Valuation and Scenario Analysis

LEARNING SECTION 01

A dealer can report a profit when entering a trade and still carry a large future loss exposure. Valuation estimates a price today. Scenario analysis examines what a position could be worth, or what it could cost, under future conditions. They address different decisions.

Question	Valuation	Scenario analysis
Purpose	Estimate today’s no-arbitrage derivative value	Understand possible future values and losses
Probability measure	Risk-neutral measure Q for the pricing model	Real-world measure P for probabilistic forecasts
Stock drift in this guide	r, for a non-dividend-paying stock	μ, the real-world model drift
Discounting	Discount the Q expected payoff at r	Report value or loss at the chosen horizon. Specify any funding benchmark separately.
Typical result	A derivative price or mark-to-market	A percentile, VaR estimate or stress result

WORKED EXAMPLE

A million written calls

A desk sells one million one-year calls for $5 million. Their theoretical value is $4.5 million, so the entry valuation difference is $0.5 million. The strike is $55.

At S_T = $80: payoff = 1,000,000 × (80 − 55) = $25 million
Payoff less original premium = $25m − $5m = $20 million

If the assumed real-world probability of finishing above $80 is 5%, that tail contains costs above $20 million on this simplified basis. The $0.5 million initial valuation difference does not measure that tail risk. This calculation ignores financing, hedging and transaction costs.

A deterministic stress scenario, such as a specified market crash, need not have an assigned probability. A percentile estimate does require a probability model.

LEARNING SECTION 02

The real and risk-neutral worlds

The real-world measure P describes the probabilities used for forecasting and risk analysis. Investors’ required returns, data and economic assumptions inform the drift μ. There is no general requirement that every asset’s μ must exceed r.

The risk-neutral measure Q is a pricing construction. In the standard frictionless, arbitrage-free model, discounted gains are martingales under Q. For the non-dividend-paying stock considered here, its expected proportional growth rate under Q is r.

Real world: dS/S = μ dt + σ dW^P
Risk-neutral world: dS/S = r dt + σ dW^Q

Both models use the same volatility coefficient σ. Changing the measure changes the drift. It does not make investors risk-neutral or turn the pricing probability into an observed frequency.

Scope of the formulasConstant rates and volatility, no dividends, continuous trading and the standard GBM assumptions. With a continuous dividend yield δ, the stock-price drift under Q is r − δ. General markets can require richer models, and incomplete markets need additional pricing choices.

LEARNING SECTION 03

Why stock prices are lognormal

Under geometric Brownian motion, the log return over T years is normal. The price is the exponential of that log return, so it remains strictly positive and has a right-skewed distribution.

ln(S_T/S₀) ~ N((g − σ²/2)T, σ²T)
ln S_T ~ N(ln S₀ + (g − σ²/2)T, σ²T)
Use g = μ under P, or g = r under Q.

Quantity	Formula	Meaning
Mean of ln S_T	ln S₀ + (g − σ²/2)T	Location on the log scale
SD of ln S_T	σ√T	Dispersion on the log scale
Median S_T	S₀ exp((g − σ²/2)T)	The 50th percentile
Mean S_T	S₀ exp(gT)	The arithmetic expected price

WORKED EXAMPLE

Mean and median are different

With S₀=$50, μ=12%, σ=20% and T=2 years: the log mean is 4.1120 and log SD is 0.2828. The median price is $61.07, while the mean is $63.56. The right tail raises the mean.

The notation N(m,v) here uses variance as its second parameter. Excel’s NORM.DIST and LOGNORM.DIST functions take standard deviation. Their TRUE argument returns cumulative probability; FALSE returns density height.

