Option Pricing Using the Black – Scholes Model[10].


30


For American-style options (both calls and puts), the longer the time to expiration, the higher the price of the option, because the price of the underlying asset has the potential to move enough to put the option in the money and make the owner a profit. For European-style options, the effect of time to expiration depends on whether the option is a call or a put.

1.4.3.4. Exchange rate volatility


The more volatile the market rate, the greater the probability that the spot rate will exceed the trading rate at contract expiration, and thus the higher the call option premium.

1.4.3.5. American or European style options and interest differentials

First, American options are more flexible than European options, so the buyer of an American option is willing to pay a higher premium for an American option with the same call rate and expiration as a European option. Indeed, given the call rate, the volatility of the exchange rate, and the time to expiration, an American option will always be priced higher (have a higher premium) than a European option. Furthermore, if the call currency has a higher interest rate than the put currency, the holder of the call option will be interested in exercising the option before expiration to earn a higher interest rate. This interest rate advantage makes the American option premium higher than the European option premium. If the call currency has a lower interest rate than the put currency, the interest rate advantage of the American option for the buyer is no longer there, but this advantage becomes a potential advantage for the seller if the buyer enters into the transaction. Therefore, in principle, if the currency chosen to buy has a lower interest rate, the seller can reduce the premium for the American option contract compared to the European option contract. However, the seller does not necessarily have to reduce the premium, because: first, because there is no interest rate advantage, the buyer will not execute the transaction and therefore the potential advantage


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The seller's potential does not materialize; second, interest rates are market factors and can change at any time in the future. This makes the interest rate of the currency option higher at a certain time in the future and thus the potential advantage of interest rates will be transferred from the seller to the buyer. Therefore, the impact of interest rate fluctuations on the option premium can be considered from the following perspectives:

If the option seller expects the interest rate of the currency in the call contract to always be lower than the currency in the put contract, then the option buyer charges the same premium for American-style contracts as for European-style contracts.

If the seller expects that the potential interest rate advantage will shift from the seller to the buyer in the future, the option premium will be higher. In addition, determining whether an option contract is ITM, ATM, or OTM is an integral part of pricing an option. Therefore, comparing the option price with the spot price alone will miss the impact of the forward point on foreign exchange prices. As a result, the option premium is affected by the forward point, or in other words, the interest rates of the two currencies involved in the option have an impact on the option premium.

1.4.3.6. Alternative option strike price


At each spot rate level, the higher the strike rate of a call currency option, the lower the probability of exercising the option, and thus the lower the call option premium.

1.4.4. Currency option pricing model


1.4.4.1. Option pricing using the Black – Scholes model[10].


This model was researched and presented by two economists Fischer Black and Myron Scholes in 1973. This model makes the following assumptions:


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- Assets are traded continuously with random and continuous price fluctuations. Investors can continue to trade in the market and the assets are traded continuously.

- The risk-free interest rate applied to borrowings and lending remains constant throughout the life of the option. Applying a log-normal distribution to the asset's forward price and a normal distribution to the asset's spot price, the variance of this distribution function remains constant throughout the life of the asset.

- No taxes and transaction costs.


Black-Scholes formula:


C = SN(d1) – EN(d2)e


-rT


P = e


-rT

EN(-d2) – SN(-d1)


In there:


C: call option value P: put option value S: spot rate

E: strike rate


T: expiration date (%/year) e: exponential function e=2.71828

N(x): standard cumulative density function r: risk-free interest rate

:% fluctuation of exchange rate


In ( S / E ) ( r 2 / 2) T

d 1 ; d2=d1- T

2 T 1/ 2


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According to the Black-Scholes model, there are 5 factors affecting the option price: spot exchange rate, option strike rate, option term, interest rates of the 2 currencies and most importantly, exchange rate volatility.

1.4.4.2 Limitations of the Black-Scholes formula and the additional formula


However, this model also has some limitations, namely: the assumptions of the model are almost those of an efficient capital market. This is often not possible in reality when financial crises occur. At that time, the inflation of the national currency in crisis will increase and can become hyperinflationary, making the assumption of the stability of the risk-free interest rate no longer exist. In addition, if a devaluation or appreciation of the currency occurs, it can lead to a sudden fluctuation, which is contrary to the hypothesis of gradual exchange rate fluctuations.

