# Actuarial Mathematics

Actuarial Mathematics Course Notes – Pricing Options

Spring 2021 TAMUC

Let’s recall some definitions (you can read in McDonald’s Derivatives Markets, which this follows closely, for more details). A European option is an option, either call or put, which can be exercised only on a fixed date. These are the types of options discussed here unless stated otherwise. A futures contract is an obligation to buy or sell a commodity at an agreed upon price at the contract expiration. A call option is an option to buy a commondity at an agreed upon strike price at the contract expiration, while a put option is an option to sell a commodity at an agreed upon strike price at the contract expiration.

An important assumption is that there are no arbitrages in the markets, no ‘free lunch’, no risk-free way of making money, usually defined as taking advantage of a difference in the price of a commodity between markets. In this case the law of one price is violated, meaning the commodity does not sell at the same price in all markets. If there are any arbitrages, they certainly don’t last for long, as those who discover an arbitrage will act to benefit from it, causing the price difference leading to the arbitrage to disappear.

Example: Commodity A can be purchased in Commerce for $5 and sold in Greenville for $10. As a result, increased ammounts of Commodity A are bought in Commerce, reducing supply and increasing the price in Commerce, and shipped to Greenville and sold there, increasing the supply and reducing the price in Greenville. This will continue until the prices in Commerce and Greenville are the same, at which time there is no longer a financial incentive to ship products from Commerce to Greenville. �

0.1 Put-call parity at a fixed strike price Put and call options are very frequently traded in the financial markets. An important question then is how to determine market prices for call and put options. These prices have to satisfy the principle that no (sus- tainable) arbitrage is possible in the market. The most basic idea behind pricing options is put-call parity, which ensures that an arbitrage cannot exist between the prices of puts and calls versus the prices of futures contracts. Consider the following examples.

Example: The investor buys a futures contract for 1 share of TomCo stock which is actionable in one year at the strike price of $50 per share. There is no premium on the contract. Let S denote the spot price of 1 share of TomCo stock at the option maturity. The investor’s payoff on the futures contract is then S − 50 whether S < 50 or S > 50. �

Example: An investor buys a call option for 1 share of TomCo stock which is actionable in one year at the strike price of $50 per share. The investor pays a premium for the call option. The investor also sells a put option for 1 share of TomCo stock due in one year at the strike price of $50 per share. The investor collects a premium for the put option.

If the spot price S satisfies S > 50 after one year, the call option is exercised and the investor makes a profit of S−50 on the call option. The put option is not exercised. The investor’s payoff is then S−50 > 0.

If the spot price S ≤ 50 in one year, the call option is not exercised, the put option is exercised so that the investor is forced to buy the stock at the spot price S and the investor takes a loss of S − 50 < 0 on the put option. So regardless of the spot price, one of the options is exercised, one is not, leading to a payoff of S−50 for the investor. To this point the profit from the futures contract equals the profit from the combination of buying a call and selling a put. The difference between this portfolio and the futures contract are the call and put options paid and collected. Let FVCall,FVPut denote the future values of the call and put option premiums. The investor’s payoff on the two positions combined is then S − 50 + FVPut −FVCall. �

Example: Now suppose an investor sells a call option for 1 share of TomCo stock which is actionable in one year at the strike price of $50 per share. The investor collects a premium for the call option. The investor also buys a put option for 1 share of TomCo stock due in one year at the strike price of $50 per share. The investor pays a premium for the put option.

If the spot price S satisfies S > 50 after one year, the sold call option is exercised, the investor is forced to sell the stock at the spot price and the investor makes a profit of 50 −S on the call option. The put option is not exercised. The investor’s payoff is then 50 −S < 0.

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MATH 503 Dr. Boucher

Actuarial Mathematics Course Notes – Pricing Options

Spring 2021 TAMUC

If the spot price S ≤ 50 in one year, the sold call option is not exercised, the bought put option is exercised so that the investor sells the stock at the strike price 50 and the investor makes a profit of 50 −S ≥ 0 on the put option. So regardless of the spot price, one of the options is exercised, one is not, leading to a payoff of 50−S for the investor. To this point the loss from the futures contract equals the profit from the combination of selling a call and buying a put. The difference between this portfolio and the futures contract are the call and put options paid and collected. Let FVCall,FVPut denote the future values of the call and put option premiums. The investor’s payoff on the two positions combined is then 50 −S + FVCall −FVPut. So the profit from selling a call and buying a put is the opposite of the profit from buying a call and selling a put. �

