Latex for Financial Engineering Mathematics Formula (Forwards Puts and Calls)
rockingdingo 20230514 #Financial EngineeringLatex Code for Financial Engineering Formula and Equation part IForwards, Puts, and Calls
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In this blog, we will summarize the latex code of most popular formulas and equations for Financial Engineering Formula and Equation part IForwards, Puts, and Calls. We will cover important topics including Forwards, PutCall 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
 PutCall 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
1. Forwards, Puts, and Calls

Forwards
Financial EngineeringEquation
Latex Code
F_{t,T}(S) = S_{t}e^{r(Tt)} = S_{t}e^{r(Tt)}  FV_{t,T}(\text{Dividends}) = S_{t}e^{(r\delta)(Tt)}
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.
 : Forward Contract at strike price S
 : Interest Rate
 : Future Value
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PutCall Parity
Financial EngineeringEquation
Latex Code
c(S_{t}, K, t, T)  p(S_{t}, K, t, T) = F^{P}_{t,T}(S)  Ke^{r(Tt)}
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 putcall parity formula for European options.
 : Price of call option c
 : Price of put option p
 : the present value of the strike price (x),
 :
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Calls and Puts with Different Strikes
Financial EngineeringEquation
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.
 : Call option of strike price K_{1}
 : Call option of strike price K_{2}
American options, For three different options with strike prices K1 < K2 < K3:
 : Call option of strike price K_{1}
 : Call option of strike price K_{2}
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Calls and Puts Arbitrage
Financial EngineeringEquation
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.
 : Strike price of option 1
 : Strike price of option 2
 : Strike price of option 3
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Call and Put Price Bounds
Financial EngineeringEquation
Latex Code
(F^{P}_{t,T}(S)  Ke^{r(Tt)})_{+} \le c(S_{t},K,t,T) \le F^{P}_{t,T}(S) \\ (Ke^{r(Tt)}  F^{P}_{t,T}(S))_{+} \le p(S_{t},K,t,T) \le Ke^{r(Tt)} \\ 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.
 : European Call Option Price
 : European Put Option Price
 : American Call Option Price
 : American Put Option Price
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Varying Times to Expiration
Financial EngineeringEquation
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.
 : American Call Option Price
 : American Put Option Price
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Early Exercise for American Options
Financial EngineeringEquation
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:
 : 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.
 : Early Exercise getting the stock's dividend payments
 : Pay the strike earlier and therefore miss the interest on that money
 : put protection if the stock price should fall.
 : 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.
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