Valuation of Contingent Claims Numerical Problems

Derivatives > Valuation of Contingent Claims Numerical Problems

Variant: Read a One-Period Binomial Tree and Find Terminal Payoffs

Abstract: A binomial tree is just two possible prices one step from now; build those prices first, then apply the option payoff rule.

A stock is $80 today. Over one period it can rise by a factor of $1.25$ or fall by a factor of $0.75$. A European call and put both have an exercise price of $85. Find the two future stock prices and every terminal option payoff.

1. Turn the factors into prices

The up factor is not a 1.25% return. It means the stock becomes 1.25 times its current price.

\[ S^+=Su=80(1.25)=\boxed{100} \]

\[ S^-=Sd=80(0.75)=\boxed{60} \]

2. Apply the call payoff rule

A call lets us buy for $85. We use it only when the stock is worth more than $85.

\[ c^+=\max(0,S^+-X)=\max(0,100-85)=\boxed{15} \]

\[ c^-=\max(0,S^--X)=\max(0,60-85)=\boxed{0} \]

3. Apply the put payoff rule

A put lets us sell for $85. We use it only when the stock is worth less than $85.

\[ p^+=\max(0,X-S^+)=\max(0,85-100)=\boxed{0} \]

\[ p^-=\max(0,X-S^-)=\max(0,85-60)=\boxed{25} \]

Note

First build the stock tree. Only then put $\max(0,S-X)$ on a call and $\max(0,X-S)$ on a put.

Variant: Calculate Risk-Neutral Probability and Catch an Impossible Tree

Abstract: Risk-neutral probability is a pricing weight forced by no-arbitrage, not our forecast of whether the stock will actually rise.

A one-period tree has $u=1.25$, $d=0.80$, and a 5% per-period risk-free rate. Find the risk-neutral up probability. Then test a second tree with the same $u$ and $d$ but a 30% risk-free rate.

1. Put risk-free growth between the branches

The pricing weights must make the stock grow at the risk-free rate on average:

\[ \pi u+(1-\pi)d=1+r \]

Rearranging gives:

\[ \pi=\frac{(1+r)-d}{u-d} \]

For the first tree:

\[ \pi=\frac{1.05-0.80}{1.25-0.80} =\frac{0.25}{0.45} =\boxed{0.5556} \]

So the risk-neutral down weight is:

\[ 1-\pi=\boxed{0.4444} \]

2. Check the no-arbitrage condition

A valid tree needs:

\[ d<1+r<u \]

The first tree passes because $0.80<1.05<1.25$.

For the second tree:

\[ \pi=\frac{1.30-0.80}{1.25-0.80}=1.1111 \]

A probability above 1 is the calculator screaming that the tree is broken. Risk-free growth of $1.30$ beats even the stock's up branch of $1.25$.

\[ \boxed{\text{The second tree violates no-arbitrage.}} \]

Note

Before valuing anything, check $d<1+r<u$. If $\pi$ falls outside $[0,1]$, do not keep calculating.

Variant: Value a One-Period Call Two Ways

Abstract: The expected-payoff route and the replication route must land on the same call value because they manufacture the same future cash flows.

A stock is $100. In one period, $u=1.35$, $d=0.74$, the exercise price is $100, and the per-period risk-free rate is 5.15%. Value the European call using both risk-neutral expectation and replication.

1. Build the end values

\[ S^+=100(1.35)=135,\qquad S^-=100(0.74)=74 \]

\[ c^+=\max(0,135-100)=35,\qquad c^-=0 \]

2. Find the pricing weight

\[ \pi=\frac{1.0515-0.74}{1.35-0.74}=\boxed{0.510656} \]

3. Weight the payoffs and discount once

\[ c=\frac{\pi c^++(1-\pi)c^-}{1+r} \]

\[ c=\frac{0.510656(35)+0.489344(0)}{1.0515} =\boxed{16.998} \]

4. Build the same payoff with stock and borrowing

The hedge ratio asks how many shares reproduce the call's change across the two states:

\[ h_c=\frac{c^+-c^-}{S^+-S^-} =\frac{35-0}{135-74} =\boxed{0.573770} \]

At expiration, $0.573770$ shares are worth $77.459$ in the up state. The call pays only $35$, so the remaining $42.459$ must be a loan repayment.

\[ \text{Borrow today}=\frac{42.459}{1.0515}=40.380 \]

\[ \text{Replication cost}=0.573770(100)-40.380=\boxed{16.998} \]

Note

Risk-neutral expectation is not a second theory. It is the same no-arbitrage replication written as weighted cash flows.

Variant: Value a One-Period Put and Check Put-Call Parity

Abstract: A put is replicated with a short stock position plus lending because its value rises when the stock falls.