Unless FVPut = FVCall there is a difference among the payoffs of the put/call combinations and the fu- tures contract. If in addition E[S] = 50 there is a difference between the payoffs of the put/call combinations – but none of these can hold without violating the ‘no arbitrage’ condition, as the futures contract and the sell put/buy call combination are really the same position with the same profit S − 50, excluding premiums, or the put/call combinations are opposite sides of the same bet. This is what is meant by put/call parity – the premiums of put options and call options for the same commodity at the same strike price and the same option duration must be the same, else an arbitrage exists in the market: if FVPut − FVCall > 0 then the investor is guaranteed to make money by choosing the buy call/sell put option versus the futures contract, while if FVPut −FVCall < 0 then the investor is guaranteed to make money by choosing a futures contract over the sell call/buy put option.

Letting C(K) denote the call option premium and P(K) denote the put option premium at strike price K and futures contract strike price F , then FVPut = FVCall and K = F implies

C(K) = P(K) ⇔ C(K) −P(K) = 0.

Think of the futures contract price F as the overall market opinion of what the commodity price will be at expiration. If the futures contract strike price F is different from the options strike price K (so the individual investor who believes K is betting that the market is wrong), then we have the put-call parity equation

C(K) −P(K) = F −K,

so that a call option has a higher premium than a put option if the strike price K is less than the futures contract price F (in the market’s opinion F − K > 0 so the call option position is a profitable one as the investor is able to buy the commodity for less than the market thinks it will be worth), and a call option has a lower premium than a put option if the strike price K is greater than the futures contract price F (in the market’s opinion F − K < 0 so the put option position is a profitable one as the investor is able to sell the commodity for less than the market thinks it will be worth).

Example: The investor considers buying a futures contract for 1 share of TomCo stock which is actionable in one year at the futures price of F = $50 per share. Other investment possibilities are a call option for 1 share of TomCo stock which is actionable in one year at the strike price of K = $40 per share and a put option for 1 share of TomCo stock due in one year at the strike price of K = $40 per share. The difference between the call option and put option premiums will then be C(K)−P(K) = F −K = $50−$40 = $10, meaning the call option premium will be $10 more than the put option premium. The call option is more valuable since the strike price is less than the futures price. �

Example: The investor considers buying a futures contract for 1 share of TomCo stock which is actionable in one year at the futures price of F = $50 per share. Other investment possibilities are a call option for 1 share of TomCo stock which is actionable in one year at the strike price of K = $60 per share and a put option for 1 share of TomCo stock due in one year at the strike price of K = $60 per share. The difference between the call option and put option premiums will then be C(K) −P(K) = F −K = $50 − $60 = −$10, meaning the put option premium will be $10 more than the call option premium. The put option is more valuable since the strike price is more than the futures price. �

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MATH 503 Dr. Boucher

Actuarial Mathematics Course Notes – Pricing Options

Spring 2021 TAMUC

0.2 Letting strike prices vary Until now we have assumed a single strike price K that is shared by the call and put options. We can make some basic statements about how option prices vary with the strike price. Consider three strike prices, K1 < K2 < K3, with corresponding call option prices C(K1),C(K2),C(K3) and put option prices P(K1),P(K2),P(K3). We will state the following without formal proof but will include a heuristic ar- gument – the basic idea is that no arbitrage can exist:

1. A call option with a lower strike price cannot be cheaper than the same call option with a higher strike price: that is C(K1) ≥ C(K2) if K1 ≤ K2 or equivalently the cost C()̇ of a call option is a decreasing function in the strike price K. (If not true, buy the low-strike call and sell the high-strike call = arbitrage.)

2. A put option with a higher strike price cannot be cheaper than the same put option with a lower strike price: that is P(K1) ≤ P(K2) if K1 ≤ K2 or equivalently the cost P()̇ of a put option is an increasing function in the strike price K. (If not true, buy the high-strike put and sell the low-strike put = arbitrage)

3. The difference in premiums between identical call options with different strike prices cannot be greater than the difference in strike prices: if K1 ≤ K2 then 0 ≤ C(K1) −C(K2) ≤ K2 −K1. (if not true, sell the low-strike call and buy the high-strike call = arbitrage)

4. The difference in premiums between identical put options with different strike prices cannot be greater than the difference in strike prices: if K1 ≤ K2 then 0 ≤ P(K2) −P(K1) ≤ K2 −K1. (If not true, buy the low-strike put and sell the high-strike put = arbitrage)

5. The prices of call options are convex functions of the strike price: that is, if K1 < K2 < K3 then

C(K1) −C(K2) K2 −K1

≥ C(K2) −C(K3)

K3 −K2 .

