(S_0,Ke^{-(r-\delta )(T-t)}, t) + C(S_0, K , T)$

#### Exam FM/2

Financial MathematicsDerivative Markets