Use the same tree as the previous problem: $S=100$, $X=100$, $u=1.35$, $d=0.74$, $r=5.15\%$, and $\pi=0.510656$. Find the European put value, hedge ratio, and parity check.

1. Find terminal put payoffs

\[ p^+=\max(0,100-135)=0 \]

\[ p^-=\max(0,100-74)=26 \]

2. Price the payoff

\[ p=\frac{0.510656(0)+0.489344(26)}{1.0515} =\boxed{12.100} \]

3. Find the put hedge ratio

\[ h_p=\frac{0-26}{135-74}=\boxed{-0.426230} \]

The negative sign means short stock. The lending leg supplies the rest of the payoff.

At expiration, lending $57.541$ offsets the short-stock loss in the up state and leaves the $26 put payoff in the down state. Its cost today is:

\[ \frac{57.541}{1.0515}=54.723 \]

So the replicated put costs:

\[ -0.426230(100)+54.723=\boxed{12.100} \]

4. Check put-call parity

For a one-period European option with no carry benefit:

\[ c-p=S-\frac{X}{1+r} \]

Left side:

\[ 16.998-12.100=4.898 \]

Right side:

\[ 100-\frac{100}{1.0515}=4.898 \]

\[ \boxed{\text{Parity holds.}} \]

Note

A long call has a non-negative hedge ratio; a long put has a non-positive hedge ratio. The sign is a fast error check.

Variant: Arbitrage an Overpriced or Underpriced Call

Abstract: If the market option and its replicating portfolio have different prices, buy the cheap copy and sell the expensive copy.

The fair one-period call value is $16.998. First suppose the market call trades at $20. Then suppose it trades at $15. State the arbitrage and today’s locked-in profit in each case.

1. Market call at $20: the option is expensive

Sell the call for $20 and buy its exact replication for $16.998.

\[ \text{Cash today}=20-16.998=\boxed{3.002} \]

At expiration, the replication pays whatever the short call owes. Future net payoff is zero in both states, so the $3.002 is genuinely locked in today.

2. Market call at $15: the option is cheap

Buy the call for $15 and short the replicating portfolio worth $16.998.

\[ \text{Cash today}=16.998-15=\boxed{1.998} \]

Again, the long call and short replication cancel in every future state.

Note

Arbitrage needs no forecast: opposite future cash flows cancel, while the price gap is collected immediately.

Variant: Roll Back a Two-Period European Call

Abstract: A two-period tree is just three one-period problems: value the two Time-1 nodes, then value today.

A stock is $7.35, with $u=1.445$, $d=0.715$, $X=8$, and a 4.35% risk-free rate per period. Value a two-period European call.

1. Find the risk-neutral weight

\[ \pi=\frac{1.0435-0.715}{1.445-0.715}=\boxed{0.45} \]

2. Build terminal stock prices and call payoffs

\[ S^{++}=7.35(1.445)^2=15.347,\quad c^{++}=7.347 \]

\[ S^{+-}=7.35(1.445)(0.715)=7.594,\quad c^{+-}=0 \]

\[ S^{--}=7.35(0.715)^2=3.758,\quad c^{--}=0 \]

3. Roll back to Time 1

\[ c^+=\frac{0.45(7.347)+0.55(0)}{1.0435}=\boxed{3.168} \]

\[ c^-=\frac{0.45(0)+0.55(0)}{1.0435}=\boxed{0} \]

4. Roll back once more

\[ c_0=\frac{0.45(3.168)+0.55(0)}{1.0435} =\boxed{1.366} \]

Note

Discount one step at a time. Do not discount a Time-1 option value by two periods again.

Variant: Value a Two-Period European Put and Cross-Check with Parity

Abstract: Work backward exactly as for the call, then use put-call parity as an independent checksum.

Keep $S=7.35$, $u=1.445$, $d=0.715$, $X=8$, $r=4.35\%$, and $\pi=0.45$. Value the matching two-period European put.

1. Find terminal put payoffs

\[ p^{++}=0,\qquad p^{+-}=8-7.594=0.406 \]

\[ p^{--}=8-3.758=4.242 \]

2. Roll back to Time 1

\[ p^+=\frac{0.45(0)+0.55(0.406)}{1.0435}=\boxed{0.214} \]

\[ p^-=\frac{0.45(0.406)+0.55(4.242)}{1.0435}=\boxed{2.411} \]

3. Roll back to today

\[ p_0=\frac{0.45(0.214)+0.55(2.411)}{1.0435} =\boxed{1.363} \]

4. Use two-period parity

\[ p=c-S+\frac{X}{(1+r)^2} \]

\[ p=1.366-7.35+\frac{8}{1.0435^2}=\boxed{1.363} \]

Note

If backward induction and parity disagree beyond rounding, a payoff, probability, or discounting step is wrong.

