Probability Theory in Finance - Personal WWW Pages

ProblemEuropean call option

Option valueStochastic models in finance

Probability Theory in Finance

Xuerong Mao

Department of Statistics and Modelling ScienceUniversity of Strathclyde

Glasgow, G1 1XH

Xuerong Mao Probability in Finance



Outline

1 Problem

2 European call option

3 Option value

4 Stochastic models in finance




Outline

1 Problem


3 Option value





Outline

1 Problem


3 Option value





Outline

1 Problem


3 Option value





Assume that on 1 January 2007, Mr King has $100K to investfor 1 year and he has two choices:

(a) invest the money in a bank saving account to receive arisk-free interest.

(b) buy a $100K house and then sell it on 1 January 2008.

Which choice should Mr King take?


















Assume the annual interest rate r = 3% and let X denote theprice of the house on 1 January 2008. Consider cases:

(i) P(X = $110K ) = 0.5 and P(X = $90K ) = 0.5.

In this case, EX = $100K so it is better to deposit themoney in the bank which gives Mr King $103K by 1January 2008.

(ii) P(X = $110K ) = 0.9 and P(X = $90K ) = 0.1.

In this case, EX = $108K which is $5K better than thereturn of the saving account.





(i) P(X = $110K ) = 0.5 and P(X = $90K ) = 0.5.


(ii) P(X = $110K ) = 0.9 and P(X = $90K ) = 0.1.






(i) P(X = $110K ) = 0.5 and P(X = $90K ) = 0.5.


(ii) P(X = $110K ) = 0.9 and P(X = $90K ) = 0.1.






(i) P(X = $110K ) = 0.5 and P(X = $90K ) = 0.5.


(ii) P(X = $110K ) = 0.9 and P(X = $90K ) = 0.1.





Assume that you trust the housing market will certainly obeyCase (ii). Should you have $100K available, you would haveinvested it into the house to obtain the expected profit of $8K.

The problem is that you do NOT have the capital of $100K andyou just feel unfair to give the opportunity to rich people like MrKing.

However, Professor Mao would like to help. On 1st January2007, Professor Mao (the writer) writes a European call optionthat gives you (the holder) the right to buy 1 house for $100Kon 1st January 2008.




European call option

Definition

A European call option gives its holder the right (but not theobligation) to purchase from the writer a prescribed asset for aprescribed price at a prescribed time in the future.

The prescribed purchase price is know as the exercise price orstrike price, and the prescribed time in the future is known asthe expiry date.




On 1st January 2008 you would then take one of two actions:

(a) if the actual value of a house turns out to be $110K youwould exercise your right to buy 1 house from ProfessorMao at the cost $100 and immediately sell it for $100Kgiving you a profit of $10K.

(b) if the actual value of a house turns out to be $90K youwould not exercise your right to buy the house fromProfessor Mao—the deal is not worthwhile.




On 1st January 2008 you would then take one of two actions:

(a) if the actual value of a house turns out to be $110K youwould exercise your right to buy 1 house from ProfessorMao at the cost $100 and immediately sell it for $100Kgiving you a profit of $10K.

(b) if the actual value of a house turns out to be $90K youwould not exercise your right to buy the house fromProfessor Mao—the deal is not worthwhile.




Note that because you are not obliged to purchase the house,you do not lose money. Indeed, in case (a) you gain $10K whilein case (b) you neither gain nor lose.

Professor Mao on the other hand will not gain any money on1st January 2008 and may lose an unlimited amount.

To compensate for this imbalance, when the option is agreedon 1st January 2007 you would be expected to pay ProfessorMao an amount of money known as the value of the option.

Question : Should Professor Mao charge you $2K, do you wantto sign the option?




Let C denote the payoff of the option on 1 January 2008. Then

C =

{$10K if X = $110K ;$0 if X = $90K .