Mathematically, this says that the slopes of secant lines of C are increasing from left to right. Combined with the fact that C()̇ is a decreasing function of the strike price K, the price of call options declines at a decreasing rate as we consider calls with higher strike prices. (Else the call price would go to zero, meaning you could acquire a call option for free = arbitrage)

The prices of put options are also convex functions of the strike price: that is, if K1 < K2 < K3 then

P(K2) −P(K1) K2 −K1

≤ P(K3) −P(K2)

K3 −K2 .

The slopes of secant lines of P are increasing from left to right. Combined with the fact that P()̇ is an increasing function of the strike price S, the price of put options increases at an increasing rate as we consider calls with higher strike prices. (Else buy the higher priced put and sell the lower priced put = arbitrage).

To reiterate, these mathematical properties of price functions eliminate different ways that an arbitrage – a risk-free method of making money – could come into existence in the market (in an ideal world!).

Arbitraging a Mispriced Option: In practice, markets are not perfect. Information flow is not instant or even uniform. So let’s say you spot an arbitrage opportunity – What if the observed option price differs from the theoretical price? How can an investor take advantage of this? Because we have a way to replicate the option using the stock, it is possible to take advantage of the mispricing.

Example: Suppose a call option with a strike price of K1 = $50 is selling at a C(K1) = $20 premium and a call option with a strike price of K2 = $55 is selling at a C(K2) = $10 premium. These creates an arbitrage opportunity since the difference in strikes is K2 − K1 = $5 and the difference in the premiums

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MATH 503 Dr. Boucher

Actuarial Mathematics Course Notes – Pricing Options

Spring 2021 TAMUC

is C(K1) − C(K2) = $10. Note this violates the 3rd property listed above, indicating the existence of an arbitrage opportuity. We could engage in arbitrage by buying the 55-strike call and selling the 50-strike call. Note that we receive C(K1)−C(K2) = $10 initially and never have to pay more than K2 −K1 = $5 in the future by exercising the higher-priced call whenever the spot price S > 50. Neither option will be exercised when S ≤ 50. �

Example: Suppose a put option with a strike price of K1 = $60 is selling at a P(K1) = $10 premium and a put option with a strike price of K2 = $55 is selling at a P(K2) = $20 premium. These creates an arbitrage opportunity since the difference in strikes is K2 − K1 = −$5 and the difference in the premiums is P(K2) − P(K1) = $10. Note this violates the 4th property listed above, indicating the existence of an arbitrage opportuity. We could engage in arbitrage by selling the 55-strike put and buying the 60-strike put. Note that we receive P(K2)−P(K1) = $10 initially and never have to pay more than K1 −K2 = $5 in the future by exercising the higher-priced put whenever the spot price S > 55. Neither option will be exercised when S ≤ 55. �

0.3 Binomial Option Pricing To this point we have described how the price of one option is related to another, but we still need methods for determining the price of an option relative to the price of the underlying asset. One method for doing so is the binomial option pricing model. The binomial option pricing model assumes that, over a period of time, the price of the underlying asset can move only up or down by a specified amount-that is, the asset price follows a Bernoulli (Binomial with a single trial) distribution. With this assumption it is possible to determine a no-arbitrage price for the option that expires at the end of this period of time. The approach is very simplistic, but can be used to price options, and helps to illustrate the concepts involved. We focus on pricing call options initially.

We have two instruments to use in replicating a call option: buying shares of stock and buying/selling bonds. We need to find a combination of stock and bonds that mimics the option. To be specific, we wish to find a portfolio consisting of 0 ≤ ∆ ≤ 1 fractional shares of stock and a dollar amount B in bonds at continuously compounded interest rate i, such that the portfolio payoff replicates the option payoff whether the stock rises or falls. B can be positive or negative; think of B > 0 as buying bonds (loaning money) and B < 0 as borrowing money (selling bonds). We will suppose that the stock has no dividend.