Variant: Calculate Node-by-Node Hedge Ratios

Abstract: The hedge ratio changes as the tree moves, so a multiperiod replication must be rebalanced at every node.

Use the preceding two-period tree. Find the call and put hedge ratios at each Time-1 node and at Time 0.

1. Call hedge ratios at Time 1

After an up move:

\[ h_c^+=\frac{7.347-0}{15.347-7.594}=\boxed{0.9476} \]

After a down move:

\[ h_c^-=\frac{0-0}{7.594-3.758}=\boxed{0} \]

2. Call hedge ratio today

\[ h_{c,0}=\frac{3.168-0}{10.621-5.255}=\boxed{0.5905} \]

3. Put hedge ratios at Time 1

\[ h_p^+=\frac{0-0.406}{15.347-7.594}=\boxed{-0.0524} \]

\[ h_p^-=\frac{0.406-4.242}{7.594-3.758}=\boxed{-1.0000} \]

4. Put hedge ratio today

\[ h_{p,0}=\frac{0.214-2.411}{10.621-5.255}=\boxed{-0.4095} \]

The hedge is not “set it and forget it.” The option’s exposure changes after the stock moves.

Note

A multiperiod option is dynamically replicated: calculate a fresh hedge ratio from the two next-node values at every node.

Variant: Compare European and American Put Values

Abstract: At every American-option node, compare the value of waiting with the cash from exercising right now and keep the larger number.

A stock is $26, with $X=25$, $u=1.466$, $d=0.656$, $r=2.05\%$ per period, and two periods remaining. Value the European and American puts.

1. The pricing weight and terminal payoffs

\[ \pi=\frac{1.0205-0.656}{1.466-0.656}=0.45 \]

Terminal stock prices are $55.878$, $25.004$, and $11.189$, so terminal put payoffs are $0$, $0$, and $13.811$.

2. European continuation values at Time 1

\[ p^+=0 \]

\[ p^-=\frac{0.45(0)+0.55(13.811)}{1.0205}=\boxed{7.4436} \]

Therefore:

\[ p_{E,0}=\frac{0.45(0)+0.55(7.4436)}{1.0205} =\boxed{4.0117} \]

3. Check immediate exercise at the down node

The stock there is $26(0.656)=17.056$.

\[ \text{Exercise value}=25-17.056=\boxed{7.9440} \]

Exercise value $7.9440$ beats continuation value $7.4436$, so replace the node value with $7.9440$.

\[ p_{A,0}=\frac{0.45(0)+0.55(7.9440)}{1.0205} =\boxed{4.2814} \]

\[ \text{Early-exercise premium}=4.2814-4.0117=\boxed{0.2697} \]

Note

For an American option, every non-terminal node is $\max(\text{continuation value},\text{exercise value})$.

Variant: Reject Early Exercise for a Non-Dividend-Paying American Call

Abstract: An American call’s extra exercise freedom is worthless when the stock pays no dividend because exercising early throws away time value and pays the strike too soon.

In the earlier $S=7.35$, $X=8$, two-period tree, the Time-1 stock values are $10.621$ after an up move and $5.255$ after a down move. The corresponding call continuation values are $3.168$ and $0$. Should an American call be exercised at either node, and what is its value today?

1. Test the up node

\[ \text{Exercise value}=\max(0,10.621-8)=\boxed{2.621} \]

\[ \text{Continuation value}=\boxed{3.168} \]

Waiting wins because $3.168>2.621$.

2. Test the down node

\[ \text{Exercise value}=\max(0,5.255-8)=\boxed{0} \]

\[ \text{Continuation value}=\boxed{0} \]

There is nothing to gain from exercising there either.

3. Keep the European node values

Because no node is replaced, the American call and European call have the same value:

\[ c_A=c_E=\boxed{1.366} \]

Note

“American” does not automatically mean “more valuable.” The right to do something useless has zero value.

Variant: Exercise an American Call Before a Known Dividend

Abstract: A non-dividend-paying call is not exercised early, but a known dividend can make grabbing the shares before ex-dividend worthwhile.

A stock is $100, a two-period American call has $X=95$, the per-period risk-free rate is 1%, and $u=1.224$, $d=0.796$. A $3 dividend is paid at Time 1. At the Time-1 up node, compare exercise with continuation. The official tree gives the European value today as $12.3438$.

1. Remove the dividend before growing the stock

The escrow method subtracts the dividend's present value from today's stock price:

\[ PV(D)=\frac{3}{1.01}=\boxed{2.9703} \]

The ex-dividend stock value at the up node is:

\[ (100-2.9703)(1.224)=\boxed{118.7644} \]

Immediately before the stock goes ex-dividend, add the $3 dividend back:

\[ 118.7644+3=121.7644 \]

2. Compare exercise with waiting

\[ \text{Exercise value}=121.7644-95=\boxed{26.7644} \]

The tree's continuation value is $24.9344$.

\[ 26.7644>24.9344 \]

Exercise wins at this node. After inserting that larger node value and rolling back, the American call is:

\[ c_A=\boxed{13.2497} \]

\[ \text{Early-exercise premium}=13.2497-12.3438=\boxed{0.9059} \]

Note

No dividend generally means no early call exercise. A known dividend creates the important exception: test the node just before ex-dividend.