Based on the results achieved by the Black-Scholes pricing model, in 1983, two economists Garman and Kohlhagen added to the above formula the interest rate factor of the two currencies:

r : foreign currency interest rate (%/year calculated for a term equal to the term of the option)

f


r : domestic interest rate (%/year for a term equal to the term of the option)

d


C = e


-rfT

SN(d1) – EN(d2)e


-rdT


P = e


-rdT

EN(-d2) – SN(-d1)e


-rfT



d 1

In ( S / E ) ( r d

r f

2 / 2) T

;

2 T 1 / 2


d2 = d1 - T


In fact, people have computerized the above formulas for ease of use. Nowadays, we just need to enter the data, the program will automatically run and give the option price results.

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Nameitem

Present value

S T X

S T > X

A

S 0

S T

S T

B

C e (S 0 ,T,X) + X(1+r) -

T

X

(S T – X) +X

= S T

The return from portfolio B is always at least equal to the return from portfolio A andsometimes even larger. Investors will notice this and price portfolio Bat least equal to category a, which means:

C e (S 0 ,T,X) + X(1+r) -T S 0

C e (S 0 ,T,X) S 0- X(1+r) -T

If S 0- X(1+r) -Tis negative, we consider the lowest value of the call option to be0. Combining these results gives us a lower bound:

C e (S 0 ,T,X) Max(0, S 0- X(1+r) -T )

If the call option price is lower than the underlying stock price minus the present value of the priceBy doing so, we can build an arbitrage portfolio.We buy the call option and the risk-free bond and short the underlying asset.This portfolio has a positive initial cash flow because the option price plus the bond price iscoupon is lower than the stock price. At maturity, the return on the portfolio is X – S Tif C >S Tand zero otherwise. The portfolio has positive cash flow now and has positive cash flowmoney is either zero or positive at maturity.

Maybe you are interested!

Option Pricing Using the Black – Scholes Model[10].



35


Call option price

Figure 5 : Price curve of European call option


C e (S 0 ,T,X)

Max(0,S0 – X(1+r) -T )

Max(0,S 0 – X)

X(1+r) -TX

0 Stock price (S 0 )


Figure 1.5 shows the results of a European call option. The curve is the price.call option, lies above the lower limit line. As the expiration time approaches, thetime to maturity decreases to the extent that the lower boundary moves to the sideRight. Time value also decreases for options and moves with it.lower bound, all gradually return to intrinsic value, Max(0,S 0 – X), at maturity.

American vs. European Call Options


C a (S 0 ,T,X) C e (S 0 ,T,X)

1.6.2. Principles of pricing put options


1.6.2.1. Minimum value of a put option


A put option is an option that gives the right to sell a stock. AThe owner of a put option is not obligated to exercise the option and willnot do so if exercise would reduce the value of the asset. Thus an optionsale never has negative value:

P(S 0 ,T,X) 0

An American put option can be exercised early. Therefore


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P a (S 0 ,T,X) Max(0,X - S 0 )

The value Max(0,X - S 0 ) is called the intrinsic value of the put option.The difference between the put option price and the intrinsic value is the time value or speculative value.The time value is defined as P a (S 0 ,T,X) - Max(0,X - S 0 ). The time valuereflects what an investor is willing to pay for the uncertainty of the outcome.final result

Figure 6 : Minimum value of European and American put options



(a) European-style option price


Put option price

0 Stock price (S 0 )



(b) American option price

X

Max(0,X -S 0 )


Put option price

0 X


Stock price (S 0 )


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In Figure 1.6, the price of a European put option lies in the shaded area.dark color in graph a. The price of an American put option lies in the shaded area.dark color in graph b.

The intrinsic value property, Max(0,X - S 0 ) is not true for a put option.European style because the option must be exercisable for the actual investorpresent interest rate arbitrage transactions.

1.6.2.2. Maximum value of a put option


At expiration, the payoff from a European put option is Max(0,X - S T ).The present value of the strike price is the maximum possible value of the put option.European style. Since American style put options can be exercised early at any timeAt any time, its maximum value is the strike price.

P e (S 0 ,T,X ) X (1 r ) T

P a (S 0 ,T,X) X

Figure 7 : Minimum and maximum values ​​of European and American put options



X(1+r) -T

(a) European-style options



Put option price

0 Stock price (S 0 )

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