Recalling the probability distribution of X

P(X = $110K ) = 0.9, P(X = $90K ) = 0.1.

we obtain the expected payoff

EC = 0.9× $10K + 0.1× $0 = $9K .

But $2K saved in a bank for a year will only grow to

(1 + 3%)× $2K = $2.06.

Therefore, the option produces the expected profit

$9K − $2.06 = $6.94.




It is significant to compare your profit with Mr King’s.

Mr King invests his $100K in the house and makes $5K moreprofit than saving his money in a bank.

You pay only $2K for the option but make $6.94 more profit thansaving your $2K in a bank.

It is even more significant to observe that you only need $2K,rather than $100K, in order to get into the market.




However, should Professor Mao charge you $8.8K, do youwant to sign for the option?

If you save your $8.8K in a bank, you will have

(1 + 3%)× $8.8K = $9, 064

which is $64 better off than EC = $9K , the expected payoffof the option. You should therefore not sign the option.




However, should Professor Mao charge you $8.8K, do youwant to sign for the option?

If you save your $8.8K in a bank, you will have

(1 + 3%)× $8.8K = $9, 064

which is $64 better off than EC = $9K , the expected payoffof the option. You should therefore not sign the option.




Key question

How much should the holder pay for the privilege of holding theoption? In other words, how do we compute a fair price for thevalue of the option?




In the simple problem discussed above, the fair price of theoption is

EC1 + r

=$9K

1 + 3%= $8738

However, the idea can be developed to cope with morecomplicated distribution.




Example

Assume that the house price will increase by $6K per half ayear with probability 60% but decrease by $6K per half a yearwith probability 40%. Then the house price X on 1 January2008 will have the probability distribution:

X (in K$) | 88 100 112P | 0.16 0.48 0.36

HenceEC = 0.36× $12K = $4320

and the option value is

EC1 + r

=$43201 + 3%

= $4194




Example

Assume that the house price will increase by $3K per quarterwith probability 60% but decrease by $3K per quarter withprobability 40%. Then the house price X on 1 January 2008 willhave the probability distribution:

X (in K$) | 88 94 100 106 112P | 0.0256 0.1536 0.3456 0.3456 0.1296

Hence

EC = 0.3456× $6K + 0.1296× $12K = $3629

and the option value is

EC1 + r

=$36291 + 3%

= $3523




However, the housing price, or more generally, an asset price ismuch more complicated than the probability distributionsassumed above. In their Nobel-prize winning model, Black andScholes showed that an asset price follows a log-normaldistribution. Applying their theory to the housing price yieldsthat the house price X on 1 January 2008 will obey alog-normal distribution. That is

log(X ) ∼ N(µ, σ2).




Assume µ = 4.63 and σ = 0.1 (we will explain how they comelater). Then

log(X ) = 4.63 + 0.1Z ,

where Z ∼ N(0, 1). That is

X = e4.63+0.1Z = 102.514e0.1Z .

Note that the payoff

C =

{X − 100 if X > 100,

0 if X ≤ 100.

But X > 100 iff Z > −0.248 and Z has the p.d.f

f (z) =1√2π

e−0.5z2, z ∈ R,




Thus

EC =

∫ ∞

−0.248

1√2π

e−0.5z2(

102.514e0.1z − 100)

dz

= 102.514e0.005∫ ∞

−0.248

1√2π

e−0.5(z−0.1)2dz

− 100∫ ∞

−0.248

1√2π

e−0.5z2dz

= 103.028∫ ∞

−0.348

1√2π

e−0.5z2dz − 59.79

= 5.735(K$)

So the option value is $5.735K/1.03 = $5568.




Question : How did Black and Scholes get

log(X ) ∼ N(µ, σ2)?

One of the main problems in financial mathematics is to modelthe asset price.




Stochastic models in finance

The Black–Scholes geometric Brownian motion.

The mean reverting process.

The square root process.

The mean reverting square root process.

The stochastic volatility model.






































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