Variant: Exercise an American Call on a Coupon Bond

Abstract: A bond coupon is carry just like a stock dividend, so an American call on a coupon bond can rationally be exercised early.

A two-year call on a 7% annual-coupon bond has exercise price $100 and can be exercised after Year 1. At the Year-1 up node, the bond is $100.57 and the European continuation value is $0.29. At the Year-1 down node, the bond is $103.80 and continuation value is $1.35. The one-year rate is 3%, and the risk-neutral branch weights are 50/50. Find the American call value today.

1. Compare at the up node

\[ \text{Exercise value}=100.57-100=\boxed{0.57} \]

Because $0.57>0.29$, exercise replaces continuation.

2. Compare at the down node

\[ \text{Exercise value}=103.80-100=\boxed{3.80} \]

Because $3.80>1.35$, exercise replaces continuation here too.

3. Roll the chosen values back

\[ c_A=\frac{0.5(0.57)+0.5(3.80)}{1.03} =\boxed{2.12} \]

The comparable European call was worth only $0.80$, so the early-exercise feature adds about $1.32$.

Note

For any carry-paying underlying—dividend stock or coupon bond—test exercise before the carry leaves the asset.

Variant: Value an Interest Rate Option with a Two-Period Rate Tree

Abstract: The rollback logic is unchanged, but each rate-tree node has its own discount factor because the interest rate itself is moving.

A two-year European call and put are written on the one-year spot rate, with a $1,000,000 notional and 3.25% exercise rate. The risk-neutral up probability is 50%. Terminal rates are 3.9706%, 3.2542%, and 2.2593%. Time-1 discount factors are 0.962386 and 0.974627, and today’s factor is 0.970446. Value both options.

1. Terminal payoffs per $1 of notional

\[ c^{++}=0.039706-0.0325=0.007206 \]

\[ c^{+-}=0.032542-0.0325=0.000042,\qquad c^{--}=0 \]

\[ p^{++}=0,\qquad p^{+-}=0,\qquad p^{--}=0.0325-0.022593=0.009907 \]

2. Roll back using the discount factor at each node

\[ c^+=0.962386[0.5(0.007206)+0.5(0.000042)]=0.003488 \]

\[ c^-=0.974627[0.5(0.000042)+0.5(0)]=0.000020 \]

\[ p^+=0 \]

\[ p^-=0.974627[0.5(0)+0.5(0.009907)]=0.004828 \]

3. Roll back to today and scale the notional

\[ c_0=0.970446[0.5(0.003488)+0.5(0.000020)]=0.00170216 \]

\[ p_0=0.970446[0.5(0)+0.5(0.004828)]=0.00234266 \]

\[ \text{Call value}=1{,}000{,}000(0.00170216)=\boxed{\$1{,}702.16} \]

\[ \text{Put value}=1{,}000{,}000(0.00234266)=\boxed{\$2{,}342.66} \]

Note

This simplified tree cash-settles at Time 2. The later Black-model problem handles the real-world deferred-settlement adjustment.

Variant: Calculate BSM Call and Put Values and Replicating Positions

Abstract: BSM is still stock minus financing for a call, and financing minus stock for a put; the normal-distribution terms tell us the quantities.

A European stock option has $S=100$, $X=100$, continuously compounded $r=5\%$, $T=1$, and $\sigma=30\%$. Calculate $d_1$, $d_2$, the call, the put, and both initial replicating positions.

1. Calculate the two standardized inputs

\[ d_1=\frac{\ln(S/X)+(r+\sigma^2/2)T}{\sigma\sqrt{T}} =\frac{0+(0.05+0.30^2/2)}{0.30} =\boxed{0.3167} \]

\[ d_2=d_1-\sigma\sqrt{T}=0.3167-0.30=\boxed{0.0167} \]

Using the normal distribution:

\[ N(d_1)=0.6241,\qquad N(d_2)=0.5066 \]

2. Value the call

\[ c=SN(d_1)-Xe^{-rT}N(d_2) \]

\[ c=100(0.6241)-100e^{-0.05}(0.5066)=\boxed{14.23} \]

3. Value the put

\[ p=Xe^{-rT}N(-d_2)-SN(-d_1) \]

\[ p=100e^{-0.05}(0.4934)-100(0.3759)=\boxed{9.35} \]

4. Read the replication directly from BSM

Call: buy $N(d_1)=0.6241$ shares and short $N(d_2)=0.5066$ zero-coupon bonds whose price is $Xe^{-rT}=95.123$.