Let S denote the current spot price. We can denote the stock price as Su when the stock price is up at the end of the expiration period and as Sd when the stock price is down at the end of the expiration period. Sometimes analysts represent the stock price as a tree. Let Cu = max(0,Su−K), and Cd = max(0,Sd−K) represent the value of the call option at expiration when the stock goes up or down, respectively. These will be the payoffs at expiration of the option. The tree for the stock price movements will imply a corresponding tree for the value of the call option.

If the length of a period is h , the interest factor per period is eih. The problem is to solve for ∆ and B such that our portfolio of ∆ shares and B in bonds duplicates the option payoff. The value of the replicating portfolio at time h, with stock price Sh, is ∆Sh + eihB. To mimic the option, the portfolio of stocks and lending has to satisfy

∆Sd + e ihB = Cd

∆Su + e ihB = Cu,

so that the movements up and down of the portfolio and the option mirror each other. This system of equations

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MATH 503 Dr. Boucher

Actuarial Mathematics Course Notes – Pricing Options

Spring 2021 TAMUC

in 2 unknowns has solutions

∆ = Cu −Cd Su −Sd

B = e−ih ( CdSu −CuSd Su −Sd

) .

Given the expressions for ∆ and B, the premium for the option is the cash required to buy the shares and bonds (or the law of one price is violated). Thus, if the current spot price is S the cost of the option is then

Option Cost = ∆S + B,

which would be the amount of money needed to buy the fractional shares and bonds.

Example: Consider a European call option of TomCo stock, with a K = $40 strike and 1 year to expiration. TomCo pays no dividends and its current price is S = $41. The continuously compounded risk-free interest rate is i = 0.08. Today the price is $41 , and in 1 year the price can be either Su = 60 or Sd = 30. We want to determine the option price. The payoffs on the call option are Cu = max(0,Su−K) = max(0, 60−40) = 20 and Cd = max(0,Sd −K) = max(0, 30 − 40) = 0 since the strike price of $40 is less than Sd. Then

∆ = 20 − 0 60 − 30

= 2

3

B = e−0.08 (

0 × 60 − 20 × 30 60 − 30

) ≈−$18.47.

Since B is negative we are borrowing the $18.47 to put towards the purchase of the 2/3 fractional share of the stock. The option price is

∆S + B = 2

3 × 41 + (−18.47) ≈ $8.87.

�

Some observations:

• Clearly 0 ≤ ∆ ≤ 1, Sd ≤ Su, Cd ≤ Cu.

• If Su ≤ K then Cu = Cd = 0 and ∆ = 0,B = 0 and the option should be free when there is a certainty of it NOT being exercised due to the spot price always being less than the strike price.

• If Sd ≥ K then the spot price is always greater than the strike price and the option is certain to be exercised. In this case Cu = Su −K, Cd = Sd −K,

∆ = Cu −Cd Su −Sd

= Su −K − (Sd −K)

Su −Sd = 1

B = e−ih (

(Sd −K)Su − (Su −K)Sd Su −Sd

) = −Ke−ih,

and the price of the option is S−Ke−ih which is the current price of the stock minus the present value of the option strike price, the present value of the estimated profit when the option is exercised.

• If Sd ≤ K, Su ≥ K then Cd = 0,Cu > 0 and B < 0, ∆ > 0 which means the price of the option ∆S + B < S is less than the current spot price of the stock as the option may or may not be exercised. Exactly, the option price is

∆S + B = Su −K Su −Sd

( S −Sde−ih

) which increases in K and contains the cases above as K → Sd from above and K → Su from below.

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MATH 503 Dr. Boucher

Actuarial Mathematics Course Notes – Pricing Options

Spring 2021 TAMUC

We’ve assumed the future prices Su,Sd of the stock are known. Of course they won’t be known exactly so how to estimate them? The current spot price S of the stock is known. The forward price of the stock for a futures contact is then Seih, which is the current stock price increased by the risk-free interest rate i. We also need to incorporate risk in the form of uncertainty about the future price of the stock. A natural measure of uncertainty about the stock return is the annualized standard deviation of the continuously compounded stock return, which we will denote σ. The standard deviation measures how sure we are that the stock return will be close to the expected return. Stocks with a larger σ will have a greater chance of a return far from the expected return. If the annual standard deviation is σ then the standard deviation over a period of length h is σ √ h. The stock prices are then calculated using one standard deviation times the forward price

Su = Se ih+σ

√ h

Sd = Se ih−σ

√ h.

