## Latex for Financial Engineering Mathematics Formula (Forwards Puts and Calls)

rockingdingo 2023-05-14 #Financial Engineering

Latex Code for Financial Engineering Formula and Equation part I-Forwards, Puts, and Calls

In this blog, we will summarize the latex code of most popular formulas and equations for Financial Engineering Formula and Equation part I-Forwards, Puts, and Calls. We will cover important topics including Forwards, Put-Call Parity, Calls and Puts with Different Strikes, Calls and Puts Arbitrage, Call and Put Price Bounds, Varying Times to Expiration, Early Exercise for American Options, etc.

### 1. Forwards, Puts, and Calls

• #### Forwards

Financial Engineering

#### Equation

$F_{t,T}(S) = S_{t}e^{r(T-t)} = S_{t}e^{r(T-t)} - FV_{t,T}(\text{Dividends}) = S_{t}e^{(r-\delta)(T-t)}$

#### Latex Code

            F_{t,T}(S) = S_{t}e^{r(T-t)} = S_{t}e^{r(T-t)} - FV_{t,T}(\text{Dividends}) = S_{t}e^{(r-\delta)(T-t)}


#### Explanation

Latex code for the Forwards Contracts. I will briefly introduce the notations in this formulation. A forward contract is an agreement in which the buyer agrees at time t to pay the seller at time T and receive the asset at time T.

• $F_{t,T}(S)$: Forward Contract at strike price S
• $r$: Interest Rate
• $FV$: Future Value

• #### Put-Call Parity

Financial Engineering

#### Equation

$c(S_{t}, K, t, T) - p(S_{t}, K, t, T) = F^{P}_{t,T}(S) - Ke^{-r(T-t)}$

#### Latex Code

            c(S_{t}, K, t, T) - p(S_{t}, K, t, T) = F^{P}_{t,T}(S) - Ke^{-r(T-t)}


#### Explanation

Latex code for the Forwards Contracts. I will briefly introduce the notations in this formulation. Call options give the owner the right, but not the obligation, to buy an asset at some time in the future for a predetermined strike price. Put options give the owner the right to sell. The price of calls and puts is compared in the following put-call parity formula for European options.

• $c(S_{t}, K, t, T)$: Price of call option c
• $p(S_{t}, K, t, T)$: Price of put option p
• $F^{P}_{t,T}(S)$: the present value of the strike price (x),
• $Ke^{-r(T-t)}$:

• #### Calls and Puts with Different Strikes

Financial Engineering

#### Equation

$c(S_{t}, K, t, T) - p(S_{t}, K, t, T) = F^{P}_{t,T}(S) - Ke^{-r(T-t)}$

#### Latex Code

            K_{1} < K_{2} \\
0 \le c(K_{1}) - c(K_{2}) \le (K_{2} - K_{1})e^{-rT} \\
0 \le p(K_{2}) - p(K_{1}) \le (K_{2}) - K_{1})e^{-rT} \\
\frac{c(K_{1}) - c(K_{2})}{K_{2} - K_{1}} \ge \frac{c(K_{2}) - c(K_{3})}{K_{3} - K_{2}} \\
\frac{p(K_{1}) - p(K_{2})}{K_{2} - K_{1}} \le \frac{p(K_{3}) - p(K_{2})}{K_{3} - K_{2}}


#### Explanation

Latex code for the Calls and Puts with Different Strikes. For European calls and puts, with strike prices K_{1} and K_{2}, where K_{1} < K_{2}, we know the following.

• $c(K_{1})$: Call option of strike price K_{1}
• $c(K_{2})$: Call option of strike price K_{2}

American options, For three different options with strike prices K1 < K2 < K3:

• $c(K_{1})$: Call option of strike price K_{1}
• $c(K_{2})$: Call option of strike price K_{2}

• #### Calls and Puts Arbitrage

Financial Engineering

#### Equation

$K_{2} = \lambda K_{1} + (1 - \lambda) K_{3} \\ \lambda = \frac{K_{3} - K_{2}}{K_{3} - K_{1}}$

#### Latex Code

            K_{1} < K_{2} < K_{3}
K_{2} = \lambda K_{1} + (1 - \lambda) K_{3} \\
\lambda = \frac{K_{3} - K_{2}}{K_{3} - K_{1}}


#### Explanation

Latex code for the Calls and Puts Arbitrage. Three different options have strike prices K1, K2, K3 and K1 < K2 < K3 holds. An important formula for determining arbitrage opportunities comes from the following equations.

