Factors Affecting Currency Option Prices

1
Factors Affecting Currency Option Prices

• factors from contract specification
• strike price
• maturity date
• time value of money (risk-free rate)
• factors from currency price distribution
• currency price
• foreign risk-free rate
• volatility

2
Factors Affecting Currency Option Prices

• call options put
  option
• strike price -ve
  ve
• maturity ve
  ve
• domestic risk-free ve -ve
• currency price ve -ve
• foreign risk-free -ve ve
• volatility ve
  ve

3
Notation

• St currency price at time t
• K strike price
• T expiration date
• R continuously compounded risk-free rate
• c value of European call option
• p value of European put option
• R continuously compounded risk-free rate

4
Digression on Interest Rates

• interest rates are just a quotation mechanism
• they are not prices and in fact are derived from
  prices
• annualized rate
• semi-annual rate
• continuously compounded rate

5
Digression on Interest Rates

• relationship
• suppose the 6-month annualized rate is 6
• solving we find

6
Put-Call Parity

• holds only for European options
• put and call must have the same strike and
  maturity
• put-call parity for currency
• ct Ke-R(T-t) - pt - Ste-R(T-t) 0
• think of payoff at maturity

7
Put-Call Parity
lending
long call
underlying price
K
short put
short underlying
8
Put-Call Parity

• suppose the current exchange rate is 0.6782
• K 0.6900, R .06, T-t 0.5, c 0.10, R
  .07
• we must have

9
Put-Call Parity

• suppose the put price is 0.09
• arbitrage strategy
• long put
  -0.09
• short call
  0.10
• borrow Ke-R(T-t)
  0.6696
• long e-R(T-t) of currency -(.9656)(0.6782)
  -0.6549
• cash inflow time t
  0.0247

10
Put-Call Parity

• cash flow at maturity
• ST lt K
  ST gt K
• long put K - ST
  0
• short call 0
  K - ST
• borrow Ke-R(T-t) -K
  -K
• long e-R(T-t) of currency ST
  ST
• cash inflow time T 0.00
  0.00

11
Put-Call Parity

• put-call parity must hold because it is such a
  simple strategy
• remember that strike and maturity must be the
  same
• also must be European options
• usefulness
• if we determine the call price we can use
  put-call parity to find the put price

12
Black-Scholes Assumptions

• currency returns are normally distributed
• no transactions costs or taxes
• securities are perfectly divisible
• security trading is continuous
• borrow and lend at same rate per country
• risk-free rate is constant
• no arbitrage opportunities

13
Currency Price Distribution

• properties
• expected change in price increases as currency
  price increases
• volatility of price changes increases as currency
  price increases
• current price gives all information for future
  changes
• Markov property

14
Black-Scholes Formula

• where

15
Standard Normal Cumulative

• N(x) is the probability that a standard normal
  random variable is less than x
• standard normal
• mean is 0.0
• standard deviation is 1.0
• symmetric about mean
• N(0.0) 0.50
• table gives you these values

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Standard Normal Cumulative
example x0.80 N(x) 0.7881
AreaN(x)
0.0
x
17
Finding Volatility

• take a period of 100 days
• on each day define return as
• calculate the mean
• calculate the standard deviation
• this is the volatility

18
Black-Scholes Formula

• call example U.S./pound
• S 1.6000
• K 1.6000
• R 0.08
• T-t 0.3333
• R 0.11
• volatility 0.141

19
Black-Scholes Formula

• first calculate d1 and d2
• now use the tables to calculate N(d1) and N(d2)

20
Black-Scholes Formula

• the call price is
• on the Philadelphia exchange this contract is for
  31,250 pounds
• therefore you would pay 31,250(0.042) 1,312.50

21
Put Price

• to find the put price we can use put-call parity
• therefore you would pay 31,250(0.0575) 1,796.88

22
Forward Price

• the forward price is straightforward
• you could convert rates to annualized or period
  rates and would get the same answer

23
Hedging

• a U.S. firm needs to pay 31,250 pounds in 4
  months
• the unhedged cost is 31,250 (ST)
• with a forward hedge the cost would be 49,503.13
• with a long call hedge the cost is
• MIN (ST, K)31,250
• maximum cost is 50,000 paid 1,312.50 for this
  protection

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Summary

• completed option from Ch. 5 and appendix
• note, B-S formula in appendix is wrong (d1 is
  wrong)
• use above equation
• good questions 6, 7, 10-15
• next lecture
• managing transaction exposure ch. 10, 11
• managing economic exposure ch.12