Put: short $N(-d_1)=0.3759$ shares and buy $N(-d_2)=0.4934$ of those bonds.

Note

BSM uses continuous compounding. If given a discrete annual rate $r_d$, convert with $r=\ln(1+r_d)$.

Variant: Apply BSM to a Dividend-Paying Stock

Abstract: A dividend yield is a carry benefit: it lowers the stock piece of a call and raises the relative value of a put.

A European option has $S=60$, $X=60$, $r=2\%$, $T=0.5$, continuous dividend yield $\delta=2\%$, and $\sigma=45\%$. Value the call and put.

1. Put the dividend yield inside $d_1$

\[ d_1=\frac{\ln(S/X)+(r-\delta+\sigma^2/2)T}{\sigma\sqrt{T}} =\boxed{0.15910} \]

\[ d_2=d_1-\sigma\sqrt{T}=\boxed{-0.15910} \]

\[ N(d_1)=0.56320,\qquad N(d_2)=0.43680 \]

2. Value the call

\[ c=Se^{-\delta T}N(d_1)-Xe^{-rT}N(d_2) \]

\[ c=60e^{-0.02(0.5)}(0.56320)-60e^{-0.02(0.5)}(0.43680) =\boxed{7.5091} \]

3. Value the put

\[ p=Xe^{-rT}N(-d_2)-Se^{-\delta T}N(-d_1) =\boxed{7.5091} \]

They are equal here because $S=X$ and $r=\delta$, so carry-adjusted parity has a zero right-hand side:

\[ c-p=Se^{-\delta T}-Xe^{-rT}=0 \]

Note

Do not subtract a continuous dividend yield from $S$. Use $Se^{-\delta T}$ and replace $r$ with $r-\delta$ inside $d_1$.

Variant: Apply BSM to a Currency Option

Abstract: Treat the foreign risk-free rate as the currency’s dividend yield because holding foreign currency earns the foreign rate.

The spot rate is JPY135 per EUR. A six-month European option on EUR has $X=135$, Japanese risk-free rate $r_d=0.25\%$, euro risk-free rate $r_f=1.00\%$, and volatility $\sigma=12\%$. Find the call and put values in yen per euro.

1. Map the rates before touching the formula

The domestic rate discounts yen. The foreign rate acts like carry on the underlying euro.

\[ d_1=\frac{\ln(S/X)+(r_d-r_f+\sigma^2/2)T}{\sigma\sqrt{T}} =\boxed{-0.001768} \]

\[ d_2=d_1-\sigma\sqrt{T}=\boxed{-0.086621} \]

\[ N(d_1)=0.499295,\qquad N(d_2)=0.465487 \]

2. Value the EUR call

\[ c=Se^{-r_fT}N(d_1)-Xe^{-r_dT}N(d_2) =\boxed{\text{JPY }4.3064\text{ per EUR}} \]

3. Value the EUR put

\[ p=Xe^{-r_dT}N(-d_2)-Se^{-r_fT}N(-d_1) =\boxed{\text{JPY }4.8111\text{ per EUR}} \]

Note

First identify the quote. For JPY per EUR, EUR is the underlying and its rate is the carry yield; JPY is domestic and supplies the discount rate.

Variant: Check BSM Lower and Upper Bounds at Zero Volatility

Abstract: When volatility collapses toward zero, an option collapses toward its no-arbitrage lower bound rather than automatically becoming worthless.

A one-year European option has $S=100$, $X=95$, $r=4\%$, no dividends, and volatility approaching zero. Find the call and put lower bounds and the upper bounds.

1. Present-value the strike

\[ PV(X)=95e^{-0.04}=\boxed{91.2750} \]

2. Find the lower bounds

\[ c\geq\max[0,S-PV(X)] =\max(0,100-91.2750) =\boxed{8.7250} \]

\[ p\geq\max[0,PV(X)-S] =\max(0,91.2750-100) =\boxed{0} \]

With zero volatility, the future is effectively deterministic under the model, so these are also the limiting model values.

3. Check the upper bounds

\[ c\leq S=\boxed{100} \]

\[ p\leq PV(X)=\boxed{91.2750} \]

Note

“Volatility goes to zero” does not mean “option goes to zero.” Discounted intrinsic value can remain positive.

Variant: Value Calls and Puts on Futures with Black’s Model

Abstract: For a futures option, Black’s model uses the futures price—not the spot price—and discounts the whole expected payoff.

An index spot is 1,860, but its 0.25-year futures price is 1,851.65. A European futures option has $X=1{,}860$, $r=0.2\%$, $T=0.25$, and $\sigma=15\%$. The contract multiplier is 250. Find the call and put values.