• $K_{1}$: Strike price of option 1
• $K_{2}$: Strike price of option 2
• $K_{3}$: Strike price of option 3

• #### Call and Put Price Bounds

Financial Engineering

#### Equation

$(F^{P}_{t,T}(S) - Ke^{-r(T-t)})_{+} \le c(S_{t},K,t,T) \le F^{P}_{t,T}(S) \\ (Ke^{-r(T-t)} - F^{P}_{t,T}(S))_{+} \le p(S_{t},K,t,T) \le Ke^{-r(T-t)} \\ c(S_{t},K,t,T) \le C(S_{t},K,t,T) \le S_{t} \\ p(S_{t},K,t,T) \le P(S_{t},K,t,T) \le K$

#### Latex Code

            (F^{P}_{t,T}(S) - Ke^{-r(T-t)})_{+} \le c(S_{t},K,t,T) \le F^{P}_{t,T}(S) \\
(Ke^{-r(T-t)} - F^{P}_{t,T}(S))_{+} \le p(S_{t},K,t,T) \le Ke^{-r(T-t)} \\
c(S_{t},K,t,T) \le C(S_{t},K,t,T) \le S_{t} \\
p(S_{t},K,t,T) \le P(S_{t},K,t,T) \le K


#### Explanation

Latex code for the Calls and Puts Arbitrage. The following equations give the bounds on the prices of European calls and puts. Note that the lower bounds are no less than zero. We can also compare the prices of European and American options using the following inequalities.

• $c(S_{t},K,t,T)$: European Call Option Price
• $p(S_{t},K,t,T)$: European Put Option Price
• $C(S_{t},K,t,T)$: American Call Option Price
• $P(S_{t},K,t,T)$: American Put Option Price

• #### Varying Times to Expiration

Financial Engineering

#### Equation

$T_{2} \ge T_{1} \\ C(S_{t},K,t,T_{2}) \ge C(S_{t},K,t,T_{1}) \le S_{t} \\ P(S_{t},K,t,T_{2}) \ge P(S_{t},K,t,T_{1}) \le S_{t}$

#### Latex Code

            T_{2} \ge T_{1} \\
C(S_{t},K,t,T_{2}) \ge C(S_{t},K,t,T_{1}) \le S_{t} \\
P(S_{t},K,t,T_{2}) \ge P(S_{t},K,t,T_{1}) \le S_{t}


#### Explanation

For American options, when expiration T2 > T1, the above equations holds.

• $C(S_{t},K,t,T)$: American Call Option Price
• $P(S_{t},K,t,T)$: American Put Option Price

• #### Early Exercise for American Options

Financial Engineering

#### Equation

$T_{2} \ge T_{1} \\ C(S_{t},K,t,T_{2}) \ge C(S_{t},K,t,T_{1}) \le S_{t} \\ P(S_{t},K,t,T_{2}) \ge P(S_{t},K,t,T_{1}) \le S_{t}$

#### Latex Code

            P_{V_{t},T}(dividends) > p(S_{t}, K) + K(1 â?? e^{â??r(T â??t)}) \\
K(1 â?? e^{â??r(T â??t)}) > c(S_{t}, K) + P_{V_{t},T}(dividends)


#### Explanation

So we exercise the call option if the pros are greater than the cons, specifically, we exercise if:

• $P_{V_{t},T}(dividends) > p(S_{t}, K) + K(1 â?? e^{â??r(T â??t)})$: The cons are that we have to pay the strike earlier and therefore miss the interest on that money and we lose the put protection if the stock price should fall. So we exercise the call option if the pros are greater than the cons.
• $P_{V_{t},T}(dividends)$: Early Exercise getting the stock's dividend payments
• $K(1 â?? e^{â??r(T â??t)})$: Pay the strike earlier and therefore miss the interest on that money
• $p(S_{t}, K)$: put protection if the stock price should fall.
• $K(1 â?? e^{â??r(T â??t)}) > c(S_{t}, K) + P_{V_{t},T}(dividends)$: For puts options, the pros are the interest earned on the strike. The cons are the lost dividends on owning the stock and the call protection should the stock price rise.