LEARNING SECTION 04

Four probability and percentile formulas

Let N(z) denote the standard normal cumulative distribution function and N⁻¹(p) its inverse. For a positive price threshold V, define:

d₂(g) = [ln(S₀/V) + (g − σ²/2)T] / (σ√T)

Result	Formula	Interpretation
Finish above V	P(S_T > V) = N(d₂)	Probability of exceeding the threshold
Finish below V	P(S_T < V) = N(−d₂)	Probability of ending below the threshold
Upper-tail threshold	V = S₀ exp((g − σ²/2)T − N⁻¹(q)σ√T)	Only q of outcomes exceed this value
Lower-tail percentile	V = S₀ exp((g − σ²/2)T + N⁻¹(q)σ√T)	Only q of outcomes fall below this value

WORKED EXAMPLE

A price above $70 in two years

S₀=$50, μ=12%, σ=20%, T=2 and V=$70.

d₂ = [ln(50/70) + (0.12 − 0.20²/2) × 2] / (0.20√2)
d₂ = −0.4825; N(d₂) = 31.47%

The price exceeded with probability 5% is the 95th percentile, $97.25. It is an upper-tail threshold, not a downside risk estimate.

Class workbook: P1 Prob Above and P2 Percentile

WORKED EXAMPLE

A fall below $30

With S₀=$40, μ=9%, σ=25%, T=1.5 years and V=$30, d₂=1.2274. The lower-tail probability is N(−1.2274)=10.98%.

Class workbook: P3 Prob Below

Use =NORM.S.DIST(d2,TRUE) and =NORM.S.INV(q) in current Excel. NORMSDIST and NORMSINV in the supplied class workbook are older names for these functions.

LEARNING SECTION 05

Risk-neutral valuation and forward contracts

For a payoff H_T and a constant risk-free rate, the standard pricing relation is:

V₀ = exp(−rT) × E^Q[H_T]

The expected payoff must use Q probabilities when the discount rate is r. Combining a real-world expected payoff with the risk-free discount rate generally gives the wrong derivative price.

A long forward

The maturity payoff is S_T − K. Under Q, E[S_T]=S₀exp(rT), giving:

f_long = S₀ − K exp(−rT)
f_short = K exp(−rT) − S₀
Fair delivery price for a new zero-value forward: F_0,T = S₀exp(rT)

WORKED EXAMPLE

The short forward is worth $1.80 million

For one million shares, S₀=$50, K=$55, r=3% and T=2:

Value per share = 55 exp(−0.03 × 2) − 50 = $1.7970
Total value = $1.797 million

This is the value of an existing contract. It differs from the forward delivery price of a newly initiated contract. No estimate of μ is needed.

Class workbook: P5 Forward

LEARNING SECTION 06

Binary options: one price, two expectations

A cash-or-nothing binary call pays a fixed amount Q if the terminal price exceeds a trigger V, and zero otherwise. Its payoff size does not increase with the amount by which the price exceeds V. Equality is governed by the contract, though it has zero probability in the continuous model.

Binary price = Q exp(−rT) N(d₂(r))
Real-world expected payoff = Q N(d₂(μ))

WORKED EXAMPLE

The $100 binary

S₀=$30, V=$40, Q=$100, r=3%, μ=10%, σ=30%, T=1 year.

Calculation	Result
Q probability of payment	15.6502%
Q expected payoff	$15.6502
Price today: 15.6502 × exp(−0.03)	$15.1876
P probability of payment	21.8991%
P expected payoff	$21.8991
Equivalent rate: ln(21.8991/15.1876)	36.5962% p.a.

The real-world expected payoff is higher, but it requires a correspondingly higher equivalent discount rate to recover the same price. The stock’s 10% drift is not the binary option’s discount rate.

Class workbook: P6 Binary Two Worlds

This equivalent rate summarises one payoff and maturity. It is not a universal, constant discount rate for all derivatives, and the simple logarithmic calculation requires a positive price and expected payoff.

Two more classroom cases

Inputs	Price under Q	Probability under P
Q=$500, S₀=$61, V=$60, r=2%, μ=8%, σ=20%, T=0.25	$281.41	62.37%
Q=$100, S₀=$25, V=$30, r=3%, μ=10%, σ=30%, T=0.25	$10.66	13.05%

Class workbook: P7 Binary 7.12 and P8 Binary 7.6.

INTERACTIVE LAB

Interactive lab: probabilities and binary value

Change the assumptions to compare real-world and risk-neutral probabilities. Rates and volatility are entered as annual percentages. The chart shows the two terminal-price densities using the same volatility.