These are the possible binomial moves up and down. Note that probabilities of these price movements were not used anywhere in the option price calculations.

Since the strategy of holding ∆ shares and B bonds replicates the option whichever way the stock moves, the probability of an up or down movement in the stock is irrelevant in this strategy for pricing the option.

Example: Consider a European call option of TomCo stock, with a K = $40 strike and 1 year to expiration. TomCo pays no dividends and its current price is S = $41. The continuously compounded risk-free interest rate is i = 0.08. Suppose volatility is σ = 30% = 0.3. Since the period is 1 year, h = 1, so that σ

√ h = 0.30.

Then the up and down price changes are

Su = Ste ih+σ

√ h = $41e0.08×1+0.30 ≈ $59.95

Sd = Ste ih−σ

√ h = $41e0.08×1−0.30 ≈ $32.90.

With a strike price of K = $40, we have Cu = max(0,Su − K) = max(0, $59.954 − $40) ≈ 19.95, and Cd = max(0,Sd −K) = max(0, 32.90 − 40) = 0. Then

∆ = 19.954 − 0

59.95 − 32.90 ≈ 0.738

B = e−0.08×1 (

0 × 59.95 − 19.95 × 32.90 59.95 − 32.90

) ≈−22.40

Hence, the option price is given by ∆S + B = 0.738 × $41 − $22.40 ≈ $7.86 �

Put options can be priced similarly, the only difference between the two being that values at contract expiration are priced using Cu = 0 when S ≥ K, and Cd = K − S when the spot price S falls below the strike price K

Example: Suppose that the market price for the call option is $9.00, instead of $7.86, so that the option is mispriced. We can sell the option and collect the premium, but this leaves us with the risk that the stock price at expiration will be $59.95 and we will be required to deliver the stock at the $40 strike price, leading to a loss of $19.95. We can address this risk by buying a synthetic option at the same time we sell the actual option. We have already seen how to create the synthetic option by buying ∆S = 0.738 shares and loaning B = $22.40. If we simultaneously sell the actual option and buy the synthetic, the initial cash flow is

$9.00 − 0.738 × $41 + $22.40 = $9.00 − $7.86 = $1.14.

We earn $1.14 upfront, the amount by which the option is mispriced. Now we verify that there is no risk at expiration. We calculated that the price in 1 year could either be Su = $59.95 or Sd = $32.90.

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MATH 503 Dr. Boucher

Actuarial Mathematics Course Notes – Pricing Options

Spring 2021 TAMUC

If the stock price at expiration is S = $32.90 the call option we sold is not exercised and there is no further exchange of money. The 0.738 shares we bought are worth 0.738 × $32.90 ≈ $24.28. The cost of repaying the borrowed money B = $22.40 is $22.40e0.08 = 24.27, so this is a wash up to rounding error.

If the stock price is S = $59.95 the call option we sold is exercised and the loss is $59.95 − $40 = $19.95. The cost of repaying the borrowed money B = $22.40 is still $22.40e0.08 = 24.27, so we are out $19.95 + $24.27 ≈ 44.22. The 0.738 shares we bought are worth 0.738×$59.95 ≈ $44.24 so this is a wash up to rounding error.

And we still have the $1.14 we pocketed up front. �

0.4 Two or More Binomial Periods The model above is very limited in that it allows for only 2 possible prices of the stock at expiration Sd,Su. However, the model can be expanded to create more realistic models by adding more Binomial periods between the present time and the time of option expiration. We once again limit discussion to European options. Two or more binomial periods gives a tree with 2 or more levels. The tree is grown iteratively by repeating the process for a single Binomial period on each node of the previous period.

We illustrate the process by adding an additional period to the previous example and using it to price a 2-year option with a K = $40 strike when the current stock price is $41, assuming all other variables are the same as before.

To see how to construct the tree, suppose that we move up in Year 1 to Su = $59.95 as before. If we reach this price, then we can move further up or down by repeating the Binomial splitting process beginning at Su:

Suu = Sue ih+σ

√ h = $59.95e0.08+0.3 ≈ $87.66

Sud = Sue ih−σ

√ h$59.95e0.08−0.3 ≈ $48.11.

Similarly if the price in one year is Sd = $32.90, we have the movements up or down from here as

Sdu = Sde ih+σ

√ h = $32.90e0.08+0.3 ≈ $48.11

Sdd = Sde ih−σ

√ h$32.90e0.08−0.3 ≈ $26.40.