1. Use the futures price in $d_1$

\[ d_1=\frac{\ln[F_0(T)/X]+(\sigma^2/2)T}{\sigma\sqrt{T}} =\boxed{-0.02249} \]

\[ d_2=d_1-\sigma\sqrt{T}=\boxed{-0.09749} \]

\[ N(d_1)=0.491028,\qquad N(d_2)=0.461168 \]

2. Price each option in index points

\[ c=e^{-rT}[F_0(T)N(d_1)-XN(d_2)] =\boxed{51.41} \]

\[ p=e^{-rT}[XN(-d_2)-F_0(T)N(-d_1)] =\boxed{59.76} \]

3. Apply the multiplier last

\[ \text{Call contract}=51.4136(250)=\boxed{\$12{,}853.41} \]

\[ \text{Put contract}=59.7595(250)=\boxed{\$14{,}939.86} \]

Note

Spot 1,860 and dividend yield are distractions once the futures price is supplied. Scale by the contract multiplier only after finding points.

Variant: Value a Deferred-Settlement Interest Rate Call and Put

Abstract: An interest rate option uses the FRA rate as its underlying, multiplies by the accrual period, and discounts to payment—not merely to option expiry.

On 15 May, a three-month loan beginning 15 June has FRA rate 0.68%. A call and put expire on 15 June with exercise rate 0.60%, notional SGD10,000,000, accrual period 0.25, volatility 25%, and continuously compounded discount rate 0.55%. Use $T=31/365$ and pay on 15 September, $T+0.25$ years from valuation.

1. Use decimal rates inside the logarithm

\[ d_1=\frac{\ln(0.0068/0.0060)+(0.25^2/2)(31/365)}{0.25\sqrt{31/365}} =\boxed{1.75435} \]

\[ d_2=d_1-0.25\sqrt{31/365}=\boxed{1.68149} \]

\[ N(d_1)=0.960314,\qquad N(d_2)=0.953666 \]

2. Discount to the payment date

\[ DF=e^{-0.0055[(31/365)+0.25]}=\boxed{0.998160} \]

3. Value the call

\[ c=N_0(AP)(DF)[FRA\,N(d_1)-R_XN(d_2)] \]

\[ c=10{,}000{,}000(0.25)(0.998160) [0.0068(0.960314)-0.0060(0.953666)] =\boxed{\text{SGD }2{,}016.64} \]

4. Value the put

\[ p=N_0(AP)(DF)[R_XN(-d_2)-FRA\,N(-d_1)] =\boxed{\text{SGD }20.32} \]

The parity difference is a useful checksum:

\[ c-p=N_0(AP)(DF)(FRA-R_X)=\boxed{\text{SGD }1{,}996.32} \]

Note

Three classic traps: use the FRA, enter 0.0068 rather than 0.68, and discount through the underlying deposit’s maturity.

Variant: Show Caplet-Floorlet Parity at an At-Market Strike

Abstract: When strike equals the current FRA rate, a matching caplet and floorlet have the same value.

Keep the previous problem’s inputs, but set the exercise rate equal to the FRA rate of 0.68%. Find the caplet and floorlet values.

1. The log-moneyness term disappears

Because $FRA=R_X$:

\[ \ln(FRA/R_X)=\ln(1)=0 \]

\[ d_1=\frac{(0.25^2/2)(31/365)}{0.25\sqrt{31/365}} =0.036429 \]

\[ d_2=-0.036429 \]

2. Calculate both values

Using notional SGD10,000,000, $AP=0.25$, and $DF=0.998160$:

\[ \text{Caplet}=\boxed{\text{SGD }493.10} \]

\[ \text{Floorlet}=\boxed{\text{SGD }493.10} \]

Parity explains the equality:

\[ c-p=N_0(AP)(DF)(FRA-R_X)=0 \]

Note

A cap is a strip of rate calls; a floor is a strip of rate puts. At the at-market swap strike, equal-value cap and floor positions create swap equivalences.

Variant: Value Payer and Receiver Swaptions

Abstract: A payer swaption is call-like on the forward fixed swap rate; a receiver swaption is put-like, and the annuity factor carries the discounting.

A three-month option enters a five-year swap. The forward five-year swap rate is 2.65%, exercise rate is 2.50%, volatility is 20%, accrual period is 0.5, present value of the unit-payment annuity is 8.9, and notional is $10,000,000. Value payer and receiver swaptions.