One asset, two probability measures

Current price S₀Trigger VReal-world drift μ (%)Risk-free rate r (%)Volatility σ (%)Maturity T (years)Fixed payout Q

Green: real-world density PGold: pricing density Q

Currency units match S₀, V and Q. Chart range is finite; density height is not probability at a point. Numerical probabilities use the full model.

LEARNING SECTION 07

State prices and default probabilities

A state price π_i is the price today of one currency unit paid only when state i occurs. If the states are exhaustive, buying one claim for every state creates a sure unit payment.

Σπ_i = exp(−rT)
Risk-neutral probability p_i^Q = π_i / Σπ_j
Value = exp(−rT) Σ p_i^Q × payoff_i

WORKED EXAMPLE

Infer a risk-free rate and a pricing PD

A one-year claim paying $100 if a company defaults is worth $3. A complementary claim paying $100 if it does not default is worth $95.

Step	Calculation	Result
State prices	3/100 and 95/100	0.03 and 0.95
Discount factor	0.03 + 0.95	0.98
Continuously compounded rate	−ln(0.98)	2.0203%
Risk-neutral PD	0.03 / 0.98	3.0612%

Repricing the default claim gives 0.98 × 3.0612% × $100 = $3.

Class workbook: P9 State Prices

Use market-implied Q default probabilities for pricing within the chosen credit model. Historical or forward-looking P probabilities address actual default likelihood and expected credit losses. Market-implied and historical probabilities can differ because of risk premia, liquidity and modelling assumptions; Q PD being higher is common, not a mathematical law.

LEARNING SECTION 08

Scenario analysis, percentiles and VaR

For a long holding of n shares, a low terminal share price is adverse. At confidence c, take the lower-tail probability q=1−c and use the real-world drift μ.

S_q = S₀ exp((μ − σ²/2)T + N⁻¹(q)σ√T)
Portfolio value at that percentile = nS_q
Loss relative to today = n(S₀ − S_q)

WORKED EXAMPLE

10,000 shares at $30

Take μ=8%, σ=25% and a one-year horizon. Current position value is $300,000.

Confidence	Price percentile	Portfolio value	Loss relative to today
95%	$20.8788	$208,787.93	$91,212.07
99%	$17.6081	$176,081.24	$123,918.76

A 99% VaR of about $123,919 means the model assigns 1% probability to losses greater than this threshold. It is not the maximum possible loss and does not measure the average loss beyond the threshold.

Class workbook: P10 VaR Shares

WORKED EXAMPLE

A six-month portfolio value

For 10,000 shares at $80, μ=8%, σ=20% and T=0.5 years, the 1st-percentile price is $59.3250. Portfolio value is $593,250.33, compared with $800,000 today. The corresponding loss is $206,749.67.

Class workbook: P11 Worst Case

Here “worst-case value” means a specified percentile. GBM still permits more adverse outcomes. A stress test may instead specify an extreme but plausible scenario without claiming a probability.

LEARNING SECTION 09

Indian share illustrations: PNB and SBI

These are classroom assumptions, not current prices or forecasts. The two positions have different sizes, horizons and confidence levels, so their rupee losses should not be read as an isolated comparison of bank quality or volatility.

Input or result	PNB illustration	SBI illustration
Shares	10,000	1,000
Initial price	₹140	₹800
Real-world drift μ	10%	8%
Volatility σ	5%	15%
Horizon	1 year	0.5 years
Lower-tail probability	1%	2%
Price percentile	₹137.5620	₹665.9095
Portfolio value	₹13,75,620.42	₹6,65,909.49
Loss relative to today	₹24,379.58	₹1,34,090.51

The workbook’s PNB sensitivity table holds other inputs constant. At μ=10%, increasing σ from 5% to 20% raises its 99% one-year loss from about ₹24,380 to ₹4,47,626. This within-position comparison isolates the volatility change.

Class workbook: P12 PNB SBI. All share prices and parameters are illustrative.