We now have our Binomial tree of 2 periods with terminal nodes Sdd,Sdu,Sud,Suu, interior nodes Sd,Su, and root note S.

Note that in this example an up move followed by a down move Sud generates the same stock price as a down move followed by an up move Sdu. While not always the case, this situation is called a recombining tree. If an up move followed by a down move led to a different price than a down move followed by an up move, this is called a nonrecombining tree. Adding more Binomial periods creates more complicated and realistic models; these can easily be implemented on a computer.

So now we want to price the call option at the different time periods in the tree. Note that the value of the 2-year option will vary between Year 0 and Year 2 depending on the price of the stock in Year 1. An investor who buys a 2-year call option in Year 0 may decide to sell it in Year 1 before the option expires so it is necessary to keep the option price updated. Our Binomial tree gives us the possible stock prices at future times. However, to price an option at a given time requires also knowing the option prices resulting from up and down moves in the next period. For example, to price the option in the single Binomial period model, calculating the option price required knowing not only the stock prices Su,Sd but also the value of the option Cu,Cd at expiration. Initially the only period where the option price for a given spot price is known is at expiration (the option price here is the payoff on the option). The call option is then priced for other periods by working backward through the Binomial tree. Knowing the option price at expiration, we can determine the option price in Year 1. Having determined the price in Year 1, we can work back to the option price in Year 0.

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MATH 503 Dr. Boucher

Actuarial Mathematics Course Notes – Pricing Options

Spring 2021 TAMUC

The spot price at expiration is limited to one of three possibilities in a Binomial pricing model with 2 periods. The option prices for Year 2 can then be determined for each spot price:

1. Year 2, Stock Price Suu = $87.669. Since the option expires, the option value is max(0,Suu − 40) = $47.669.

2. Year 2, Stock Price Sud = Sdu = $48.114. The option value is max(0,Sud − 40) = $8.114.

3. Year 2, Stock Price Sdd = $26.405 The option will not be exercised so the value is 0.

From Year 2 work back to Year 1 and the nodes for Su,Sd. At Su there are up and down movements Suu,Sud. From Sd there are up and down movements Sdu,Sdd. We need to calculate the option price for each inner node Su and Sd using its respective up and down price movements. We do so by repeating the calculations for a single Binomial period at each inner node.

1. Year 1, Stock Price Su = $59.954. We already know from the Binomial tree that

Suu ≈ 87.66 Sud ≈ 48.11

and use these to calculate

Cuu = max(0,Suu −K) = max(0, 87.66 − 40) = 47.66 Cud = max(0,Sud −K) = max(0, 48.114 − 40) = 8.114

∆u = Cuu −Cud Suu −Sud

= 47.66 − 8.114 87.66 − 48.11

= 0.9998989

Bu = e −ih

( CudSuu −CuuSud

Suu −Sud

) = e−0.08

( 8.114 × 87.66 − 47.66 × 48.11

87.66 − 48.11

) = −36.91647

resulting in an option value of Cu = ∆uSu +Bu = 0.9998989×59.954 + (−36.91647) = $23.03147

2. Year 1, Stock Price Sd = $32.903. We already know from the Binomial tree that

Sdu ≈ 48.11 Sdd ≈ 26.40

and use these to calculate

Cdu = max(0,Sdu −K) = max(0, 48.114 − 40) = 8.114 Cdd = max(0,Sdd −K) = max(0, 26.40 − 40) = 0

∆d = Cdu −Cdd Sdu −Sdd

= 8.114 − 0

48.11 − 26.40 = 0.3737448

Bd = e −ih

( CddSdu −CduSdd

Sdu −Sdd

) = e−0.08

( 0 × 48.11 − 8.114 × 26.40

48.11 − 26.40

) = −9.108263

resulting in an option value of Cd = ∆dSd +Bd = 0.3737448×32.903 + (−9.108263) = $3.189062.

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MATH 503 Dr. Boucher

Actuarial Mathematics Course Notes – Pricing Options

Spring 2021 TAMUC

Now that the option prices for Year 1 are known we work backwards one more time to get to the overall option price in Year 0 where the Stock Price S = $41 with Su = $59.954, Sd = $32.903, using Cu = $23.03147 and Cd = $3.189062 from Year 1 as the option prices yields the option price in Year 0 = $10.74.