1. Use option expiry, not swap tenor, in $d_1$

\[ d_1=\frac{\ln(0.0265/0.0250)+(0.20^2/2)(0.25)}{0.20\sqrt{0.25}} =\boxed{0.632689} \]

\[ d_2=0.632689-0.20\sqrt{0.25}=\boxed{0.532689} \]

\[ N(d_1)=0.736532,\qquad N(d_2)=0.702876 \]

2. Value the payer swaption

\[ PAYSWN=N_0(AP)(PVA)[R_{FIX}N(d_1)-R_XN(d_2)] \]

\[ PAYSWN=10{,}000{,}000(0.5)(8.9) [0.0265(0.736532)-0.0250(0.702876)] =\boxed{\$86{,}605.86} \]

3. Value the receiver swaption

\[ RECSWN=N_0(AP)(PVA)[R_XN(-d_2)-R_{FIX}N(-d_1)] =\boxed{\$19{,}855.86} \]

4. Check swaption parity

\[ PAYSWN-RECSWN=N_0(AP)(PVA)(R_{FIX}-R_X) \]

\[ =10{,}000{,}000(0.5)(8.9)(0.0015)=\boxed{\$66{,}750} \]

Note

Do not add another discount factor: $PVA$ already contains the payment-date discounting. Use the forward swap rate, not today’s spot swap rate.

Variant: Approximate an Option Price Change with Delta

Abstract: Delta is the option’s local speed: multiply it by a small underlying move to estimate the option’s value change.

A call is worth $7.80 and has delta $0.42. A matching put has no dividend yield. Estimate both new option values if the stock falls by $1.50 and the put is currently worth $5.20.

1. Call change

\[ \Delta c\approx\Delta_c\Delta S=0.42(-1.50)=\boxed{-0.63} \]

\[ \widehat{c}=7.80-0.63=\boxed{7.17} \]

2. Get put delta from call delta

With no dividends:

\[ \Delta_p=\Delta_c-1=0.42-1=\boxed{-0.58} \]

3. Put change

\[ \Delta p\approx(-0.58)(-1.50)=\boxed{+0.87} \]

\[ \widehat{p}=5.20+0.87=\boxed{6.07} \]

Note

Delta is reliable for small moves. A large move bends away from the straight-line estimate, which is exactly what gamma measures.

Variant: Make an Option Position Delta Neutral

Abstract: Add a hedge whose delta exactly cancels the portfolio delta; the sign tells you whether to buy or sell the hedge.

You are short puts on 10,000 shares. Each put has delta $-0.419$. Find a stock hedge. Then find a hedge using calls with delta $0.532$.

1. Find the short-put position delta

A long put has negative delta, so a short put has positive delta:

\[ \Delta_{portfolio}=10{,}000(+0.419)=\boxed{+4{,}190} \]

2. Hedge with stock

Stock delta is $+1$ per share.

\[ N_H=-\frac{\Delta_{portfolio}}{\Delta_H} =-\frac{4{,}190}{1} =\boxed{-4{,}190\text{ shares}} \]

The minus sign means short 4,190 shares.

3. Hedge with calls

\[ N_H=-\frac{4{,}190}{0.532} =-7{,}875.94 \]

Round to the nearest whole option:

\[ \boxed{\text{Sell }7{,}876\text{ calls}} \]

Note

Position sign comes first. Shorting a negative-delta put creates positive delta; the hedge must carry negative delta. If one listed contract covers several underlying units, multiply option delta by that contract multiplier before sizing the hedge.

Variant: Improve a Price Estimate with Delta Plus Gamma

Abstract: Delta draws a straight tangent line; the gamma term adds the missing bend in the option-price curve.

A call is worth $8.00, delta is $0.55$, gamma is $0.018$, and the stock rises by $6. Estimate the new call value using delta only and then delta plus gamma.

1. Delta-only estimate

\[ \widehat{c}=c+\Delta_c(\Delta S) \]

\[ \widehat{c}=8.00+0.55(6)=\boxed{11.30} \]

2. Add curvature

\[ \widehat{c}=c+\Delta_c(\Delta S)+\frac{1}{2}\Gamma_c(\Delta S)^2 \]

\[ \widehat{c}=8.00+0.55(6)+\frac{1}{2}(0.018)(6^2) \]

\[ =8.00+3.30+0.324=\boxed{11.624} \]

The gamma correction is positive because a long option has positive gamma and the stock move is squared.

Note

Do not forget the $\tfrac12$, and square only the stock-price change—not delta or gamma.

Variant: Neutralize Gamma First and Delta Second

Abstract: Stock can repair delta but has zero gamma, so use an option to repair gamma first and stock to clean up the remaining delta.

A portfolio has delta $+2,400$ and gamma $-600$. A traded call has delta $0.40$ and gamma $0.15$ per option. Make the portfolio gamma neutral and then delta neutral.

1. Fix gamma with the call

\[ N_C=-\frac{\Gamma_{portfolio}}{\Gamma_C} =-\frac{-600}{0.15} =\boxed{+4{,}000\text{ calls}} \]

Buying calls adds positive gamma $4{,}000(0.15)=600$, exactly cancelling $-600$.