LEARNING SECTION 10

A derivatives portfolio needs both worlds

A derivative’s value at a future risk horizon depends on the market state then and its remaining contractual life. First generate future states under P. Then value the remaining derivative in each state using the pricing model under Q.

STEP 1 · REAL-WORLD SCENARIO

Project the market to horizon t

Use μ and σ to determine the market state S_t. Select the tail adverse to the position.

STEP 2 · RISK-NEUTRAL REVALUATION

Value the remaining contract

Use r, the scenario market state and remaining maturity T−t. Repeat across scenarios to obtain the portfolio-value distribution.

WORKED EXAMPLE

A short forward on one million shares

S₀=$50, K=$55, r=3%, μ=10%, σ=30%, contract maturity T=2 years and risk horizon t=0.5 years. A rising share price hurts the short position, so use the upper 1% share-price threshold.

Step	Calculation	Result
Today’s short value	1m × (55e^−0.03×2 − 50)	$1.7970m
Adverse market state under P	50 × exp((0.10−0.30²/2)×0.5 − N⁻¹(0.01)×0.30√0.5)	$84.1849
Remaining maturity	2 − 0.5	1.5 years
Horizon value under Q	1m × (55e^−0.03×1.5 − 84.1849)	−$31.6051m

The model assigns 1% probability to a contract value below −$31.6051 million at six months. The simple change from today’s +$1.7970 million value is a loss of $33.4021 million. Value at the horizon and loss from today are different quantities. A funded or hedged P&L measure would require its own cash-flow benchmark.

Class workbook: P13 Both Worlds

INTERACTIVE LAB

Interactive lab: stress a short forward

Change the drift, volatility and horizon. The adverse stock-price state uses P; the contract is then revalued with r. This lab follows the analytical example above.

Market scenario and remaining contract value

Current price S₀Delivery price KNumber of sharesReal-world drift μ (%)Risk-free rate r (%)Volatility σ (%)Contract maturity T (years)Risk horizon t (years)Upper-tail probability (%)

Tail probability must be between 0% and 50%; horizon must be positive and shorter than maturity. No dividends, margin cash flows, financing or hedging costs are included. A negative reported loss means a gain relative to today’s value.

LEARNING SECTION 11

Estimating drift without double-counting variance

For equally spaced daily adjusted prices, calculate log returns u_i=ln(P_i/P_i−1). With 252 trading days per year:

Annual log-return mean a = 252 × AVERAGE(daily log returns)
Annual volatility σ = √252 × STDEV.S(daily log returns)
GBM proportional drift μ = a + σ²/2

The price equation uses μ−σ²/2. If a is estimated directly from mean log returns, use a in that position. Calling a “μ” and subtracting σ²/2 again double-counts the variance adjustment.

WORKED EXAMPLE

ITC: reconcile the historical classroom example

The supplied workbook uses 250 prices for 23 September 2024 to 23 September 2025, with last price ₹403.20. Its annualised mean log return is −19.5337%, and volatility is 16.6041%.

Quantity	Consistent estimate
Annual mean log return a	−19.5337%
Annual volatility σ	16.6041%
GBM drift μ = a + σ²/2	-18.1552%
Probability of S_T > ₹450 in one year	3.30%

The original P4 worksheet reports 2.74% because they label a as μ and subtract half the variance once more. The log-return-consistent calculation is about 3.30%. These historical inputs are a calibration exercise, not a current investment forecast.

Class workbook: P4 ITC Data

Short samples give noisy estimates of expected returns. Volatility estimates also have uncertainty and may change through time. Forward-looking economic assumptions, longer samples and CAPM-based estimates can be useful, provided their return and compounding conventions are consistent.

Market price of risk

For a non-dividend-paying GBM stock: λ = (μ − r) / σ
Hence μ − r = λσ

The classroom CAPM illustration gives β=0.6×30%/20%=0.90 and an expected return of 6.5%+0.90×6%=11.9%. The separate commodity illustration uses λ=0.3×6%/20%=0.09, so λσ=0.09×40%=3.6 percentage points. These use the stated model conventions; they do not establish a universal sign for λ.