∆ = Cu −Cd Su −Sd

= 23.03147 − 3.189062

59.954 − 32.903 = 0.7335185

B = e−ih ( CdSu −CuSd Su −Sd

) = e−0.08

( 3.189062 × 59.954 − 23.03147 × 32.903

59.954 − 32.903

) = −19.3355

resulting in an option value of C = ∆S + B = 0.7335185 × 41 + (−19.3355) = $10.73876. This process can be generalized to price options of more than 2 binomial periods. If the periods are near

enough together in time, this can begin to match a more realistic model. The calculations are tedious by hand but can easily be scripted and performed with a computer.

0.5 Relation of Binomial Pricing Model to Other Models Do stock prices follow a random walk? An example of a random walk is St where

St =

t∑ i=1

Yi,

with IID P(Yi = −1) = P(Yi = 1) = 1/2 so that E[Yi] = 0, V ar(Yi) = 1. Note mean and variance of St.

E[St] = 0 = tE[Yi]

V ar(St) = t = tV ar(Yi)

Issues with stock price St as random walk:

1. The stock price St could become negative.

2. The magnitude of the price move (V ar(St)) should depend upon how quickly the coin flips occur and the level of the stock price.

3. The stock on average should have a positive return. The random walk model has E[St] = 0

The binomial option pricing model is a variant of the random walk that solves these problems. The binomial model assumes that the continuously compounded returns are a random walk. Some properties of continuously compounded returns.

1. The logarithmic function computes returns from stock prices as the log price ratio. Let St and St+h be stock prices at times t and t + h. The continuously compounded return between tand t + h , rt,t+h is then

rt,t+h = ln

( St+h St

) 2. The exponential function computes prices from returns. If we know the continuously compounded

return rt,t+h and the current price St, we can compute St+h as

St+h = Ste rt,t+h.

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MATH 503 Dr. Boucher

Actuarial Mathematics Course Notes – Pricing Options

Spring 2021 TAMUC

3. Continuously compounded returns rt,t+h are additive over disjoint time periods. Suppose we have continuously compounded returns over a number of periods, for example, rt,t+h,rt+h,t+2h, . . .. The continuously compounded return over a long period is the sum of continuously compounded returns over the shorter periods, i.e.,

rt,t+nh =

n∑ i=1

rt+(i−1)h,t+ih

4. Continuously compounded returns can be less than -100%. A continuously compounded return that is a large negative number still gives a positive stock price. The reason is that er is positive for any r. Thus, if the log of the stock price follows a random walk, the stock price cannot become negative.

Then assuming no dividends there is the relationship with the Binomial model

Sdt+h = Ste ih−σ

√ h

Sut+h = Ste ih+σ

√ h

so that rt,t+h has two possible values

rt,t+h = ln

( St+h St

) = ih−σ

√ h

rt,t+h = ln

( St+h St

) = ih + σ

√ h

and the issues noted above are taken care of:

1. The stock price cannot become negative.

2. As stock price moves occur more frequently, h gets smaller, therefore up and down moves get smaller.

3. There is a ih term, so with an appropriate probability of an up move we can guarantee that the expected change in the stock price is positive.

0.6 Lognormal McDonald denotes Normal PDF as φ(x; µ,σ). CH 12 – Normal distributions, standardizing, sums of Normals, CLT. CH18…

A random variable Y , is said to be lognormally distributed if it can be written as ln(Y ) = X is Normal or Y = eX when X is Normal. This last equation links normally distributed continuously compounded returns with lognormality of the stock price as

rt,t+h = ln

( St+h St

) .

Lognormal PDF is

g(y; µ,σ) = 1

yσ √

2π e−(ln(y)−µ)

2/(2σ2, y ≥ 0

Assumption is that returns

rt,t+h = ln

( St+h St

) are Normal with mean αh, standard deviation σh and stock prices St are lognormal, so that since

rt,t+nh =

n∑ i=1

rt+(i−1)h,t+ih

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MATH 503 Dr. Boucher

Actuarial Mathematics Course Notes – Pricing Options

Spring 2021 TAMUC

where

E[rt+(i−1)h,t+ih] = αh

V ar(rt+(i−1)h,t+ih) = σ 2 h

then the rates over longer periods are random walks of moves over shorter periods,

E[rt,t+nh] = E

[ n∑ i=1

rt+(i−1)h,t+ih

] = nαh

V ar(rt,t+nh) = E

( n∑ i=1

rt+(i−1)h,t+ih

) = nσ2h

Let h = 1 and let α, σ be the annual mean and annual standard deviation of the stock price St. Then since St is Lognormal,

ln

( St S0

) ∼ N

( (α− σ

2

2 )t,σ2t

) or equivalently

ln (St) ∼ N (

ln(S0) + (α− σ 2

2 )t,σ2t

) as S0 is assumed known. This can be used for calculations of probabilities involving St.