2. Recalculate delta after the option hedge

\[ \Delta_{new}=2{,}400+4{,}000(0.40)=\boxed{4{,}000} \]

3. Fix delta with stock

\[ N_S=-\frac{4{,}000}{1}=\boxed{-4{,}000\text{ shares}} \]

Shorting stock removes delta but adds no gamma, so both targets remain satisfied.

Note

Order matters: option first for gamma, stock second for delta. Stock cannot alter gamma because its delta is always 1.

Variant: Translate Theta, Vega, and Rho into Profit and Loss

Abstract: Each Greek is a rate of change; multiply by the correctly measured input move and keep the position sign straight.

One long option has daily theta $-0.0327$, vega $0.4231$ per one percentage-point volatility change, and rho $0.3705$ per one percentage-point rate change. Estimate the separate effects of five calendar days, volatility rising from 24% to 27%, and the risk-free rate rising from 4.0% to 4.5%.

1. Theta: five days pass

\[ \Delta V_{theta}\approx5(-0.0327)=\boxed{-0.1635} \]

2. Vega: volatility rises by 3 percentage points

\[ \Delta V_{vega}\approx3(0.4231)=\boxed{+1.2693} \]

3. Rho: the rate rises by 0.5 percentage point

\[ \Delta V_{rho}\approx0.5(0.3705)=\boxed{+0.1853} \]

If all three changes are treated as independent first-order effects:

\[ \Delta V\approx-0.1635+1.2693+0.1853=\boxed{+1.2911} \]

Note

Confirm the vendor’s units: a move from 24% to 27% is 3 units when vega is quoted per percentage point. Theta is usually negative, but a deep-in-the-money European put near expiry can have positive theta.

Variant: Infer Implied Volatility and Trade Relative Value

Abstract: Implied volatility is the volatility input that makes the model equal the market price; a higher option price means a higher implied volatility, all else equal.

With every BSM input except volatility fixed, a put is worth $6.40 at 20% volatility and $7.49 at 24%. The market put price is $7.20. Bracket its implied volatility with a linear interpolation. Then decide what to do if your own fair-volatility forecast is 19% for an option quoted at 24% implied volatility.

1. Bracket the answer

$7.20 lies between $6.40 and $7.49, so implied volatility lies between 20% and 24%.

2. Interpolate

The market price has covered this fraction of the model-price gap:

\[ \frac{7.20-6.40}{7.49-6.40} =\frac{0.80}{1.09} =0.7339 \]

Apply that fraction to the four-point volatility interval:

\[ \sigma_{imp}\approx20\%+0.7339(4\%) =\boxed{22.94\%} \]

This is an approximation; an exact answer comes from numerically inverting the pricing model.

3. Make the relative-value decision

The market asks 24% implied volatility, but your fair estimate is 19%. You think the option is priced with too much volatility and is therefore expensive.

\[ \boxed{\text{Sell the option, subject to hedge and risk limits.}} \]

The opposite rule also holds: if your fair volatility is above quoted implied volatility, you see the option as cheap and would buy it.

Note

Historical volatility looks backward. Implied volatility is backed out from today’s option price and is the market’s common comparison unit across strikes and maturities.

On This Page

Variant: Read a One-Period Binomial Tree and Find Terminal Payoffs Variant: Calculate Risk-Neutral Probability and Catch an Impossible Tree Variant: Value a One-Period Call Two Ways Variant: Value a One-Period Put and Check Put-Call Parity Variant: Arbitrage an Overpriced or Underpriced Call Variant: Roll Back a Two-Period European Call Variant: Value a Two-Period European Put and Cross-Check with Parity Variant: Calculate Node-by-Node Hedge Ratios Variant: Compare European and American Put Values Variant: Reject Early Exercise for a Non-Dividend-Paying American Call Variant: Exercise an American Call Before a Known Dividend Variant: Exercise an American Call on a Coupon Bond Variant: Value an Interest Rate Option with a Two-Period Rate Tree Variant: Calculate BSM Call and Put Values and Replicating Positions Variant: Apply BSM to a Dividend-Paying Stock Variant: Apply BSM to a Currency Option Variant: Check BSM Lower and Upper Bounds at Zero Volatility Variant: Value Calls and Puts on Futures with Black’s Model Variant: Value a Deferred-Settlement Interest Rate Call and Put Variant: Show Caplet-Floorlet Parity at an At-Market Strike Variant: Value Payer and Receiver Swaptions Variant: Approximate an Option Price Change with Delta Variant: Make an Option Position Delta Neutral Variant: Improve a Price Estimate with Delta Plus Gamma Variant: Neutralize Gamma First and Delta Second Variant: Translate Theta, Vega, and Rho into Profit and Loss Variant: Infer Implied Volatility and Trade Relative Value