LEARNING SECTION 12

Girsanov, Monte Carlo and model limits

Girsanov’s theorem provides the formal change of probability measure behind the drift adjustment. In the standard continuous diffusion model, changing from P to Q changes the drift and retains the diffusion coefficient. This statement does not imply that separately calibrated historical and option-implied volatility estimates will coincide in market data.

Under P: S_t+Δt = S_t exp((μ − σ²/2)Δt + σ√Δt × Z)
Z ~ N(0,1); in Excel, Z = NORM.S.INV(U), with 0 < U < 1

For the short-forward example, simulate S_t under P and calculate Kexp(−r(T−t))−S_t for each draw. Sort the resulting contract values to estimate the lower tail.

The supplied workbook retains 1,000 normal draws. Its 10th-worst simulated value is about −$30.46 million, while PERCENTILE interpolation produces about −$30.05 million. The analytical answer is −$31.61 million. Sampling error and quantile conventions explain why a finite simulation need not match the closed-form result exactly.

A 1% tail from 1,000 scenarios contains only about ten observations.Use more scenarios and convergence checks before relying on an extreme quantile. Multiple market factors need a defensible dependence model. Interest rates, volatility and some commodities may need mean reversion; jumps, changing volatility, liquidity, funding and counterparty exposure need additional modelling.

LEARNING SECTION 13

Check your understanding

1. Which world should produce an equity holding’s 99% one-year percentile?

Use P with a defensible real-world drift. This is a probability forecast of future values, not derivative pricing.

2. Which probability values a three-month cash binary?

Use its risk-neutral payment probability and discount the expected payment at r, within the pricing model.

3. Why does a short forward use an upper-tail stock price?

Its value Kexp(−r(T−t))−S falls as S rises. An upper stock-price tail therefore maps into a lower contract-value tail.

4. Is −$31.61 million the short forward’s loss from today?

It is the adverse horizon contract value. Relative to today’s +$1.797 million value, the simple decline is about $33.402 million, before funding or hedge cash flows.

5. Does 99% VaR place a ceiling on losses?

No. The model assigns 1% probability to larger losses. VaR does not describe how large losses beyond the threshold become.

6. Is the average daily log return × 252 the GBM drift μ?

It estimates a=μ−σ²/2. Add half the annual variance to estimate μ, or use a directly as the log-return mean.

7. Must risk-neutral default probabilities always exceed historical ones?

No. They often differ because of risk premia and other market effects, but there is no universal mathematical ordering.

8. What remains unchanged when switching measures in the standard GBM model?

The volatility coefficient. The drift changes from μ to r for the non-dividend-paying stock. In richer models, state-variable drift adjustments require their own specification.

CLASS MATERIALS

Class materials and further reading

Download the solved-problems workbook. The workbook contains fifteen numerical problems and a chapter question-and-answer tab. Work through the steps before changing its yellow input cells.

Download solved problems (.xlsx)

This download retains the original classroom workbook. For the historical ITC worksheet, apply the drift convention explained in Section 11. Historical examples and illustrative bank inputs are not current market data.

Workbook tabs	Use them to practise
P1–P3	Upper and lower probabilities, plus a percentile threshold
P4 and P15	Historical estimation, CAPM and the market price of risk
P5–P9	Forward value, binary options and state prices
P10–P12	Shareholding loss percentiles and Indian bank illustrations
P13–P14	Real-world scenarios followed by risk-neutral revaluation

Reading and related topics

John C. Hull, Risk Management and Financial Institutions, 5th edition (2018), Chapter 7. The article explains the supplied lecture material in original teaching prose.
NYU: Girsanov’s theorem and risk-neutral dynamics — mathematical background.
NIST: the lognormal distribution — distribution definitions and moments.

Educational material. The numerical results are conditional on the stated model and assumptions. They are not forecasts or security recommendations.

Valuation andScenario Analysis

Valuation & scenarios at a glance

What is it worth today?

What could happen?

COMMON VOLATILITY, DIFFERENT DRIFT

A DERIVATIVES BOOK USES BOTH