# Synthetic Options and Rationale

The prices of put and call options have an identity relationship through the concept of put-call parity.

c0 + X/(1+rF)T = p0 + S0

- c0 = Current price of the European call
- p0 = Current price of the European put
- X = Strike price of the put and the call
- T = Time to expiration
- rF = Risk free rate
- S0 = Current spot price of the underlying asset

The formula translation is: the price of a call with strike X plus the present value of strike price X equals the price of the put with strike X plus the current spot price.

**Synthetic Call Option:**If an investor believes that a call option is over-priced, then he/she can sell the call on the market and replicate a synthetic call.Borrow the present value of the strike price at the risk free rate and purchase the underlying stock and a put.

c0 = p0 + S0 - X/(1+rF)T

**Synthetic Put Option:**Similar to the synthetic call option. A synthetic put can be created by re-arranging the put-call parity relationship, if the trader believes the put is overvalued.**Synthetic Stock:**A synthetic stock can also be created by rearranging the put-call parity identity. In this case, the investor will buy the call, sell the put, and lend the present value of the strike at the current risk free rate.

S0 = c0 - p0 + X/(1+rF)T

If the stock rises in value, then the long call will provide the upside; if the stock falls, then the short put will replicate the downside.

Rationale for a "Synthetic"

Rational investors would not arbitrarily enter into "synthetic" position; it is done to exploit a perceived mispricing. For example, if the investor believes that put call parity is showing that a stock's call is overvalued, then he/she may execute a synthetic call strategy.

The key to a synthetic strategy is to buy the undervalued asset, sell the overvalued asset and invest or borrow the difference at the risk free rate.

- CFA Level 2: Derivatives Part 2 – Introduction
- Introduction to Options
- Synthetic Options and Rationale
- One Period Binomial Option Pricing Model
- Call Option Price Formula
- Binomial Interest Rate Options Pricing
- Black-Scholes-Merton (BSM) Option Pricing Model
- Black-Scholes-Merton Model and the Greeks
- Dynamic Delta Hedging & Gamma Related Issues
- Estimating Volatility for Option Pricing
- Put-Call Parity for Options on Forwards
- Introduction to Swaps
- Plain Vanilla Interest Rate Swap
- Equity Swaps
- Currency Swaps
- Swap Pricing vs. Swap Valuing
- Pricing and Valuing a Plain Vanilla Interest Rate Swap
- Pricing and Valuing Currency Swaps
- Pricing and Valuing Equity Swaps
- Swaps as Theoretical Equivalents of Other Derivatives
- Swaptions and their Valuation
- Swap Credit Risk and Swap Spread
- Interest Rate Derivatives - Caps and Floors
- Credit Default Swaps (CDS)
- Credit Derivative Trading Strategies

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