The lognormal distribution is the probability distribution that arises from the assumption that continuously compounded retums on the stock are normally distributed. When we traverse the binomial tree, we are implicitly adding up binomial random return components of ih ± σ

√ h. In the limit (as n → ∞ or, the

same thing, h → 0), the sum of binomial random variables is normally distributed. Thus, continuously compounded returns in a binomial tree are (approximately) normally distributed, which means that the stock is lognormally distributed.

Is the Binomial model realistic? Any option pricing model relies on an assumption about the behavior of stock prices. As we have seen in this section, the binomial model is a form of the random walk model, adapted to modeling stock prices. The lognormal random walk model in this section assumes, among other things, that volatility is constant, that ”large” stock price movements do not occur, and that returns are independent over time. All of these assumptions appear to be violated in the data.

0.7 Black-Scholes Prelims In the study of derivative, stock and other asset prices are commonly assumed to follow a stochastic process called geometric Brownian motion. This chapter explains what this means and develops the notation and assumptions underlying the Black-Scholes model. Second, given that a stock price follows geometric Brow- nian motion, we want to characterize the behavior of a claim-such as an option-that has a payoff dependent upon the stock price. For this purpose we need Ito’s Lemma, which tells us the process followed by a claim that is a function of the stock price.

In the absence of any randomness we can express the risk-free continuously compounded price as the differential equation

d

dt St = µSt ⇔

d dt St

St = µ ⇔

∫ d dt St

St dt =

∫ µdt = µt ⇒ ln(St) = µt + C ⇒ St = S0eµt.

However, many assets are subject to random fluctuations in their prices. Most option pricing, including the original paper by Black and Scholes, begin by assuming that the price of the underlying asset St satisfies the following stochastic differential equation:

dSt St

= µdt + σdZ(t).

11

MATH 503 Dr. Boucher

Actuarial Mathematics Course Notes – Pricing Options

Spring 2021 TAMUC

This augments the previous deterministic model with a ‘noise’ term σdZ(t) where Z(t) is a normally dis- tributed random variable that follows a process called Brownian motion.

This noise term models the random fluctuations in the stock price. Also, dSt is the instantaneous change in the stock price, µ is the continuously compounded expected return on the stock, σ is the continuously compounded standard deviation (volatility). This implies the solution is

St = S0e µt+σZ(t) = S0e

µteσZ(t)

with St being called geometric Brownian motion. Brownian motion definition. Zt is Brownian motion if

1. Z0 = 0

2. Zt is (almost surely) continuous

3. Zt −Zs = is N(0, t−s) when s ≤ t

Brownian motion is the continuous version of a random walk. Compare St above with the solution to the ordinary differential equation without the randomness

St = S0e µt.

The solution to the stochastic differential equation is the deterministic solution to the ordinary differential equation (constant) multiplied by a term with a Normal random variable in the exponent so that St is lognor- mal as we have already seen:

ln (St) ∼ N (

ln(S0) + (α− σ 2

2 )t,σ2t

) 0.8 Black-Scholes Formula It is possible to show that as the number of steps in a Binomial option pricing mode approaches infinity, the option price is given by the Black-Scholes formula. Thus, the Black-Scholes formula is a limiting case of the binomial formula for the price of a European option.

More generally, the Black-Scholes formula applies when the price of the underlying asset is a geometric Brownian motion.

Assuming that the price of the underlying asset St satisfies the stochastic differential equation discussed earlier:

dSt St

= µdt + σdZ(t).

the Black-Scholes formula for a European call option on a stock that pays no dividends is

SN(d1) −Ke−itN(d2)

with

d1 = ln(S/K) + (i + 1

2 σ2)t

σ √ t

d2 = d1 −σ √ t

The six inputs to the Black-Scholes formula are the same as those for the binomial option pricing model (assuming the stock pays no dividends): S = the current price of the stock, K =the strike price of the option; σ = the standard deviation of continuous compounded returns on the stock, i = the continuously

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